Approximate mixed synchronization by groups for a coupled system of wave equations

Abstract

We first show that under a suitable balanced repartition of the mixed controls within the system, Kalman’s rank condition is still necessary and sufficient for the uniqueness of solution to the adjoint system associated with incomplete internal and boundary observations, therefore for the approximate controllability of the primary system by means of mixed controls. Then we study the stability of the approximately synchronizable state by groups with respect to applied controls.

Keywords

Mixed controls stability of synchronizable state wave equations

1. Introduction

Let $Ω \subset R^{m}$ be a bounded domain with smooth boundary Γ and $ω \subset Ω$ be a neighbourhood of Γ. Let A be a matrix of order N, $D_{1}$ and $D_{2}$ be full column-rank matrices of order $N \times M_{1}$ and $N \times M_{2}$ respectively, all the matrices are of constant elements. Consider the following system for the state variable $U = {(u^{(1)}, \dots, u^{(N)})}^{T}$ , controlled by the internal control $H = {(h^{(1)}, \dots, h^{(M_{1})})}^{T}$ and the boundary control $G = {(g^{(1)}, \dots, g^{(M_{2})})}^{T}$ : $\begin{matrix} (1.1) & \{\begin{array}{ll} U^{″} - Δ U + A U = D_{1} χ_{ω} H & in (0, + \infty) \times Ω, \\ U = D_{2} G & on (0, + \infty) \times Γ \end{array} \end{matrix}$ associated with the initial condition: $\begin{matrix} (1.2) & t = 0 : U = {\hat{U}}_{0}, U^{'} = {\hat{U}}_{1} in Ω, \end{matrix}$ where $χ_{ω}$ denotes the characteristic function of ω. We will show that when the controls are well distributed within the system, Kalman’s rank condition $\begin{aligned} (1.3) & rank (D, A D, \dots, A^{N - 1} D) = N with D = (D_{1}, D_{2}) \end{aligned}$ is not only necessary but also sufficient for the approximate controllability of system (1.1).

By duality approach as in [10], the approximate controllability of system (1.1) is equivalent to the uniqueness of solution to the following adjoint system for the variable $Φ = {(ϕ^{(1)}, \dots, ϕ^{(N)})}^{T}$ : $\begin{matrix} (1.4) & \{\begin{array}{ll} Φ^{″} - Δ Φ + A^{T} Φ = 0 & in (0, T) \times Ω, \\ Φ = 0 & on (0, T) \times Γ \end{array} \end{matrix}$ associated with the initial data $\begin{matrix} (1.5) & t = 0 : Φ = {\hat{Φ}}_{0}, Φ^{'} = {\hat{Φ}}_{1} in Ω, \end{matrix}$ the locally distributed internal observation $\begin{matrix} (1.6) & D_{1}^{T} χ_{ω} Φ = 0 in (0, T) \times Ω \end{matrix}$ and the Neumann boundary observation $\begin{matrix} (1.7) & D_{2}^{T} \partial_{ν} Φ = 0 on (0, T) \times Γ . \end{matrix}$

As $D_{2} = 0$ , it was shown in [8] that the following Kalman’s rank condition: $\begin{matrix} (1.8) & rank (D_{1}, A D_{1}, \dots, A^{N - 1} D_{1}) = N \end{matrix}$ is sufficient for the approximate controllability of system (1.1). Similarly, as $D_{1} = 0$ , Kalman’s rank condition $\begin{matrix} (1.9) & rank (D_{2}, A D_{2}, \dots, A^{N - 1} D_{2}) = N \end{matrix}$ is still sufficient for the approximate controllability for nilpotent matrix A and star-shaped domain Ω [10], these conditions are not necessary, but technically indispensable in the proof.

In the case of mixed controls, we will project problem onto suitable subspaces and split it into two subsystems, among which one is subjected to the internal observation, while another to the boundary observation. Since Kalman’s rank condition (1.3) only announces that there are enough observations, some additional algebraic conditions have to be required to ensure a balanced repartition of observations according to the decomposition of the system. In this way, we will guarantee the uniqueness of solution for each subsystem, therefore for the whole system.

The study generalizes the consideration in [8] for the internal control and the results given by [10] for the boundary control. However, it is not a simple collection of the known results on internal controllability and boundary controllability, but rather than the coordination of several compositions in a complex system! The work raises many interesting questions and opens up a new direction on this topic.

2. A uniqueness theorem

We first recall some algebraic tools.

Lemma 2.1 ([6, Lemma 2.5]).

Let $d ⩾ 0$ be an integer. The Kalman’s rank condition $\begin{matrix} (2.1) & rank (D, A D, \dots, A^{N - 1} D) = N - d \end{matrix}$ holds with the control matrix D if and only if d is the dimension of the largest subspace which is invariant for $A^{T}$ and contained in $Ker (D^{T})$ . Moreover, the largest subspace invariant for $A^{T}$ and contained in $Ker (D^{T})$ is given by $\begin{matrix} (2.2) & V = Ker {(D, A D, \dots, A^{N - 1} D)}^{T} . \end{matrix}$

Lemma 2.2.
Let $V_{1}$ , $V_{2}$ and V denote the largest subspaces invariant for $A^{T}$ and contained in $Ker (D_{1}^{T})$ , $Ker (D_{2}^{T})$ and $Ker (D^{T})$ , respectively. We have $\begin{matrix} (2.3) & V_{1} \cap V_{2} = V . \end{matrix}$
Lemma 2.3 ([10, Proposition 2.7]).

The non trivial subspaces $V = span {E_{1}, \dots, E_{d}}$ and $W = span {e_{1}, \dots, e_{d}}$ are bi-orthonormal in $R^{N}$ , namely, $\begin{matrix} (2.4) & E_{k}^{T} e_{l} = δ_{k l}, 1 ⩽ k, l ⩽ d, \end{matrix}$ where $δ_{k l}$ is the Kronecker symbol, if and only if $\begin{matrix} (2.5) & dim (V) = dim (W) and V \cap W^{⊥} = {0} \end{matrix}$ or equivalently if and only if V is a supplement of $W^{⊥}$ .

Lemma 2.4 ([10, Proposition 2.11]).

We have $\begin{matrix} (2.6) & rank (C_{p} D) = rank (D) if and only if Ker (C_{p}) \cap Im (D) = {0}, \end{matrix}$ or equivalentely, $\begin{matrix} (2.7) & rank (C_{p} D) = rank (C_{p}) if and only if Ker (D^{T}) \cap Im (C_{p}^{T}) = {0} . \end{matrix}$

Lemma 2.5 ([10, Proposition 2.15]).

The following assertions are equivalent:

A satisfies the condition of $C_{p}$ -compatibility: $\begin{matrix} (2.8) & A Ker (C_{p}) \subseteq Ker (C_{p}); \end{matrix}$

there exists a unique matrix $A_{p}$ of order $(N - p)$ , such that $\begin{matrix} (2.9) & C_{p} A = A_{p} C_{p} . \end{matrix}$

Furthermore, let $D_{p} = C_{p} D$ . Then we have $\begin{aligned} (2.10) & rank (D_{p}, A_{p} D_{p}, \dots, A_{p}^{N - p - 1} D_{p}) = rank C_{p} (D, A D, \dots, A^{N - 1} D) . \end{aligned}$

By classic theory of semigroups [13], system (1.4) forms a $C^{0}$ -semigroup in the space ${(H_{0}^{1} (Ω) \times L^{2} (Ω))}^{N}$ .

Theorem 2.1 ([8, Theorem 10]).

Assume that the Kalman’s rank condition $\begin{aligned} (2.11) & rank (D_{1}, A D_{1}, \dots, A^{N - 1} D_{1}) = N \end{aligned}$ holds with the control mnatrix $D_{1}$ . Then for any given initial data $({\hat{Φ}}_{0}, {\hat{Φ}}_{1}) \in {(H_{0}^{1} (Ω) \times L^{2} (Ω))}^{N}$ , system ( 1.4 ) associated with the internal observation ( 1.6 ) has only the trivial solution, provided that $T > 2 d (Ω)$ , where $d (Ω)$ denotes the geodesic diameter of Ω.

Theorem 2.2 ([10, Theorem 8.16]).

Let $Ω \subset R^{m}$ be a bounded domain with smooth boundary Γ satisfying the usual multiplier control condition. Assume that A is a nilpotent matrix. Assume furthermore that $\begin{aligned} (2.12) & rank (D_{2}, A D_{2}, \dots, A^{N - 1} D_{2}) = N . \end{aligned}$ Then, for any given initial data $({\hat{Φ}}_{0}, {\hat{Φ}}_{1}) \in {(H_{0}^{1} (Ω) \times L^{2} (Ω))}^{N}$ , system ( 1.4 ) associated with the boundary observation ( 1.7 ) has only the trivial solution, provided that the time $T > 0$ is large enough.

Proposition 2.1.
For any given initial data $({\hat{Φ}}_{0}, {\hat{Φ}}_{1}) \in {(H_{0}^{1} (Ω) \times L^{2} (Ω))}^{N}$ , if system ( 1.4 ) associated with ( 1.6 ) and ( 1.7 ) has only the trivial solution, then we have Kalman’s rank condition ( 1.3 ).
Proof.
If Kalman’s rank condition (1.3) fails, by Lemma 2.1, the largest subspace V invariant for $A^{T}$ and contained in $Ker {(D_{1}, D_{2})}^{T}$ is not reduced to ${0}$ . The subspace $\begin{array}{r} V = {ϕ E : ϕ \in D (Ω), E \in V} \end{array}$ is invariant for the operator $- Δ + A^{T}$ and system (1.4) admits non trivial solutions which lie in $V \subseteq Ker {(D_{1}, D_{2})}^{T}$ , therefore satisfy the conditions of observations (1.6) and (1.7). □
Theorem 2.3.
Let $Ω \subset R^{m}$ be a bounded domain with smooth boundary Γ satisfying the usual multiplier control condition and ω be a subdomain of Ω. Assume that
Kalman’s rank condition ( 1.3 ) holds with the control matrix $D = (D_{1}, D_{2})$ ;

the largest subspace $V_{1}$ invariant for $A^{T}$ and contained in $Ker (D_{1}^{T})$ admits a supplement $V_{s}$ which is also invariant for $A^{T}$ ;

the restriction of $A^{T}$ on $V_{1}$ is a nilpotent matrix.

Then, for any given initial data $({\hat{Φ}}_{0}, {\hat{Φ}}_{1}) \in {(H_{0}^{1} (Ω) \times L^{2} (Ω))}^{N}$ , system ( 1.4 ) associated with ( 1.6 ) and ( 1.7 ) has only the trivial solution, provided that the time $T > 0$ is large enough.
Proof.
Let $\begin{aligned} (2.13) & rank (D_{1}, A D_{1}, \dots, A^{N - 1} D_{1}) = N - p \end{aligned}$ for some p with $0 ⩽ p ⩽ N$ . The case $p = 0$ corresponds to Theorem 2.1, while the case $p = N$ implies $D_{1} = 0$ , therefore corresponds to Theorem 2.2. We then need only to consider the case p with $0 < p < N$ .

Firstly, let $V_{1} = Ker {(D_{1}, A D_{1}, \dots, A^{N - 1} D_{1})}^{T}$ be the p-dimensional subspace. By Lemma 2.1, we have $\begin{aligned} (2.14) & A^{T} V_{1} \subseteq V_{1}, V_{1} \subseteq Ker (D_{1}^{T}) . \end{aligned}$ Define a matrix $E_{2}$ of order $p \times N$ by $Im (E_{2}^{T}) = V_{1}$ . By the first condition in (2.14), $Ker (E_{2})$ is invariant for A. By Lemma 2.5 with $C_{p}$ replaced by $E_{2}$ , there exists a matrix $A_{2}$ of order p, such that $\begin{aligned} (2.15) & E_{2} A = A_{2} E_{2} . \end{aligned}$ Similarly, define a matrix $E_{1}$ of order $(N - p) \times N$ by $Im (E_{1}^{T}) = V_{s}$ . Since $V_{s}$ is invariant for $A^{T}$ , $Ker (E_{1})$ is invariant for A. Still by Lemma 2.5 with $C_{p}$ replaced by $E_{1}$ , there exists a matrix $A_{1}$ of order $(N - p)$ , such that $\begin{aligned} (2.16) & E_{1} A = A_{1} E_{1} . \end{aligned}$ Since $Im (E_{1}^{T})$ and $Im (E_{2}^{T})$ are mutually supplementary, for any given $Φ \in R^{N}$ , we can write $\begin{aligned} (2.17) & Φ = E_{1}^{T} Φ_{1} + E_{2}^{T} Φ_{2}, where Φ_{1} \in R^{N - p} and Φ_{2} \in R^{p} . \end{aligned}$ Projecting system (1.4) on the subspaces $Im (E_{1}^{T})$ and $Im (E_{2}^{T})$ , we get, respectively, $\begin{aligned} (2.18) & \{\begin{array}{ll} Φ_{1}^{″} - Δ Φ_{1} + A_{1}^{T} Φ_{1} = 0 & in (0, + \infty) \times Ω, \\ Φ_{1} = 0 & on (0, + \infty) \times Γ \end{array} \end{aligned}$ and $\begin{aligned} (2.19) & \{\begin{array}{ll} Φ_{2}^{″} - Δ Φ_{2} + A_{2}^{T} Φ_{2} = 0 & in (0, + \infty) \times Ω, \\ Φ_{2} = 0 & on (0, + \infty) \times Γ . \end{array} \end{aligned}$

Secondly, noting the second condition in (2.14), we have $Im (E_{2}^{T}) = V_{1} \subseteq Ker (D_{1}^{T})$ , then $D_{1}^{T} E_{2}^{T} = 0$ and the $D_{1}$ -observation (1.6) becomes $\begin{aligned} (2.20) & D_{1}^{T} (E_{1}^{T} Φ_{1} + E_{2}^{T} Φ_{2}) = {(E_{1} D_{1})}^{T} Φ_{1} = 0 in (0, T) \times ω . \end{aligned}$ Using (2.10) in Lemma 2.5, we get $\begin{aligned} rank E_{1} (D_{1}, A D_{1}, \dots, A^{N - 1} D_{1}) \\ (2.21) & = rank (E_{1} D_{1}, A_{1} E_{1} D_{1} \dots, A_{1}^{N - p - 1} E_{1} D_{1}) . \end{aligned}$ Since $\begin{aligned} (2.22) & Ker {(D_{1}, A D_{1}, \dots, A^{N - 1} D_{1})}^{T} \cap Im (E_{1}^{T}) = V_{1} \cap V_{s} = {0}, \end{aligned}$ by Lemma 2.4, we have $\begin{aligned} (2.23) & rank (E_{1} D_{1}, A_{1} E_{1} D_{1} \dots, A_{1}^{N - p - 1} E_{1} D_{1}) = rank (E_{1}) = N - p . \end{aligned}$ So the pair $(A_{1}, E_{1} D_{1})$ satisfies the corresponding Kalman’s rank condition (2.11). By Theorem 2.1, subsystem (2.18) associated with the internal $E_{1} D_{1}$ -observation (2.20) has only the trivial solution $Φ_{1} = 0$ , provided that $T > 2 d (Ω)$ .

