Controllability of some semilinear shadow reaction-diffusion systems

Abstract

The shadow limit is a versatile tool used to study the reduction of reaction-diffusion systems into simpler PDE-ODE models by letting one of the diffusion coefficients tend to infinity. This reduction has been used to understand different qualitative properties and their interplay between the original model and its reduced version. The aim of this work is to extend previous results about the controllability of linear reaction-diffusion equations and how this property is inherited by the corresponding shadow model. Defining a suitable class of nonlinearities and improving some uniform Carleman estimates, we extend the results to the semilinear case and prove that the original model is null-controllable and that the shadow limit preserves this important feature.

Keywords

Shadow limit semilinear reaction-diffusion equations uniform null-controllability Carleman estimates

1. Introduction

1.1. Motivation

Reaction-diffusion systems have been established as one of the main tools for modeling biological, chemical, and biological processes. These systems are commonly composed by coupled semi-linear parabolic equations endowed with zero-flux boundary conditions. To fix ideas, let us consider the following model

\begin{aligned} {\begin{cases} y_{t} - σ_{1} Δ y = f (y, z) & in (0, T) \times Ω, \\ z_{t} - σ_{2} Δ z = g (y, z) & in (0, T) \times Ω, \\ \frac{\partial y}{\partial ν} = \frac{\partial z}{\partial ν} = 0 & on (0, T) \times \partial Ω, \\ y (0, \cdot) = y^{0}, z (0, \cdot) = z^{0} & in Ω . \end{cases} \end{aligned}

(1.1)

Here, $T > 0$ is given, $Ω \subset R^{N}$ ( $N \in N^{*}$ ) is a nonempty, bounded set with smooth boundary $\partial Ω$ and ν denotes the unit outward normal vector to $\partial Ω$ . In the remainder of this document, to abridge the notation, we set $Q_{T} := (0, T) \times Ω$ and $Σ_{T} = (0, T) \times \partial Ω$ .

In system (1.1), f and g are two suitable nonlinear functions, $σ_{1}, σ_{2} > 0$ represent the diffusion rates of the variables $y = y (t, x)$ and $z = z (t, x)$ , respectively, and $y^{0}$ , $z^{0}$ are their corresponding initial data.

The paradigmatic model (1.1) has been studied extensively from many different angles: nonnegativity properties, monotonicity, entropy inequalities, long-time behavior, blow-up, pattern formation, among others. The literature is very rich so we refer the reader (for instance) to the monographs [36,38], and [39] and the references therein for further material.

As pointed out in [24], reaction-difussion systems like (1.1) can exhibit quite complex structures and from a mathematical point of view it makes sense to reduce their complexity to understand the dynamics of the full system while preserving its main properties.

In this direction, a successful model reduction tool can be found in the seminal papers [23] and [37]. In such works, the idea is to study the limit of system (1.1) when one diffusion coefficient is fixed and the other tends to infinity. For instance, if $σ_{1} = 1$ and $σ := σ_{2} \to + \infty$ in (1.1), due to the boundary conditions, the variable z becomes spatially homogeneous, i.e. $z (t, x) = ξ (t)$ , and it can be proved that system (1.1) becomes

\begin{aligned} {\begin{cases} y_{t} - Δ y = f (y, ξ) & in Q_{T}, \\ \dot{ξ} = - - \int_{Ω} g (y (t, x), ξ (t)) d x & in (0, T), \\ \frac{\partial y}{\partial ν} = 0 & on Σ_{T}, \\ y (0, \cdot) = y^{0} & in Ω, ξ (0) = - - \int_{Ω} z^{0} d x, \end{cases} \end{aligned}

(1.2)

where for any

u \in L^{1} (Ω)

we write

\begin{aligned} - - \int_{Ω} u d x := \frac{1}{| Ω |} \int_{Ω} u (x) d x \end{aligned}

and where

| Ω |

denotes the measure of the set Ω.

The resulting equation (1.2) is the so-called shadow system and the limit behavior between (1.1) and (1.2) and some related properties have been studied in numerous works, see e.g. [17,22,25,28,31,32,34,41].

As a model reduction, system (1.2) has been successfully used to understand some properties of the original system (1.1) and vice versa. In [41], under appropriate assumptions on f and g, a stationary solution of the shadow problem allows to find a stationary solution of the original one. Similarly, in [34] it is shown that stability properties of those stationary states is preserved between the models. In the same spirit, [17] establishes a close relationship between the compact attractors of the original reaction-diffusion system and its shadow limit.

Other works explore particular phenomena related to the shadow limit. In [22] and [31], the authors analyze diffusion-driven instability, a phenomenon that appears while coupling ordinary differential equations and PDEs, as in the case for the shadow limit. The long-time existence of a shadow model is studied in [25], while [32] approaches the problem from the perspectives of renormalization groups and the center manifold theorem.

To conclude this part, it is noteworthy to mention that the dynamics and properties of the shadow system and the original one do not always align: for instance, it may happen that under suitable conditions the solutions of the shadow system blow-up in finite time while the original one has global solutions in time (see [28]).

1.2. Problem formulation and main result

Motivated by the discussion above, the aim of this paper is to study some controllability properties of (1.2) by using the shadow-model reduction of the original system (1.1).

To formulate this problem, let $ω \subset Ω$ be a nonempty, possibly small, open set and denote by $1_{ω}$ its characteristic function. We begin by considering the system

\begin{aligned} {\begin{cases} y_{t} - Δ y = f (y, z) + h 1_{ω} & in Q_{T}, \\ z_{t} - σ Δ z = g (y, z) & in Q_{T}, \\ \frac{\partial y}{\partial ν} = \frac{\partial z}{\partial ν} = 0 & on Σ_{T}, \\ y (0, \cdot) = y^{0}, z (0, \cdot) = z^{0} & in Ω, \end{cases} \end{aligned}

(1.3)

where

h = h (t, x)

is an external control force.

One of the classical goals in control theory is to find a control h such that a given system is steered to rest. In our case, this translates into the following.

Definition 1.1.

System ( 1.3 ) is said to be null-controllable at time T if for any $(y^{0}, z^{0}) \in [L^{2} (Ω)]^{2}$ , there exists a control $h \in L^{2} (0, T; L^{2} (ω))$ such that the associated controlled states $(y, z)$ satisfy

\begin{aligned} y (T, \cdot) = z (T, \cdot) = 0 in Ω . \end{aligned}

(1.4)

Notice that in (1.3), the action of the control h enters directly onto the first equation of the system, but the second one only sees the action indirectly through the reaction term $g (y, z)$ . In this direction, there is an extensive literature on the controllability of parabolic systems with less controls than equations, see e.g. the survey [2] for results up to 2011 and [26] for a more recent one.

For our purposes, we shall make the following instrumental assumptions on the nonlinear reaction terms: $f, g \in C^{1} (R^{2}; R)$ are such that

(H1)

f and g are globally Lipschitz functions with constants $C_{f} > 0$ and $C_{g} > 0$ , respectively, i.e.,

\begin{aligned} \begin{aligned} \forall (y_{1}, z_{1}, y_{2}, z_{2}) \in R^{4}, | f (y_{1}, z_{1}) - f (y_{2}, z_{2}) | & ⩽ C_{f} (| y_{1} - y_{2} | + | z_{1} - z_{2} |), \\ | g (y_{1}, z_{1}) - g (y_{2}, z_{2}) | & ⩽ C_{g} (| y_{1} - y_{2} | + | z_{1} - z_{2} |) . \end{aligned} \end{aligned}

(H2)

$f (0, 0) = 0$ and $g (0, 0) = 0$ .

(H3)

There is a constant ${\hat{a}}_{21} > 0$ , such that $\frac{\partial g}{\partial y} (y, z) ⩾ {\hat{a}}_{21}$ or $- \frac{\partial g}{\partial y} (y, z) ⩾ {\hat{a}}_{21}$ for all $(y, z) \in R^{2}$ .

Hypothesis (H1) is related to the well-posedness of (1.3). Indeed, under this assumption, for any $(y^{0}, z^{0}) \in [L^{2} (Ω)]^{2}$ and $h \in L^{2} (0, T; L^{2} (ω))$ , system (1.3) has a unique solution in the class

\begin{aligned} (y, z) \in L^{2} (0, T; [H^{1} (Ω)]^{2}) \cap C ([0, T]; [L^{2} (Ω)]^{2}) . \end{aligned}

On the other hand, (H2) ensures that the state

(0, 0)

is an equilibrium point for (1.3) whenever

h \equiv 0

, while (H3) will be useful later on during the implementation of some control techniques (see Section 3).

Remark 1.2.

Some remarks are in order.

Hypothesis (H1) and (H2) are standard assumptions in controllability problems for scalar semilinear heat equations, see e.g. [ 9 ] and [ 12 ]. The globally Lipschitz condition (H1) is linked to a global controllability result (see Theorem 1.4).

Conversely, (H3) is specific to the coupled nature of our problem, enabling the implementation of well-known strategies (see [ 15 , Lemma 2.2]) for controlling many equations with few controls (cf. Step 2 in the proof of Proposition 2.4 below). Similar conditions can be found in other works addressing control problems for semilinear coupled systems (with Lipschitz nonlinearities), see for instance [ 1 ] and [ 10 ]. Unlike those works, which impose smoothness/smallness assumptions ([ 1 , Remark 3 and Theorem 6]) or linear simplifications ([ 10 , Eqs (1.6)–(1.8)]), here we only require the lower bound in (H3), preserving the nonlinear nature of the function while achieving a global controllability result.

We provide a simple example of a function g verifying (H1)–(H3). Let

\begin{aligned} g (y, z) := g_{0} (y) + g_{1} (z), \end{aligned}

(1.5)

where

g_{0} : R \to R

is defined as

g_{0} (y) := \sqrt{1 + y^{2}} \arctan (y)

and

g_{1} : R \to R

is any globally Lipschitz function with constant

C_{g_{1}}

with

g_{1} (0) = 0

. With these definitions, clearly g satisfies (H2). A direct computation gives that

\begin{aligned} g_{0}^{^{'}} (y) = \frac{y \arctan (y) + 1}{\sqrt{1 + y^{2}}} . \end{aligned}

We note that

g_{0}^{^{'}}

is an even function,

g_{0}^{^{'}} (y) ⩾ 0

for all

y \in R

and

lim_{y \to + \infty} g_{0}^{^{'}} (y) = π / 2

. Analyzing the critical points of

g_{0}^{^{'}}

, we can see that there is only one (global) minimum located at

y = 0

, therefore

1 ⩽ g_{0}^{^{'}} (y) ⩽ π / 2

for all

y \in R

. This implies that g in (1.5) is globally Lipschitz with constant

C_{g} = max {π / 2, C_{g_{1}}}

and that

\frac{\partial g (y, z)}{\partial y} ⩾ 1

for all

(y, z) \in R^{2}

, thus fulfilling (H1) and (H3).

One can find more general examples verifying (H3). For instance, taking the parameter-dependent functions $g_{0} (y; k, r) = \sqrt{r + y^{2}} \arctan (k y)$ or $g_{0} (y; k) = \frac{y (1 + y^{2})^{1 / 2}}{(1 + | y |^{k})^{1 / k}}$ with $r \in [1, \infty)$ and $k \in (2, + \infty)$ yield (after long but straightforward computations) that ( 1.5 ) verifies (H3) for some ${\hat{a}}_{21} > 0$ depending on k and r.

For further comments regarding the nonlinear functions f and g, please refer to Section 5.

In our particular case, it is well-known that for fixed $σ > 0$ and assuming that (H1)–(H3) hold, system (1.3) is null-controllable,1 so it is natural to ask if this property is inherited by its shadow limit, namely, we wonder if there exists a control $h \in L^{2} (0, T; L^{2} (ω))$ (by abuse of notation) such that the system

\begin{aligned} {\begin{cases} y_{t} - Δ y = f (y, ξ) + h 1_{ω} & in Q_{T}, \\ \dot{ξ} = - - \int_{Ω} g (y (t, x), ξ (t)) d x & in (0, T), \\ \frac{\partial y}{\partial ν} = 0 & on Σ_{T}, \\ y (0, \cdot) = y^{0} & in Ω, ξ (0) = - - \int_{Ω} z^{0} d x . \end{cases} \end{aligned}

(1.6)

verifies

\begin{aligned} y (T, \cdot) = 0 in Ω and ξ (T) = 0. \end{aligned}

(1.7)

In the linear case, i.e., $f (y, z) = a y + b z$ and $g (y, z) = c y + d z$ with $a, b, c, d \in R$ and $c \neq 0$ , the controllability of (1.6) in the sense (1.7) has been proved in [20, Theorem 2]. In that work, the controllability of the linear shadow model was obtained by studying the observability of the corresponding adjoint system, namely

\begin{aligned} {\begin{cases} - φ_{t} - Δ φ = a φ + c θ & in Q_{T}, \\ - \dot{θ} = b - - \int_{Ω} φ (t, x) d x + d θ & in (0, T), \\ \frac{\partial φ}{\partial ν} = 0 & on Σ_{T}, \\ φ (T, \cdot) = φ^{T} & in Ω, θ (T) = θ^{T}, \end{cases} \end{aligned}

(1.8)

where

φ^{T} \in L^{2} (Ω)

and

θ^{T} \in R

The main ideas to prove such result are to use Carleman estimates for parabolic equations for the first equation of (1.8) and define $ξ (t) := - - \int_{Ω} φ (t, x) d x$ for obtaining the following ode-ode system (thanks to the homogeneous Neumann boundary condition)

\begin{aligned} {\begin{cases} - \dot{ζ} = a ζ + c θ & in (0, T), \\ - \dot{θ} = b ζ + d θ & in (0, T), \end{cases} \end{aligned}

(1.9)

for which Kalman rank condition holds as long as

c \neq 0

. All in all, this showed in the linear case the suitability of the shadow model reduction in the context of controllability. So a natural open question is to verify that such reduction is still meaningful in the semilinear case.

In this direction, the main contribution of this paper is the following.

Theorem 1.3.

Assume that hypotheses (H1)–(H3) hold. There exists a control $h \in L^{2} (0, T; L^{2} (ω))$ such that the corresponding solution $(y, ξ)$ to ( 1.6 ) satisfies ( 1.7 ).

As usual in other semilinear control problems, the methodology roughly consists in obtaining a linearization of the considered system, then studying the observability of the corresponding adjoint equation and finally use a fixed point method. However, as it has been pointed out in [20, Section 6.1], linearizing direclty system (1.6) yields

\begin{aligned} {\begin{cases} y_{t} - Δ y = a (t, x) y + b (t) ξ + h 1_{ω} & in Q_{T}, \\ \dot{ξ} = - - \int_{Ω} c (t, x) y (t, x) d x + d (t) ξ & in (0, T), \\ \frac{\partial y}{\partial ν} = 0 & on Σ_{T}, \\ y (0, \cdot) = y^{0} & in Ω, ξ (0) = - - \int_{Ω} z^{0} d x, \end{cases} \end{aligned}

(1.10)

for some coefficients a, b, c, d, and the dependency of c on x invalidates many ideas used for the linear constant-coefficient case. In particular, the shadow system (1.10) cannot longer be reduced to (1.9) which is crucial in the proof shown in [20].

To circumvent this, we shall use instead the approach of [18] which uses as a starting point the controllability properties of the original system and then the convergence towards the shadow system. In more detail, the first step consists in proving the following uniform result.

Theorem 1.4.

