Stabilization with Decay Estimate for Second-Order Bilinear Systems with Time Delay

Abstract

This paper is concerned with stabilization with energy decay rates of a class of second-order bilinear systems with time delay in a Hilbert space. We study strong and exponential stabilization with bilinear feedback controls. First, we rely on the cosine family to establish the well-posedness of the resulting closed-loop system. Then, we provide sufficient conditions under which the chosen control strategy achieves strong or exponential stabilization of the system. In addition, in the case of the strong stabilization, an explicit decay estimate is established. The obtained results are finally illustrated by wave and beam equations with simulation as applications.

Keywords

second-order systems semilinear systems time delay strong stabilization exponential stabilization

1. Introduction

Time delays play a critical role in various fields of engineering and science (Artstein, 1982; Fridman, 2014; Richard, 2003; Smith, 2011). They occur due to inherent delays in system components and the need for feedback controls to stabilize systems, which introduce inevitable delays. These delays can have a significant impact on system performance and stability. In the case of hyperbolic partial differential equation control systems, it has been well known for several years that even a small delay in the boundary or internal feedback could destabilize a system, which is uniformly asymptotically stable without delays (Datko, 1988; Datko et al., 1986; Nicaise & Pignotti, 2006).

In the context of second-order systems, extensive research has been dedicated over the years to the development of stabilizing controllers in the presence of time delay (Bayili et al., 2020; Hamidi et al., 2021; Komornik & Pignotti, 2022; Logemann et al., 1996; Nicaise & Pignotti, 2011; Nicaise et al., 2011; Paolucci & Pignotti, 2022; Pignotti, 2023; Rebarber & Townley, 1998; Xu et al., 2006) to name but a few.

For abstract evolution equations with delay, in Ait Benhassi et al. (2009) the authors study the feedback stabilization of second-order systems with time delay, where the delayed damping term is smaller than the undelayed one. Using Ammari and Tucsnak’s method (Ammari & Tucsnak, 2001), they reduced the exponential stability problem of a closed-loop system to an observability estimate for an uncontrolled system and a boundedness property of the transfer function. Moreover, the authors in Paolucci and Pignotti (2021) studied a class of semilinear wave-type equations with viscoelastic damping and delay feedback with time-variable coefficients. By combining semigroup arguments, careful energy estimates, and an iterative approach, they are able to prove, under suitable assumptions on the delay feedback coefficient and on the memory kernel, a well-posedness result and an exponential decay estimate for solutions corresponding to small initial data. This extends and concludes the analysis (Nicaise & Pignotti, 2015, 2018) and further developed in Komornik and Pignotti (2022) and Continelli and Pignotti (2023). In Paolucci (2022) the author study a nonlinear abstract evolution equation with an infinite number of time-dependent time delays and the exponential decay of the solution is proved.

In the specific context of boundary delayed feedback for wave-type equations, Gugat (2010) studied the stabilization of a vibrating string with constant delay. In particular, the author considered the case where the delay coincides with the intrinsic period of the uncontrolled system, namely $δ = 2 L / c$ , with $L$ the string length and $c$ the wave speed. It was shown that for such resonant delays, exponential stability can be guaranteed for feedback parameters belonging to a certain bounded interval. This result highlights the deep connection between delay effects, spectral properties, and the periodicity of solutions.

In this work, we consider the following second order system with time delay:

{\begin{cases} y_{t t} (t) + A y (t) = u (t) B y_{t} (t - δ), t \geq 0, \\ y (t) = ψ (t), y_{t} (t) = ψ_{t} (t), t \in [- δ, 0], \end{cases}

(1.1)

where

A

is a strictly positive and self-adjoint operator on a Hilbert space

H

endowed with the inner product

⟨ \cdot, \cdot ⟩_{H}

and the corresponding norm

‖ \cdot ‖_{H}

, the operator

A

is associated with a bilinear form defined on

V \times V

where

V

is a another Hilbert space such that

V \subset H \subset V^{'}

(dual space of

V

B : H \to H

is a linear bounded operator from

H

into itself,

u (\cdot)

denote the control function, the positive real number

δ

denotes the delay. We associate to the Hilbert space

X = V \times H

the Banach space of all continuous functions

C = C ([- δ, 0], V \times H)

equipped with the supremum norm,

‖ ϕ ‖_{C} = sup_{t \in [- δ, 0]} ‖ ϕ (t) ‖_{X}

, for all

ϕ \in C

. Furthermore, we consider the time evolution of the history segments, if

\tilde{y} = (y, y_{t}) \in C ([- δ, \infty], X)

and

t \geq 0

. Define

{\tilde{y}}_{t} \in C

as follows:

{\tilde{y}}_{t} (θ) = \tilde{y} (t + θ) for all θ \in [- δ, 0]

Equivalently,

y \in C ([- δ, \infty), V) \cap C^{1} ([- δ, \infty), H)

and

t \geq 0,

then

y^{t} \in C ([- δ, 0], V) \cap C^{1} ([- δ, 0), H)

is defined by

y^{t} (θ) = y (t + θ), with θ \in [- δ, 0],

(1.2)

and

y_{t}^{t} (θ) = y_{t} (t + θ) = y_{θ} (t + θ) = y_{θ}^{t} (θ) .