Finally, noting $Φ_{1} = 0$ , the boundary $D_{2}$ -observation (1.7) becomes $\begin{aligned} (2.24) & D_{2}^{T} (E_{1}^{T} \partial_{ν} Φ_{1} + E_{2}^{T} \partial_{ν} Φ_{2}) = {(E_{2} D_{2})}^{T} \partial_{ν} Φ_{2} = 0 on (0, T) \times Γ . \end{aligned}$ Let $V_{2} = Ker {(D_{2}, A D_{2}, \dots, A^{N - 1} D_{2})}^{T}$ denote the largest subspace invariant for $A^{T}$ and contained in $Ker (D_{2}^{T})$ , and V denote the largest subspace invariant for $A^{T}$ and contained in $Ker (D^{T})$ . By Lemma 2.1, condition (1.3) implies $V = {0}$ . Then it follows from Lemma 2.2 that $\begin{matrix} (2.25) & Im (E_{2}^{T}) \cap Ker {(D_{2}, A D_{2}, \dots, A^{N - 1} D_{2})}^{T} = V_{1} \cap V_{2} = V = {0} . \end{matrix}$ By Lemma 2.4, it follows that $\begin{aligned} (2.26) & rank E_{2} (D_{2}, A D_{2}, \dots, A^{N - 1} D_{2}) = rank (E_{2}) = p . \end{aligned}$ Using (2.10) in Lemma 2.5, we get $\begin{aligned} rank E_{2} (D_{2}, A D_{2}, \dots, A^{N - 1} D_{2}) \\ (2.27) & = rank (E_{2} D_{2}, A_{2} E_{2} D_{2}, \dots, A_{2}^{N - p - 1} E_{2} D_{2}) . \end{aligned}$ It follows from (2.26) and (2.27) that $\begin{aligned} (2.28) & rank (E_{2} D_{2}, A_{2} E_{2} D_{2}, \dots, A_{2}^{N - p - 1} E_{2} D_{2}) = p . \end{aligned}$ So the pair $(A_{2}, E_{2} D_{2})$ satisfies the corresponding Kalman’s rank condition (2.12). Moreover, since $A_{2}$ is nilpotent and Ω satisfies the usual multiplier control condition, by Theorem 2.2, the subsystem (2.19) associated with the boundary $E_{2} D_{2}$ -observation (2.24) has only the trivial solution $Φ_{2} = 0$ , provided that $T > 0$ is large enough. Then, noting $Φ_{1} = 0$ , it follows from (2.17) that $Φ = 0$ , provided that the time $T > 0$ is large enough. □
Remark 2.1.
The usual multiplier control condition is technically required for the uniqueness of solution to subsystem (2.19) by boundary observation (2.24). Moreover, the observation time T depends on the number p of equations in (2.19) and on the rank of matrix $E_{2} D_{2}$ . It might be large enough and indeterminable explicitly in general ([5,10,15]).
Corollary 2.1.
Let $Ω \subset R^{m}$ be a bounded domain with smooth boundary Γ satisfying the multiplier control condition and ω be a subdomain of Ω. Assume that
Kalman’s rank condition ( 1.3 ) holds with the control matrix $D = (D_{1}, D_{2})$ ;

we have the following condition $\begin{aligned} (2.29) & rank (D_{1}, A D_{1}, \dots, A^{N - 1} D_{1}) + rank (D_{2}, A D_{2}, \dots, A^{N - 1} D_{2}) = N; \end{aligned}$

the restriction of $A^{T}$ on $V_{1}$ is a nilpotent matrix, where $V_{1}$ is the largest subspace invariant for $A^{T}$ and contained in $Ker (D_{1}^{T})$ .

Then, for any given initial data $({\hat{Φ}}_{0}, {\hat{Φ}}_{1}) \in {(H_{0}^{1} (Ω) \times L^{2} (Ω))}^{N}$ , system ( 1.4 ) associated with ( 1.6 ) and ( 1.7 ) has only the trivial solution for $T > 0$ large enough.
Proof.
For $i = 1, 2$ , let $V_{i} = Ker {(D_{i}, A D_{i}, \dots, A^{N - 1} D_{i})}^{T}$ denote the largest subspace invariant for $A^{T}$ and contained in $Ker (D_{i}^{T})$ , and V denote the largest subspace invariant for $A^{T}$ and contained in $Ker (D^{T})$ . We have $V_{1} \cap V_{2} = V = {0}$ . Moreover, condition (2.29) implies $dim (V_{1}) + dim (V_{2}) = N$ . Therefore, $V_{2}$ is a supplement of $V_{1}$ , and both $V_{1}$ and $V_{2}$ are invariant for $A^{T}$ . We can thus apply Theorem 2.3 for getting the uniqueness of solution to system (1.4) with the internal observation (1.6) and the boundary observation (1.7). □
Remark 2.2.
The subspace $V_{1}$ is contained in $Ker (D_{1}^{T})$ , therefore unobservable by the $D_{1}$ -observation, hence, it should be naturally observed by the $D_{2}$ -observation. Accordingly, the supplement $V_{s}$ should be observed by the $D_{1}$ -observation. So the balanced distribution of observations is implemented by a coordination of the subspace $V_{1}$ and its supplement $V_{s}$ .
Theorem 2.4.
Let $Ω \subset R^{m}$ be a bounded domain with smooth boundary Γ and ω be a subdomain of Ω. Assume that
Kalman’s rank condition ( 1.3 ) holds with the control matrix $D = (D_{1}, D_{2})$ ;

the largest subspace $V_{1}$ invariant for $A^{T}$ and contained in $Ker (D_{1}^{T})$ admits a supplement $V_{s}$ which is also invariant for $A^{T}$ ;

the boundary control matrix $D_{2}$ satisfies the following condition $\begin{aligned} (2.30) & V_{1} \cap Ker (D_{2}^{T}) = {0} . \end{aligned}$

Then, for any given initial data $({\hat{Φ}}_{0}, {\hat{Φ}}_{1}) \in {(H_{0}^{1} (Ω) \times L^{2} (Ω))}^{N}$ , system ( 1.4 ) associated with ( 1.6 ) and ( 1.7 ) has only the trivial solution, provided that $T > 2 d (Ω)$ .
Proof.
By the same procedure as in Theorem 2.3, we project system (1.4) on the subspaces $V_{1}$ and $V_{s}$ into two subsystems (2.18) and (2.19). Since the subsystem (2.18) is observed by internal $(E_{1} D_{1})$ -observation (2.20), under the corresponding Kalman’s rank condition (2.23) for the pair $(A_{1}, E_{1} D_{1})$ , we can still deduce $Φ_{1} = 0$ .

We next show that $Φ_{2} = 0$ . Since $Φ_{1} = 0$ , the boundary $D_{2}$ -observation (1.7) becomes $\begin{aligned} (2.31) & D_{2}^{T} \partial_{ν} Φ = {(E_{2} D_{2})}^{T} \partial_{ν} Φ_{2} = 0 on (0, T) \times Γ . \end{aligned}$ Noting $V_{1} = Im (E_{2}^{T})$ , condition (2.30) implies $\begin{array}{r} Ker (D_{2}^{T}) \cap Im (E_{2}^{T}) = {0} . \end{array}$ Then, by Lemma 2.4, we get $\begin{aligned} (2.32) & rank (E_{2} D_{2}) = rank (E_{2}) = p . \end{aligned}$ Since the matrix $E_{2} D_{2}$ is of order $p \times M_{2}$ , so ${(E_{2} D_{2})}^{T}$ is a full column-rank matrix, then it follows from (2.31) that $\begin{aligned} (2.33) & \partial_{ν} Φ_{2} = 0 on (0, T) \times Γ . \end{aligned}$ By Holmgren’s uniqueness Theorem, the subsystem (2.19) associated with the full boundary observation (2.33) has only the trivial solution $Φ_{2} = 0$ , provided that $T > 2 d (Ω)$ . □
Corollary 2.2.
Let $Ω \subset R^{m}$ be a bounded domain with smooth boundary Γ and ω be a subdomain of Ω. Assume that conditions ( 1.3 ), ( 2.29 ) and ( 2.30 ) hold with $D = (D_{1}, D_{2})$ . Then, for any given initial data $({\hat{Φ}}_{0}, {\hat{Φ}}_{1}) \in {(H_{0}^{1} (Ω) \times L^{2} (Ω))}^{N}$ , system ( 1.4 ) associated with ( 1.6 ) and ( 1.7 ) has only the trivial solution, provided that $T > 2 d (Ω)$ .
Proof.
Under condition (1.3), Lemma 2.2 implies $V_{1} \cap V_{2} = V = {0}$ . Moreover, condition (2.29) implies $dim (V_{1}) + dim (V_{2}) = N$ , then $V_{2}$ is a supplement of $V_{1}$ , and both $V_{1}$ and $V_{2}$ are invariant for $A^{T}$ . We can then apply Theorem 2.4 for getting the uniqueness of solution. □
Remark 2.3.
Under condition (2.30), the uniqueness of solution to subsystem (2.19) with boundary observation (2.31) directly follows from Holmgren’s unique continuation theorem. So the multiplier control condition on Ω is no longer needed for Theorem 2.4 and Corollary 2.2, moreover, the observation time can be explicitly given by $T > 2 d (Ω)$ . This should be interesting for the engineering application.

3. Approximate mixed controllability

The following result can be easily shown by transposition (see [1,2,12,13]).

Proposition 3.1.
Let $Ω \subset R^{m}$ be a bounded domain with smooth boundary Γ. For any given initial data $({\hat{U}}_{0}, {\hat{U}}_{1}) \in {(L^{2} (Ω) \times H^{- 1} (Ω))}^{N}$ and any given mixed controls $(H, G) \in {(L^{2} (0, T; H^{- 1} (Ω)))}^{M_{1}} \times {(L^{2} (0, T; L^{2} (Γ)))}^{M_{2}}$ , system ( 1.1 ) admits a unique solution $\begin{matrix} (3.1) & U \in {(C^{0} ([0, T]; L^{2} (Ω)) \cap C^{1} ([0, T]; H^{- 1} (Ω)))}^{N} \end{matrix}$ with continuous dependence.
Definition 3.1.
System (1.1) is approximately controllable at the time $T > 0$ if for any given initial data $({\hat{U}}_{0}, {\hat{U}}_{1}) \in {(L^{2} (Ω) \times H^{- 1} (Ω))}^{N}$ , there exists a sequence ${(H_{n}, G_{n})}_{n \in N}$ of controls in ${(L^{2} (0, + \infty; H^{- 1} (Ω)))}^{M_{1}} \times {(L^{2} (0, + \infty; L^{2} (Γ)))}^{M_{2}}$ with compact support in $[0, T]$ , such that the sequence ${U_{n}}_{n \in N}$ of the corresponding solutions to system (1.1) satisfies $\begin{matrix} (3.2) & U_{n} \to 0 as n \to + \infty \end{matrix}$ in the space $\begin{matrix} (3.3) & (C_{loc}^{0} ([T, + \infty); L^{2} (Ω)) \cap C_{loc}^{1} ([T, + \infty); H^{- 1} {(Ω)))}^{N} . \end{matrix}$

By the duality approach, the approximate controllability of system (1.1) is equivalent to the uniqueness of solution to system (1.4) associated with the mixed observations (1.6) and (1.7). As a direct consequence of Theorem 2.3, we have
Theorem 3.1.
Let $Ω \subset R^{m}$ be a bounded domain with smooth boundary Γ satisfying the multiplier control condition and ω be a subdomain of Ω. Assume that
Kalman’s rank condition ( 1.3 ) holds with the control matrix $D = (D_{1}, D_{2})$ ;

the largest subspace $V_{1}$ invariant for $A^{T}$ and contained in $Ker (D_{1}^{T})$ admits a supplement $V_{s}$ which is also invariant for $A^{T}$ ;

the restriction of $A^{T}$ on $V_{1}$ is a nilpotent matrix.

Then system ( 1.1 ) is approximately controllable by mixed controls $(H, G)$ , provided that $T > 0$ is large enough.
Corollary 3.1.
Let $Ω \subset R^{m}$ be a bounded domain with smooth boundary Γ satisfying the multiplier control condition and ω be a subdomain of Ω. Assume that
the conditions ( 1.3 ) and ( 2.29 ) hold with the control matrix $D = (D_{1}, D_{2})$ ;

the restriction of $A^{T}$ on $V_{1}$ is a nilpotent matrix, where $V_{1}$ denotes the largest subspace invariant for $A^{T}$ and contained in $Ker (D_{1}^{T})$ .

Then system ( 1.1 ) is approximately controllable by mixed controls $(H, G)$ , provided that $T > 0$ is large enough.
Remark 3.1.
As what has been explained in Remark 2.1, the controllability time $T > 0$ is not explicitly determinable in these cases.
Remark 3.2.
When $D_{1} = 0$ or $D_{2} = 0$ , we find again the previous results on the approximate boundary controllability in [7,10] or the approximate internal controllability in [8].

As a direct consequence of Theorem 2.4, we have the following result which is more practically useful for applications. Theorem 3.2.
Let $Ω \subset R^{m}$ be a bounded domain with smooth boundary Γ and ω be a subdomain of Ω. Assume that
conditions ( 1.3 ) holds with the control matrix $D = (D_{1}, D_{2})$ ;

the largest subspace $V_{1}$ invariant for $A^{T}$ and contained in $Ker (D_{1}^{T})$ admits a supplement $V_{s}$ which is also invariant for $A^{T}$ ,

condition ( 2.30 ) holds.