Let $σ ⩾ 1$ and assume that (H1)–(H3) hold. There exists a control $h = h (σ) \in L^{2} (0, T; L^{2} (ω))$ such that the corresponding solution $(y, z)$ to (1.3) satisfies (1.4). Moreover, we have the following uniform bound on the control

\begin{aligned} ‖ h (σ) ‖_{L^{2} (0, T; L^{2} (ω))} ⩽ C (‖ y^{0} ‖_{L^{2} (Ω)} + ‖ z^{0} ‖_{L^{2} (Ω)}), \end{aligned}

(1.11)

where

C > 0

is a constant independent of σ,

y^{0}

and

z^{0}

. Moreover, the solution

(y, z)

satisfies the uniform estimate

\begin{aligned} ‖ y ‖_{L^{2} (0, T; H^{1} (Ω))}^{2} + ‖ z ‖_{L^{2} (0, T; H^{1} (Ω))}^{2} + ‖ y_{t} ‖_{L^{2} (0, T; (H^{1} (Ω))^{'})}^{2} + ‖ z_{t} ‖_{L^{2} (0, T; (H^{1} (Ω))^{'})}^{2} + σ \iint_{Q_{T}} | \nabla z |^{2} d x d t \\ ⩽ C (‖ y^{0} ‖_{L^{2} (Ω)}^{2} + ‖ z^{0} ‖_{L^{2} (Ω)}^{2}) . \end{aligned}

(1.12)

for a constant

C > 0

depending at most on Ω, ω, T,

C_{f}

, and

C_{g}

At this point, we will follow a very conventional route, that is, we linearize system (1.3) (see Eq. (2.1) below) and prove a uniform controllability result with respect to σ. Then, by means of a fixed point argument, we will obtain a uniform result for the nonlinear system. Here, it is important to mention that the Carleman estimate used in [20] to address the linear case cannot be directly used and some improvements had to be implemented to obtain Theorem 1.4 (see the discussion below Proposition 2.1).

With Theorem 1.4 at hand, the second step is to build a sequence of controls $(h (σ))_{σ ⩾ 1}$ for system (1.3) and use the uniform bound (1.11) to extract a subsequence such that

\begin{aligned} h (σ) ⇀ h weakly in L^{2} (0, T; L^{2} (ω)) as σ \to + \infty, \end{aligned}

(1.13)

for some

h \in L^{2} (0, T; L^{2} (ω))

. Then, we will use the uniform estimates (1.12) and the convergence (1.13) to implement a shadow limit to reduce the controlled system (1.3) into (1.6) while preserving the null-controllability property. For that, we will make an adaptation of the shadow model reduction techniques shown in [21] and [31], each of them relying on different tools – compactness and variational results for the first and semigroup theory for the second – but complementing each other allowing us to prove Theorem 1.3. Remark 1.5.

We shall note that Theorem 1.4 remains valid if we change boundary conditions in the third equation (1.3) to be $y = z = 0$ on $Σ_{T}$ or $y = \frac{\partial z}{\partial ν} = 0$ on $Σ_{T}$ (other combinations including Robin boundary conditions might work as well, but this has to be checked). However, the shadow limit will just work for purely Neumann boundary conditions or when $y = \frac{\partial z}{\partial ν} = 0$ . In the latter case, the control of the shadow limit has been explored in a different context in [ 19 ].

1.3. Organization of the paper

The rest of the paper is organized as follows. In Section 2 we focus on deriving a uniform controllability result with respect to σ for a linearized version of system (1.3). This result will be used later in Section 3 to perform a fixed point method yielding, in particular, the proof of Theorem 1.4. Section 4 is devoted to implementing the shadow limit and proving Theorem 1.3. Finally, in Section 5, we provide additional discussion, remarks and results concerning the controllability of shadow systems.

2. Null controllability of a linear coupled system

2.1. An observability inequality

In this part of the paper, we study the null controllability of the linear reaction-diffusion system with Neumann boundary conditions

\begin{aligned} {\begin{cases} y_{t} - Δ y = a_{11} y + a_{12} z + h 1_{ω} & in Q_{T}, \\ z_{t} - σ Δ z = a_{21} y + a_{22} z & in Q_{T}, \\ \frac{\partial y}{\partial ν} = \frac{\partial z}{\partial ν} = 0 & on Σ_{T} \\ y (0, \cdot) = y^{0}, z (0, \cdot) = z^{0} & in Ω, \end{cases} \end{aligned}

(2.1)

where

a_{i j} \in L^{\infty} (Q_{T})

i, j = 1, 2

. This will be the first step towards the proof of Theorem 1.4.

First, we prove the following observability inequality for the adjoint system, given by

\begin{aligned} {\begin{cases} - ϕ_{t} - Δ ϕ = a_{11} ϕ + a_{21} ψ & in Q_{T}, \\ - ψ_{t} - σ Δ ψ = a_{12} ϕ + a_{22} ψ & in Q_{T}, \\ \frac{\partial ϕ}{\partial ν} = \frac{\partial ψ}{\partial ν} = 0 & on Σ_{T}, \\ ϕ (T, \cdot) = ϕ_{T}, ψ (T, \cdot) = ψ_{T} & in Ω . \end{cases} \end{aligned}

(2.2)

The result reads as follows.

Proposition 2.1.

Let $σ ⩾ 1$ and assume that there is a constant ${\hat{a}}_{21} > 0$ such that $a_{21} (t, x) > {\hat{a}}_{21}$ or $- a_{21} (t, x) > {\hat{a}}_{21}$ for all $(t, x) \in (0, T) \times ω$ . Then, for all $(ϕ_{T}, ψ_{T}) \in [L^{2} (Ω)]^{2}$ , the solutions $(ϕ, ψ)$ of the adjoint system ( 2.2 ) verify

\begin{aligned} \int_{Ω} (| ϕ (0, x) |^{2} + | ψ (0, x) |^{2}) d x ⩽ e^{C K} \iint_{(0, T) \times ω} | ϕ |^{2} d x d t, \end{aligned}

(2.3)

where

\begin{aligned} K = 1 + T + \frac{1}{T} + T \sum_{i, j = 1}^{2} ‖ a_{i j} ‖_{L^{\infty}} + max_{i, j = 1, 2} ‖ a_{i j} ‖_{L^{\infty}}^{\frac{2}{3}}, \end{aligned}

(2.4)

and

C > 0

is a constant independent of σ and T.

We postpone the proof of this result to the end of the section. For its proof, we improve a Carleman estimate that appears in [20, Theorem 16] in two ways: first, we allow that the coefficients $a_{i j}$ , $i, j = 1, 2$ , depend on the variables t and x; secondly, we keep track of the dependency of the norms of $a_{i j}$ allowing us to obtain an explicit expression of the constant K. We shall note that this was not done even for the constant coefficient case in [20].

To prove Proposition 2.1, we begin by recalling some instrumental definitions and results about Carleman estimates for heat equations with homogeneous Neumann boundary conditions.

Lemma 2.2 ([12, Lemma 1.1])

Let $B \subset\subset Ω$ be a non-empty open subset. Then, there exists $η^{0} \in C^{2} (\bar{Ω})$ such that $η^{0} > 0$ in Ω, $η_{0} = 0$ on $\partial Ω$ , and $| \nabla η^{0} | > 0$ in $\bar{Ω ∖ B}$ .

For a parameter $λ > 0$ , we define

\begin{aligned} α (t, x) := \frac{e^{2 λ ‖ η^{0} ‖_{\infty}} - e^{λ η^{0} (x)}}{t (T - t)}, ξ (t, x) := \frac{e^{λ η^{0} (x)}}{t (T - t)}, \end{aligned}

(2.5)

\begin{aligned} \begin{aligned} \hat{α} (t) := max_{x \in \bar{Ω}} α (t, x), ξ^{*} (t) := max_{x \in \bar{Ω}} ξ (t, x), \\ α^{*} (t) := min_{x \in \bar{Ω}} α (t, x), \hat{ξ} (t) := min_{x \in \bar{Ω}} ξ (t, x) . \end{aligned} \end{aligned}

(2.6)

The Carleman estimate for heat equations with homogeneous boundary conditions we will use reads as follows.

Lemma 2.3.

There exists $C = C (Ω, ω)$ and $λ_{0} = λ_{0} (Ω, ω)$ such that, for every $λ ⩾ λ_{0}$ there exists $s_{0} = s_{0} (Ω, ω, λ)$ such that the solution q to

\begin{aligned} {\begin{cases} - \hat{σ} q_{t} - Δ q = g & in Q_{T}, \\ \frac{\partial q}{\partial ν} = 0 & on Σ_{T}, \\ q (x, T) = q_{T} (x) & in Ω, \end{cases} \end{aligned}

satisfies for any

s ⩾ s_{0} (T + T^{2})

q_{T} \in L^{2} (Ω)

g \in L^{2} (Q_{T})

, and

\hat{σ} \in (0, 1]

, the following estimate

\begin{aligned} I (s, \hat{σ}; q) ⩽ C (\iint_{Q_{T}} e^{- 2 s α} | g |^{2} d x d t + s^{3} \iint_{(0, T) \times B} e^{- 2 s α} ξ^{3} | q |^{2} d x d t), \end{aligned}

(2.7)

where

\begin{aligned} I (s, \hat{σ}; q) & := s^{- 1} \iint_{Q_{T}} e^{- 2 s α} ξ^{- 1} ({\hat{σ}}^{2} | q_{t} |^{2} + \sum_{i, j = 1}^{N} {| \frac{\partial^{2} q}{\partial x_{i} x_{j}} |}^{2}) d x d t \\ + s \iint_{Q_{T}} e^{- 2 s α} ξ | \nabla q |^{2} d x d t + s^{3} \iint_{Q_{T}} e^{- 2 s α} ξ^{3} | q |^{2} d x d t . \end{aligned}

Now, we are in position to present one of the main results of this section.

Proposition 2.4.

Let $σ ⩾ 1$ and assume that there is a constant ${\hat{a}}_{21} > 0$ such that $a_{21} (t, x) > {\hat{a}}_{21}$ or $- a_{21} (t, x) > {\hat{a}}_{21}$ for all $(t, x) \in (0, T) \times ω$ . Then, there is a constant $C > 0$ depending at most on Ω, ω and ${\hat{a}}_{21}$ , such that for any $(ϕ_{T}, ψ_{T}) \in [L^{2} (Ω)]^{2}$ , the solutions $(ϕ, ψ)$ of the adjoint (2.2) verify

\begin{aligned} s^{3} \iint_{Q_{T}} e^{- 2 s α} ξ^{3} | ϕ |^{2} d x d t + s^{3} \iint_{Q_{T}} e^{- 2 s α} ξ^{3} | ψ |^{2} d x d t ⩽ C s^{8} \iint_{(0, T) \times ω} e^{- 4 s α^{*} + 2 s \hat{α}} (ξ^{*})^{8} | ϕ |^{2} d x d t, \end{aligned}

for any

\begin{aligned} s ⩾ C (T + T^{2} + T^{2} max_{i, j = 1, 2} ‖ a_{i j} ‖_{L^{\infty}}^{\frac{2}{3}}) . \end{aligned}

Proof.

Let us consider sets $ω_{i} \subset Ω$ , $i = 0, 1$ such that

\begin{aligned} ω_{0} \subset\subset ω_{1} \subset\subset ω . \end{aligned}

(2.8)

All along the proof, C will be a generic positive constant that depends at most on Ω, ω and

{\hat{a}}_{21}

and that may change from line to line. For readability, we have divided the proof in four steps.

Step 1: initial estimates. We fix λ to a value large enough and apply estimate (2.7) to the first equation of (2.2) with $\hat{σ} = 1$ , $B = ω_{0}$ , and $g (t, x) = a_{11} (t, x) ϕ + a_{21} (t, x) ψ$ to get

\begin{aligned} I (s, 1; ϕ) & ⩽ C \iint_{Q_{T}} e^{- 2 s α} {| a_{11} (t, x) ϕ + a_{21} (t, x) ψ |}^{2} d x d t + C s^{3} \iint_{(0, T) \times ω_{0}} e^{- 2 s α} ξ^{3} | ϕ |^{2} d x d t . \end{aligned}

(2.9)

Similarly, we divide over σ the second equation of (2.2) and once again apply estimate (2.7) with

\hat{σ} = σ^{- 1}

B = ω_{0}

, and

g (t, x) = σ^{- 1} (a_{12} (t, x) ϕ + a_{22} (t, x) ψ)

\begin{aligned} I (s, σ^{- 1}; ψ) & ⩽ C \iint_{Q_{T}} e^{- 2 s α} | σ^{- 1} (a_{12} (t, x) ϕ + a_{22} (t, x) ψ) |^{2} d x d t + C s^{3} \iint_{(0, T) \times ω_{0}} e^{- 2 s α} ξ^{3} | ψ |^{2} d x d t . \end{aligned}

(2.10)

Now, adding the inequalities (2.9) and (2.10), using that

σ ⩾ 1

, the right-hand side can be estimated as follows

\begin{aligned} I (s, 1; ϕ) + I (s, σ^{- 1}; ψ) \\ ⩽ C \iint_{Q_{T}} e^{- 2 s α} (‖ a_{11} ‖_{L^{\infty}}^{2} + ‖ a_{12} ‖_{L^{\infty}}^{2}) | ϕ |^{2} d x d t + C \iint_{Q_{T}} e^{- 2 s α} (‖ a_{21} ‖_{L^{\infty}}^{2} + ‖ a_{22} ‖_{L^{\infty}}^{2}) | ψ |^{2} d x d t \\ + C s^{3} \iint_{(0, T) \times ω} e^{- 2 s α} ξ^{3} (| ϕ |^{2} + | ψ |^{2}) d x d t . \end{aligned}

Note that

ξ^{- 1} (t, x) ⩽ \frac{T^{2}}{4}

. This implies

\begin{aligned} I (s, 1; ϕ) + I (s, σ^{- 1}; ψ) \\ ⩽ C {(T^{2} max_{i, j = 1, 2} ‖ a_{i j} ‖_{L^{\infty}}^{\frac{2}{3}})}^{3} \iint_{Q_{T}} e^{- 2 s α} ξ^{3} (| ϕ |^{2} + | ψ |^{2}) d x d t + C s^{3} \iint_{(0, T) \times ω} e^{- 2 s α} ξ^{3} (| ϕ |^{2} + | ψ |^{2}) d x d t . \end{aligned}

(2.11)

Under the assumptions of Lemma 2.3, we have that

s ⩾ C (T + T^{2})

for some

C > 0

depending on Ω and ω. If we choose

\begin{aligned} s ⩾ C (T + T^{2} + T^{2} max_{i, j = 1, 2} ‖ a_{i j} ‖_{L^{\infty}}^{\frac{2}{3}}), \end{aligned}

(2.12)

for a (possibly larger) constant

C > 0

independent of

a_{i j}

and T, we can absorb the first term in the right-hand side of (2.11), that is

\begin{aligned} I (s, 1; ϕ) + I (s, σ^{- 1}; ψ) ⩽ C s^{3} \iint_{(0, T) \times ω_{0}} e^{- 2 s α} ξ^{3} (| ϕ |^{2} + | ψ |^{2}) d x d t . \end{aligned}

(2.13)

Step 2: local energy estimates. In this step we estimate the local integral term of ψ in the right-hand side of (2.13). Let $η \in C_{0}^{\infty} (ω_{1})$ be such that

\begin{aligned} \begin{aligned} 0 ⩽ η ⩽ 1 in ω_{1}, \\ η (x) = 1 for all x \in ω_{0}, \end{aligned} \end{aligned}

(2.14)

and assume (without loss of generality) that

- a_{21} ⩾ {\hat{a}}_{21} > 0

(0, T) \times ω

. Using the first equation of (2.2), we obtain

\begin{aligned} s^{3} \iint_{(0, T) \times ω_{0}} e^{- 2 s α} ξ^{3} | ψ |^{2} d x d t & ⩽ s^{3} \iint_{(0, T) \times ω_{1}} η (x) e^{- 2 s α} ξ^{3} | ψ |^{2} d x d t \\ = - s^{3} \iint_{(0, T) \times ω_{1}} η (x) e^{- 2 s α} ξ^{3} \frac{ψ}{a_{21} (t, x)} (ϕ_{t} + Δ ϕ + a_{11} (t, x) ϕ) d x d t \\ ⩽ | \frac{s^{3}}{{\hat{a}}_{21}} \iint_{Q_{T}} η (x) e^{- 2 s α} ξ^{3} ψ (ϕ_{t} + Δ ϕ + a_{11} (t, x) ϕ) d x d t | . \end{aligned}

(2.15)