(1.3)

In the case when this system is free of time delay (i.e.,

δ = 0

), the paper Haraux (1989) considered the exponential stability of system (1.1) in which the control function

u (t) = 1

and the corresponding uncontrollable system (i.e.,

B = 0

), satisfy the observability inequality. Also in Ghisi et al. (2016) the authors studied the exponential stabilization of a special case of system (1.1) where the operator

B = I

, the used method requires that the operator

B

commutes with

A

. Furthermore, exponential and strong stabilization has been achieved for a suitable function

u (t)

in Ezzaki and Zerrik (2021) such that the solution

z (t)

of the uncontrolled (i.e.,

B = 0

) system satisfies

E_{y} (0) \leq C \int_{0}^{T} | ⟨ B z_{t} (t), z_{t} (t) ⟩ | d t, for some T, C > 0.

(1.4)

Moreover, El Kazoui et al. (2025) build upon Ezzaki and Zerrik (2021), achieving stabilization with precise decay rates through a more general class of control functions.

When $δ > 0$ , stabilization requires an adapted observability inequality involving the delayed velocity term:

‖ ψ_{t} (0) ‖_{H}^{2} + ‖ ψ (0) ‖_{V}^{2} \leq C \int_{δ}^{T} | ⟨ B z_{t} (t - δ), z_{t} (t) ⟩ | d t, for some T, C > 0.

(1.5)

Noting that in the case where the uncontrolled solution

z

δ

-periodic, inequality (1.5) reduces to the classical observability inequality (1.4). This is in agreement with the results of Piskarev (1983), who established a spectral characterization of periodic and almost periodic cosine operator functions arising in abstract second-order problems.

The stabilization problem for system (1.1) then consists in designing a feedback control $u$ such that the corresponding closed-loop system satisfies

E_{y} (t) ⟶ 0 as t \to + \infty,

where the energy functional is defined by

E_{y} (t) = \frac{1}{2} (‖ y (t) ‖_{V}^{2} + ‖ y_{t} (t) ‖_{H}^{2}) .

(1.6)

Formally differentiating and assuming that the operator

A

is self-adjoint and strictly positive, we obtain

\frac{d}{d t} E_{y} (t) = u (t) ⟨ B y_{t} (t - δ), y_{t} (t) ⟩_{H} .

(1.7)

This motivates the following feedback law:

u_{γ} (t) = {\begin{cases} - \frac{⟨ B y_{t} (t - δ), y_{t} (t) ⟩_{H}}{r_{γ} (y^{t}, y_{t}^{t})}, & r_{γ} (y^{t}, y_{t}^{t}) \neq 0 \\ 0, & otherwise, \end{cases} with γ = {0, 1},

(1.8)

where

r_{γ}

is defined by

\begin{aligned} r_{γ} : C & ⟶ R^{+}, \\ (φ, ϕ) & ⟼ sup_{θ \in [- δ, 0]} {(‖ φ (θ) ‖_{V}^{2} + ‖ ϕ (θ) ‖_{H}^{2})}^{γ} . \end{aligned}

(1.9)

Hence the resulting closed-loop system is

{\begin{cases} y_{t t} (t) + A y (t) = u_{γ} (t) B y_{t} (t - δ), \\ y (t) = ψ (t), y_{t} (t) = ψ_{t} (t), t \in [- δ, 0], \\ y_{0} = y (0), y_{1} = y^{'} (0) . \end{cases}

(1.10)

In this paper, we study the stabilization of the bilinear system (1.1) under the observability inequality for the corresponding uncontrolled system using the controls (1.8). Moreover, we obtain exponential and polynomial decay rates. This paper is organized as follows. In Section 2, we give some definitions and fundamental properties of a strongly continuous cosine family. In Section 3, we analyze the existence and uniqueness of the global mild solution of the resulting closed-loop system using the cosine family. Section 4 is devoted to the stabilization of system (1.1) together with an energy decay rate estimate. Finally, in Section 5, we provide some illustrating examples with simulations.