Then system ( 1.1 ) is approximately controllable by mixed controls $(H, G)$ at the time $T > 2 d (Ω)$ .
Corollary 3.2.
Let $Ω \subset R^{m}$ be a bounded domain with smooth boundary Γ and ω be a subdomain of Ω. Assume that conditions ( 1.3 ), ( 2.29 ) and ( 2.30 ) hold with the control matrix $D = (D_{1}, D_{2})$ . Then system ( 1.1 ) is approximately controllable by mixed controls $(H, G)$ at the time $T > 2 d (Ω)$ .
Remark 3.3.
As what has been explained in Remark 2.3, under the stronger condition (2.30), the multiplier control condition is no longer necessary in Theorem 3.2 and Corollary 3.2, and the controllability time can be specified by $T > 2 d (Ω)$ .
Remark 3.4.
In order to better fit with the exact controllability and the exact synchronization, we have chosen ${(H_{0}^{1} (Ω) \times L^{2} (Ω))}^{N}$ as working space for system (1.4), accordingly, ${(L^{2} (Ω) \times H^{- 1} (Ω))}^{N}$ as working space for the primary system (1.1). Of course, the choice of working spaces is abundant.

4. Approximate mixed synchronization by groups

Let $p ⩾ 1$ be an integer and $\begin{matrix} (4.1) & 0 = n_{0} < n_{1} < \dots < n_{p} = N \end{matrix}$ be a partition with $n_{r} - n_{r - 1} ⩾ 2$ for $1 ⩽ r ⩽ p$ . We rearrange the components of the state variable U into p groups $\begin{matrix} (4.2) & (u^{(1)}, \dots, u^{(n_{1})}), (u^{(n_{1} + 1)}, \dots, u^{(n_{2})}), \dots, (u^{(n_{p - 1} + 1)}, \dots, u^{(n_{p})}) . \end{matrix}$

Definition 4.1.
System (1.1) is approximately synchronizable by p-groups at $T > 0$ if for any given initial data $({\hat{U}}_{0}, {\hat{U}}_{1}) \in {(L^{2} (Ω) \times H^{- 1} (Ω))}^{N}$ , there exists a sequence of controls ${(H_{n}, G_{n})}_{n \in N}$ in ${(L^{2} (0, + \infty; H^{- 1} (Ω)))}^{M_{1}} \times {(L^{2} (0, + \infty; L^{2} (Γ)))}^{M_{2}}$ with compact support in $[0, T]$ , such that the sequence ${U_{n}}_{n \in N}$ of the corresponding solutions to system (1.1) satisfies $\begin{aligned} (4.3) & u_{n}^{(k)} - u_{n}^{(l)} \to 0 in C_{loc}^{0} ([T, + \infty); L^{2} (Ω)) \cap C_{loc}^{1} ([T, + \infty); H^{- 1} (Ω)) \end{aligned}$ as $n \to + \infty$ for all $n_{r - 1} + 1 ⩽ k, l ⩽ n_{r}$ and $1 ⩽ r ⩽ p$ . Furthermore, if there exists a vector-valued function ${(u_{1}, \dots, u_{p})}^{T}$ , called the approximately synchronizable state by p-groups, such that $\begin{aligned} (4.4) & u_{n}^{(k)} \to u_{r} in C_{loc}^{0} ([T, + \infty); L^{2} (Ω)) \cap C_{loc}^{1} ([T, + \infty); H^{- 1} (Ω)) \end{aligned}$ as $n \to + \infty$ for all $n_{r - 1} + 1 ⩽ k ⩽ n_{r}$ and $1 ⩽ r ⩽ p$ , Then system (1.1) will be called approximately synchronizable by p-groups in the pinning sense with the approximately synchronizable state by p-groups ${(u_{1}, \dots, u_{p})}^{T}$ .

Let $S_{r}$ be a full row-rank matrix of order $(n_{r} - n_{r - 1} - 1) \times (n_{r} - n_{r - 1})$ : $\begin{matrix} S_{r} = (\begin{array}{cc} 1 & - 1 \\ 1 & - 1 \\ ⋱ & ⋱ \\ 1 & - 1 \end{array}), 1 ⩽ r ⩽ p . \end{matrix}$ Define the $(N - p) \times N$ matrix $C_{p}$ of synchronization by p-groups as $\begin{matrix} (4.5) & C_{p} = (\begin{array}{c} S_{1} \\ S_{2} \\ ⋱ \\ S_{p} \end{array}) . \end{matrix}$ Let $\begin{array}{r} Ker (C_{p}) = Span {e_{1}, \dots, e_{p}}, \end{array}$ where $\begin{matrix} (4.6) & e_{r} = {(0, \dots, 0, \overset{(n_{r - 1} + 1)}{1}, \dots, \overset{(n_{r})}{1}, 0, \dots, 0)}^{T}, 1 ⩽ r ⩽ p . \end{matrix}$ (4.3) can be written as $\begin{aligned} (4.7) & C_{p} U_{n} \to 0 as n \to + \infty \end{aligned}$ in the space $\begin{aligned} (4.8) & (C_{loc}^{0} ([T, + \infty); L^{2} (Ω)) \cap C_{loc}^{1} ([T, + \infty); H^{- 1} (Ω)))^{N - p}, \end{aligned}$ respectively, (4.4) can be written as $\begin{aligned} (4.9) & U_{n} \to \sum_{r = 1}^{p} e_{r} u_{r} as n \to + \infty \end{aligned}$ in the space $\begin{aligned} (4.10) & (C_{loc}^{0} ([T, + \infty); L^{2} (Ω)) \cap C_{loc}^{1} ([T, + \infty); H^{- 1} (Ω)))^{N} . \end{aligned}$

Assume that A satisfies the condition of $C_{p}$ -compatibility: $\begin{aligned} (4.11) & A Ker (C_{p}) \subseteq Ker (C_{p}) . \end{aligned}$ By Lemma 2.5, there exists a unique matrix $A_{p}$ of order $(N - p)$ , such that $\begin{aligned} (4.12) & C_{p} A = A_{p} C_{p} . \end{aligned}$ Applying $C_{p}$ to system (1.1) and setting $W_{p} = C_{p} U$ , we get the following reduced system: $\begin{matrix} (4.13) & \{\begin{array}{ll} W_{p}^{″} - Δ W_{p} + A_{p} W_{p} = D_{p 1} χ_{ω} H & in (0, + \infty) \times Ω, \\ W_{p} = D_{p 2} G & on (0, + \infty) \times Γ, \end{array} \end{matrix}$ where $\begin{matrix} (4.14) & D_{p} = (D_{p 1}, D_{p 2}) with D_{p 1} = C_{p} D_{1} and D_{p 2} = C_{p} D_{2} . \end{matrix}$ Obviously, the approximate synchronization by p-groups of system (1.1) is equivalent to the approximate controllability of the reduced system (4.13).

As a direct application of Theorem 3.1, we have
Theorem 4.1.
Let $Ω \subset R^{m}$ be a bounded domain with smooth boundary Γ satisfying the multiplier control condition and ω be a subdomain of Ω. Assume that A satisfies the condition of $C_{p}$ -compatibility ( 4.11 ). Assume furthermore that

the rank condition $\begin{matrix} (4.15) & rank (D_{p}, A_{p} D_{p}, \dots, A_{p}^{N - p - 1} D_{p}) = N - p \end{matrix}$ holds with $D_{p} = (D_{p 1}, D_{p 2})$ ;

the largest subspace $V_{p 1}$ invariant for $A_{p}^{T}$ and contained in $Ker (D_{p 1}^{T})$ admits a supplement $V_{p s}$ which is also invariant for $A_{p}^{T}$ ;

the restriction of $A_{p}^{T}$ on the subspace $V_{p 1}$ is a nilpotent matrix.

Then system ( 1.1 ) is approximately synchronizable by p-groups at the time $T > 0$ large enough.

Similarly, by Theorem 3.2, we have
Theorem 4.2.
Let $Ω \subset R^{m}$ be a bounded domain with smooth boundary Γ and ω be a subdomain of Ω. Assume that A satisfies the condition of $C_{p}$ -compatibility ( 4.11 ). Assume furthermore that

the condition ( 4.15 ) holds with $D_{p} = (D_{p 1}, D_{p 2})$ ;

the largest subspace $V_{p 1}$ invariant for $A_{p}^{T}$ and contained in $Ker (D_{p 1}^{T})$ admits a supplement $V_{p s}$ which is also invariant for $A_{p}^{T}$ ;

the boundary control matrix $D_{p 2}$ satisfies the condition $\begin{matrix} (4.16) & V_{p 1} \cap Ker (D_{p 2}^{T}) = {0} . \end{matrix}$

Then system ( 1.1 ) is approximately synchronizable by p-groups at the time $T > 2 d (Ω)$ .

For the convenience of applications, we will reformulate Theorem 4.1 in terms of original matrices A, $D_{1}$ and $D_{2}$ . For a better presentation, we first make some elementary preparation.
Lemma 4.1 ([4, p.19]).

Let $f : X \to Y$ , $A \subseteq X$ and $B \subseteq Y$ . The preimage of B under f is defined by $\begin{aligned} (4.17) & f^{- 1} (B) = {x \in X : f (x) \in B} . \end{aligned}$ The following properties hold: $\begin{aligned} (4.18) & A \subseteq f^{- 1} (f (A)), equality holds if f is injective, \\ (4.19) & f (f^{- 1} (B)) \subseteq B, equality holds if f is surjective, \\ (4.20) & f^{- 1} (B) \subseteq f^{- 1} (B \cap f (X)), equality holds if f is injective \\ (4.21) & f^{- 1} (B_{1} \oplus B_{2}) = f^{- 1} (B_{1}) \oplus f^{- 1} (B_{2}), \\ (4.22) & f^{- 1} (B_{1} \cap B_{2}) = f^{- 1} (B_{1}) \cap f^{- 1} (B_{2}) . \end{aligned}$

Proposition 4.1.
Let $A_{p}$ be given by ( 4.12 ) and $D_{p}$ be given by ( 4.14 ), respectively.
We have $\begin{aligned} (4.23) & Ker (D_{p}^{T}) = {(C_{p}^{T})}^{- 1} Ker (D^{T}) . \end{aligned}$

For any given subspace V invariant for $A^{T}$ , the subspace ${(C_{p}^{T})}^{- 1} V$ is invariant for $A_{p}^{T}$ .

Let V be the largest subspace invariant for $A^{T}$ and contained in $Ker (D^{T})$ . $\begin{aligned} (4.24) & V_{p} = {(C_{p}^{T})}^{- 1} V \end{aligned}$ is the largest subspace invariant for $A_{p}^{T}$ and contained in $Ker (D_{p}^{T})$ .

The restriction of $A_{p}^{T}$ on $V_{p}$ is the same as the restriction of $A^{T}$ on V.

Let V be the largest subspace invariant for $A^{T}$ and contained in $Ker (D^{T})$ and $V_{s}$ be a supplement of V. Both $V_{s}$ and V are invariant for $A^{T}$ .

Then ${(C_{p}^{T})}^{- 1} V_{s}$ is a supplement of ${(C_{p}^{T})}^{- 1} V$ , moreover, both ${(C_{p}^{T})}^{- 1} V_{s}$ and ${(C_{p}^{T})}^{- 1} V$ are invariant for $A_{p}^{T}$ .
Proof.
(a) Let $x \in Ker (D_{p}^{T}) = Ker (D^{T} C_{p}^{T})$ , namely, $C_{p}^{T} x \in Ker (D^{T})$ , or equivalently, $x \in {(C_{p}^{T})}^{- 1} Ker (D^{T})$ . This gives (4.23).

(b) Using Lemma 4.1, we easily get $\begin{aligned} A_{p}^{T} {(C_{p}^{T})}^{- 1} V \\ = {(C_{p}^{T})}^{- 1} C_{p}^{T} A_{p}^{T} {(C_{p}^{T})}^{- 1} V ((4.18)) \\ = {(C_{p}^{T})}^{- 1} A^{T} C_{p}^{T} {(C_{p}^{T})}^{- 1} V_{s} ((4.12)) \\ \subseteq {(C_{p}^{T})}^{- 1} A^{T} V \subseteq {(C_{p}^{T})}^{- 1} V ((4.19) and A^{T} V \subseteq V) \end{aligned}$ So ${(C_{p}^{T})}^{- 1} V$ is invariant for $A_{p}^{T}$ .

(c) Still by (4.19) in Lemma 4.1, we have $\begin{array}{r} D_{p}^{T} {(C_{p}^{T})}^{- 1} V = D^{T} C_{p}^{T} {(C_{p}^{T})}^{- 1} V \subseteq D^{T} V = {0} . \end{array}$ So ${(C_{p}^{T})}^{- 1} V$ is invariant for $A_{p}^{T}$ and contained in $Ker (D_{p}^{T})$ . Therefore, $\begin{aligned} (4.25) & {(C_{p}^{T})}^{- 1} V \subseteq V_{p}, \end{aligned}$ where $V_{p}$ is the largest subspace invariant for $A_{p}^{T}$ and contained in $Ker (D_{p}^{T})$ .

On the other hand, noting (4.12), we get $\begin{array}{r} A^{T} C_{p}^{T} V_{p} = C_{p}^{T} A_{p}^{T} V_{p} \subseteq C_{p}^{T} V_{p} \end{array}$ and $\begin{array}{r} D^{T} C_{p}^{T} V_{p} = D_{p}^{T} V_{p} = {0} . \end{array}$ So $C_{p}^{T} V_{p}$ is invariant for $A^{T}$ and contained in $Ker (D^{T})$ , therefore $C_{p}^{T} V_{p} \subseteq V$ . Since $C_{p}^{T}$ is injective, by (4.18) in Lemma 4.1, we have $\begin{array}{r} V_{p} = {(C_{p}^{T})}^{- 1} C_{p}^{T} V_{p} \subseteq {(C_{p}^{T})}^{- 1} V, \end{array}$ which together with (4.25) implies (4.24).

(d) Noting (4.12), we have $\begin{matrix} (4.26) & A^{T} V = A^{T} C_{p}^{T} V_{p} = C_{p}^{T} A_{p}^{T} V_{p} . \end{matrix}$ Since $V_{p}$ is invariant for $A_{p}^{T}$ , there exists a matrix $B^{T}$ of order $(N - p)$ , such that $\begin{matrix} (4.27) & A_{p}^{T} V_{p} = V_{p} B^{T} . \end{matrix}$ Then it follows from (4.26) and (4.27) that $\begin{matrix} (4.28) & A^{T} V = C_{p}^{T} V_{p} B^{T} = V B^{T} . \end{matrix}$ So the matrix $B^{T}$ is also the restriction of $A^{T}$ on V.