Integrating by parts in time and using triangle inequality yields

\begin{aligned} s^{3} \iint_{(0, T) \times ω_{0}} e^{- 2 s α} ξ^{3} | ψ |^{2} d x d t \\ ⩽ | \frac{s^{3}}{{\hat{a}}_{21}} \iint_{(0, T) \times Ω} (η (x) (e^{- 2 s α} ξ^{3})_{t} ψ ϕ + η (x) e^{- 2 s α} ξ^{3} ψ_{t} ϕ) d x d t | \\ + | \frac{s^{3}}{{\hat{a}}_{21}} \iint_{(0, T) \times Ω} η (x) e^{- 2 s α} ξ^{3} ψ Δ ϕ d x d t | + | \frac{s^{3}}{{\hat{a}}_{21}} \iint_{(0, T) \times Ω} a_{11} (t, x) η (x) e^{- 2 s α} ξ^{3} ψ ϕ d x d t |, \end{aligned}

whence, substituting the second equation of (2.2) in the first term of the right-hand side of the previous inequality we get

\begin{aligned} s^{3} \iint_{(0, T) \times ω_{1}} e^{- 2 s α} ξ^{3} | ψ |^{2} d x d t & ⩽ | \frac{s^{3}}{{\hat{a}}_{21}} \iint_{Q_{T}} η (x) {(e^{- 2 s α} ξ^{3})}_{t} ψ ϕ d x d t | \\ + | \frac{s^{3}}{{\hat{a}}_{21}} \iint_{Q_{T}} η (x) e^{- 2 s α} ξ^{3} σ Δ ψ ϕ d x d t | \\ + | \frac{s^{3}}{{\hat{a}}_{21}} \iint_{Q_{T}} η (x) e^{- 2 s α} ξ^{3} a_{12} (t, x) | ϕ |^{2} d x d t | \\ + | \frac{s^{3}}{{\hat{a}}_{21}} \iint_{Q T} η (x) e^{- 2 s α} ξ^{3} a_{22} (t, x) ψ ϕ d x d t | \\ + | \frac{s^{3}}{{\hat{a}}_{21}} \iint_{Q_{T}} η (x) e^{- 2 s α} ξ^{3} ψ Δ ϕ d x d t | \\ + | \frac{s^{3}}{{\hat{a}}_{21}} \iint_{Q_{T}} a_{11} (t, x) η (x) e^{- 2 s α} ξ^{3} ψ ϕ d x d t | = \sum_{i = 1}^{6} | K_{i} | . \end{aligned}

(2.16)

Before estimating the terms $K_{i}$ for $i \in {1, \dots, 6}$ , we establish the following estimates for the derivatives of the Carleman weights

\begin{aligned} | {(e^{- 2 s α} ξ^{3})}_{t} | ⩽ C s^{2} ξ^{5} e^{- 2 s α}, | {(e^{- 2 s \hat{α}} {\hat{ξ}}^{- 2})}_{t} | ⩽ C s^{2} e^{- 2 s \hat{α}}, \end{aligned}

(2.17)

\begin{aligned} | \nabla (e^{- 2 s α} ξ^{3}) | ⩽ C e^{- 2 s α} s ξ^{4}, | Δ (e^{- 2 s α} ξ^{3}) | ⩽ C e^{- 2 s α} s^{2} ξ^{5} . \end{aligned}

(2.18)

From (2.17), together with Cauchy–Schwarz and Young inequalities, we have

\begin{aligned} | K_{1} | & ⩽ C δ s^{3} \iint_{Q_{T}} e^{- 2 s α} ξ^{3} | ψ |^{2} d x d t + C_{δ} s^{7} \iint_{(0, T) \times ω_{1}} e^{- 2 s α} ξ^{7} | ϕ |^{2} d x d t, \end{aligned}

(2.19)

for any

δ \in (0, 1)

By definition of the functions $\hat{α}$ , $α^{*}$ , $\hat{ξ}$ y $ξ^{*}$ given in (2.6) and applying Cauchy–Schwarz and Young inequalities

\begin{aligned} | K_{2} | & ⩽ C σ^{2} s^{- 2} \iint_{Q_{T}} e^{- 2 s \hat{α}} {\hat{ξ}}^{- 2} | Δ ψ |^{2} d x d t + C s^{8} \iint_{(0, T) \times Ω} η^{2} (x) e^{- 4 s α + 2 s \hat{α}} {\hat{ξ}}^{2} ξ^{6} | ϕ |^{2} d x d t \\ ⩽ C σ^{2} s^{- 2} \iint_{Q_{T}} e^{- 2 s \hat{α}} {\hat{ξ}}^{- 2} | Δ ψ |^{2} d x d t + C s^{8} \iint_{(0, T) \times ω_{1}} e^{- 4 s α^{*} + 2 s \hat{α}} (ξ^{*})^{8} | ϕ |^{2} d x d t . \end{aligned}

(2.20)

To estimate $K_{3}$ , we recall that (2.12) holds. This implies that

\begin{aligned} ‖ a_{i j} ‖_{L^{\infty}} ⩽ {(\frac{s}{C T^{2}})}^{\frac{3}{2}} ⩽ C s^{\frac{3}{2}} ξ^{\frac{3}{2}}, \end{aligned}

(2.21)

for

i, j = 1, 2

. Using the previous estimate for the coefficient

a_{12}

we get

\begin{aligned} | K_{3} | ⩽ C s^{\frac{9}{2}} \iint_{(0, T) \times ω_{1}} e^{- 2 s α} ξ^{\frac{9}{2}} | ϕ |^{2} d x d t . \end{aligned}

(2.22)

In the same spirit, using (2.21) and applying the Cauchy-Schwarz and Young inequalities, we can estimate

\begin{aligned} | K_{i} | & ⩽ C \iint_{(0, T) \times Ω} η (x) e^{- 2 s α} s^{\frac{9}{2}} ξ^{\frac{9}{2}} | ψ | | ϕ | d x d t \\ ⩽ C δ s^{3} \iint_{Q_{T}} e^{- 2 s α} ξ^{3} | ψ |^{2} d x d t + C_{δ} s^{6} \iint_{(0, T) \times ω_{1}} e^{- 2 s α} ξ^{6} | ϕ |^{2} d x d t, \end{aligned}

(2.23)

for

i = 4, 6

and any

δ \in (0, 1)

Now, we estimate the term $| K_{5} |$ . Firstly, we integrate by parts the term $K_{5}$

\begin{aligned} K_{5} & = \frac{s^{3}}{{\hat{a}}_{21}} (- \iint_{Q_{T}} \nabla {η (x) e^{- 2 s α} ξ^{3} ψ} \nabla ϕ d x d t + \iint_{Σ_{T}} \frac{\partial ϕ}{\partial ν} η (x) e^{- 2 s α} ξ^{3} ψ d S d t) \\ = - \frac{s^{3}}{{\hat{a}}_{21}} \iint_{Q} \nabla {η (x) e^{- 2 s α} ξ^{3}} ψ \nabla ϕ d x d t - \frac{s^{3}}{{\hat{a}}_{21}} \iint_{Q} η (x) e^{- 2 s α} ξ^{3} \nabla ψ \nabla ϕ d x d t, \end{aligned}

(2.24)

where we have used that ϕ has homogeneous Neumann boundary conditions. Once again, integrating by parts in

Q_{T}

\begin{aligned} K_{5} = & \frac{s^{3}}{{\hat{a}}_{21}} \iint_{Q_{T}} Δ {η (x) e^{- 2 s α} ξ^{3}} ψ ϕ d x d t + \frac{s^{3}}{{\hat{a}}_{21}} \iint_{Q_{T}} \nabla {η (x) e^{- 2 s α} ξ^{3}} \nabla ψ ϕ d x d t \\ - \frac{s^{3}}{{\hat{a}}_{21}} \iint_{Σ} ϕ ψ \frac{\partial (η e^{- 2 s α} ξ^{3})}{\partial ν} d S d t - \frac{s^{3}}{{\hat{a}}_{21}} \iint_{Σ} ϕ \frac{\partial ψ}{\partial ν} e^{- 2 s α} ξ^{3} d S d t \\ + \frac{s^{3}}{{\hat{a}}_{21}} \iint_{Q_{T}} \nabla {η (x) e^{- 2 s α} ξ^{3}} \nabla ψ ϕ d x d t + \frac{s^{3}}{{\hat{a}}_{21}} \iint_{Q_{T}} η (x) e^{- 2 s α} ξ^{3} Δ ψ ϕ d x d t \\ = & \frac{s^{3}}{{\hat{a}}_{21}} \iint_{Q_{T}} Δ {η (x) e^{- 2 s α} ξ^{3}} ψ ϕ d x d t + \frac{2 s^{3}}{{\hat{a}}_{21}} \iint_{Q_{T}} \nabla {η (x) e^{- 2 s α} ξ^{3}} \nabla ψ ϕ d x d t \\ + \frac{s^{3}}{{\hat{a}}_{21}} \iint_{Q_{T}} η (x) e^{- 2 s α} ξ^{3} Δ ψ ϕ d x d t, \end{aligned}

where using that η is compactly supported in Ω (recall Eq. (2.14)) we have that

η = \nabla η = 0

\partial Ω

and the two integrals on the boundary vanish.

Finally, using (2.18) we obtain

\begin{aligned} | K_{5} | & ⩽ C \iint_{Q} | Δ η | e^{- 2 s α} s^{3} ξ^{3} | ψ | | ϕ | d x d t + C \iint_{Q} | \nabla η | e^{- 2 s α} s^{4} ξ^{4} | ψ | | ϕ | d x d t \\ + C \iint_{Q} η (x) C e^{- 2 s α} s^{5} ξ^{5} | ψ | | ϕ | d x d t + C \iint_{Q} | \nabla η | e^{- 2 s α} s^{3} ξ^{3} | \nabla ψ | | ϕ | d x d t \\ + C \iint_{Q} η (x) C e^{- 2 s α} s^{4} ξ^{4} | \nabla ψ | | ϕ | d x d t + C \iint_{Q} η (x) e^{- 2 s α} s^{3} ξ^{3} | Δ ψ | | ϕ | d x d t . \end{aligned}

By Young inequality and the fact that

η \in C_{0}^{\infty} (ω_{1})

we get

\begin{aligned} | K_{5} | & ⩽ C_{δ} s^{3} \iint_{(0, T) \times ω_{1}} e^{- 2 s α} ξ^{3} | ϕ |^{2} d x d t + 2 C_{δ} s^{5} \iint_{(0, T) \times ω_{1}} e^{- 2 s α} ξ^{5} | ϕ |^{2} d x d t \\ + 3 C_{δ} s^{7} \iint_{(0, T) \times ω_{1}} e^{- 2 s α} ξ^{7} | ϕ |^{2} d x d t + 3 C δ s^{3} \iint_{Q_{T}} e^{- 2 s α} ξ^{3} | ψ |^{2} d x d t \\ + 2 C δ s \iint_{Q_{T}} e^{- 2 s α} ξ | \nabla ψ |^{2} d x d t + C δ s^{- 1} \iint_{Q_{T}} e^{- 2 s α} ξ^{- 1} | Δ ψ |^{2} d x d t, \end{aligned}

(2.25)

for any

δ \in (0, 1)

Putting together (2.15), (2.16) and the estimates for $| K_{i} |$ , for $i = 1, \dots, 6$ , obtained in (2.19), (2.20), (2.22), (2.23) and (2.25) yields

\begin{aligned} s^{3} \iint_{(0, T) \times ω_{0}} e^{- 2 s α} ξ^{3} {| ψ |}^{2} d x d t \\ ⩽ 6 C δ s^{3} \iint_{Q_{T}} e^{- 2 s α} ξ^{3} {| ψ |}^{2} d x d t + 4 C_{δ} s^{7} \iint_{(0, T) \times ω_{1}} e^{- 2 s α} ξ^{7} {| ϕ |}^{2} d x d t \\ + C σ^{2} s^{- 2} \iint_{Q_{T}} e^{- 2 s \hat{α}} {\hat{ξ}}^{- 2} {| Δ ψ |}^{2} d x d t + C s^{8} \iint_{(0, T) \times ω_{1}} e^{- 4 s α^{*} + 2 s \hat{α}} (ξ^{*})^{8} {| ϕ |}^{2} d x d t \\ + C s^{\frac{9}{2}} \iint_{(0, T) \times ω_{1}} e^{- 2 s α} ξ^{\frac{9}{2}} {| ϕ |}^{2} d x d t + 2 C_{δ} s^{6} \iint_{(0, T) \times ω_{1}} e^{- 2 s α} ξ^{6} {| ϕ |}^{2} d x d t \\ + C_{δ} s^{3} \iint_{(0, T) \times ω_{1}} e^{- 2 s α} ξ^{3} {| ϕ |}^{2} d x d t + 2 C_{δ} s^{5} \iint_{(0, T) \times ω_{1}} e^{- 2 s α} ξ^{5} {| ϕ |}^{2} d x d t \\ + 2 C δ s \iint_{Q_{T}} e^{- 2 s α} ξ | \nabla ψ |^{2} d x d t + C δ s^{- 1} \iint_{Q_{T}} e^{- 2 s α} ξ^{- 1} | Δ ψ |^{2} d x d t . \end{aligned}

We obtain by the definition of

\hat{α}

α^{*}

in (2.6) and

ξ^{- 1} ⩽ C T^{2}

that

s^{p} ξ^{p} e^{- 2 s α} ⩽ C s^{8} ξ^{8} e^{- 4 s α^{*} + 2 s \hat{α}}

for any

p < 8

. Then, using this in the localized terms, we get

\begin{aligned} s^{3} \iint_{(0, T) \times ω_{0}} e^{- 2 s α} ξ^{3} | ψ |^{2} d x d t \\ ⩽ 6 C δ s^{3} \iint_{Q_{T}} e^{- 2 s α} ξ^{3} | ψ |^{2} d x d t + 2 C δ s \iint_{Q_{T}} e^{- 2 s α} ξ | \nabla ψ |^{2} d x d t \\ + C δ s^{- 1} \iint_{Q_{T}} e^{- 2 s α} ξ^{- 1} | Δ ψ |^{2} d x d t + C σ^{2} s^{- 2} \iint_{Q_{T}} e^{- 2 s \hat{α}} {\hat{ξ}}^{- 2} | Δ ψ |^{2} d x d t \\ + C s^{8} \iint_{(0, T) \times ω_{1}} e^{- 4 s α^{*} + 2 s \hat{α}} {(ξ^{*})}^{8} | ϕ |^{2} d x d t, \end{aligned}

(2.26)

for any

δ \in (0, 1)

Step 3: a uniform global estimate. Note that the first three terms in the right-hand side of (2.26) are multiplied by a small parameter δ but the fourth one has the large parameter σ in front of it. Thus, the idea is to estimate it uniformly with respect to σ.