2. Preliminary

We start with a review of the essential background needed for this paper. In particular, we recall the definition and some fundamental properties of a strongly continuous cosine family, which are essential for our subsequent analysis.

Definition 2.1
A family ${C (t)}_{t \in R}$ of operators in $L (H)$ is a strongly continuous cosine family in $H$ if
$C (0) = I$ .

$C (t + s) + C (t - s) = 2 C (t) C (s)$ .

$C (t) x$ is continuous in $t$ on $R$ , for all $x \in H$ .

The corresponding strongly continuous sine family is ${S (t)}_{t \in R} \subset L (H)$ defined by
$S (t) x = \int_{0}^{t} C (s) x d s, t \in R, x \in H .$
(2.1)

The generator of a strongly continuous cosine family ${C (t)}_{t \in R}$ is a linear operator $A : D (A) \subset H \to H$ given by
$A x = {\frac{d^{2}}{d t^{2}} C (t) x |}_{t = 0}, for all x \in D (A),$
(2.2)
where the domain $D (A)$ is defined as
$D (A) = {x \in H, C (\cdot) x \in C^{2} (R, H)} .$
Also, we define the set $V$ as follows:
$V = {x \in H, C (\cdot) x \in C^{1} (R, H)} .$
(2.3)
Proposition 2.2
Let ${C (t)}_{t \in R}$ be a strongly continuous cosine family with an infinitesimal generator $A$ and an associated sine family ${S (t)}_{t \in R}$ . Then, we have the following properties: (i)
There exist $M \geq 1$ and $ω \geq 0$ such that
$‖ C (t) ‖ \leq M e^{ω | t |} .$

(ii)
$D (A)$ is dense in $H$ and $A$ is closed operator in $H$ ,
(ii)
if $x \in H,$ then $S (t) x \in V$ ,
(iii)
if $x \in V,$ then $C (t) x \in V$ ,
(iv)
if $x \in V,$ then $S (t) x \in D (A)$ and $\frac{d}{d t} C (t) x = A S (t) x$ ,
(v)
if $x \in D (A),$ then $C (t) x \in D (A)$ and $\frac{d^{2}}{d t^{2}} C (t) x = A C (t) x = C (t) A x$ .

Note that if $A$ is self-adjoint and strictly positive, then $- A$ is a generator of a cosine family. In addition, we can introduce the square root of $A$ that commutes with every bounded operator in $H$ . This leads to define the Hilbert space $D (A^{1 / 2})$ endowed with the inner product:
$⟨ \cdot, \cdot ⟩_{V} = ⟨ A^{1 / 2} \cdot, A^{1 / 2} \cdot ⟩_{H} = ⟨ \tilde{A} \cdot, \cdot ⟩_{V^{'}, V},$
(2.4)
where $\tilde{A} : V \to V^{'}$ , with $V = D (A^{1 / 2}) .$
3. Well-Posedness

First, we recall the definition of the mild solution of system (1.1).

Definition 3.1
Let $T > 0$ . A function $y \in C^{1} ([- δ, T], H) \cap C ([- δ, T], V)$ is said to be a mild solution of system (1.1), if it satisfies the following integral equation:
${\begin{aligned} y (t) = C (t) ψ (0) + S (t) ψ_{t} (0) + \int_{0}^{t} S (t - s) u (s) B y_{s}^{s} (- δ) d s, & t \in [0, T], \\ ψ \in C^{1} ([- δ, 0], H) \cap C ([- δ, 0], V), & t \in [- δ, 0] . \end{aligned}$
(3.1)

In the following section, before turning our attention to study the well-posedness, let us first rewrite the closed-loop system (1.10) as follows:
${\begin{cases} y_{t t} (t) + A y (t) = G_{γ} (y^{t}, y_{t}^{t}), t \geq 0, \\ y (t) = ψ (t), y_{t} (t) = ψ_{t} (t), t \in [- δ, 0] . \end{cases}$
(3.2)
where the function $G_{γ} : C \to H$ is given by
$G_{γ} (φ, ϕ) = - {\begin{cases} \frac{⟨ B ϕ (- δ), ϕ (0) ⟩_{H}}{r_{γ} (φ, ϕ)} B ϕ (- δ), with γ \in {0, 1} . \end{cases}$
(3.3)

Now, we are in a position to study the well-posedness of the closed-loop system (3.2). Our approach relies on the one hand on cosine family theory and on the other hand on a particular Lipschitz estimate for the nonlinear term, which is essential to guarantee the existence and uniqueness of a mild solution.