(e) Using (4.21) and (4.22) in Lemma 4.1, we get $\begin{aligned} {(C_{p}^{T})}^{- 1} V \cap {(C_{p}^{T})}^{- 1} V_{s} \\ = {(C_{p}^{T})}^{- 1} (V \cap V_{s}) = {(C_{p}^{T})}^{- 1} {0} = {0} \end{aligned}$ and $\begin{aligned} {(C_{p}^{T})}^{- 1} V \oplus {(C_{p}^{T})}^{- 1} V_{s} \\ = {(C_{p}^{T})}^{- 1} (V \oplus V_{s}) = {(C_{p}^{T})}^{- 1} R^{N} = R^{N - p} . \end{aligned}$ □

The following theorem is a version of Theorem 4.1 in terms of original matrices.
Theorem 4.3.
Let $Ω \subset R^{m}$ be a bounded domain with smooth boundary Γ satisfying the multiplier control condition and ω be a subdomain of Ω. Assume that A satisfies the condition of $C_{p}$ -compatibility ( 4.11 ). Assume furthermore that
the rank condition $\begin{matrix} (4.29) & rank C_{p} (D, A D, \dots, A^{N - 1} D) = N - p \end{matrix}$ holds with control matrix $D = (D_{1}, D_{2})$ ;

the largest subspace $V_{1}$ invariant for $A^{T}$ and contained in $Ker {(D_{1})}^{T}$ admits a supplement $V_{s}$ which is also invariant for $A^{T}$ ;

the restriction of $A^{T}$ on the subspace $V_{1}$ is a nilpotent matrix.

Then system ( 1.1 ) is approximately synchronizable by p-groups for $T > 0$ large enough.
Proof.
It is not difficult to check all the assumptions in Theorem 4.1. □

Similarly, the following theorem is a version of Theorem 4.2 in terms of the original matrices. Theorem 4.4.
Let $Ω \subset R^{m}$ be a bounded domain with smooth boundary Γ and ω be a subdomain of Ω. Assume that A satisfies the condition of $C_{p}$ -compatibility ( 4.11 ). Assume furthermore that
the rank condition ( 4.29 ) holds with the control matrix $D = (D_{1}, D_{2})$ ;

the largest subspace $V_{1}$ invariant for $A^{T}$ and contained in $Ker {(D_{1})}^{T}$ admits a supplement $V_{s}$ which is also invariant for $A^{T}$ ;

the boundary control matrix $D_{2}$ satisfies the following condition $\begin{matrix} (4.30) & V_{1} \cap Ker (D_{2}^{T}) \cap Im (C_{p}^{T}) = {0} . \end{matrix}$

Then system ( 1.1 ) is approximately synchronizable by p-groups at the time $T > 2 d (Ω)$ .
Remark 4.1.
Under the stronger condition (4.30), the multiplier control condition is no longer necessary for Theorem 4.4, and the controllability time is specified by $T > 2 d (Ω)$ .

5. Condition of $C_{p}$ -compatibility

Let $V = span {E_{1}, \dots, E_{d}}$ denote the largest subspace invariant for $A^{T}$ and contained in $Ker {(D_{1}, D_{2})}^{T}$ . We define the projection ${(ψ_{1}, \dots, ψ_{d})}^{T}$ of system (1.1) on V by $\begin{matrix} (5.1) & ψ_{r} = E_{r}^{T} U, r = 1, \dots, d . \end{matrix}$

Proposition 5.1.
The projection ${(ψ_{1}, \dots, ψ_{d})}^{T}$ is independent of applied controls $(H, G)$ .
Proof.
Let $β_{r s}$ ( $1 ⩽ r, s ⩽ d$ ) be some coefficients such that $\begin{matrix} (5.2) & A^{T} E_{r} = \sum_{s = 1}^{d} β_{r s} E_{s}, D_{1}^{T} E_{r} = 0 and D_{2}^{T} E_{r} = 0, r = 1, \dots, d . \end{matrix}$ For $r = 1, \dots, d$ , applying $E_{r}^{T}$ to system $(I)$ , we get $\begin{matrix} (5.3) & \{\begin{array}{ll} ψ_{r}^{″} - Δ ψ_{r} + \sum_{s = 1}^{d} β_{r s} ψ_{s} = 0 & in (0, + \infty) \times Ω, \\ ψ_{r} = 0 & on (0, + \infty) \times Γ, \end{array} \end{matrix}$ which is independent of applied controls. □
Proposition 5.2.
Let V denote the largest subspace invariant for $A^{T}$ and contained in $Ker (D^{T})$ . Assume that system ( 1.1 ) is approximately synchronizable by p-groups. Then we have $\begin{aligned} (5.4) & Im (C_{p}^{T}) \cap V = {0} . \end{aligned}$
Proof.
For any given $({\hat{U}}_{0}, {\hat{U}}_{1}) \in {(L^{2} (Ω) \times H^{- 1} (Ω))}^{N}$ , there exists a sequence ${(H_{n}, G_{n})}_{n \in N}$ in ${(L^{2} (0, + \infty; H^{- 1} (Ω)))}^{M_{1}} \times {(L^{2} (0, + \infty; L^{2} (Γ)))}^{M_{2}}$ with compact support in $[0, T]$ , such that the sequence of the corresponding solutions ${U_{n}}_{n \in N}$ to system (1.1) satisfies (4.7).

Let $x_{1}, \dots, x_{d} \in R$ and $y \in R^{N - p}$ , such that $\begin{array}{r} \sum_{s = 1}^{d} x_{r} E_{r} = C_{p}^{T} y . \end{array}$ We have $\begin{matrix} (5.5) & \sum_{s = 1}^{d} x_{r} ψ_{r} = y^{T} C_{p} U_{n} . \end{matrix}$ By (4.7), the right-hand side of (5.5) goes to zero as $n \to + \infty$ , but the left-hand side is independent of applied controls because of Proposition 5.1. We get thus $x_{1} = \dots = x_{d} = 0$ . Then $V \cap Im (C_{p}^{T}) = {0}$ . □
Proposition 5.3.
Assume that system ( 1.1 ) is approximately synchronizable by p-groups. Then the rank condition $\begin{matrix} (5.6) & rank C_{p} (D, A D, \dots, A^{N - 1} D) = N - p \end{matrix}$ holds with control matrix $D = (D_{1}, D_{2})$ , or equivalentely, $\begin{matrix} (5.7) & rank (D_{p}, A_{p} D_{P}, \dots, A_{p}^{N - p - 1} D_{p}) = N - p, \end{matrix}$ where $D_{p}$ is given by ( 4.14 ). Assume furthermore that we have $\begin{matrix} (5.8) & rank (D, A D, \dots, A^{N - 1} D) = N - p . \end{matrix}$ Then A satisfies the condition of $C_{p}$ -compatibility ( 4.11 ).
Proof.
By Lemma 2.1, $V = Ker {(D, A D, \dots, A^{N - 1} D)}^{T}$ is the largest subspace invariant for $A^{T}$ and contained in $Ker (D^{T})$ . By Proposition 5.2, $\begin{array}{r} Ker {(D, A D, \dots, A^{N - 1} D)}^{T} \cap Im (C_{p}^{T}) = {0} . \end{array}$ Then by Lemma 2.4, we get (5.6).

Noting (4.7) and applying successively $C_{p}, C_{p} A, \dots$ to system (1.1) with $H = H_{n}$ and $G = G_{n}$ , we get $\begin{matrix} C_{p} A U_{n} \to 0, C_{p} A^{2} U_{n} \to 0, \dots in D^{'} ((T, + \infty) \times Ω), \end{matrix}$ namely, $\begin{matrix} C_{\tilde{p}} U_{n} \to 0 in D^{'} ((T, + \infty) \times Ω), \end{matrix}$ where $C_{\tilde{p}}$ is the extension matrix of order $(N - \tilde{p}) \times N$ with $\tilde{p} ⩽ \tilde{p}$ defined by $\begin{matrix} Im (C_{\tilde{p}}^{T}) = span {C_{p}^{T}, A^{T} C_{p}^{T}, \dots, {(A^{T})}^{N - 1} C_{p}^{T}} . \end{matrix}$ By Cayley-Hamilton’s Theorem, $Im (C_{\tilde{p}}^{T})$ is invariant for $A^{T}$ , namely, A satisfies the condition of $C_{\tilde{p}}$ -compatibility. By Proposition 5.2 with $C_{p}$ replaced by $C_{\tilde{p}}$ , we get $\begin{matrix} V \cap Im (C_{\tilde{p}}^{T}) = {0} . \end{matrix}$ By Lemma 2.4, we get $\begin{matrix} N - p = ran (D, A D, \dots, A^{N - 1} D) ⩾ rank (C_{\tilde{p}} (D, A D, \dots, A^{N - 1} D)) = N - \tilde{p} . \end{matrix}$ It follows that $p = \tilde{p}$ , namely, $C_{p}^{T} = C_{\tilde{p}}^{T}$ , therefore, A satisfies the condition of $C_{p}$ -compatibility (4.11). □

6. Induced mixed synchronization

Definition 6.1.
Let $C_{q}$ be a matrix of order $(N - q) \times N$ . System (1.1) is approximately $C_{q}$ -synchronizable at the time $T > 0$ if for any given initial data $({\hat{U}}_{0}, {\hat{U}}_{1}) \in {(L^{2} (Ω) \times H^{- 1} (Ω))}^{N}$ , there exists a sequence ${(H_{n}, G_{n})}_{n \in N}$ of controls in ${(L^{2} (0, + \infty; H^{- 1} (Ω)))}^{M_{1}} \times {(L^{2} (0, + \infty; L^{2} (Γ)))}^{M_{2}}$ with compact support in $[0, T]$ , such that the sequence ${U_{n}}_{n \in N}$ of corresponding solutions to system (1.1) satisfies $\begin{matrix} (6.1) & C_{q} U_{n} \to 0 as n \to + \infty \end{matrix}$ in the space $\begin{matrix} (6.2) & (C_{loc}^{0} ([T, + \infty); L^{2} (Ω)) \cap C_{loc}^{1} ([T, + \infty); H^{- 1} (Ω)))^{N - q} . \end{matrix}$ Moreover, if there exist a vector-valued function ${(v_{1}, \dots, v_{q})}^{T}$ , called the approximately $C_{q}$ -synchronizable state, such that $\begin{matrix} (6.3) & U_{n} \to \sum_{s = 1}^{q} v_{s} ϵ_{s}, where Ker (C_{q}) = span {ϵ_{1}, \dots, ϵ_{q}}, \end{matrix}$ as $n \to + \infty$ in the space $\begin{matrix} (6.4) & (C_{loc}^{0} ([T, + \infty); L^{2} (Ω)) \cap C_{loc}^{1} ([T, + \infty); H^{- 1} (Ω)))^{N}, \end{matrix}$ system (1.1) will be approximately $C_{q}$ -synchronizable in the pinning sense.

When $Im (C_{p}^{T}) \subseteq Im (C_{q}^{T})$ with $0 ⩽ q < p$ , and system (1.1) is already approximately synchronizable by p-groups, the approximate $C_{q}$ -synchronization will be called the induced synchronization. In particular, the matrix $C_{q}$ will be defined by the following external extension.

For $1 ⩽ i ⩽ m$ , let $λ_{i}$ be the eigenvalue of $A^{T}$ and denote by (see [3,14]) $\begin{aligned} (6.5) & E_{i 0} = 0, A^{T} E_{i j} = λ_{i} E_{i j} + E_{i, j - 1}, 1 ⩽ j ⩽ d_{i} \end{aligned}$ the corresponding Jordan chain. Let I denote the set of indices i such that $\begin{aligned} (6.6) & I = {i : E_{i {\overline{d}}_{i}} \in Im (C_{p}^{T}) with 1 ⩽ {\overline{d}}_{i} ⩽ d_{i}} . \end{aligned}$

Let $C_{q}$ be the extension matrix of order $(N - q) \times N$ given by $\begin{matrix} (6.7) & Im (C_{q}^{T}) = ⨁_{i \in I} span {E_{i 1}, \dots, E_{i {\overline{d}}_{i}}, \dots, E_{i d_{i}}} \end{matrix}$ and $\begin{matrix} (6.8) & q = N - \sum_{i \in I} d_{i} . \end{matrix}$

Define the control matrix $D_{q}$ of order $N \times (N - p)$ by $\begin{aligned} (6.9) & Ker (D_{q}^{T}) = ⨁_{i \in I^{c}} span {E_{i 1}, \dots, E_{i d_{i}}} \oplus ⨁_{i \in I} span {E_{i {\overline{d}}_{i} + 1}, \dots, E_{i d_{i}}}, \end{aligned}$ where $I^{c}$ denotes the supplement of I.
Lemma 6.1 (see [9]).

Let $C_{q}$ and $D_{q}$ be defined by ( 6.7 ) and ( 6.9 ), respectively. We have $\begin{aligned} (6.10) & A Ker (C_{q}) \subseteq Ker (C_{q}), \\ (6.11) & rank (D_{q}, A D_{q}, \dots, A^{N - 1} D_{q}) = N - q, \\ (6.12) & rank C_{p} (D_{q}, A D_{q}, \dots, A^{N - 1} D_{q}) = N - p, \\ (6.13) & rank C_{q} (D_{q}, A D_{q}, \dots, A^{N - 1} D_{q}) = N - q . \end{aligned}$

Lemma 6.2 (see [9]).

Let A satisfy the condition of $C_{p}$ -compatibility ( 2.19 ). Assume that $\begin{aligned} (6.14) & rank C_{p} (D, A D, \dots, A^{N - 1} D) = N - p . \end{aligned}$ Then we have $\begin{aligned} (6.15) & rank C_{q} (D, A D, \dots, A^{N - 1} D) = N - q, \end{aligned}$ where $C_{q}$ is defined by ( 6.7 ).

We will develop the approximate $C_{q}$ -synchronization with $C_{q}$ given by (6.7).