Using the second equation of (2.2), multiplying by $e^{- 2 s \hat{α}} {\hat{ξ}}^{- 2} Δ ψ$ , integrating by parts, and applying the Cauchy-Schwarz inequality we obtain

\begin{aligned} σ \iint_{Q_{T}} e^{- 2 s \hat{α}} {\hat{ξ}}^{- 2} {| Δ ψ |}^{2} d x d t & ⩽ \iint_{Q_{T}} e^{- 2 s \hat{α}} {\hat{ξ}}^{- 2} \nabla ψ \nabla ψ_{t} d x d t - \iint_{Σ} \frac{\partial ψ}{\partial ν} ψ_{t} e^{- 2 s \hat{α}} {\hat{ξ}}^{- 2} d S d t \\ + \iint_{Q_{T}} e^{- 2 s \hat{α}} {\hat{ξ}}^{- 2} ‖ a_{12} ‖_{L^{\infty}} | Δ ψ | | ϕ | d x d t + \iint_{Q_{T}} e^{- 2 s \hat{α}} {\hat{ξ}}^{- 2} ‖ a_{22} ‖_{L^{\infty}} | Δ ψ | | ψ | d x d t . \end{aligned}

Taking into account (2.21) and the fact that

\hat{ξ} ⩽ ξ ⩽ C \hat{ξ}

(0, T) \times \bar{Ω}

, where C is a constant only depending on Ω and ω, we deduce

\begin{aligned} σ \iint_{Q_{T}} e^{- 2 s \hat{α}} {\hat{ξ}}^{- 2} | Δ ψ |^{2} d x d t & ⩽ \frac{1}{2} \iint_{Q_{T}} e^{- 2 s \hat{α}} {\hat{ξ}}^{- 2} {(| \nabla ψ |^{2})}_{t} d x d t + 2 δ σ \iint_{Q_{T}} e^{- 2 s \hat{α}} {\hat{ξ}}^{- 2} | Δ ψ |^{2} d x d t \\ + C_{δ} σ^{- 1} s^{3} \iint_{Q_{T}} e^{- 2 s \hat{α}} \hat{ξ} | ϕ |^{2} d x d t + C_{δ} σ^{- 1} s^{3} \iint_{Q_{T}} e^{- 2 s \hat{α}} \hat{ξ} | ψ |^{2} d x d t . \end{aligned}

(2.27)

In order to estimate the first integral at the right-hand side of inequality (2.27), we integrate on

(0, T)

and continue by integrating by parts on Ω, to obtain

\begin{aligned} \frac{1}{2} \iint_{Q_{T}} e^{- 2 s \hat{α}} {\hat{ξ}}^{- 2} {(| \nabla ψ |^{2})}_{t} d x d t \\ = \frac{1}{2} (\int_{Ω} e^{- 2 s \hat{α}} {\hat{ξ}}^{- 2} (| \nabla ψ |^{2}) d x |_{0}^{T} - \iint_{Q_{T}} {(e^{- 2 s \hat{α}} {\hat{ξ}}^{- 2})}_{t} | \nabla ψ |^{2} d x d t) \\ = \frac{1}{2} \iint_{Q_{T}} {(e^{- 2 s \hat{α}} {\hat{ξ}}^{- 2})}_{t} ψ Δ ψ d x d t \\ ⩽ C s^{2} \iint_{Q_{T}} e^{- 2 s \hat{α}} | ψ | | Δ ψ | d x d t, \end{aligned}

(2.28)

where we have used that

| (e^{- 2 s \hat{α}} {\hat{ξ}}^{- 2})_{t} | ⩽ C s^{2} e^{- 2 s \hat{α}}

to get the last line of the above expression. Substituting (2.28) into (2.27) and applying Young’s inequality yields

\begin{aligned} σ \iint_{Q_{T}} e^{- 2 s \hat{α}} {\hat{ξ}}^{- 2} | Δ ψ |^{2} d x d t \\ ⩽ 3 δ σ \iint_{Q_{T}} e^{- 2 s \hat{α}} {\hat{ξ}}^{- 2} | Δ ψ |^{2} d x d t + C_{δ} σ^{- 1} s^{4} \iint_{Q_{T}} e^{- 2 s \hat{α}} {\hat{ξ}}^{2} | ψ |^{2} d x d t \\ + C_{δ} σ^{- 1} s^{3} \iint_{Q_{T}} e^{- 2 s \hat{α}} \hat{ξ} | ϕ |^{2} d x d t + C_{δ} σ^{- 1} s^{3} \iint_{Q_{T}} e^{- 2 s \hat{α}} \hat{ξ} | ψ |^{2} d x d t, \end{aligned}

(2.29)

for

δ \in (0, 1)

. If we choose δ sufficiently small, observe that in (2.29) we can absorb the first term in the right-hand side to obtain

\begin{aligned} σ \iint_{Q_{T}} e^{- 2 s \hat{α}} {\hat{ξ}}^{- 2} | Δ ψ |^{2} d x d t & ⩽ C σ^{- 1} s^{4} \iint_{Q_{T}} e^{- 2 s \hat{α}} {\hat{ξ}}^{2} | ψ |^{2} d x d t \\ + C σ^{- 1} s^{3} \iint_{Q_{T}} e^{- 2 s \hat{α}} \hat{ξ} | ϕ |^{2} d x d t + C σ^{- 1} s^{3} \iint_{Q_{T}} e^{- 2 s \hat{α}} \hat{ξ} | ψ |^{2} d x d t . \end{aligned}

(2.30)

Multiplying the inequality (2.30) by

σ s^{- 2}

on both sides we get

\begin{aligned} σ^{2} s^{- 2} \iint_{Q_{T}} e^{- 2 s \hat{α}} {\hat{ξ}}^{- 2} | Δ ψ |^{2} d x d t & ⩽ C s^{2} \iint_{Q_{T}} e^{- 2 s \hat{α}} {\hat{ξ}}^{2} | ψ |^{2} d x d t \\ + C s \iint_{Q_{T}} e^{- 2 s \hat{α}} \hat{ξ} | ϕ |^{2} d x d t + C s \iint_{Q_{T}} e^{- 2 s \hat{α}} \hat{ξ} | ψ |^{2} d x d t, \end{aligned}

(2.31)

and using the properties of the functions defined in (2.5) and (2.6) we have that

e^{- 2 s \hat{α}} ⩽ e^{- 2 s α}

and

\hat{ξ} ⩽ ξ

which in combination with (2.31), yield

\begin{aligned} σ^{2} s^{- 2} \iint_{Q_{T}} e^{- 2 s \hat{α}} {\hat{ξ}}^{- 2} | Δ ψ |^{2} d x d t & ⩽ C s^{2} \iint_{Q_{T}} e^{- 2 s α} ξ^{2} | ψ |^{2} d x d t \\ + C s \iint_{Q_{T}} e^{- 2 s α} ξ | ϕ |^{2} d x d t + C s \iint_{Q_{T}} e^{- 2 s α} ξ | ψ |^{2} d x d t . \end{aligned}

(2.32)

Step 4: conclusion. Recalling (2.8), we put put together estimates (2.13), (2.26) and (2.32). Then taking $δ \in (0, 1)$ small enough and choosing the parameter s as in (2.12) we get

\begin{aligned} s^{3} \iint_{Q} e^{- 2 s α} ξ^{3} | ϕ |^{2} d x d t + s^{3} \iint_{Q} e^{- 2 s α} ξ^{3} | ψ |^{2} d x d t ⩽ C s^{8} \iint_{(0, T) \times ω} e^{- 4 s α^{*} + 2 s \hat{α}} (ξ^{*})^{8} {| ϕ |}^{2}, \end{aligned}

which is the desired result.

We recall the following result concerning Carleman weights.

Proposition 2.5 ([35, Lemma 6.1])

For any $ϵ > 0$ there exists $λ_{0} > 0$ such that, for every $λ > λ_{0}$ such that

\begin{aligned} e^{\hat{α} s} ⩽ e^{(1 + ϵ) α^{*} s}, \end{aligned}

(2.33)

for all

s > 0

Now, we are in position to prove Proposition 2.1. Proof of Proposition 2.1. Proof of Proposition 2.1.

From Proposition 2.4, we have that for any $(ϕ_{T}, ψ_{T}) \in [L^{2} (Ω)]^{2}$ , the solutions $(ϕ, ψ)$ of (2.2) verify

\begin{aligned} s^{3} \iint_{Q_{T}} e^{- 2 s α} ξ^{3} | ϕ |^{2} d x d t + s^{3} \iint_{Q_{T}} e^{- 2 s α} ξ^{3} | ψ |^{2} d x d t \\ ⩽ C s^{8} \iint_{(0, T) \times ω} e^{- 4 s α^{*} + 2 s \hat{α}} {(ξ^{*})}^{8} | ϕ |^{2} d x d t, \end{aligned}

(2.34)

for any

\begin{aligned} s ⩾ C (T + T^{2} + T^{2} max_{i, j = 1, 2} ‖ a_{i j} ‖_{L^{\infty}}^{\frac{2}{3}}) . \end{aligned}

(2.35)

where

C > 0

is a constant independent of T and

a_{i j}

Now, our goal is to bound the exponential weights in both sides of (2.34). To simplify the notation, we introduce the following

\begin{aligned} \begin{aligned} {\tilde{M}}_{0} := max_{x \in \bar{Ω}} (e^{2 λ ‖ η^{0} ‖_{L^{\infty}}} - e^{λ η^{0} (x)}), M_{0} := max_{x \in \bar{Ω}} e^{λ η^{0} (x)}, \\ {\tilde{m}}_{0} := min_{x \in \bar{Ω}} (e^{2 λ ‖ η^{0} ‖_{L^{\infty}}} - e^{λ η^{0} (x)}), m_{0} := min_{x \in \bar{Ω}} e^{λ η^{0} (x)} . \end{aligned} \end{aligned}

Let $s_{0} := max {C, \frac{1}{2^{4} m_{0}}}$ where C is the constant appearing in (2.35) and fix

\begin{aligned} s = s_{0} (T + T^{2} + T^{2} max_{i, j = 1, 2} ‖ a_{i j} ‖_{L^{\infty}}^{\frac{2}{3}}) . \end{aligned}

(2.36)

We prove first that

\begin{aligned} s^{8} (ξ^{*})^{8} e^{- 4 s α^{*} + 2 s \hat{α}} ⩽ {(\frac{2^{3} M_{0}}{{\tilde{m}}_{0} e})}^{8}, \end{aligned}

(2.37)

(0, T)

. Applying (2.33) with

ϵ = \frac{1}{2}

to the left-hand term of (2.37) we see that

\begin{aligned} s^{8} {ξ^{*}}^{8} e^{- 4 s α^{*} + 2 s \hat{α}} & ⩽ s^{8} {ξ^{*}}^{8} e^{- α^{*} s} = s^{8} \frac{M_{0}^{8}}{t^{8} (t - T)^{8}} e^{- \frac{{\tilde{m}}_{0}}{t (T - t)} s} \\ ⩽ s^{8} max_{t \in (0, T)} {\frac{M_{0}^{8}}{t^{8} (t - T)^{8}} e^{- \frac{{\tilde{m}}_{0}}{t (T - t)} s}} = s^{8} M_{0}^{8} \frac{2^{16}}{T^{16}} e^{- \frac{2^{2} {\tilde{m}}_{0}}{T^{2}} s} \\ ⩽ M_{0}^{8} \frac{2^{16}}{T^{16}} max_{s > 0} {s^{8} e^{- \frac{2^{2} {\tilde{m}}_{0}}{T^{2}} s}} = {(\frac{2^{3} M_{0}}{{\tilde{m}}_{0} e})}^{8}, \end{aligned}

(2.38)

where we have used that

max_{x > 0} x^{a} e^{- b x} = (\frac{a}{e b})^{a}

x = \frac{a}{b}

On the other hand

\begin{aligned} s^{3} ξ^{3} e^{- 2 s α} ⩾ \frac{1}{3^{3}} e^{- \frac{C s}{T^{2}}} in (\frac{T}{4}, \frac{3 T}{4}) \times Ω . \end{aligned}

(2.39)

Indeed,

\begin{aligned} s^{3} ξ^{3} e^{- 2 s α} & ⩾ \frac{s^{3} m_{0}^{3}}{t^{3} (T - t)^{3}} e^{- \frac{2 {\tilde{M}}_{0}}{t (T - t)} s} \\ ⩾ s^{3} m_{0}^{3} min_{t \in [\frac{T}{4}, \frac{3 T}{4}]} {\frac{1}{t^{3} (t - t)^{3}} e^{- \frac{2 {\tilde{M}}_{0}}{t (T - t)} s}} \\ = s^{3} {(\frac{2^{4} m_{0}}{3 T^{2}})}^{3} e^{- \frac{2^{5} {\tilde{M}}_{0}}{3 T^{2}} s} . \end{aligned}

(2.40)

From the particular choice of s in (2.36),

s ⩾ \frac{T^{2}}{2^{4} m_{0}}

and this yields (2.39). Using the bounds (2.38) and (2.40) in (2.34) gives

\begin{aligned} \iint_{(T / 4, 3 T / 4) \times Ω} | ϕ |^{2} d x d t + \iint_{(T / 4, 3 T / 4) \times Ω} | ψ |^{2} d x d t ⩽ C e^{\frac{C s}{T^{2}}} \iint_{(0, T) \times ω} | ϕ |^{2} d x d t, \end{aligned}

(2.41)

for some

C > 0

independent of T and

a_{i j}

Now, multiplying the first equation of (2.2) by ϕ and integrating over Ω (resp. multiplying the second equation of (2.2) by ψ and integrating over Ω), we can deduce after some direct computations that

\begin{aligned} - \frac{d}{d t} \int_{Ω} ({| ϕ |}^{2} + {| ψ |}^{2}) d x + 2 \int_{Ω} ({| \nabla ϕ |}^{2} + {| \nabla ψ |}^{2}) d x \\ ⩽ 2 (1 + \sum_{i, j} ‖ a_{i j} ‖_{L^{\infty}}) \int_{Ω} ({| ϕ |}^{2} + {| ψ |}^{2}) d x . \end{aligned}

Then

\begin{aligned} \frac{d}{d t} (e^{2 (1 + \sum_{i, j} ‖ a_{i j} ‖_{L^{\infty}}) t} \int_{Ω} (| ϕ |^{2} + | ψ |^{2}) d x) ⩾ 0. \end{aligned}

(2.42)

Let us now integrate this inequality on

[\frac{T}{4}, t]

with

t \in [T / 4, 3 T / 4]

\begin{aligned} \int_{Ω} (| ϕ (t) |^{2} + | ψ (t) |^{2}) d x ⩾ e^{- T (1 + \sum_{i, j} ‖ a_{i j} ‖_{L^{\infty}})} \int_{Ω} (| ϕ (T / 4, x) |^{2} + | ψ (T / 4, x) |^{2}) d x . \end{aligned}

Integrating on

[T / 4, 3 T / 4]

and after some straightforward estimates we have

\begin{aligned} \int_{Ω} ({| ϕ (T / 4, x) |}^{2} + {| ψ (T / 4, x) |}^{2}) d x ⩽ C e^{C (T^{- 1} + T \sum_{i, j} ‖ a_{i j} ‖_{L^{\infty}})} \iint_{(T / 4, 3 T / 4) \times Ω} ({| ϕ |}^{2} + {| ψ |}^{2}) d x . \end{aligned}

(2.43)

On the other hand, using again the inequality (2.42) and integrating on

[0, T / 4]

we have

\begin{aligned} \int_{Ω} ({| ϕ (0, x) |}^{2} + {| ψ (0, x) |}^{2}) d x ⩽ e^{C T (1 + \sum_{i, j} ‖ a_{i j} ‖_{L^{\infty}})} \int_{Ω} ({| ϕ (T / 4, x) |}^{2} + {| ψ (T / 4, x) |}^{2}) d x . \end{aligned}

(2.44)

Finally, combining the estimates (2.41), (2.43) and (2.44), we obtain the observability inequality (2.3).

2.2. Controllability of the linear system with explicit control cost

In order to find a null control for (2.1) with explicit control cost and uniform with respect to diffusion coefficient σ, we follow the penalized Hilbert Uniqueness Method (HUM) (see e.g. [14] or [4]).

We begin by stating a uniform energy estimate (w.r.t. σ) for the solutions of system (2.1). The result reads as follows.

Lemma 2.6.
Let $σ ⩾ 1$ , $(y^{0}, z^{0}) \in [L^{2} (Ω)]^{2}$ and $h \in L^{2} (0, T; L^{2} (ω))$ be given. There exists a constant $C > 0$ depending only Ω such that $(y, z)$ the solution of system (2.1) satisfies the following estimate
$\begin{aligned} ‖ y ‖_{L^{2} (0, T; H^{1} (Ω))}^{2} + ‖ z ‖_{L^{2} (0, T; H^{1} (Ω))}^{2} + ‖ y_{t} ‖_{L^{2} (0, T; {(H^{1} (Ω))}^{'})}^{2} + ‖ z_{t} ‖_{L^{2} (0, T; (H^{1} (Ω))^{'})}^{2} + σ \iint_{Q_{T}} | \nabla z |^{2} d x d t \\ ⩽ e^{C \tilde{K}} ({‖ y^{0} ‖}_{L^{2} (Ω)}^{2} + {‖ z^{0} ‖}_{L^{2} (Ω)}^{2} + ‖ h ‖_{L^{2} (0, T; L^{2} (ω))}^{2}), \end{aligned}$
(2.45)
where
$\begin{aligned} \tilde{K} = (1 + T) (1 + \sum_{i, j = 1}^{2} ‖ a_{i j} ‖_{L^{\infty}}) \end{aligned}$
(2.46)

For the proof, we can follow a classical methodology (see e.g. [8, Section 7.1]), just by taking care that in each step the estimates are independent of σ. For brevity, we omit it.