Then, we have the following well-posedness result.
Theorem 3.2 Well-posedness

Assume that $A$ is a strictly positive self-adjoint operator and let $γ \in {0, 1}$ . Then, for any $φ \in C ([- δ, 0], V) \cap C^{1} ([- δ, 0], H)$ , there exists a unique mild solution to the closed-loop systems (3.2) that exhibits the regularity

y \in C ([- δ, + \infty), V) \cap C^{1} ([- δ, + \infty), H) .

(3.4)

Furthermore, the energy of system (3.2), defined as

E_{y} (t) = \frac{1}{2} (‖ y (t) ‖_{V}^{2} + ‖ y_{t} (t) ‖_{H}^{2}),

(3.5)

is monotonically decreasing for all

t \geq 0

and satisfies

E_{y} (t) \leq E_{y} (0), t \geq 0.

(3.6)

Proof.

To establish well-posedness, we first derive specific Lipschitz estimates for function (3.3). Consider $R > 0$ , and let $(φ_{1}, ϕ_{1}), (φ_{2}, ϕ_{2}) \in B (0, R) = {(φ, ϕ) \in C, ‖ (φ, ϕ) ‖_{C} \leq R}$ , with $0 < ‖ (φ_{1}, ϕ_{1}) ‖ \leq ‖ (φ_{2}, ϕ_{2}) ‖ \leq R$ . Consequently, we obtain

\begin{aligned} G_{γ} (φ_{1}, ϕ_{1}) - G_{γ} (φ_{2}, ϕ_{2}) & = \frac{⟨ B ϕ_{2} (- δ), ϕ_{2} (0) ⟩}{r_{γ} (φ_{2}, ϕ_{2})} B ϕ_{2} (- δ) - \frac{⟨ B ϕ_{1} (- δ), ϕ_{1} (0) ⟩}{r_{γ} (φ_{1}, ϕ_{1})} B ϕ_{1} (- δ) \\ = (\frac{⟨ B ϕ_{2} (- δ), ϕ_{2} (0) ⟩}{r_{γ} (φ_{2}, ϕ_{2})} - \frac{⟨ B ϕ_{1} (- δ), ϕ_{1} (0) ⟩}{r_{γ} (φ_{1}, ϕ_{1})}) B ϕ_{2} (- δ) \\ + \frac{⟨ B ϕ_{1} (- δ), ϕ_{1} (0) ⟩}{r_{γ} (φ_{1}, ϕ_{1})} B (ϕ_{2} (- δ) - ϕ_{1} (- δ)) \\ = \frac{⟨ B ϕ_{1} (- δ), ϕ_{1} (0) ⟩}{r_{γ} (φ_{1}, ϕ_{1})} B (ϕ_{2} (- δ) - ϕ_{1} (- δ)) \\ + \frac{r_{γ} (φ_{1}, ϕ_{1}) - r_{γ} (φ_{2}, ϕ_{2})}{r_{γ} (φ_{1}, ϕ_{1}) r_{γ} (φ_{2}, ϕ_{2})} ⟨ B ϕ_{1} (- δ), ϕ_{1} (0) ⟩ B ϕ_{2} (- δ) \\ + \frac{⟨ B ϕ_{2} (- δ), ϕ_{2} (0) ⟩ - ⟨ B ϕ_{1} (- δ), ϕ_{1} (0) ⟩}{r_{γ} (φ_{2}, ϕ_{2})} B ϕ_{2} (- δ) . \end{aligned}

By using the fact that

⟨ B x_{1}, x_{1} ⟩ - ⟨ B x_{2}, x_{2} ⟩ = ⟨ B (x_{1} - x_{2}), x_{1} ⟩ - ⟨ B x_{2}, x_{2} - x_{1} ⟩, \forall x_{1}, x_{2} \in H,

(3.7)

and the triangle inequality, we can estimate

| ⟨ B ϕ_{2} (- δ), ϕ_{2} (0) ⟩ - ⟨ B ϕ_{1} (- δ), ϕ_{1} (0) ⟩ | \leq 2 ‖ B ‖ ‖ (φ_{2}, ϕ_{2}) ‖_{C} ‖ (φ_{1} - φ_{2}, ϕ_{1} - ϕ_{1}) ‖_{C} .