Theorem 6.1.
Let $Ω \subset R^{m}$ be a bounded domain with smooth boundary Γ and ω be a subdomain of Ω. Assume that A satisfies the condition of $C_{p}$ -compatibility ( 4.11 ). Assume furthermore that
Kalman’s rank condition ( 4.29 ) holds with the control matrix $D = (D_{1}, D_{2})$ ;

the largest subspace $V_{1}$ invariant for $A^{T}$ and contained in $Ker (D_{1}^{T})$ admits a supplement $V_{s}$ which is also invariant for $A^{T}$ ;

the control matrix $D_{2}$ satisfies condition ( 4.30 ) and $Ker (D_{2}^{T})$ is invariant for $A^{T}$ .

Then system ( 1.1 ) is approximately $C_{q}$ -synchronizable at the time $T > 2 d (Ω)$ .
Proof.
First, by Lemma 6.2, condition (4.29) implies $\begin{matrix} (6.16) & rank C_{q} (D, A D, \dots, A^{N - 1} D) = rank (C_{q}), \end{matrix}$ where $D = (D_{1}, D_{2})$ and $C_{q}$ is given by (6.7). By Lemma 2.4 with $C_{p}$ replaced by $C_{q}$ , it follows from (6.16) that $\begin{matrix} (6.17) & V \cap Im (C_{q}^{T}) = {0}, \end{matrix}$ where by Lemma 2.1, $V = Ker {(D, A D, \dots, A^{N - 1} D)}^{T}$ is the largest subspace invariant for $A^{T}$ and contained in $Ker (D^{T})$ . Noting that $Ker (D_{2}^{T})$ is invariant for $A^{T}$ , so $V_{2} = Ker (D_{2}^{T})$ is the largest subspace invariant for $A^{T}$ and contained in $Ker (D_{2}^{T})$ . By Lemma 2.2, $V_{1} \cap Ker (D_{2}^{T}) = V_{1} \cap V_{2} = V$ . Then it follows from (6.17) that $\begin{matrix} (6.18) & V_{1} \cap Ker (D_{2}^{T}) \cap Im (C_{q}^{T}) = V \cap Im (C_{q}^{T}) = {0} . \end{matrix}$

Finally, since A satisfies the condition of $C_{q}$ -compatibility (6.10) in Lemma 6.1, applying Theorem 4.4 with $C_{p}$ replaced by $C_{q}$ , we get the approximate $C_{q}$ -synchronization for system (1.1). □
Remark 6.1.
Under the stronger condition (c), the multiplier control condition is no longer necessary for Theorem 6.1, and the controllability time is specified by $T > 2 d (Ω)$ .

In order to guarantee $\begin{matrix} (6.19) & V_{1} \cap Ker (D_{2}^{T}) \cap Im (C_{q}^{T}) = {0} \end{matrix}$ for getting the approximate $C_{q}$ -synchronization, we require $Ker (D_{2}^{T})$ to be invariant for $A^{T}$ in Theorem 6.1.

We denote by $\begin{array}{r} D_{p}^{} = {D = (D_{1}, D_{2}) : (a)–(c) in Theorem 6.1 hold} . \end{array}$ Theorem 6.2.
Let* $Ω \subset R^{m}$ be a bounded domain with smooth boundary Γ and ω be a subdomain of Ω. Assume that A satisfies the condition of $C_{p}$ -compatibility ( 4.11 ). Then we have $\begin{matrix} (6.20) & min_{D \in D_{p}^{}} rank (D, A D, \dots, A^{N - 1} D) = N - q . \end{matrix}$
Proof.
First, by Theorem 6.1, for any given $D \in D_{p}^{}$ , system (1.1) is approximately $C_{q}$ -synchronizable. By Proposition 5.3 with $C_{p}$ replaced by $C_{q}$ , we have $\begin{aligned} (6.21) & rank C_{q} (D, A D, \dots, A^{N - 1} D) = N - q . \end{aligned}$ In particular, we have $\begin{aligned} (6.22) & rank (D, A D, \dots, A^{N - 1} D) ⩾ N - q . \end{aligned}$

We next show that the control matrix $D = (D_{q}, 0) \in D_{p}^{}$ , where $D_{q}$ is defined by (6.9). Then, combining (6.11) and (6.22), we get (6.20).

(a) Noting (6.12), the rank condition (4.29) holds for $D = (D_{q}, 0)$ .

(b) Noting $\begin{aligned} (6.23) & V_{1} = ⨁_{i \in I^{c}} span {E_{i 1}, \dots, E_{i d_{i}}}; V_{s} = ⨁_{i \in I} span {E_{i 1}, \dots, E_{i d_{i}}} \end{aligned}$ $V_{s}$ and $V_{1}$ are mutually supplementary, moreover, both $V_{1}$ and $V_{s}$ are invariant for $A^{T}$ .

(c) Noting $D_{2} = 0$ and $Ker (D_{2}^{T}) = R^{N}$ , we have $\begin{matrix} (6.24) & V_{1} \cap Ker (D_{2}^{T}) \cap Im (C_{p}^{T}) = V_{1} \cap Im (C_{p}^{T}) \subseteq V_{1} \cap V_{s} = {0} . \end{matrix}$ Moreover, $Ker (D_{2}^{T}) = R^{N}$ is obviously invariant for $A^{T}$ . Then the control matrix $D = (D_{q}, 0) \in D_{p}^{}$ . The proof is complete. □
Remark 6.2.
The approximate $C_{q}$ -synchronization (6.1) provides $(p - q)$ more convergences than the approximate synchronization by p-groups (4.7). We recover then the $(p - q)$ controls lost in the reduced system (4.13). This is the best thing that we can expect.

7. Stability of approximate mixed synchronization by groups

We first characterize Kalman’s rank condition (5.8) from an algebraic point of view.

Theorem 7.1.
Let $Ω \subset R^{m}$ be a bounded domain with smooth boundary Γ satisfying the multiplier control condition and ω be a subdomain of Ω. Assume that system ( 1.1 ) is approximately synchronizable by p-groups. The following assertions are equivalent:
Kalman’s rank condition ( 5.8 ) holds with the control matrix $D = (D_{1}, D_{2})$ ;

$Im (C_{p}^{T})$ is invariant for $A^{T}$ and admits a supplement V, on which the projection of system ( 1.1 ) is independent of applied controls;

$Im (C_{p}^{T})$ is invariant for $A^{T}$ and admits a supplement V, which is invariant for $A^{T}$ and contained in $Ker (D^{T})$ .

Proof.
(a) ⟹ (b). By Proposition 5.3, we have $A Ker (C_{p}) \subseteq Ker (C_{p})$ , then, $A^{T} Im (C_{p}^{T}) \subseteq Im (C_{p}^{T})$ .

By Proposition 5.2, $V \cap Im (C_{p}^{T}) = {0}$ , where V is the largest subspace invariant for $A^{T}$ and contained in $Ker (D^{T})$ . By Lemma 2.1, condition (5.8) implies $dim (V) = p$ . Noting $dim Ker (C_{p}) = p$ , Lemma 2.3 implies that V is a supplement of $Im (C_{p}^{T})$ . Then, applying Proposition 5.1 with $d = p$ , the projection ${(ψ_{1}, \dots, ψ_{p})}^{T}$ of system (1.1) is independent of applied controls.

(b) ⟹ (c). Let $({\hat{U}}_{0}, {\hat{U}}_{1}) = (0, 0)$ . By Proposition 3.1, the linear mapping $\begin{aligned} (7.1) & F : (H, G) \to (U, U^{'}) \end{aligned}$ is continuous from the space ${(L^{2} (0, T; H^{- 1} (Ω)))}^{M_{1}} \times {(L^{2} (0, T; L^{2} (Γ)))}^{M_{2}}$ to the space ${(C^{0} ([0, T]; L^{2} (Ω) \times H^{- 1} (Ω)))}^{N}$ . The Fréchet derivative $\hat{U} \overset{△}{=} F^{'} (0) \hat{H}$ satisfies $\begin{aligned} (7.2) & \{\begin{array}{ll} {\hat{U}}^{″} - Δ \hat{U} + A \hat{U} = D_{1} χ_{ω} \hat{H} & in (0, + \infty) \times Ω, \\ \hat{U} = D_{2} \hat{G} & on (0, + \infty) \times Γ, \\ t = 0 : \hat{U} = {\hat{U}}^{'} = 0 & in Ω . \end{array} \end{aligned}$ Noting $S \oplus Im (C_{p}^{T}) = R^{N}$ , for any given $E \in S$ , let $\begin{aligned} (7.3) & A^{T} E = E_{1} + E_{2}, where E_{1} \in V and E_{2} \in Im (C_{p}^{T}) . \end{aligned}$ Applying $E^{T}$ to (7.2), we get $\begin{aligned} (7.4) & \{\begin{array}{ll} E^{T} {\hat{U}}^{″} - Δ E^{T} \hat{U} + E_{1}^{T} \hat{U} + E_{2}^{T} \hat{U} = E^{T} D_{1} χ_{ω} \hat{H} & in (0, + \infty) \times Ω, \\ E^{T} \hat{U} = E^{T} D_{2} \hat{G} & on (0, + \infty) \times Γ . \end{array} \end{aligned}$ Since the projections $E^{T} U$ and $E_{1}^{T} U$ on V are independent of applied controls, their Fréchet derivatives $E^{T} \hat{U} = 0$ and $E_{1}^{T} U = 0$ . It follows from (7.4) that $\begin{aligned} (7.5) & E^{T} D_{2} \hat{G} = 0, \forall \hat{G} \in {(L^{2} (0, T; L^{2} (Γ)))}^{M_{2}}, \\ (7.6) & E_{2}^{T} \hat{U} = E^{T} D_{1} χ_{ω} \hat{H}, \forall \hat{H} \in {(L^{2} (0, T; H^{- 1} (Ω)))}^{M_{1}} . \end{aligned}$ From (7.5), we get $D_{2}^{T} E = 0$ for all $E \in S$ , then $S \subseteq Ker (D_{2}^{T})$ .

Let $\hat{H} = D_{1}^{T} E h$ and $\hat{G} = 0$ in (7.2). By Proposition 3.1, the mapping $\begin{aligned} (7.7) & R : h \to (\hat{U}, {\hat{U}}^{'}) \end{aligned}$ is continuous from $L^{2} (0, T; H^{- 1} (Ω))$ to ${(L^{2} (0, T; L^{2} (Ω) \times H^{- 1} (Ω)))}^{N}$ , therefore compact from ${(L^{2} (0, T; H^{- 1} (Ω)))}^{M_{1}}$ to ${(L^{2} (0, T; H^{- 1} (Ω)))}^{N}$ by [11, Theorem 4.1]. On the other hand, by (7.6), there exists a positive constant c such that $\begin{aligned} (7.8) & {‖ D_{1}^{T} E ‖}^{2} ‖ h ‖_{L^{2} (0, T; H^{- 1} (Ω))} ⩽ c ‖ R h ‖_{{(L^{2} (0, T; H^{- 1} (Ω)))}^{M}} . \end{aligned}$ We get thus $D_{1}^{T} E = 0$ , namely, $S \subseteq Ker (D_{1}^{T})$ , which together with $S \subseteq Ker (D_{2}^{T})$ implies $S \subseteq Ker (D^{T})$ with $D = (D_{1}, D_{2})$ .

Now, writing $E_{2} = C_{p}^{T} x$ , it follows from (7.6) that $\begin{aligned} (7.9) & 0 ⩽ t ⩽ T : E_{2}^{T} \hat{U} = x^{T} C_{p} \hat{U} = 0 . \end{aligned}$ Since A satisfies the condition of $C_{p}$ -compatibility (4.11), by Lemma 2.5, there exists a matrix $A_{p}$ of order $(N - p)$ satisfying (4.12). Applying $C_{p}$ to system (7.2) and setting ${\hat{W}}_{p} = C_{p} \hat{U}$ , we get the following reduced system: $\begin{matrix} (7.10) & \{\begin{array}{ll} {\hat{W}}_{p}^{″} - Δ {\hat{W}}_{p} + A_{p} {\hat{W}}_{p} = C_{p} D_{1} χ_{ω} \hat{H} & in (0, + \infty) \times Ω, \\ {\hat{W}}_{p} = C_{p} D_{2} \hat{G} & on (0, + \infty) \times Γ, \end{array} \end{matrix}$ which is approximately controllable. Then the state variable ${\hat{W}}_{p} = C_{p} \hat{U}$ is dense in ${(H_{0}^{1} (Ω))}^{N}$ at the time $T > 0$ as the control $(\hat{H}, \hat{G})$ runs through the space ${(L^{2} (0, T; H^{- 1} (Ω)))}^{M_{1}} \times {(L^{2} (0, T; L^{2} (Γ)))}^{M_{2}}$ . Then it follows from (7.9) that $x = 0$ , namely, $E_{2} = 0$ . It follows from (7.3) that V is invariant for $A^{T}$ .

(c) ⟹ (a). Noting $dim (S) = p$ , by Lemma 2.1, we get $\begin{aligned} (7.11) & rank (D, A D, \dots, A^{N - 1} D) ⩽ N - p . \end{aligned}$ By Proposition 5.3, we have $\begin{aligned} (7.12) & rank (D, A D, \dots, A^{N - 1} D) ⩾ N - p . \end{aligned}$ We get thus (5.8). □
Remark 7.1.
In order to realize the approximate synchronization by p-groups under Kalman’s rank condition (5.8), $A^{T}$ should be diagonalizable by blocks following the decomposition $Im (C_{p}^{T}) \oplus Ker (D^{T})$ . This is an important algebraic condition on the matrix A.
Lemma 7.1 (see [9]).