Now, we are in position to prove the following. Theorem 2.7.
Under the assumptions of Proposition 2.1, for every $(y^{0}, z^{0}) \in [L^{2} (Ω)]^{2}$ there exists a control $h \in L^{2} (0, T; L^{2} (ω))$ such that the solution $(y, z)$ of the system (2.1) satisfies $y (T, \cdot) = z (T, \cdot) = 0$ in Ω. Moreover, we have that the control cost is given by
$\begin{aligned} ‖ h ‖_{L^{2} (0, T; L^{2} (ω))} ⩽ e^{C K} ({‖ y^{0} ‖}_{L^{2} (Ω)}^{2} + {‖ z^{0} ‖}_{L^{2} (Ω)}^{2}), \end{aligned}$
(2.47)
where $C > 0$ is a constant independent of σ and K is defined in (2.4).
Remark 2.8.
There are at least two different ways to prove Theorem 2.7. The first one is through some functional analysis arguments (see e.g., [ 7 , Section 2.3, Theorem 2.44 and Lemma 2.48]) which are valid in a very general abstract setting. We have decided to include a proof using the penalized HUM method since it relies on easy and direct arguments that can be useful for future reference.
Proof of Theorem 2.7. Proof of Theorem 2.7.

For readability, we have divided the proof in three steps.

Step 1: Let $σ ⩾ 1$ be fixed and consider the following minimization problem

\begin{aligned} min_{h \in L^{2} ((0, T) \times ω)} F_{ε} (h) \end{aligned}

where, for every

ϵ > 0

, we write

\begin{aligned} F_{ε} (h) = \frac{1}{2} \iint_{(0, T) \times ω} | h |^{2} d x d t + \frac{1}{2 ε} (\int_{Ω} {| y (T) |}^{2} d x + \int_{Ω} {| z (T) |}^{2} d x) . \end{aligned}

(2.48)

Since

F_{ε}

is continuous, strictly convex, and coercive, there exists a unique minimizer that we denote by

h_{ε}

. By a classical procedure (see, for instance, [29]), i.e., obtaining the Euler-Lagrange equation for (2.48) at the minimum

h_{ε}

and a duality argument, the control

h_{ε}

can be characterized as

\begin{aligned} h_{ε} = - ϕ_{ε} |_{ω} in Q_{T} \end{aligned}

(2.49)

where

ϕ_{ε}

is the solution of the first component of

(ϕ_{ε}, ψ_{ε})

verifying

\begin{aligned} {\begin{cases} - {ϕ_{ε}}_{t} - Δ ϕ_{ε} = a_{11} ϕ_{ε} + a_{21} ψ_{ε} & in Q_{T}, \\ - {ψ_{ε}}_{t} - σ Δ ψ_{ε} = a_{12} ϕ_{ε} + a_{22} ψ_{ε} & in Q_{T}, \\ \frac{\partial ϕ_{ε}}{\partial ν} = \frac{\partial ψ_{ε}}{\partial ν} = 0 & on Σ_{T}, \\ ϕ_{ε} (T, \cdot) = ε^{- 1} y_{ε} (T), ψ_{ε} (T, \cdot) = ε^{- 1} z_{ε} (T) & in Ω, \end{cases} \end{aligned}

and where

(y_{ε} (T), z_{ε} (T))

can be extracted from the solution to system (2.1) with control

h = h_{ε}

, more precisely,

\begin{aligned} {\begin{cases} y_{ε t} - Δ y_{ε} = a_{11} y_{ε} + a_{12} z_{ε} + h_{ε} 1_{ω} & in Q_{T}, \\ z_{ε t} - σ Δ z_{ε} = a_{21} y_{ε} + a_{22} z_{ε} & in Q_{T}, \\ \frac{\partial y_{ε}}{\partial ν} = \frac{\partial z_{ε}}{\partial ν} = 0 & on Σ_{T}, \\ y_{ε} (0, \cdot) = y^{0}, z_{ε} (0, \cdot) = z^{0} & in Ω . \end{cases} \end{aligned}

(2.50)

Let us prove the following convergences

\begin{aligned} \begin{aligned} h_{ε} ⇀ h in L^{2} ((0, T) \times ω), \\ (y_{ε} (T), z_{ε} (T)) \to (0, 0) in {[L^{2} (Ω)]}^{2}, \end{aligned} \end{aligned}

(2.51)

for some h in

L^{2} ((0, T) \times ω)

. Indeed, by duality between the solutions of

(y_{ε}, z_{ε})

and

(ϕ_{ε}, ψ_{ε})

, we have

\begin{aligned} \frac{1}{ε} \int_{Ω} ({| y_{ε} (T) |}^{2} + {| z_{ε} (T) |}^{2}) d x + \iint_{(0, T) \times ω} | ϕ_{ε} |^{2} d x d t = \int_{Ω} (y^{0} ϕ_{ε} (0) + z^{0} ψ_{ε} (0)) d x . \end{aligned}

(2.52)

Applying Cauchy-Schwarz inequality and using estimate (2.3) in the right-hand side of (2.52) we see that

\begin{aligned} \int_{Ω} (y^{0} ϕ_{ε} (0) + z^{0} ψ_{ε} (0)) d x ⩽ {‖ (y^{0}, z^{0}) ‖}_{{(L^{2} (Ω))}^{2}} {(e^{C K} \iint_{(0, T) \times ω} | ϕ_{ε} |^{2} d x d t)}^{\frac{1}{2}} . \end{aligned}

(2.53)

and combining (2.52), (2.53), and applying Young’s inequality with

δ > 0

to the right-hand side of (2.53) we get

\begin{aligned} \frac{1}{ε} \int_{Ω} ({| y_{ε} (T) |}^{2} + {| z_{ε} (T) |}^{2}) d x + \iint_{(0, T) \times ω} | h_{ε} |^{2} d x d t ⩽ e^{C K} {‖ (y^{0}, z^{0}) ‖}_{{(L^{2} (Ω))}^{2}}^{2} + δ \iint_{(0, T) \times ω} | h_{ε} |^{2} d x, \end{aligned}

(2.54)

where we have used (2.49). Taking

δ > 0

sufficiently small, we can eliminate the local term in the right-hand side of (2.54) to obtain

\begin{aligned} \frac{1}{ε} \int_{Ω} ({| y_{ε} (T) |}^{2} + {| z_{ε} (T) |}^{2}) d x + \iint_{(0, T) \times ω} | h_{ε} |^{2} d x d t ⩽ e^{C K} {‖ (y^{0}, z^{0}) ‖}_{{(L^{2} (Ω))}^{2}}^{2} . \end{aligned}

(2.55)

Note that this estimate is independent of the diffusion coefficient σ and the parameter

ε

. Therefore, the estimate (2.55) implies the convergences of (2.51). To conclude this step, we obtain a uniform bound on the control h with respect to σ as follows. Since

\begin{aligned} h_{ε} ⇀ h in L^{2} ((0, T) \times ω), \end{aligned}

(2.56)

holds in (2.51) we can apply Fatou’s Lemma (see [5, Colloraly II.2.8]) to get

\begin{aligned} ‖ h ‖_{L^{2} (0, T; L^{2} (ω))} ⩽ \underset{ε \to 0}{\lim \inf} ‖ h_{ε} ‖_{L^{2} (0, T; L^{2} (ω))} . \end{aligned}

(2.57)

Using (2.55) in the right-hand side of (2.57) we obtain

\begin{aligned} ‖ h ‖_{L^{2} (0, T; L^{2} (ω))} ⩽ e^{C K} ({‖ y^{0} ‖}_{L^{2} (Ω)}^{2} + {‖ z^{0} ‖}_{L^{2} (Ω)}^{2}), \end{aligned}

(2.58)

where C is independent of σ,

ε

and where K is defined in (2.4). This proves the control estimate in (2.47).

Step 2: Here we prove that for any $σ ⩾ 1$ fixed

\begin{aligned} \begin{aligned} y_{ε} ⇀ y en L^{2} (0, T; H^{1} (Ω)), \\ z_{ε} ⇀ z en L^{2} (0, T; H^{1} (Ω)), \end{aligned} \end{aligned}

(2.59)

uniformly with respect

ε > 0

, where

(y, z) \in L^{2} (0, T; H^{1} (Ω))

satisfies (2.1) with the control h defined in Step 1.

To this end, using Lemma 2.6, the solution $(y_{ε}, z_{ε})$ to (2.50) satisfies

\begin{aligned} ‖ y_{ε} ‖_{L^{2} (0, T; H^{1} (Ω))}^{2} + ‖ z_{ε} ‖_{L^{2} (0, T; H^{1} (Ω))}^{2} ⩽ e^{C \tilde{K}} ({‖ y^{0} ‖}_{L^{2} (Ω)}^{2} + {‖ z^{0} ‖}_{L^{2} (Ω)}^{2} + ‖ h_{ε} ‖_{L^{2} (0, T; L^{2} (ω))}^{2}), \end{aligned}

where

C > 0

is independent of σ and

ε

, and where we recall that

\tilde{K}

is defined in (2.46). Using (2.55) on the right-hand of the above expression yields

\begin{aligned} ‖ y_{ε} ‖_{L^{2} (0, T; H^{1} (Ω))}^{2} + ‖ z_{ε} ‖_{L^{2} (0, T; H^{1} (Ω))}^{2} ⩽ e^{C \bar{K}} ({‖ y^{0} ‖}_{L^{2} (Ω)}^{2} + {‖ z^{0} ‖}_{L^{2} (Ω)}^{2}), \end{aligned}

with

\bar{K} = 1 + T + \frac{1}{T} + (1 + T) \sum_{i, j = 1}^{2} ‖ a_{i j} ‖_{L^{\infty}} + max_{i, j = 1, 2} ‖ a_{i j} ‖_{L^{\infty}}^{\frac{2}{3}}

Therefore, we can extract a subsequence (still denoted) ${(y_{ε}, z_{ε})}_{ε ⩾ 0}$ such that it verifies the weak convergences in (2.59). To check that the limits in (2.59) satisfy (2.1) with the control h obtained in the previous step, we argue as follows.

Let us denote by $(\hat{y}, \hat{z})$ the solution to

\begin{aligned} {\begin{cases} {\hat{y}}_{t} - Δ \hat{y} = a_{11} \hat{y} + a_{12} \hat{z} + h 1_{ω} & in Q_{T}, \\ {\hat{z}}_{t} - σ Δ \hat{z} = a_{21} \hat{y} + a_{22} \hat{z} & in Q_{T}, \\ \frac{\partial \hat{y}}{\partial ν} = \frac{\partial \hat{z}}{\partial ν} = 0 & on Σ_{T}, \\ \hat{y} (0, \cdot) = y^{0}, \hat{z} (0, \cdot) = z^{0} & in Ω, \end{cases} \end{aligned}

(2.60)

where h is a control provided by (2.51). For any

(F_{1}, F_{2}) \in [L^{2} (Q_{T})]^{2}

and any

(Φ_{T}, Ψ_{T}) \in [L^{2} (Ω)]^{2}

, let us introduce the following adjoint system

\begin{aligned} {\begin{cases} - Φ_{t} - Δ Φ = a_{11} Φ + a_{21} Ψ + F_{1} & in Q_{T}, \\ - Ψ_{t} - σ Δ Ψ = a_{12} Φ + a_{22} Ψ + F_{2} & in Q_{T}, \\ \frac{\partial Φ}{\partial ν} = \frac{\partial Ψ}{\partial ν} = 0 & on Σ_{T}, \\ Φ (T, \cdot) = Φ_{T}, Ψ (T, \cdot) = Ψ_{T} & in Ω . \end{cases} \end{aligned}

(2.61)

Setting $(Φ_{T}, ψ_{T}) = (0, 0)$ in (2.61), we obtain by duality between (2.60) and (2.61) that

\begin{aligned} - \int_{Ω} (y^{0} Φ (0) + z^{0} Ψ (0)) d x = - \iint_{Q_{T}} (\hat{y} F_{1} + \hat{z} F_{2}) d x d t + \iint_{Q_{T}} h Φ d x d t . \end{aligned}

(2.62)

In the same spirit, from (2.50), (2.61), and recalling that we have set zero initial data, we have

\begin{aligned} - \int_{Ω} (y^{0} ϕ (0) + z^{0} ψ (0)) d x = - \iint_{Q_{T}} (y_{ε} F_{1} + z_{ε} F_{2}) d x d t + \iint_{Q_{T}} h_{ε} ϕ d x d t . \end{aligned}

(2.63)

From (2.56) and (2.59), we can pass to the limit as

ε \to 0

in (2.63) and this, together with (2.62), yields

\begin{aligned} \iint_{Q_{T}} (y - \hat{y}) F_{1} d x d t + \iint_{Q_{T}} (z - \hat{z}) F_{2} d x d t = 0, \end{aligned}

for all

(F_{1}, F_{2}) \in [L^{2} (Q_{T})]^{2}

. This implies that

(y, z) = (\hat{y}, \hat{z})

which proves our initial claim.

Step 3: To conclude, let us see that the limit h obtained in Step 1 is in fact a null control for (2.1). From the conclusion of Step 2, by duality between $(y, z)$ solution to (2.1) and (2.61) with $(F_{1}, F_{2}) = (0, 0)$ , we deduce

\begin{aligned} \int_{Ω} (y (T) Φ_{T} + z (T) Ψ_{T}) d x = \iint_{(0, T) \times ω} h Φ d x d t + \int_{Ω} (y^{0} Φ (0) + z^{0} Φ (0)) d x d t \end{aligned}

(2.64)

Similarly, we can readily check that the solutions

(y_{ε}, z_{ε})

and

(Φ, Ψ)

to (2.50) and (2.61) (with

(F_{1}, F_{2}) = (0, 0)

) respectively, verify

\begin{aligned} \int_{Ω} (y_{ε} (T) Φ_{T} + z_{ε} (T) Ψ_{T}) d x = \iint_{Q_{T}} h_{ε} Φ d x d t + \int_{Ω} (y^{0} Φ (0) + z^{0} Φ (0)) d x d t . \end{aligned}

(2.65)

Recalling the convergences provided in (2.51) and passing the limit as $ε \to 0$ in (2.65) yield

\begin{aligned} \iint_{(0, T) \times ω} h Φ d x d t + \int_{Ω} (y^{0} Φ (0) + z^{0} Φ (0)) d x d t = 0. \end{aligned}

(2.66)

Substituting (2.66) in (2.64) gives

\begin{aligned} \int_{Ω} (y (T) Φ_{T} + z (T) Ψ_{T}) d x = 0, \end{aligned}

for all

(Φ_{T}, Ψ_{T}) \in [L^{2} (Ω)]^{2}

. Then

y (T, \cdot) = z (T, \cdot) = 0

in Ω as claimed. This ends the proof.

3. Controllability of the nonlinear system: Proof of Theorem 1.4

In this section, we prove Theorem 1.4. This result relies on three key components: a suitable linearization of system (1.3), the uniform controllability result presented in Theorem 2.7, and the application of Schaefer’s fixed point theorem. For clarity, we recall the latter.

Theorem 3.1 ([8, Section 9.2])

Let X be a real Banach space. Suppose that $A : X \to X$ is a continuous and compact mapping. Assume further that the set ${u \in X : u = λ A [u] for some 0 ⩽ λ ⩽ 1}$ is bounded. Then A has a fixed point.

Proof of Theorem 1.4. Proof of Theorem 1.4.

Let $σ ⩾ 1$ be fixed. By means of Taylor formula, the nonlinear system (1.3) can be rewritten as

\begin{aligned} {\begin{cases} y_{t} - Δ y = a_{11} (y, z) y + a_{12} (y, z) z + h 1_{ω} & in Q_{T}, \\ z_{t} - σ Δ z = a_{21} (y, z) y + a_{22} (y, z) z & in Q_{T}, \\ \frac{\partial y}{\partial ν} = \frac{\partial z}{\partial ν} = 0 & on Σ_{T}, \\ y (0, \cdot) = y^{0}, z (0, \cdot) = z^{0}, & in Ω, \end{cases} \end{aligned}

(3.1)

where

\begin{aligned} \begin{aligned} a_{11} (y, z) = \int_{0}^{1} \frac{\partial f}{\partial y} (δ y, δ z) d δ, a_{12} (y, z) = \int_{0}^{1} \frac{\partial f}{\partial z} (δ y, δ z) d δ, \\ a_{21} (y, z) = \int_{0}^{1} \frac{\partial g}{\partial y} (δ y, δ z) d δ, a_{22} (y, z) = \int_{0}^{1} \frac{\partial g}{\partial z} (δ y, δ z) d δ . \end{aligned} \end{aligned}

For each $(\bar{y}, \bar{z}) \in [L^{2} (Q_{T})]^{2}$ , let us consider the linearized system

\begin{aligned} {\begin{cases} y_{t} - Δ y = {\bar{a}}_{11} y + {\bar{a}}_{12} z + h 1_{ω} & in Q_{T}, \\ z_{t} - σ Δ z = {\bar{a}}_{21} y + {\bar{a}}_{22} z & in Q_{T}, \\ \frac{\partial y}{\partial ν} = \frac{\partial z}{\partial ν} = 0, & on Σ_{T}, \\ y (0, \cdot) = y^{0}, z (0, \cdot) = z^{0}, & in Ω, \end{cases} \end{aligned}

(3.2)

where

{\bar{a}}_{11} = \int_{0}^{1} \frac{\partial f}{\partial y} (δ \bar{y}, δ \bar{z}) d δ

, and similarly,

{\bar{a}}_{12}

{\bar{a}}_{21}

, and

{\bar{a}}_{22}

are defined with analogous notation.

Note that hypothesis (H1) guarantees that there is a constant C that only depending on T, Ω, $C_{f}$ and $C_{g}$ such that

\begin{aligned} ‖ {\bar{a}}_{i j} ‖_{L^{\infty} (Q_{T})} ⩽ C, i, j = 1, 2. \end{aligned}

(3.3)

Moreover, we note that the assumption (H3) implies that there exist a constant

{\hat{a}}_{21} > 0

such that

{\bar{a}}_{21} (t, x) ⩾ {\hat{a}}_{21}

- {\bar{a}}_{21} (t, x) > {\hat{a}}_{21}

for all

(t, x) \in Q_{T}

and, in particular, in

(0, T) \times ω

Therefore, the hypotheses of Theorem 2.7 are satisfied and we can build a control h such that the solution $(y, z)$ to (3.2) satisfy

\begin{aligned} y (T, x; \bar{y}, \bar{z}) = z (T, x; \bar{y}, \bar{z}) = 0, x \in Ω, \end{aligned}

for each

σ ⩾ 1

and each

(\bar{y}, \bar{z}) \in [L^{2} (Q_{T})]^{2}

. By construction, such control is uniformly bounded with respect to σ, and thanks to (3.3) it is also uniformly bounded with respect to

(\bar{y}, \bar{z})

Now, we consider the following map $Λ : [L^{2} (Q_{T})]^{2} \to [L^{2} (Q_{T})]^{2}$ given by

\begin{aligned} Λ (\bar{y}, \bar{z}) = (y (t, x, \bar{y}, \bar{z}), z (t, x, \bar{y}, \bar{z})), \end{aligned}

where

(\bar{y}, \bar{z}) \in [L^{2} (Q_{T})]^{2}

and

(y, z)

is solution of (3.2). Due to the Aubin–Lions Lemma, the space

\begin{aligned} W = {u : u \in L^{2} (0, T; H^{1} (Ω)), u_{t} \in L^{2} (0, T; (H^{1} (Ω))^{^{'}})}, \end{aligned}

is compactly embedded into

L^{2} (Q_{T})

. This, together with the energy estimate given in Lemma 2.6, yields that Λ is continuous and compact mapping in

[L^{2} (Q_{T})]^{2}

into itself. Also, using the Lemma 2.6 we can see that the set

\begin{aligned} M = {(y, z) \in [L^{2} (Q_{T})]^{2} : (y, z) = λ Λ (y, z) for some 0 ⩽ λ ⩽ 1}, \end{aligned}

is bounded in

[L^{2} (Q_{T})]^{2}

. Indeed, for every

(y, z) \in M

, according to the energy estimate (2.45) and the uniform bound on the control h with respect to

σ ⩾ 1

given in (2.47) we have that

\begin{aligned} {‖ (y, z) ‖}_{[L^{2} (Q_{T})]^{2}}^{2} & = {‖ λ Λ (y, z) ‖}_{[L^{2} (Q_{T})]^{2}}^{2} = λ^{2} {‖ (y, z) ‖}_{[L^{2} (Q_{T})]^{2}}^{2} \\ ⩽ e^{C \tilde{K}} ({‖ y^{0} ‖}_{L^{2} (Ω)}^{2} + {‖ z^{0} ‖}_{L^{2} (Ω)}^{2} + ‖ h ‖_{L^{2} (0, T; L^{2} (ω))}^{2}) \\ ⩽ C ({‖ y^{0} ‖}_{L^{2} (Ω)}^{2} + {‖ z^{0} ‖}_{L^{2} (Ω)}^{2}), \end{aligned}

where in last line we have used (3.3) to obtain that the constant C only depends on T, Ω,

C_{f}

and

C_{g}

Therefore, the map Λ satisfies all the hypotheses of Theorem 3.1 and there is $(y, z) \in (L^{2} (0, T; L^{2} (Ω)))^{2}$ such that $Λ (y, z) = (y, z)$ . Consequently, $y (T) = z (T) = 0$ in Ω. Moreover, by construction is clear that $h = h (σ)$ is uniformly bounded as in (2.47) and using (3.3) we obtain

\begin{aligned} ‖ h (σ) ‖_{L^{2} ((0, T) \times ω)} ⩽ C ({‖ y^{0} ‖}_{L^{2} (Ω)}^{2} + {‖ z^{0} ‖}_{L^{2} (Ω)}^{2}), \end{aligned}

for every

σ ⩾ 1

fixed and some constant

C > 0

only depending on Ω, ω, T,

C_{f}

, and

C_{g}

Since $(y, z)$ is solution of the linearized system (3.2) for any $(\bar{y}, \bar{z}) \in [L^{2} (Q_{T})]^{2}$ we have that satisfies the energy estimate (2.45). In particular, for the fixed point $(y, z)$ obtained above we get

\begin{aligned} ‖ y ‖_{L^{2} (0, T; H^{1} (Ω))}^{2} + ‖ z ‖_{L^{2} (0, T; H^{1} (Ω))}^{2} + ‖ y_{t} ‖_{L^{2} (0, T; (H^{1} (Ω))^{^{'}})}^{2} + ‖ z_{t} ‖_{L^{2} (0, T; (H^{1} (Ω))^{^{'}})}^{2} + σ \iint_{Q_{T}} | \nabla z |^{2} d x d t \\ ⩽ C ({‖ y^{0} ‖}_{L^{2} (Ω)}^{2} + {‖ z^{0} ‖}_{L^{2} (Ω)}^{2}), \end{aligned}

where we have used again (3.3) to obtain a constant C only depending on T, Ω, ω,

C_{f}

, and

C_{g}

. Thus the proof of Theorem 1.4 is finished.

4. Passing to the shadow limit: Proof of Theorem 1.3

This section is devoted to proof our main result for controllability of semilinear shadow systems, that is, Theorem 1.3. The proof relies on the following well-known properties for the heat semigroup with Neumann boundary conditions.

Lemma 4.1.
Let ${e^{t σ Δ}}$ be the heat semigroup with the Neumann boundary conditions in a bounded domain $Ω \subset R^{N}$ with smooth boundary and such that $| Ω | = 1$ . (1)
For every $C \in R$ , we have $e^{t σ Δ} C = C$ for all $t ⩾ 0$ .
(2)
There is a constant $C > 0$ only depending on Ω such that for all $1 ⩽ q ⩽ p ⩽ + \infty$ , the estimate
$\begin{aligned} ‖ e^{t σ Δ} z_{0} ‖_{L^{p} (Ω)} ⩽ C (1 + (t σ)^{- \frac{N}{2} (\frac{1}{q} - \frac{1}{p})}) e^{- λ_{1} σ t} ‖ z_{0} ‖_{L^{q} (Ω)}, \end{aligned}$
holds for all $z \in L^{q} (Ω)$ such that $\int_{Ω} z d x = 0$ , where $λ_{1}$ denotes the first non-zero eigenvalue of $- Δ$ under zero Neumann boundary conditions.

We are in position to prove our main result. Proof of Theorem 1.3. Proof of Theorem 1.3.

During the proof, we assume without loss of generality that $| Ω | = 1$ to simplify some computations. For clarity we argue in several steps.

Step 1: Let us consider sequences ${y^{σ}, z^{σ}}_{σ ⩾ 1}$ of solutions of (1.3) and controls ${h (σ)}_{σ ⩾ 1}$ , such that the controls $h (σ)$ are provided by the Theorem 1.4. Therefore, $y^{σ} (T) = z^{σ} (T) = 0$ in Ω for every $σ ⩾ 1$ .

In this step, we will prove that

\begin{aligned} (y^{σ}, z^{σ}) \to (y, ξ) in L^{2} (Q_{T}) \times L^{2} (0, T), \end{aligned}

(4.1)

such that y and ξ satisfies the first equation of (1.6) in the weak sense. Moreover, we will check that

y (T) = 0

. We argue as follows.

For any $σ ⩾ 1$ , the solution $(y^{σ}, z^{σ})$ to (1.3) satisfies the estimate (1.12) and the control h is uniformly bounded as in (1.11). Therefore ${y^{σ}, z^{σ}}_{σ ⩾ 1}$ is a uniformly bounded sequence in $[L^{2} (0, T; H^{1} (Ω))]^{2}$ (w.r.t. σ). Then, we can extract a subsequence, still denoted ${y^{σ}, z^{σ}}_{σ ⩾ 1}$ such that

\begin{aligned} (y^{σ}, z^{σ}) ⇀ (y, z) in (L^{2} (0, T; H^{1} (Ω))^{2} . \end{aligned}

Furthermore,

{y_{t}^{σ}, z_{t}^{σ}}_{σ ⩾ 1}

is uniformly bounded in

(L^{2} (0, T; (H^{1} (Ω))^{^{'}})^{2}

with respect to the coefficient diffusion

σ ⩾ 1

. Thus, by Aubin-Lions Lemma, we get that

\begin{aligned} (y^{σ}, z^{σ}) \to (y, z) in {[L^{2} (Q_{T})]}^{2} . \end{aligned}

(4.2)

According to Fatou’s Lemma, using the strong convergence (4.2) and arguing as in the proof of theorem 2.7, we deduce

\begin{aligned} ‖ y ‖_{L^{2} (Q_{T})}^{2} + ‖ z ‖_{L^{2} (Q_{T})}^{2} ⩽ C ({‖ y^{0} ‖}_{L^{2} (Ω)}^{2} + {‖ z^{0} ‖}_{L^{2} (Ω)}^{2}), \end{aligned}

(4.3)

where the constant

C > 0

is independent of the diffusion coefficient

σ ⩾ 1

. Since f and g are Lipschitz functions, we can use the strong convergences in (4.2) to see that

\begin{aligned} f (y^{σ}, z^{σ}) \to f (y, z) in L^{2} (Q_{T}), \\ g (y^{σ}, z^{σ}) \to g (y, z) in L^{2} (Q_{T}) . \end{aligned}

Moreover, from the energy estimate (1.12) we know that ${\nabla y^{σ}}_{σ ⩾ 1}$ is uniformly bounded $L^{2} (Q_{T})$ and

\begin{aligned} ‖ \nabla y^{σ} ‖_{L^{2} (Q_{T})} + σ ‖ \nabla z^{σ} ‖_{L^{2} (Q_{T})} ⩽ C ({‖ y^{0} ‖}_{L^{2} (Ω)}^{2} + {‖ z^{0} ‖}_{L^{2} (Ω)}^{2}), \end{aligned}

whence we deduce that

\begin{aligned} \begin{aligned} \nabla y^{σ} ⇀ \nabla y L^{2} (Q_{T}), \\ \nabla z^{σ} \to 0 L^{2} (Q_{T}) . \end{aligned} \end{aligned}

(4.4)

The second convergence in (4.4) implies that the limit z only depends on t, so we can write it as

z (t, \cdot) = ξ (t)

for

t \in (0, T)

. Using this fact together with the convergence (4.2) we obtain (4.1).

Also, we recall that estimate (1.11) says that the sequence of controls ${h (σ)}_{σ ⩾ 1}$ is uniformly bounded with respect $σ ⩾ 1$ . Then we can extract a subsequence such that

\begin{aligned} h (σ) ⇀ h in~ L^{2} (0, T; L^{2} (ω)) . \end{aligned}

Therefore, the convergences above imply that the y and ξ satisfies the first equation of (1.6), in the weak sense. Since

y^{σ} (T) = 0

, we have that

y (T) = 0

in Ω. It remains to prove that the solution ξ satisfies the second equation of (1.6).

Step 2: In this step we obtain a first estimation of the difference between $z (t, \cdot)$ and $ξ (t)$ in $L^{2}$ -norm for $t \in (0, T)$ . For this, we consider the heat semigroup with homogeneous Neumann boundary conditions on a domain $Ω \subset R^{n}$ denoted by $e^{t σ Δ}$ for all $t ⩾ 0$ and write the solutions z and ξ of the second component of systems (1.3) and (1.6), respectively, as follows

\begin{aligned} z^{σ} (t, x) = e^{t σ Δ} z^{0} + \int_{0}^{t} e^{(t - s) σ Δ} g (y^{σ} (s, x), z^{σ} (s, x)) d s, \\ ξ (t) = \int_{Ω} z^{0} d x + \int_{0}^{t} \int_{Ω} g (y (s, x), ξ (s)) d x d s . \end{aligned}

Subtracting the above solutions and computing the

L^{2}

-norm we get

\begin{aligned} {‖ z^{σ} (t, \cdot) - ξ (t) ‖}_{L^{2} (Ω)} \\ ⩽ ‖ e^{t σ Δ} z^{0} - \int_{Ω} z^{0} d x + \int_{0}^{t} e^{(t - s) σ Δ} (g (y^{σ} (s, \cdot), z^{σ} (s, \cdot))) d s \\ - {\int_{0}^{t} \int_{Ω} g (y (s, x), ξ (s, x)) d x d s ‖}_{L^{2} (Ω)} . \end{aligned}

(4.5)

Applying property $i)$ of Lemma 4.1 to the terms $c_{1} = \int_{Ω} z^{0} d x$ and $c_{2} = \int_{Ω} g (y (t, x), ξ (t)) d x$ in the right-hand side of (4.5) we get

\begin{aligned} {‖ z^{σ} (t, \cdot) - ξ (t) ‖}_{L^{2} (Ω)} \\ ⩽ {‖ e^{t σ Δ} (z^{0} - \int_{Ω} z^{0} d x) ‖}_{L^{2} (Ω)} \\ + {‖ \int_{0}^{t} e^{(t - s) σ Δ} (g (y^{σ} (s, \cdot), z^{σ} (s, \cdot)) - \int_{Ω} g (y^{σ} (s, x), z^{σ} (s, x))) d s ‖}_{L^{2} (Ω)} \\ + {‖ \int_{0}^{t} e^{(t - s) σ Δ} \int_{Ω} g (y^{σ} (s, x), z^{σ} (s, x)) - g (y (s, x), ξ (s, x)) d x d s ‖}_{L^{2} (Ω)} . \end{aligned}

(4.6)