Let the matrix $C_{q}$ be defined by ( 6.7 ), such that $C_{q} C_{q}^{T} = I$ . We have $\begin{matrix} (7.13) & C_{p} = C_{p} C_{q}^{T} C_{q} . \end{matrix}$ Furthermore, the matrix $\begin{matrix} (7.14) & C_{p q} = C_{p} C_{q}^{T} \end{matrix}$ is a full row-rank matrix of order $(N - p) \times (N - q)$ , such that $\begin{matrix} (7.15) & Ker (C_{p q}) = C_{q} Ker (C_{p}) . \end{matrix}$

Lemma 7.2.
Assume that $\begin{aligned} (7.16) & rank C_{p} (D, A D, \dots, A^{N - 1} D) = N - p, \\ (7.17) & rank (D, A D, \dots, A^{N - 1} D) = N - p . \end{aligned}$ Then there exists a matrix $Q_{p}$ of order $N \times (N - p)$ , such that for any given $U \in R^{N}$ , we have $\begin{matrix} (7.18) & U = \sum_{r = 1}^{p} ψ_{r} e_{r} + Q_{p} C_{p} U, \end{matrix}$ where $V = span {E_{1}, \dots, E_{p}}$ is the largest subspace invariant for $A^{T}$ and contained in $Ker (D^{T})$ , and $ψ_{r} = E_{r}^{T} U$ for $r = 1, \dots, p$ .
Proof.
By Lemma 2.4, conditions (7.16) and (7.17) imply that $V \cap Im (C_{p}^{T}) = {0}$ . Noting $dim (V) = dim Ker (C_{p}) = p$ , by Lemma 2.3, V and $Ker (C_{p})$ , and $Im (C_{p}^{T})$ and $V^{⊥}$ are bi-orthonormal. Then we can choose $\begin{matrix} (7.19) & E_{r}^{T} e_{s} = δ_{r s}, 1 ⩽ r, s ⩽ p \end{matrix}$ and an $N \times (N - p)$ matrix $Q_{p}$ by $Im (Q_{p}) = V^{⊥}$ , such that $\begin{matrix} (7.20) & C_{p} Q_{p} = I_{N - p} . \end{matrix}$ Moreover, $Ker (C_{p})$ is a supplement of $Im (Q_{p})$ , then, for any given $U \in R^{N}$ , there exist $x_{1}, \dots, x_{p} \in R$ and $Y \in R^{N - p}$ , such that $\begin{matrix} (7.21) & U = \sum_{s = 1}^{p} x_{s} e_{s} + Q_{p} Y . \end{matrix}$ Noting (7.20) and applying $C_{p}$ to (7.21), we get $Y = C_{p} U$ . Similarly, noting (7.19) and applying $E_{r}^{T}$ to (7.21), we get $x_{r} = ψ_{r}$ for $r = 1, \dots, p$ . □

The following result reveals the consequence of Kalman’s rank condition (5.8) from the point of view of control.
Theorem 7.2.
Let $Ω \subset R^{m}$ be a bounded domain with smooth boundary Γ satisfying the multiplier control condition and ω be a subdomain of Ω. Assume that the coupling matrix A satisfies the condition of $C_{p}$ -compatibility ( 4.11 ), and the control matrix $D = (D_{1}, D_{2}) \in D_{p}^{}$ . Then the following assertions are equivalent:*
system ( 1.1 ) is approximately synchronizable by p-groups and the Kalman’s rank condition ( 5.8 ) holds with the control matrix $D = (D_{1}, D_{2})$ ;

system ( 1.1 ) is approximately pinning synchronizable by p-groups and the approximately synchronizable state by p-groups ${(u_{1}, \dots, u_{p})}^{T}$ is independent of applied controls;

system ( 1.1 ) is approximately pinning synchronizable by p-groups and the components $u_{1}, \dots, u_{p}$ of the approximately synchronizable state by p-groups are linearly independent;

system ( 1.1 ) possesses the approximate pinning synchronization by p-groups and it cannot be extended to another approximate synchronization by groups.

Proof.
(a) ⟹ (b). By Proposition 5.2, $Im (C_{p}^{T}) \cap V = {0}$ , where V is the largest subspace invariant for $A^{T}$ and contained in $Ker (D^{T})$ . Noting (5.8), by Lemma 7.2, there exists a matrix $Q_{p}$ of order $N \times (N - p)$ , such that $\begin{matrix} (7.22) & U_{n} = \sum_{r = 1}^{p} ψ_{r} e_{r} + Q_{p} C_{p} U_{n} . \end{matrix}$ Noting (4.7) and passing to the limit in (7.22) as $n \to + \infty$ , we get (4.9) with $u_{r} = ψ_{r} (r = 1, \dots, p)$ , which are determined by (5.3) and independent of applied controls.

(b) ⟹ (c). Assume that $u_{1}, \dots, u_{p}$ are linearly dependent, then there exist some coefficients $c_{1}, \dots, c_{p}$ not all zero, such that $\begin{matrix} (7.23) & t ⩾ T : \sum_{r = 1}^{p} c_{r} u_{r} = 0 \end{matrix}$ for all the initial data $({\hat{U}}_{0}, {\hat{U}}_{1}) \in {(L^{2} (Ω) \times H^{- 1} (Ω))}^{N}$ . Define the extended matrix $\begin{matrix} (7.24) & C_{p - 1} = (\begin{array}{c} C_{p} \\ c_{N - p + 1} \end{array}), \end{matrix}$ where the row vector $c_{N - p + 1}$ is given by $\begin{matrix} (7.25) & c_{N - p + 1} = \frac{c_{1}}{n_{1} - n_{0}} e_{1}^{T} + \dots + \frac{c_{p}}{n_{p} - n_{p - 1}} e_{p}^{T} . \end{matrix}$ Since $Ker (C_{p}) \cap Im (C_{p}^{T}) = {0}$ , the extended matrix $C_{p - 1}$ has the full row-rank $N - p + 1$ . Noting (4.7) and (7.23), system (1.1) is approximately synchronizable by $(p - 1)$ -groups. By Theorem 6.2, system (1.1) is approximately $C_{q}$ -synchronizable at the time $T > 2 d (Ω)$ with $q < p$ .

Since $C_{q}$ satisfies the condition of compatibility (6.1), by Lemma 2.5, there exists a matrix of order $(N - q)$ , such that $C_{q} A = A_{q} C_{q}$ . Setting $W_{q} = C_{q} U$ , the following reduced system: $\begin{matrix} (7.26) & \{\begin{array}{ll} W_{q}^{″} - Δ W_{q} + A_{q} W_{q} = C_{q} D_{1} χ_{ω} H & in (0, + \infty) \times Ω, \\ W_{q} = C_{q} D_{2} G & on (0, + \infty) \times Γ \end{array} \end{matrix}$ is approximate controllable.

Let $C_{p q} = C_{p} C_{q}^{T}$ . By Lemma 7.1, $C_{p q}$ is an $(N - p) \times (N - q)$ full row-rank matrix with $q < p$ . We easily get $\begin{aligned} C_{p q} A_{q} Ker (C_{p q}) \\ = C_{p} C_{q}^{T} A_{q} C_{q} Ker (C_{p}) ((7.15)) \\ = C_{p} C_{q}^{T} C_{q} A Ker (C_{p}) (C_{q} A = A_{q} C_{q}) \\ \subseteq C_{p} Ker (C_{p}) = {0} ((7.13) and (4.11)), \end{aligned}$ then we have $\begin{matrix} (7.27) & A_{q} Ker (C_{p q}) \subseteq Ker (C_{p q}) . \end{matrix}$ By Lemma 2.5 with $C_{p}$ replaced by $C_{p q}$ , there exists a matrix $A_{p q}$ of order $(N - p)$ , such that $\begin{matrix} (7.28) & C_{p q} A_{q} = A_{p q} C_{p q} . \end{matrix}$ Applying $C_{p q}$ to system (7.26), we get $\begin{matrix} (7.29) & \{\begin{array}{ll} C_{p q} W_{q}^{″} - Δ C_{p q} W_{q} + A_{p q} C_{p q} W_{q} = 0 & in (T, + \infty) \times Ω, \\ C_{p q} W_{q} = 0 & on (T, + \infty) \times Γ . \end{array} \end{matrix}$ It follows that $\begin{matrix} (7.30) & t ⩾ T : C_{p q} W_{q} = 0, \end{matrix}$ provided that $\begin{matrix} (7.31) & t = T : C_{p q} W_{q} = 0, C_{p q} W_{q}^{'} = 0 . \end{matrix}$ By the approximate $C_{q}$ -synchronization, for any given initial data $(W (T), W^{'} (T)) \in {(L^{2} (Ω) \times H^{- 1} (Ω))}^{N - q}$ , there exists a sequence ${(H_{n}, G_{n})}_{n \in N}$ of mixed controls in ${(L^{2} (0, + \infty; H^{- 1} (Ω)))}^{M_{1}} \times {(L^{2} (0, + \infty; L^{2} (Γ)))}^{M_{2}}$ with compact support in $[0, T]$ , such that the sequence ${U_{n}}_{n \in N}$ of solutions to system (1.1) associated with the initial data $({\hat{U}}_{0}, {\hat{U}}_{1}) = (0, 0)$ satisfies $\begin{matrix} (7.32) & t ⩾ T : C_{q} (U_{n}, U_{n}^{'}) \to (W_{q}, W_{q}^{'}) as n \to + \infty, \end{matrix}$ or, by Lemma 7.1, we get $\begin{matrix} (7.33) & t ⩾ T : C_{p} (U_{n}, U_{n}^{'}) \to C_{p q} (W_{q}, W_{q}^{'}) as n \to + \infty . \end{matrix}$ Noting (7.30), we have $\begin{matrix} (7.34) & t ⩾ T : C_{p} (U_{n}, U_{n}^{'}) \to 0 as n \to + \infty . \end{matrix}$ Since the approximately synchronizable state by p-groups is independent of applied controls, so $U \equiv 0$ is the unique approximately synchronizable state by p-groups for system (1.1) with the initial data $({\hat{U}}_{0}, {\hat{U}}_{1}) = (0, 0)$ , then we get $\begin{matrix} (7.35) & t ⩾ T : (U_{n}, U_{n}^{'}) \to 0 as n \to + \infty . \end{matrix}$ Passing to the limit as $n \to + \infty$ in (7.32), we get $\begin{array}{r} t ⩾ T : W_{q} = W_{q}^{'} = 0 . \end{array}$ It follows from (7.31) that $Ker (C_{p q}) = {0}$ . By (7.15), we get a contradiction $Ker (C_{p}) \subseteq Ker (C_{q})$ with $q < p$ .

$(c) ⟹ (d)$ . Assume that system (1.1) can be extended to another approximate $C_{q}$ -synchronization. Applying $C_{q}$ to (4.9), we get $\begin{array}{r} t ⩾ T : \sum_{r = 1}^{p} C_{q} e_{r} u_{r} = 0 \end{array}$ for all the initial data $({\hat{U}}_{0}, {\hat{U}}_{1}) \in {(L^{2} (Ω) \times H^{- 1} (Ω))}^{N}$ . Since $Ker (C_{q}) ⊊ Ker (C_{p})$ , there exists an integer r with $1 ⩽ r ⩽ p$ , such that $C_{q} e_{r} \neq 0$ . This contradicts the linear independence of $u_{1}, \dots, u_{p}$ .

(d) ⟹ (a). First, for $1 ⩽ i ⩽ m$ , we denote by $λ_{i}$ the eigenvalue of $A^{T}$ associated with the corresponding Jordan chain: $\begin{aligned} (7.36) & E_{i 0} = 0, A^{T} E_{i j} = λ_{i} E_{i j} + E_{i, j - 1}, 1 ⩽ j ⩽ d_{i} . \end{aligned}$ The non-extensibility of approximate synchronization by p-groups implies $C_{q} = C_{p}$ . Noting ${\bar{d}}_{i} = d_{i}$ in (6.7), we can write $\begin{aligned} (7.37) & Im (C_{p}^{T}) = ⨁_{i \in I} span {E_{i 1}, \dots, E_{i d_{i}}} . \end{aligned}$ Assume that $\begin{aligned} (7.38) & rank (D, A D, \dots, A^{N - 1} D) = N - \hat{p}, \hat{p} ⩽ p . \end{aligned}$ By Lemma 2.1, the $\hat{p}$ -dimensional subspace $\begin{array}{r} V = Ker {(D, A D, \dots, A^{N - 1} D)}^{T} \end{array}$ is the largest subspace invariant for $A^{T}$ and contained in $Ker (D^{T})$ . By Proposition 5.2, we have $\begin{aligned} (7.39) & V \cap Im (C_{p}^{T}) = V \cap ⨁_{i \in I} span {E_{i 1}, \dots, E_{i d_{i}}} = {0} . \end{aligned}$ Then, for each $i \in I^{c}$ (the complement of I), there exists an integer ${\overline{d}}_{i}$ with $1 ⩽ {\overline{d}}_{i} ⩽ d_{i}$ , such that $\begin{aligned} (7.40) & V = ⨁_{i \in I^{c}} span {E_{i 1}, \dots, E_{i {\overline{d}}_{i} - 1}} \end{aligned}$ and $\begin{aligned} (7.41) & dim (V) = \sum_{i \in I^{c}} ({\overline{d}}_{i} - 1) = \hat{p} ⩽ p . \end{aligned}$ Noting that the cardinal of $I^{c}$ is equal to p, if $dim (V) = \hat{p} < p$ , there exists at least an integer $i_{0} \in I^{c}$ , such that $\begin{aligned} (7.42) & E_{i_{0} 1}, E_{i_{0} 2}, \dots, E_{i_{0} {\overline{d}}_{i_{0}} - 1} \in V and E_{i_{0} {\overline{d}}_{i_{0}}} \notin V . \end{aligned}$