We apply once again property

i)

of Lemma 4.1 to the term

c_{3} = \int_{Ω} g (y^{σ} (s, x), z^{σ} (s, x)) - g (y (s, x), ξ (s, x)) d x

in the right-hand side of (4.6) to obtain

\begin{aligned} {‖ z^{σ} (t, \cdot) - ξ (t) ‖}_{L^{2} (Ω)} \\ ⩽ {‖ e^{σ Δ t} (z^{0} - \int_{Ω} z^{0} d x) ‖}_{L^{2} (Ω)} \\ + {‖ \int_{0}^{t} e^{σ Δ (t - s)} (g (y^{σ} (s, \cdot), z^{σ} (s, \cdot)) - \int_{Ω} g (y^{σ} (s, x), z^{σ} (s, x)) d x) d s ‖}_{L^{2} (Ω)} \\ + | \int_{0}^{t} \int_{Ω} g (y^{σ} (s, x), z^{σ} (s, x)) - g (y (s, x), ξ (s)) d x d s | =: M_{1} + M_{2} + M_{3} . \end{aligned}

(4.7)

Step 3: In this step, let us estimate each term $M_{i}$ , $1 ⩽ i ⩽ 3$ . Applying property $i i)$ of Lemma 4.1, with $p = q = 2$ and $x_{0} = z^{0} - \int_{Ω} z^{0} d x$ , we obtain the following estimate for $M_{1}$

\begin{aligned} t^{\frac{n}{2}} M_{1} & ⩽ C t^{\frac{n}{2}} e^{- λ_{1} σ t} {‖ z^{0} ‖}_{L^{2} (Ω)} \\ ⩽ C σ^{- \frac{n}{2}} {(σ t)}^{\frac{n}{2}} e^{- λ_{1} σ t} {‖ z^{0} ‖}_{L^{2} (Ω)} \\ ⩽ C σ^{- \frac{n}{2}} sup_{s ⩾ 0} s^{\frac{n}{2}} e^{- λ_{1} s} {‖ z^{0} ‖}_{L^{2} (Ω)} \\ ⩽ C σ^{- \frac{n}{2}} {‖ z^{0} ‖}_{L^{2} (Ω)}, \end{aligned}

(4.8)

for

t \in [0, T]

To estimate $M_{2}$ , let us define $R (s, x) = R_{1} (s, x) - R_{2} (s)$ where $R_{1} (s, x) = g (y^{σ} (s, x), z^{σ} (s, x))$ and $R_{2} (s) = \int_{Ω} R_{1} (s, x) d x$ . First, we show that $R (s, \cdot)$ is uniformly bounded in $L^{2} (Ω)$ with respect $σ ⩾ 1$ . Indeed, using that g is Lipschitz and $g (0, 0) = 0$ , by applying the energy estimate (1.12), we have

\begin{aligned} {‖ R_{1} (s, \cdot) ‖}_{L^{2} (Ω)}^{2} & ⩽ C ({‖ y^{σ} (s, \cdot) ‖}_{L^{2} (Ω)}^{2} + {‖ z^{σ} (s, \cdot) ‖}_{L^{2} (Ω)}^{2}) \\ ⩽ C ({‖ y^{0} ‖}_{L^{2} (Q_{T})}^{2} + {‖ z^{0} ‖}_{L^{2} (Q_{T})}^{2}), \end{aligned}

(4.9)

where C is independent of

σ ⩾ 1

. To conclude that

R (s, \cdot)

is uniformly bounded in

L^{2} (Ω)

with respect to

σ ⩾ 1

we apply Jensen inequality and (4.9) to get

\begin{aligned} {‖ R_{2} (s, \cdot) ‖}_{L^{2} (Ω)}^{2} & ⩽ \int_{Ω} {| \int_{Ω} R_{1} (s, \bar{x}) d \bar{x} |}^{2} d x ⩽ C \int_{Ω} {| R_{1} (s, x) |}^{2} d x \\ ⩽ C ({‖ y^{0} ‖}_{L^{2} (Q)}^{2} + {‖ z^{0} ‖}_{L^{2} (Q)}^{2}) . \end{aligned}

Rewriting $M_{2}$ and by Hölder inequality, we have

\begin{aligned} M_{2} = ‖ \int_{0}^{t} e^{σ Δ (t - s)} R (s, \cdot) d s ‖_{L^{2} (Ω)} ⩽ \int_{0}^{t} ‖ e^{σ Δ (t - s)} R (s, \cdot) ‖_{L^{2} (Ω)} d s . \end{aligned}

We note that

R \in L^{2} (Ω)

and by definition of R we have that

\int_{Ω} R (s, x) d x = 0

. Therefore we can use the property

i i)

of the Lemma 4.1 with

p = q = 2

and

x_{0} = R (s, x)

to obtain

\begin{aligned} \sup_{0 ⩽ t ⩽ T} t^{\frac{n}{2}} M_{2} & ⩽ \sup_{0 ⩽ t ⩽ T} t^{\frac{n}{2}} \int_{0}^{t} ‖ e^{σ Δ (t - s)} R (s, \cdot) ‖_{L^{2} (Ω)} d s \\ ⩽ \sup_{0 ⩽ t ⩽ T} t^{\frac{n}{2}} C \int_{0}^{t} e^{- (t - s) λ_{1} σ} d s ‖ R (s, \cdot) ‖_{L^{2} (Ω)} \\ ⩽ C \sup_{0 ⩽ t ⩽ T} t^{\frac{n}{2}} \frac{1 - e^{- t λ_{1} σ}}{λ_{1} σ} \sup_{0 ⩽ t ⩽ T} ‖ R (s, \cdot) ‖_{L^{2} (Ω)} \\ ⩽ \frac{C}{λ_{1} σ} \sup_{0 ⩽ t ⩽ T} ‖ R (s, \cdot) ‖_{L^{2} (Ω)} . \end{aligned}

(4.10)

To finish this step, we estimate $M_{3}$ as follows. Using that g is a Lipschitz function and applying the Hölder inequality we get

\begin{aligned} M_{3} & = | \int_{0}^{t} \int_{Ω} g (y^{σ} (s, x), z^{σ} (s, x)) - g (y (s, x), ξ (s, x)) d x d s | \\ ⩽ C_{g} \int_{0}^{t} \int_{Ω} | (y^{σ} (s, x) - y (s, x) | + | z^{σ} (s, x)) - ξ (s, x) | d x d s \\ ⩽ C \int_{0}^{t} ({‖ y^{σ} - y ‖}_{L^{2} (Ω)} + {‖ z^{σ} - ξ ‖}_{L^{2} (Ω)}) d s . \end{aligned}

(4.11)

Step 4: In this step we use the inequality (4.7) together with the estimates of $M_{1}$ , $M_{2}$ and $M_{3}$ to conclude the proof. Putting together (4.7) and (4.11) allows us to write

\begin{aligned} {‖ z^{σ} (t, \cdot) - ξ (t) ‖}_{L^{2} (Ω)} ⩽ A (t) + C \int_{0}^{t} Y (s) d s, \end{aligned}

(4.12)

where

\begin{aligned} Y (t) & = ‖ y^{σ} (s, x) - y (s, \cdot) ‖_{L^{2} (Ω)} + ‖ z^{σ} (t, \cdot) - ξ (t) ‖_{L^{2} (Ω)}, \\ A (t) & = M_{1} (t) + M_{2} (t), \end{aligned}

for

t \in (0, T)

. Moreover, for any

t_{0} \in (0, T)

, we can choose

ε \in (0, t_{0})

and using inequality (4.12) we obtain the following estimate

\begin{aligned} ‖ z^{σ} (t, \cdot) - ξ (t) ‖_{L^{2} (Ω)} & ⩽ A_{σ, ε, T} + C ε^{\frac{1}{2}} {(\int_{0}^{ε} {| Y (s) |}^{2} d s)}^{\frac{1}{2}} \\ + C \int_{ε}^{t} ‖ y^{σ} (s, x) - y (s, \cdot) ‖_{L^{2} (Ω)} d s + C \int_{ε}^{t} ‖ z^{σ} (t, \cdot) - ξ (t) ‖_{L^{2} (Ω)} d s, \end{aligned}

for

t \in (ε, T)

, where

A_{σ, ε, T} = sup_{t \in [ε, T]} A (t)

. Applying Gronwall’s inequality

\begin{aligned} ‖ z^{σ} (t, \cdot) - ξ (t) ‖_{L^{2} (Ω)} \\ ⩽ (A_{σ, ε, T} + C ε^{\frac{1}{2}} {(\int_{0}^{ε} {| Y (s) |}^{2} d s)}^{\frac{1}{2}} + C \int_{ε}^{T} ‖ y^{σ} (s, x) - y (s, \cdot) ‖_{L^{2} (Ω)} d s) e^{C (T - ε)}, \end{aligned}

for

t \in (ε, T)

Notice that estimates (4.8) and (4.10) imply that $A_{σ, ε, T}$ tends towards 0 when $σ \to \infty$ . On the other hand, using the definition of Y and applying triangle inequality give

\begin{aligned} \int_{0}^{ε} {| Y (s) |}^{2} d s & ⩽ C ({‖ y^{σ} - y ‖}_{L^{2} (0, ε; L^{2} (Ω))}^{2} + {‖ z^{σ} - ξ ‖}_{L^{2} (0, ε; L^{2} (Ω))}^{2}) \\ ⩽ C ({‖ y^{σ} ‖}_{L^{2} (0, T; L^{2} (Ω))}^{2} + ‖ y ‖_{L^{2} (0, T; L^{2} (Ω))}^{2} + {‖ z^{σ} ‖}_{L^{2} (0, T; L^{2} (Ω))}^{2} + ‖ ξ ‖_{L^{2} (0, T)}^{2}) . \end{aligned}

(4.13)

Recall from (4.1) that we have

z^{σ} \to ξ

L^{2} (0, T)

, then ξ satisfies (4.3), that is to say

\begin{aligned} ‖ ξ ‖_{L^{2} (0, T)}^{2} ⩽ C ({‖ y^{0} ‖}_{L^{2} (Ω)}^{2} + {‖ z^{0} ‖}_{L^{2} (Ω)}^{2}), \end{aligned}

(4.14)

where C is independent of the diffusion coefficient

σ ⩾ 1

Using (4.14) together with (1.12) and (4.3) allows us to estimate the right-hand side of (4.13) as follows

\begin{aligned} \int_{0}^{ε} {| Y (s) |}^{2} d s ⩽ C {‖ (y^{0}, z^{0}) ‖}_{L^{2} (Ω)}^{2} . \end{aligned}

Therefore, the term

(\int_{0}^{ε} | Y (s) |^{2} d s)^{1 / 2}

is uniformly bounded with respect to σ and

ε

From Step 1, we have that $y^{σ} \to y$ in $L^{2} (Q)$ , then the term $\int_{ε}^{T} ‖ y^{σ} (s, x) - y (s, \cdot) ‖_{L^{2} (Ω)} d s$ tend to zero when $σ \to \infty$ . Then for every $t_{0} \in (0, T)$ and one can choose $ε \in (0, t_{0})$ small enough so

\begin{aligned} lim_{σ \to \infty} sup_{t \in [t_{0}, T]} {‖ z^{σ} (t, \cdot) - ξ (t) ‖}_{L^{2} (Ω)} = 0. \end{aligned}

(4.15)

Moreover, using the limit (4.15) and the fact that

z^{σ} (T) = 0

for every

σ ⩾ 1

, we have that

ξ (T) = 0

. To conclude, we note that by the uniqueness of the limit we obtain that

z = ξ

(t_{0}, T) \times Ω

. This ends the proof of Theorem 1.3.

5. Further remarks and results

5.1. Slightly superlinear nonlinearities

It is natural to ask whether Theorem 1.3 or Theorem 1.4 can be extended to other types of nonlinearities. From the classical results in [11] for the scalar heat equation and [16] for coupled parabolic systems, a first possibility is to consider nonlinear functions f (and similarly for g) behaving as

\begin{aligned} lim_{| s |, | r | \to \infty} \frac{f (s, r)}{(| s | + | r |) \ln^{3 / 2} (1 + | s | + | r |)} = 0. \end{aligned}

However, the situation turns out to be quite delicate due to the presence of the parameter σ and we believe that it requires a careful analysis which is beyond the scope of this paper.

The main difficulty can be already seen in the case of a single equation and appears while trying to improve the usual $L^{2}$ -observability inequality to an $L^{1}$ -estimate. To illustrate this, consider the controlled linear heat equation

\begin{aligned} {\begin{cases} y_{t} - σ Δ y + a y = h 1_{ω} & in Q_{T}, \\ \frac{\partial y}{\partial ν} = 0 & on Σ_{T}, \\ y (0, \cdot) = y^{0} & in Ω, \end{cases} \end{aligned}

where, for simplicity,

a = a (x) ⩾ a_{0} > 0 \forall x \in Ω

. From Lemma 2.3 and arguing as in [11, Proposition 3.2], we can prove that the uniform

L^{2}

-observability inequality

\begin{aligned} {‖ φ (0, \cdot) ‖}_{L^{2} (Ω)}^{2} ⩽ C_{T} \iint_{(\frac{T}{3}, \frac{2 T}{3}) \times ω^{'}} | φ |^{2} d x d t, \end{aligned}

(5.1)

holds for any

φ^{T} \in L^{2} (Ω)

where φ is the solution of the adjoint

\begin{aligned} {\begin{cases} - φ_{t} - σ Δ φ + a φ = 0 & in Q_{T}, \\ \frac{\partial φ}{\partial ν} = 0 & on Σ_{T}, \\ φ (T, \cdot) = φ^{T} & in Ω, \end{cases} \end{aligned}

ω^{'} \subset\subset ω

, and the constant

C_{T}

is of the form

e^{C (1 + 1 / T + T ‖ a ‖_{L^{\infty}} + ‖ a ‖_{L^{\infty}}^{2 / 3})}

for some

C > 0

independent of σ.

Following [11, Proposition 3.2], the next task is to iterate $L^{q}$ - $L^{p}$ regularity estimates in a smart way to obtain that

\begin{aligned} \iint_{(\frac{T}{3}, \frac{2 T}{3}) \times ω^{'}} | φ |^{2} d x d t ⩽ C T^{α} {(1 + T^{1 / 2} (1 + ‖ a ‖_{L^{\infty}}))}^{β} {(\iint_{(\frac{T}{3}, \frac{2 T}{3}) \times ω} | φ | d x d t)}^{2} \end{aligned}

(5.2)

for positive constants C, α and β uniformly bounded with respect to σ, which combined with (5.1) yields the desired

L^{1}

observability estimate.

However, (5.2) cannot hold uniformly bounded with respect to σ, at least with the approach shown in [11, Proposition 3.2]. Arguing as in that work, by introducing a suitable smooth cut-off function $θ = θ (t, x)$ , the idea is to study the system verified by $ψ = θ φ$ , which in our case reads as

\begin{aligned} - ψ_{t} - σ Δ ψ = a ψ - [θ_{t} + σ Δ θ] φ + 2 σ \nabla θ \cdot \nabla φ in Q_{T}, \end{aligned}

complemented with homogenous Neumann boundary conditions and zero terminal datum. By the change of variables

\tilde{ψ} (t, x) = ψ (T - t, x)

(resp.

\tilde{φ} (t, x) = φ (T - t, x)

) and using the

L^{q}

L^{p}

regularity results in [40, Lemma 3, pp. 25] (which are valid since

a (x) > 0

) we can get

\begin{aligned} ‖ \tilde{ψ} (t, \cdot) ‖_{L^{r_{0}} (Ω)} \\ ⩽ C ‖ a ‖_{L^{\infty}} \int_{0}^{t} (t - s)^{- \frac{N}{2} (\frac{1}{r_{1}} - \frac{1}{r_{0}})} ‖ \tilde{ψ} (s, \cdot) ‖_{L^{r_{1}} (Ω)} d s \\ + C σ \int_{δ_{1}}^{t} [(t - s)^{- \frac{N}{2} (\frac{1}{r_{1}} - \frac{1}{r_{0}})} + (t - s)^{- \frac{N}{2} (\frac{1}{r_{1}} - \frac{1}{r_{0}}) - \frac{1}{2}} ‖ \tilde{φ} (s, \cdot) ‖_{L^{r_{1}} (ω)} d s], \end{aligned}

for positive parameters

r_{0}

r_{1}

and

δ_{1}

well chosen and some

C > 0

. Upon comparison with [11, Eq. (3.11)], we note that we have an extra σ in front of the last term and the iterative argument outlined in that paper yields (5.2) but with a constant depending badly on σ.