For $j = 1, \dots {\overline{d}}_{i_{0}} - 1$ , applying $E_{i_{0} j}$ to system (1.1) and setting $ψ_{j} = E_{i_{0} j}^{T} U$ , we get $\begin{matrix} (7.43) & ψ_{0} = 0, \{\begin{array}{ll} ψ_{j}^{″} - Δ ψ_{j} + λ_{i_{0}} ψ_{j} + ψ_{j - 1} = 0 & in (0, + \infty) \times Ω, \\ ψ_{j} = 0 & on (0, + \infty) \times Γ . \end{array} \end{matrix}$ Applying $E_{i_{0} {\overline{d}}_{i_{0}}}$ to system (1.1) and setting $ψ_{{\overline{d}}_{i_{0}}} = E_{i_{0} {\overline{d}}_{i_{0}}}^{T} U$ , we get $\begin{matrix} (7.44) & \{\begin{array}{ll} ψ_{{\overline{d}}_{i_{0}}}^{″} - Δ ψ_{{\overline{d}}_{i_{0}}} + λ_{i_{0}} ψ_{{\overline{d}}_{i_{0}}} + ψ_{{\overline{d}}_{i_{0}} - 1} = E_{i_{0} {\overline{d}}_{i_{0}}}^{T} D_{1} χ_{ω} H & in (0, + \infty) \times Ω, \\ ψ_{{\overline{d}}_{i_{0}}} = E_{i_{0} {\overline{d}}_{i_{0}}}^{T} D_{2} G & on (0, + \infty) \times Γ . \end{array} \end{matrix}$ Since system (7.43) is independent of controls, we can define $\overline{ψ}$ by $\begin{matrix} (7.45) & \{\begin{array}{ll} {\overline{ψ}}^{″} - Δ \overline{ψ} + λ_{i_{0}} \overline{ψ} + ψ_{{\overline{d}}_{i_{0}} - 1} = 0 & in (0, + \infty) \times Ω, \\ \overline{ψ} = 0 & on (0, + \infty) \times Γ . \end{array} \end{matrix}$ Then, inserting the new variable $\begin{matrix} (7.46) & \tilde{ψ} = ψ_{{\overline{d}}_{i_{0}}} - \overline{ψ} \end{matrix}$ into system (7.44), we get $\begin{matrix} (7.47) & \{\begin{array}{ll} {\tilde{ψ}}^{″} - Δ \tilde{ψ} + λ_{i_{0}} \tilde{ψ} = E_{i_{0} {\overline{d}}_{i_{0}}}^{T} D_{1} χ_{ω} H & in (0, + \infty) \times Ω, \\ \tilde{ψ} = E_{i_{0} {\overline{d}}_{i_{0}}}^{T} D_{2} G & on (0, + \infty) \times Γ . \end{array} \end{matrix}$ Thus, setting $\begin{aligned} (7.48) & C_{\tilde{p}} = (\begin{array}{c} C_{p} \\ E_{i_{0} {\overline{d}}_{i_{0}}}^{T} \end{array}), \tilde{W} = (\begin{array}{c} W_{p} \\ \tilde{ψ} \end{array}) \end{aligned}$ and $\begin{aligned} (7.49) & \tilde{A} = (\begin{array}{cc} A_{p} & 0 \\ 0 & λ_{i_{0}} \end{array}), {\tilde{D}}_{1} = (\begin{array}{c} D_{p 1} \\ E_{i_{0} {\overline{d}}_{i_{0}}}^{T} D_{1} \end{array}), {\tilde{D}}_{2} = (\begin{array}{c} D_{p 2} \\ E_{i_{0} {\overline{d}}_{i_{0}}}^{T} D_{2} \end{array}), \end{aligned}$ we can combine the two systems (4.13) and (7.47) to get an extended system: $\begin{matrix} (7.50) & \{\begin{array}{ll} {\tilde{W}}^{″} - Δ \tilde{W} + \tilde{A} \tilde{W} = {\tilde{D}}_{1} χ_{ω} H & in (0, + \infty) \times Ω, \\ \tilde{W} = {\tilde{D}}_{2} G & on (0, + \infty) \times Γ . \end{array} \end{matrix}$ Using Proposition 7.1 below, we can apply Theorem 3.2 to get the approximate controllability of system (7.50). Then, for any given initial data $({\hat{U}}_{0}, {\hat{U}}_{1})$ , there exists a sequence of controls ${(H_{n}, G_{n})}_{n \in N}$ such that the sequence of the corresponding solutions ${{\tilde{W}}_{n}}_{n \in N}$ to system (7.50) satisfies $\begin{matrix} (7.51) & t = T : {\tilde{W}}_{n} = (\begin{array}{c} C_{p} U_{n} \\ E_{i_{0} {\overline{d}}_{i_{0}}}^{T} U_{n} - \overline{ψ} \end{array}) \to (\begin{array}{c} 0 \\ - \overline{ψ} \end{array}) as n \to + \infty, \end{matrix}$ namely, $\begin{matrix} (7.52) & t = T : C_{p} U_{n} \to 0 and E_{i_{0} {\overline{d}}_{i_{0}}}^{T} U_{n} \to 0 as n \to + \infty . \end{matrix}$ Noting (4.13) with $W_{p} = C_{p} U$ and the support of $(H, G)$ is contained in $[0, T]$ , we get $\begin{matrix} (7.53) & t ⩾ T : C_{p} U_{n} \to 0 as n \to + \infty . \end{matrix}$ By the definition of approximate pinning synchronization by p-groups, there exist $u_{1}, \dots, u_{p}$ such that $\begin{matrix} (7.54) & t ⩾ T : U_{n} \to \sum_{r = 1}^{p} e_{r} u_{r} as n \to + \infty \end{matrix}$ in the space $\begin{matrix} (7.55) & (C_{loc}^{0} ([T, + \infty); L^{2} (Ω)) \cap C_{loc}^{1} ([T, + \infty); H^{- 1} (Ω)))^{N} . \end{matrix}$ Then, passing to the limit as $n \to + \infty$ in the second relation (7.52), we get $\begin{matrix} (7.56) & t = T : \sum_{r = 1}^{p} E_{i_{0} {\overline{d}}_{i_{0}}}^{T} e_{r} u_{r} = 0 . \end{matrix}$ Since the approximately synchronizable state by p-groups ${(u_{1}, \dots, u_{p})}^{T}$ fulfils the subspace ${(L^{2} (Ω) \times H^{- 1} (Ω))}^{p}$ , it follows from (7.56) that $\begin{aligned} (7.57) & E_{i_{0} {\overline{d}}_{i_{0}}}^{T} e_{1} = \dots = E_{i_{0} {\overline{d}}_{i_{0}}}^{T} e_{p} = 0, \end{aligned}$ namely, $E_{i_{0} {\overline{d}}_{i_{0}}} \in {Ker (C_{p}^{T})}^{⊥} = Im (C_{p}^{T})$ . Noting that $i_{0} \in I^{c}$ , this contradicts (7.37) which shows that $\hat{p} = p$ . □

In order to complete the proof of Theorem 7.2, we give the following Proposition 7.1.
Let the matrix $\tilde{A}$ of order $(N - p + 1)$ , respectively, the matrix $\tilde{D} = ({\tilde{D}}_{1}, {\tilde{D}}_{2})$ of order $(N - p + 1) \times M$ be defined by ( 7.49 ). Then,
we have Kalman’s rank condition: $\begin{aligned} (7.58) & rank (\tilde{D}, \tilde{A} \tilde{D}, \dots, {\tilde{A}}^{N - p} \tilde{D}) = N - p + 1; \end{aligned}$

the largest subspace ${\tilde{V}}_{1}$ invariant for ${\tilde{A}}^{T}$ and contained in $Ker ({\tilde{D}}_{1})$ admits a supplement ${\tilde{V}}_{s}$ which is also invariant for ${\tilde{A}}^{T}$ ;

we have $\begin{aligned} (7.59) & {\tilde{V}}_{1} \cap Ker ({\tilde{D}}_{2}^{T}) = {0} . \end{aligned}$

Proof.
(a) Let $x_{p} \in R^{N - p}$ and $y \in R$ , such that $\begin{aligned} (7.60) & (\begin{array}{c} x_{p} \\ y \end{array}) \in Ker {({\tilde{D}}_{1}, {\tilde{D}}_{2})}^{T}, (\begin{array}{cc} A_{p}^{T} & 0 \\ 0 & λ_{i_{0}} \end{array}) (\begin{array}{c} x_{p} \\ y \end{array}) = λ_{i_{0}} (\begin{array}{c} x_{p} \\ y \end{array}) . \end{aligned}$ Then, noting (7.49), we have $\begin{aligned} (7.61) & \{\begin{array}{l} {(D_{1}, D_{2})}^{T} (C_{p}^{T} x_{p} + y E_{i_{0} {\overline{d}}_{i_{0}}}) = 0, \\ A_{p}^{T} x_{p} = λ_{i_{0}} x_{p} . \end{array} \end{aligned}$ We will show that $y = 0$ . Otherwise, without loss of generality, we may take $y = 1$ . Using (4.12) and (7.36), we have $\begin{aligned} A^{T} (E_{i_{0} {\overline{d}}_{i_{0}}} + C_{p}^{T} x_{p}) \\ = A^{T} E_{i_{0} {\overline{d}}_{i_{0}}} + λ_{i_{0}} C_{p}^{T} x_{p} \\ (7.62) & = E_{i_{0} {\overline{d}}_{i_{0} - 1}} + λ_{i_{0}} (E_{i_{0} {\overline{d}}_{i_{0}}} + C_{p}^{T} x_{p}) . \end{aligned}$ Then the subspace $\begin{aligned} (7.63) & V \oplus span {E_{i_{0} {\overline{d}}_{i_{0}}} + C_{p}^{T} x_{p}} \end{aligned}$ is invariant for $A^{T}$ and contained in $Ker {(D_{1}, D_{2})}^{T}$ .

Next we show that $\begin{aligned} (7.64) & V \cap span {E_{i_{0} {\overline{d}}_{i_{0}}} + C_{p}^{T} x_{p}} = {0} . \end{aligned}$ In fact, let $a \in C$ such that $\begin{matrix} a (E_{i_{0} {\overline{d}}_{i_{0}}} + C_{p}^{T} x_{p}) \in V, \end{matrix}$ namely, $\begin{matrix} a C_{p}^{T} x_{p} \in - a E_{i_{0} {\overline{d}}_{i_{0}}} + V . \end{matrix}$ Noting (7.40) and $i_{0} \in I^{c}$ , we get $\begin{matrix} - a E_{i_{0} {\overline{d}}_{i_{0}}} + V \subseteq ⨁_{i \in I^{c}} span {E_{i 1}, \dots, E_{i d_{i}}} . \end{matrix}$ Noting (7.37), we get $a = 0$ . Then, by Lemma 2.1, we get $\begin{array}{r} rank (D, A D, \dots, A^{N - 1} D) ⩽ N - dim (W) < N - dim (V) = N - q, \end{array}$ which contradicts (7.38).

Setting $y = 0$ in (7.61), we get that $x_{p}$ is an eigenvector of $A_{p}^{T}$ contained in $Ker {(D_{p 1}, D_{p 2})}^{T}$ . On the other hand, by Proposition 5.3, condition (5.7) holds. By Lemma 2.1, the largest subspace $V_{p}$ invariant for $A_{p}^{T}$ and contained in $Ker (D_{p}^{T})$ is reduced to ${0}$ , therefore $x_{p} = 0$ . Consequently, we get $(\begin{array}{c} x_{p} \\ y \end{array}) = (\begin{array}{c} 0 \\ 0 \end{array})$ , therefore the largest subspace invariant for ${\tilde{A}}^{T}$ and contained in $Ker ({\tilde{D}}^{T})$ is reduced to ${0}$ . Still by Lemma 2.1, we get the corresponding Kalman’s rank condition (7.58).

In order to verify the conditions (b) and (c), we have to distinguish the following two cases.

I. We first consider the case $E_{i_{0} {\overline{d}}_{i_{0}}} \in Ker (D_{1}^{T})$ .

(I-b) Define $\begin{aligned} (7.65) & {\tilde{V}}_{1} = {(C_{p}^{T})}^{- 1} V_{1} \times R, {\tilde{V}}_{s} = {(C_{p}^{T})}^{- 1} V_{s} \times {0}, \end{aligned}$ where $V_{1}$ is the largest subspace invariant for $A^{T}$ and contained in $Ker (D_{1}^{T})$ , and $V_{s}$ is a supplement of $V_{1}$ and invariant for $A^{T}$ .

By Proposition 4.1, ${(C_{p}^{T})}^{- 1} V_{1}$ is the largest subspace invariant for $A_{p}^{T}$ and contained in $Ker (D_{p 1}^{T})$ , and ${(C_{p}^{T})}^{- 1} V_{s}$ is a supplement of ${(C_{p}^{T})}^{- 1} V_{1}$ and invariant for $A_{p}^{T}$ . Obviously, ${\tilde{V}}_{s}$ is a supplement of ${\tilde{V}}_{1}$ . Moreover, noting (7.49), we easily check that both ${\tilde{V}}_{1}$ and ${\tilde{V}}_{s}$ are invariant for ${\tilde{A}}^{T}$ : $\begin{array}{c} {\tilde{A}}^{T} {\tilde{V}}_{1} = A_{p}^{T} {(C_{p}^{T})}^{- 1} V_{1} \times λ_{i_{0}} R \subseteq {(C_{p}^{T})}^{- 1} V_{1} \times R = {\tilde{V}}_{1}, \\ {\tilde{A}}^{T} {\tilde{V}}_{s} = A_{p}^{T} {(C_{p}^{T})}^{- 1} V_{s} \times {0} \subseteq {(C_{p}^{T})}^{- 1} V_{s} \times {0} = {\tilde{V}}_{s} . \end{array}$ Since $E_{i_{0} {\overline{d}}_{i_{0}}} \in Ker (D_{1}^{T})$ , we have $\begin{aligned} (7.66) & {\tilde{D}}_{1} = (\begin{array}{c} D_{p 1} \\ 0 \end{array}) . \end{aligned}$ Noting that ${(C_{p}^{T})}^{- 1} V_{1} \subseteq Ker (D_{p 1}^{T})$ , we have $\begin{array}{r} {\tilde{D}}_{1}^{T} {\tilde{V}}_{1} = D_{p 1}^{T} {(C_{p}^{T})}^{- 1} V_{1} = {0} . \end{array}$ So, the subspace ${\tilde{V}}_{1}$ is contained in $Ker ({\tilde{D}}_{1}^{T})$ .

Let $d ⩾ 0$ be the dimension of ${(C_{p}^{T})}^{- 1} V_{1}$ . Noting that $A_{p}$ is of order $(N - p)$ , by Lemma 2.1, we have $\begin{aligned} (7.67) & rank (D_{p 1}, A_{p} D_{p 1}, \dots, A_{p}^{N - p - 1} D_{p 1}) = (N - p) - d . \end{aligned}$ Noting (7.66), we have $\begin{array}{r} ({\tilde{D}}_{1}, \tilde{A} {\tilde{D}}_{1}, \dots, {\tilde{A}}^{N - p} {\tilde{D}}_{1}) = (\begin{array}{cccc} D_{p 1}, & A_{p} D_{p 1}, & \dots, & A_{p}^{N - p} D_{p 1} \\ 0, & 0, & \dots, & 0 \end{array}) . \end{array}$ It follows from (7.67) that $\begin{aligned} rank ({\tilde{D}}_{1}, \tilde{A} {\tilde{D}}_{1}, \dots, {\tilde{A}}^{N - p} {\tilde{D}}_{1}) \\ = rank (D_{p 1}, A_{p} D_{p 1}, \dots, A_{p}^{N - p - 1} D_{p 1}) \\ = (N - p + 1) - (d + 1) . \end{aligned}$ Noting that $\tilde{A}$ is of order $(N - p + 1)$ , by Lemma 2.1, the largest subspace invariant for ${\tilde{A}}^{T}$ and contained in $Ker ({\tilde{D}}_{1}^{T})$ has the dimension $(d + 1)$ .

On the other hand, using (7.65), we have $\begin{matrix} dim ({\tilde{V}}_{1}) = dim ({(C_{p}^{T})}^{- 1} V_{1}) + 1 = d + 1 . \end{matrix}$ So, ${\tilde{V}}_{1}$ is indeed the largest subspace invariant for ${\tilde{A}}^{T}$ and contained in $Ker ({\tilde{D}}_{1}^{T})$ .

(I-c) Noting (7.42) and $E_{i_{0} {\overline{d}}_{i_{0}}} \in Ker (D_{1}^{T})$ , the sub-chain $E_{i_{0} 1}, \dots, E_{i_{0} {\overline{d}}_{i_{0}}}$ lies in $Ker (D_{1}^{T})$ , therefore, belongs to $V_{1}$ . For any given $(x, y) \in {\tilde{V}}_{1}$ , we have $C_{p}^{T} x \in V_{1}$ , therefore, $C_{\tilde{p}}^{T} (x, y) = C_{p}^{T} x + y E_{i_{0} {\overline{d}}_{i_{0}}} \in V_{1}$ , namely, $(x, y) \in {(C_{\tilde{p}}^{T})}^{- 1} V_{1}$ , then, ${\tilde{V}}_{1} \subseteq {(C_{\tilde{p}}^{T})}^{- 1} V_{1}$ . It follows then that $\begin{aligned} {\tilde{V}}_{1} \cap Ker ({\tilde{D}}_{2}^{T}) \\ \subseteq {(C_{\tilde{p}}^{T})}^{- 1} V_{1} \cap {(C_{\tilde{p}}^{T})}^{- 1} Ker (D_{2}^{T}) ((4.23)) \\ = {(C_{\tilde{p}}^{T})}^{- 1} (V_{1} \cap Ker (D_{2}^{T})) ((4.21)) \\ = {(C_{\tilde{p}}^{T})}^{- 1} (V_{1} \cap V_{2}) (V_{2} = Ker (D_{2}^{T})) \\ = {(C_{\tilde{p}}^{T})}^{- 1} V (V = V_{1} \cap V_{2}) \\ = {(C_{\tilde{p}}^{T})}^{- 1} (V \cap Im (C_{\tilde{p}}^{T})) ((4.20)) . \end{aligned}$ Noting (7.37), (7.40) and (7.42), we have $V \cap Im (C_{\tilde{p}}^{T}) = {0}$ . We get thus (7.59).

II. We next consider the case $E_{i_{0} {\overline{d}}_{i_{0}}} \notin Ker (D_{1}^{T})$ .

(II-b) Define ${\tilde{V}}_{1}$ and ${\tilde{V}}_{s}$ by $\begin{aligned} (7.68) & {\tilde{V}}_{1} = {(C_{p}^{T})}^{- 1} V_{1} \times {0}, {\tilde{V}}_{s} = {(C_{p}^{T})}^{- 1} V_{s} \times R . \end{aligned}$ Similarly to the previous case, we easily check that ${\tilde{V}}_{1}$ is invariant for ${\tilde{A}}^{T}$ and contained in $Ker ({\tilde{D}}_{1}^{T})$ , and ${\tilde{V}}_{s}$ is a supplement of ${\tilde{V}}_{1}$ and invariant for ${\tilde{A}}^{T}$ .

Let $d ⩾ 0$ be the dimension of ${(C_{p}^{T})}^{- 1} V_{1}$ , the largest subspace invariant for $A_{p}^{T}$ and contained in $Ker (D_{p 1}^{T})$ . Noting that $A_{p}$ is of order $(N - p)$ , by Lemma 2.1, we have $\begin{aligned} (7.69) & rank (D_{p 1}, A_{p} D_{p 1}, \dots, A_{p}^{N - p - 1} D_{p 1}) = (N - p) - d . \end{aligned}$ Noting (7.49), we have $\begin{aligned} (7.70) & ({\tilde{D}}_{1}, \tilde{A} {\tilde{D}}_{1}, \dots, {\tilde{A}}^{N - p} {\tilde{D}}_{1}) = (\begin{array}{cccc} D_{p 1}, & A_{p} D_{p 1}, & \dots, & A_{p}^{N - p} D_{p 1} \\ E_{i_{0} {\overline{d}}_{i_{0}}}^{T} D_{1}, & λ_{i_{0}} E_{i_{0} {\overline{d}}_{i_{0}}}^{T} D_{1}, & \dots, & λ_{i_{0}}^{N - p} E_{i_{0} {\overline{d}}_{i_{0}}}^{T} D_{1} \end{array}) . \end{aligned}$ We claim that the last row of (7.70) is not a linear combination of the first $(N - p)$ rows. Otherwise, there exists a non-zero vector $x \in R^{N - p}$ , such that $\begin{array}{r} E_{i_{0} {\overline{d}}_{i_{0}}}^{T} D_{1} = x^{T} D_{p 1} . \end{array}$ Noting that $D_{p 1} = C_{p} D_{1}$ in (4.14), it follows that $\begin{aligned} (7.71) & D_{1}^{T} (C_{p}^{T} x - E_{i_{0} {\overline{d}}_{i_{0}}}) = 0 . \end{aligned}$ Then, we get $\begin{aligned} (7.72) & C_{p}^{T} x \in E_{i_{0} {\overline{d}}_{i_{0}}} + Ker (D_{1}^{T}) . \end{aligned}$ Noting (7.40) and (7.42), we have $\begin{aligned} (7.73) & E_{i_{0} {\overline{d}}_{i_{0}}} \in ⨁_{i \in I^{c}} span {E_{i 1}, \dots, E_{i d_{i}}} . \end{aligned}$ Noting ${\bar{d}}_{i} = d_{i}$ , it follows from (6.9) that $\begin{aligned} (7.74) & Ker (D_{1}^{T}) \subseteq ⨁_{i \in I^{c}} span {E_{i 1}, \dots, E_{i d_{i}}} . \end{aligned}$ On the other hand, noting (7.37), we have $\begin{aligned} (7.75) & C_{p}^{T} x \in ⨁_{i \in I} span {E_{i 1}, \dots, E_{i d_{i}}} . \end{aligned}$ Then, taking (7.73)–(7.75) into account, it follows from (7.72) that $\begin{matrix} C_{p}^{T} x = 0 . \end{matrix}$ We get thus a contradiction from (7.71): $D_{1}^{T} E_{i_{0} {\overline{d}}_{i_{0}}} = 0$ .

It follows then from (7.69) and (7.70) that $\begin{aligned} rank ({\tilde{D}}_{1}, \tilde{A} {\tilde{D}}_{1}, \dots, {\tilde{A}}^{N - p} {\tilde{D}}_{1}) \\ = rank (D_{p 1}, A_{p} D_{p 1}, \dots, A_{p}^{N - p - 1} D_{p 1}) + 1 \\ (7.76) & = (N - p + 1) - d . \end{aligned}$ Noting that $\tilde{A}$ is of order $(N - p + 1)$ , by Lemma 2.1, the largest subspace invariant for ${\tilde{A}}^{T}$ and contained in $Ker ({\tilde{D}}_{1}^{T})$ has the dimension d. On the other hand, by (7.68), we have $dim ({\tilde{V}}_{1}) = d$ . Then, ${\tilde{V}}_{1}$ is indeed the largest subspace invariant for ${\tilde{A}}^{T}$ and contained in $Ker ({\tilde{D}}_{1}^{T})$ .

(II-c) For any given $(x, 0) \in {\tilde{V}}_{1}$ , we have $x \in {(C_{p}^{T})}^{- 1} V_{1}$ , namely, $C_{p}^{T} x \in V_{1}$ , therefore, $C_{\tilde{p}}^{T} (x, 0) = C_{p}^{T} x \in V_{1}$ , namely, $(x, 0) \in {(C_{\tilde{p}}^{T})}^{- 1} V_{1}$ . Consequently, we get ${\tilde{V}}_{1} \subseteq {(C_{\tilde{p}}^{T})}^{- 1} V_{1}$ . It follows that $\begin{aligned} {\tilde{V}}_{1} \cap Ker ({\tilde{D}}_{2}^{T}) \\ \subseteq {(C_{\tilde{p}}^{T})}^{- 1} V_{1} \cap {(C_{\tilde{p}}^{T})}^{- 1} Ker (D_{2}^{T}) ((4.23)) \\ = {(C_{\tilde{p}}^{T})}^{- 1} (V_{1} \cap Ker (D_{2}^{T})) ((4.21)) \\ = {(C_{\tilde{p}}^{T})}^{- 1} (V_{1} \cap V_{2}) (V_{2} = Ker (D_{2}^{T})) \\ = {(C_{\tilde{p}}^{T})}^{- 1} V (V = V_{1} \cap V_{2}) \\ = {(C_{\tilde{p}}^{T})}^{- 1} (V \cap Im (C_{\tilde{p}}^{T})) ((4.20)) . \end{aligned}$ Noting (7.37), (7.40) and (7.42), we have $V \cap Im (C_{\tilde{p}}^{T}) = {0}$ . We get again (7.59). □

8. Mixed pinning synchronization

Theorem 7.2 describes the situation when $A^{T}$ is diagonalizable by blocks on the basis $Im (C_{p}^{T}) \oplus V$ . If this is not the case, we can extend the matrix $C_{p}$ to the matrix $C_{q}$ by the procedure described by (6.7), so that $A^{T}$ can be diagonalized by blocks on the basis $Im (C_{q}^{T}) \oplus V$ . In this case, by Theorem 7.2, the rank of Kalman’s matrix attaints the minimum $(N - q)$ and system (1.1) is approximately $C_{q}$ -synchronizable.

The following result fits Theorem 7.2 to the general case, and answer the third question on what will be happened in the general situation.

Theorem 8.1.

Let $Ω \subset R^{m}$ be a bounded domain with smooth boundary Γ satisfying the multiplier control condition and ω be a subdomain of Ω. Assume that $D = (D_{1}, D_{2}) \in D_{p}^{*}$ . The following assertions are equivalent.

system ( 1.1 ) is approximately $C_{q}$ -synchronizable under the minimal rank condition $\begin{matrix} (8.1) & D \in arg min_{D \in D_{p}^{*}} rank (D, A D, \dots, A^{N - 1} D), \end{matrix}$ where the symbol $arg {min}_{D \in D_{p}^{*}}$ denotes the argument of the minimum in $D_{p}^{*}$ ;

system ( 1.1 ) is approximately pinning $C_{q}$ -synchronizable and the approximately $C_{q}$ -synchronizable state ${(v_{1}, \dots, v_{q})}^{T}$ is independent of applied controls;

system ( 1.1 ) is approximately pinning $C_{q}$ -synchronizable and the components $v_{1}, \dots, v_{q}$ of the approximately $C_{q}$ -synchronizable state are linearly independent;

system ( 1.1 ) is approximately pinning $C_{q}$ -synchronizable and the approximate pinning $C_{q}$ -synchronization cannot be extended to another approximate synchronization.

Proof.

By Theorem 6.2, the minimal rank condition (8.1) is equivalent to $\begin{matrix} (8.2) & rank (D, A D, \dots, A^{N - 1} D) = N - q, \end{matrix}$ where q is given by (6.8). Then we meet again the same situation in Theorem 7.2. □

Theorem 8.2.

Let $Ω \subset R^{m}$ be a bounded domain with smooth boundary Γ satisfying the multiplier control condition and ω be a subdomain of Ω. Assume that system ( 1.1 ) is approximately synchronizable by p-groups under the minimum rank condition ( 8.1 ). Then system ( 1.1 ) is approximately synchronizable by p-groups in the pinning sense.

Proof.

By Theorem 8.1, system (1.1) is approximately $C_{q}$ -synchronizable in the pinning sense. Since $Ker (C_{q}) \subseteq Ker (C_{p})$ , there exist some coefficients $c_{r s} (1 ⩽ r ⩽ p; 1 ⩽ s ⩽ q)$ such that $\begin{matrix} (8.3) & ϵ_{s} = \sum_{r = 1}^{p} c_{r s} e_{r}, s = 1, \dots, q, \end{matrix}$ then, by (6.3) we have $\begin{matrix} (8.4) & U_{n} \to \sum_{s = 1}^{q} ϵ_{s} v_{s} = \sum_{r = 1}^{p} (\sum_{s = 1}^{q} c_{r s} v_{s}) e_{r} as n \to + \infty, \end{matrix}$ where $e_{1}, \dots, e_{p}$ are given by (4.6). We then get the approximate synchronization by p-groups in the pinning sense with the approximately synchronizable state by p-groups given by $\begin{matrix} (8.5) & u_{r} = \sum_{s = 1}^{q} c_{r s} v_{s}, r = 1, \dots, p . \end{matrix}$ We identify thus the approximate synchronization by p-groups in the consensus sense with that in the pinning sense, however, when $q < p$ , the components of the approximately synchronizable state by p-groups given by (8.5) are not linearly independent! □

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Approximate mixed synchronization by groups for a coupled system of wave equations

Abstract

Keywords

1. Introduction

2. A uniqueness theorem

Lemma 2.1 ([6, Lemma 2.5]).

Lemma 2.2. Let V 1 , V 2 and V denote the largest subspaces invariant for A T and contained in Ker ( D 1 T ) , Ker ( D 2 T ) and Ker ( D T ) , respectively. We have (2.3) V 1 ∩ V 2 = V . Lemma 2.3 ([10, Proposition 2.7]).

Lemma 2.4 ([10, Proposition 2.11]).

Lemma 2.5 ([10, Proposition 2.15]).

Theorem 2.1 ([8, Theorem 10]).

Theorem 2.2 ([10, Theorem 8.16]).

Lemma 6.2 (see [9]).

References

Lemma 2.2.
Let $V_{1}$ , $V_{2}$ and V denote the largest subspaces invariant for $A^{T}$ and contained in $Ker (D_{1}^{T})$ , $Ker (D_{2}^{T})$ and $Ker (D^{T})$ , respectively. We have $\begin{matrix} (2.3) & V_{1} \cap V_{2} = V . \end{matrix}$
Lemma 2.3 ([10, Proposition 2.7]).