Overall, addressing this question is not straightforward and requires a detailed analysis of the equation’s regularity in norms beyond $L^{2}$ and its dependence on σ. The main difficulties can be found also in the coupled case, where very delicate regularity estimates are needed to adapt the methodology used in [16]. As a result, this issue remains an open problem for future research.

5.2. Other kind of nonlinearities: A local result

In this paper, we have highlighted the versatility of the shadow limit, a widely utilized tool in various reaction-diffusion systems. We have shown that control properties are preserved under this limit when careful attention is given to control estimates, therefore we can expect to have local controllability results for semilinear systems with specific polynomial behaviors.

While prior works like [18] and [19] have explored similar problems using the shadow limit, their frameworks and objectives differ from those in this paper. Nonetheless, we present below a local-controllability result for the Gray–Scott system, a well-known model in chemical kinetics (see [33]), that can be directly adapted (although non-trivially) from [18] and [19]. We have omitted a detailed exposition of the steps, focusing instead on conveying the applicability of the shadow model reduction.

To this end, consider spatial dimensions $N = 1, 2$ or 3 and the reaction-diffusion system given by

\begin{aligned} {\begin{cases} y_{t} - Δ y = - y z^{2} + J (1 - y) + h 1_{ω} & in Q_{T}, \\ z_{t} - σ Δ z = y z^{2} - (J + k) z & in Q_{T}, \\ \frac{\partial y}{\partial ν} = \frac{\partial z}{\partial ν} = 0 & on Q_{T}, \\ y (0, \cdot) = y^{0}, z (0, \cdot) = z^{0} & in Ω, \end{cases} \end{aligned}

(5.3)

where J and k are positive constants.

It is known that if $h \equiv 0$ and J and k satisfy suitable conditions (see [33, Section 3]), system (5.3) has two uniform steady-state solutions $(y_{+}, z_{-}) \in (0, + \infty)^{2}$ and $(y_{-}, z_{+}) \in (0, + \infty)^{2}$ characterized as

\begin{aligned} y_{\pm} = \frac{1}{2} (1 \pm \sqrt{1 - 4 γ^{2} J}), z_{\mp} = \frac{1}{2 γ} (1 \mp \sqrt{1 - 4 γ^{2} J}), with γ = \frac{J + k}{J} . \end{aligned}

(5.4)

In this direction, our first result reads as follows.

Theorem 5.1.

Let $T > 0$ and $(u_{\pm}, v_{\mp})$ be a steady-state for (5.3). There exists $δ > 0$ such that for every $σ \in (1, + \infty)$ and every initial datum $(u_{0}, v_{0}) \in [H^{1} (Ω)]^{2}$ verifying $‖ (y^{0} - u_{\pm}, z^{0} - z_{\mp}) ‖_{[H^{1} (Ω)]^{2}} ⩽ δ$ , system (5.3) is uniformly locally exact controllable to the steady-states, more precisely, there is a control $h \in L^{2} (0, T; L^{2} (ω))$ such that the solution to (5.3) (and the control h) satisfy

\begin{aligned} {‖ (y, z) ‖}_{L^{\infty} (0, T; {[H^{1} (Ω)]}^{2}) \cap H^{1} (0, T; {[L^{2} (Ω)]}^{2})} + ‖ h ‖_{L^{2} (0, T; L^{2} (ω))} ⩽ C \end{aligned}

(5.5)

for a constant

C > 0

depending at most on Ω, ω, T, F, k,

y_{0}

, and

z_{0}

and such that

\begin{aligned} y (T, \cdot) = y_{\pm}, z (T, \cdot) = z_{\mp} . \end{aligned}

The proof of this result can be divided in several parts and their proofs can be extrapolated from this and previous works, in detail, it is composed by:

A linearization procedure and a change of variable: this reduces the original problem to a null-control one and can be done exactly as in [18, Section 2.1].

The uniform Carleman estimate (w.r.t σ) shown in Proposition 2.4.

The source term method and the Banach fixed point method: this is the part that requires most adaptations, but relies on the well-known source term method introduced in [30] that has been exploited in many other works (see e.g. [13,27,42]). The proof follows the lines in [19, Section 2.8] and allows to pass from the linear case to the nonlinear terms $- y z^{2}$ and $y z^{2}$ in (5.3) while preserving the uniform bound on the control and solution. It is important to mention that here is needed that the dimension is less or equal to 3 to use standard embeddings and get good estimates on the nonlinear terms.

Once Theorem 5.1 is established, we can use the uniform bounds in (5.5) and the shadow limit to deduce a control result for the system

\begin{aligned} {\begin{cases} y_{t} - Δ y = - y ξ^{2} + J (1 - y) + h 1_{ω} & in Q_{T}, \\ \dot{ξ} = (- - \int_{Ω} y (t, x) d x) ξ^{2} - (J + k) ξ & in (0, T), \\ \frac{\partial y}{\partial ν} = 0 & on Σ_{T}, \\ y (0, \cdot) = y^{0} in Ω, ξ (0) = - - \int_{Ω} z^{0} . \end{cases} \end{aligned}

(5.6)

The exact result reads as follows.

Theorem 5.2.

Let $T > 0$ and $(u_{\pm}, v_{\mp})$ be a steady-state for ( 5.3 ). Then, system ( 5.6 ) is locally exact controllable to the steady-states, more precisely, there is a control $h \in L^{2} (0, T; L^{2} (ω))$ such that the solution to ( 5.6 ) satisfies

\begin{aligned} y (T, \cdot) = y_{\pm}, ξ (T) = z_{\mp} . \end{aligned}

The proof of Theorem 5.2 can be obtained by using the uniform estimates provided by Theorem 5.1 and a shadow limit as in [18, Proposition 3.4]. The main difference with respect to the procedure done in Section 4 is that the work [18] exploits the structure of the nonlinear terms $\pm y z^{2}$ instead of the Lipschitz estimate as employed above.

We conclude by mentioning that the Gray–Scott system serves merely as an illustrative example and there are many other models in the literature that can be studied with the same techniques, such as the Glycolysis model (refer to [3]) or the Brusselator model (refer to [6]).

5.3. Cascade systems

The shadow limit is not restricted to $2 \times 2$ reaction-diffusion systems. As noted in [24], the same ideas can be carried out to study general $n \times n$ systems having the following structure

\begin{aligned} {\begin{cases} y_{t} - D_{y} Δ y = f (y, z) & in Q_{T}, \\ z_{t} - D_{z} Δ z = g (y, z) & in Q_{T}, \\ \frac{\partial y}{\partial ν} = \frac{\partial z}{\partial ν} = 0 & on Σ_{T}, \\ y (0, \cdot) = y^{0}, z (0, \cdot) = z^{0} & in Ω, \end{cases} \end{aligned}

(5.7)

where

y : {\bar{Q}}_{T} \to R^{m}

v : {\bar{Q}}_{T} \to R^{k}

, with

m + k = n

ν

denotes the outward unit normal vector to

\partial Ω

and the Laplace operator Δ is applied component by component and multiplied by diagonal diffusion matrices

D_{u} \in R^{m \times m}

and

D_{v} \in R^{k \times k}

with positive entries. Here

f : R^{m + k} \to R^{m}

and

g : R^{m + k} \to R^{k}

are suitable (possibly nonlinear) functions.

As expected, by letting the diffusion matrix $D_{z} \to + \infty$ , we can recover the $n \times n$ PDE-ODE system given by

\begin{aligned} {\begin{cases} y_{t} - D_{y} Δ y = f (y, ξ) & in Q_{T}, \\ \dot{ξ} = - - \int_{Ω} g (y (t, x), ξ (t)) d x & in (0, T), \\ \frac{\partial y}{\partial ν} = 0 & on Σ_{T}, \\ y (0, \cdot) = y^{0}, ξ (0) = - - \int_{Ω} z^{0} & in Ω . \end{cases} \end{aligned}

(5.8)

This naturally leads the questions of whether (5.8) is null-controllable and how many controls are necessary to control it. With the tools presented in this paper, we can identify readily two examples: one where things goes well and one where it is not so clear.

Let $y = (y_{1}, y_{2})^{⊤}$ , i.e. $m = 2$ , and take $k = 1$ . Consider the following linear control system

\begin{aligned} {\begin{cases} y_{t} - Δ y = A_{1} (y, z) + (\begin{matrix} h \\ 0 \end{matrix}) 1_{ω} & in Q_{T}, \\ z_{t} - σ Δ z = A_{2} (y, z) & in Q_{T}, \\ \frac{\partial y}{\partial ν} = \frac{\partial z}{\partial ν} = 0 & on Σ_{T}, \\ y (0, \cdot) = y^{0}, z (0, \cdot) = z^{0} & in Ω, \end{cases} \end{aligned}

(5.9)

where, for readability, we have used and omitted the notation

D_{y} = I_{2 \times 2}

. In (5.9), the functions

A_{1} : R^{3} \to R^{2}

and

A_{2} : R^{3} \to R^{1}

are defined through the constant-entry matrices

\begin{aligned} A_{1} = (\begin{matrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \end{matrix}), A_{2} = (\begin{array}{ccc} 0 & a_{32} & a_{33} \end{array}) . \end{aligned}

System (5.9) resembles the classical cascade structure for control systems studied in [15], where the goal is to control as many equations with only one control force. Actually, from that work we know that if

σ \equiv 1

, the controllability of (5.9) is ensured as long as the coefficients

a_{21}

and

a_{32}

are different from zero.

We can improve such result by obtaining a uniform controllability result (w.r.t σ) for (5.9), which is the main ingredient for passing to the shadow limit. For brevity, we present only the uniform controllability result and make brief comments of the proof.

Theorem 5.3.

Let $a_{21}$ and $a_{32}$ be constants different from zero. There is a control $h \in L^{2} (0, T; L^{2} (ω))$ uniformly bounded with respect to σ such that ( 5.9 ) satisfies $y (T, \cdot) = 0$ and $z (T, \cdot) = 0$ .

The main ingredient is to obtain a Carleman estimate for the adjoint system

\begin{aligned} {\begin{cases} - φ_{t} - Δ φ = B_{1} (φ, ψ) & in Q_{T}, \\ - ψ_{t} - σ Δ ψ = B_{2} (φ, ψ) & in Q_{T}, \\ \frac{\partial φ}{\partial ν} = \frac{\partial ψ}{\partial ν} = 0 & on Σ_{T}, \\ φ (0, \cdot) = φ^{T}, ψ (T, \cdot) = ψ^{T} & in Ω, \end{cases} \end{aligned}

(5.10)

where

B_{1} = (\begin{matrix} a_{11} & a_{21} & 0 \\ a_{12} & a_{22} & a_{32} \end{matrix})

B_{2} = (\begin{array}{ccc} a_{13} & a_{23} & a_{33} \end{array})

, with only observation of

φ_{1}

. But note that the structure of the system allows us to replicate directly the procedure in the proof of Proposition 2.4 to obtain a uniform Carleman estimates for the variables

φ_{2}

and ψ in the form

\begin{aligned} I (s, 1; φ_{2}) + I (s, σ^{- 1}; ψ) \\ ⩽ C (s^{8} \iint_{(0, T) \times ω_{0}} e^{- s α} ξ^{8} | φ_{2} |^{2} d x d t + s^{3} \iint_{Q_{T}} e^{- 2 s α} ξ^{3} | φ_{1} |^{3}) \end{aligned}

(5.11)

for a constant

C > 0

not depending on σ, some

ω_{0} \subset\subset ω

, and where we have used Proposition 2.5 to bound

e^{- 4 s α^{*} + 2 s \hat{α}}

e^{- s α}

(cf. eq. (2.20)).

From here, using that the first equation of (5.10) reads as $- φ 1, t - Δ φ = a_{11} φ_{1} + a_{21} φ_{2}$ we can add the Carleman estimate for $φ_{1}$ to (5.11), absorb the (global) lower-order term and perform local energy estimates to obtain that

\begin{aligned} I (s, 1; φ_{1}) + I (s, 1; φ_{2}) + I (s, σ^{- 1}; ψ) \\ ⩽ C (s^{17} \iint_{(0, T) \times ω} e^{- s α} ξ^{17} | φ_{1} |^{2} d x d t) \end{aligned}

for a constant

C > 0

independent of σ. Following Section 2 this implies Theorem 5.3.

We would like to point out that the situation is different if we consider the following system

\begin{aligned} {\begin{cases} y_{t} - Δ y = {\tilde{A}}_{1} (y, z) + h 1_{ω} & in Q_{T}, \\ z_{t} - σ Δ z = {\tilde{A}}_{2} (y, z) & in Q_{T}, \\ \frac{\partial y}{\partial ν} = \frac{\partial z}{\partial ν} = 0 & on Σ_{T}, \\ y (0, \cdot) = y^{0}, z (0, \cdot) = z^{0} & in Ω, \end{cases} \end{aligned}

(5.12)

where

{\tilde{A}}_{1} = (\begin{array}{ccc} a_{11} & a_{12} & a_{13} \end{array})

{\tilde{A}}_{2} = (\begin{matrix} a_{21} & a_{22} & a_{23} \\ 0 & a_{32} & a_{33} \end{matrix})

and whose adjoint is given by

\begin{aligned} {\begin{cases} - φ_{t} - Δ φ = {\tilde{B}}_{1} (φ, ψ) & in Q_{T}, \\ - ψ_{t} - σ Δ ψ = {\tilde{B}}_{2} (φ, ψ) & in Q_{T}, \\ \frac{\partial φ}{\partial ν} = \frac{\partial ψ}{\partial ν} = 0 & on Σ_{T}, \\ φ (0, \cdot) = φ^{T}, ψ (T, \cdot) = ψ^{T} & in Ω, \end{cases} \end{aligned}

with

{\tilde{B}}_{1} = (\begin{array}{ccc} a_{11} & a_{21} & 0 \end{array})

{\tilde{B}}_{2} = (\begin{matrix} a_{12} & a_{22} & a_{32} \\ a_{13} & a_{23} & a_{33} \end{matrix})

The presence of the parameter σ in the both equations for $ψ_{1}$ and $ψ_{2}$ complicates the analysis and following naively the strategy in the proof of Proposition 2.4 does not eliminate the presence of the parameter σ. The main issue is the appearance of a localized term of the form

\begin{aligned} σ^{2} \iint_{(0, T) \times ω_{1}} e^{- 2 s α} s^{5} ξ^{5} | \nabla ψ_{1} |^{2} \end{aligned}

which comes from performing a computation like in (2.24), but using the equations verified by

ψ_{1}

and

ψ_{2}

Although this term can be estimated uniformly with respect to σ with a procedure similar to the one shown in Step 3 of the proof of Proposition 2.4, using the equation verified $ψ_{1}$ reintroduces a global term for $ψ_{2}$ than cannot be absorbed with the parameter s.

However, given the good convergence properties between (5.7) and (5.8), we believe that it is feasible to establish a result for systems like (5.12), where multiple equations converge to an ODE. Nonetheless, determining the appropriate methodology for addressing this problem remains open.

Footnotes

Acknowledgements

We gratefully acknowledge the invaluable contributions of the anonymous referees whose insightful comments and suggestions significantly enhanced the quality and clarity of this paper.

The first author would like to thank all members of the Departments of Mathematics and Mechanics of IIMAS-UNAM for their kind hospitality during his research stay which was useful for developing the first version of this manuscript. He would also like to thank Prof. Luz de Teresa (IM-UNAM) and Prof. Kévin Le Balc’h (INRIA) for fruitful discussions about the controllability of coupled parabolic systems.

This work has received support from grants A1-S-17475 and CBF2023-2024-116 of CONAHCYT, Mexico, and by Projects IN109522, IN104922, IA100324, and IA100923 of DGAPA-UNAM, Mexico. The work of the first author was supported by the program “Estancias Posdoctorales por México para la Formación y Consolidación de las y los Investigadores por México” of CONAHCYT while the second author was supported by the program “Becas Nacionales” of the same institution.

Notes

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