So, we have

\begin{aligned} ‖ G_{γ} (φ_{1}, ϕ_{1}) - G_{γ} (φ_{1}, ϕ_{1}) ‖ & \leq \frac{2 ‖ B ‖^{2} ‖ (φ_{2}, ϕ_{2}) ‖_{C}^{2}}{r_{γ} (φ_{2}, ϕ_{2})} ‖ (φ_{1} - φ_{2}, ϕ_{1} - ϕ_{1}) ‖_{C} \\ + \frac{| r_{γ} (φ_{1}, ϕ_{1}) - r_{γ} (φ_{2}, ϕ_{2}) |}{r_{γ} (φ_{1}, ϕ_{1}) r_{γ} (φ_{2}, ϕ_{2})} ‖ B ‖^{2} ‖ (φ_{1}, ϕ_{1}) ‖^{2} ‖ (φ_{2}, ϕ_{2}) ‖ \\ + ‖ B ‖^{2} \frac{‖ (φ_{1}, ϕ_{1}) ‖^{2}}{r_{γ} (φ_{1}, ϕ_{1})} ‖ (φ_{1} - φ_{2}, ϕ_{1} - ϕ_{1}) ‖_{C} . \end{aligned}

(3.8)

Two situations are provided:

Case 1: $γ = 1$

\begin{aligned} \frac{r_{γ} (φ_{1}, ϕ_{1}) - r_{γ} (φ_{2}, ϕ_{2})}{r_{γ} (φ_{1}, ϕ_{1}) r_{γ} (φ_{2}, ϕ_{2})} & = \frac{‖ (φ_{1}, ϕ_{1}) ‖^{2} - ‖ (φ_{2}, ϕ_{2}) ‖^{2}}{‖ (φ_{1}, ϕ_{1}) ‖^{2} ‖ (φ_{2}, ϕ_{2}) ‖^{2}} \\ \leq \frac{‖ (φ_{1}, ϕ_{1}) ‖ + ‖ (φ_{2}, ϕ_{2}) ‖}{‖ (φ_{1}, ϕ_{1}) ‖^{2} ‖ (φ_{2}, ϕ_{2}) ‖^{2}} ‖ (φ_{1}, ϕ_{1}) ‖ - ‖ (φ_{2}, ϕ_{2}) ‖ \\ \leq \frac{2 ‖ (φ_{2}, ϕ_{2}) ‖}{‖ (φ_{1}, ϕ_{1}) ‖^{2} ‖ (φ_{2}, ϕ_{2}) ‖^{2}} ‖ (φ_{1} - φ_{2}, ϕ_{1} - ϕ_{1}) ‖_{C} . \end{aligned}

The above estimate and (3.8) yield that

‖ G_{1} (φ_{1}, ϕ_{1}) - G_{1} (φ_{1}, ϕ_{1}) ‖ \leq 5 ‖ B ‖^{2} ‖ (φ_{1} - φ_{2}, ϕ_{1} - ϕ_{1}) ‖_{C} .

Case 2: $γ = 0$

Remarking that $r_{γ} (φ, ϕ) = 1$ for all $(φ, ϕ) \in C$ , thus

‖ G_{0} (φ_{1}, ϕ_{1}) - G_{0} (φ_{1}, ϕ_{1}) ‖ \leq 3 ‖ B ‖^{2} R^{2} ‖ (φ_{1} - φ_{2}, ϕ_{1} - ϕ_{1}) ‖_{C} .

Then, for any

γ \in {0, 1}

, function (3.3) is locally Lipschitz for each

ϕ_{1}, ϕ_{2} \in B_{C} (0, R)

. We conclude that system (3.2) possesses a unique mild solution on

y \in C ([- δ, T^{*}), V) \cap C^{1} ([- δ, T^{*}), V)

given by the variation of constants formula (see Chueshov & Rezounenko, 2015):

y (t) = {\begin{aligned} C (t) ψ (0) + S (t) ψ_{t} (0) + \int_{0}^{t} S (t - σ) G_{γ} (y_{σ}, y_{σ}^{σ}) d σ, for all t \in [0, T^{*}), \\ ψ \in C^{1} ([- δ, 0], H) \cap C ([- δ, 0], V) . \end{aligned}

(3.9)

To establish that $T^{*} = \infty$ , we will show that $E_{y} (t)$ is bounded on the interval $[0, T^{*})$ , where $E_{y} (t)$ denotes the energy associated with system (3.2). To do this, we consider the following function:

\begin{aligned} g : [0, T] & \to H, \\ t & \to G_{γ} (y^{t}, {y_{t}}^{t}), with 0 < T < T^{*} . \end{aligned}

Since

g \in C ([0, T], H)

, then there exists a sequence

(g_{n}) \subset C^{1} ([0, T], H)

uniformly converges to

g

C ([0, T], H)

. For each

n \in N

, let

(ξ_{n}, ξ_{n}^{'}) \subset D (A) \times V

such that

(ξ_{n}, ξ_{n}^{'})

converges to

(ψ (0), ψ_{t} (0))

V \times H

, and consider the following equation:

y_{n} (t) = C (t) ξ_{n} + S (t) ξ_{n}^{'} + \int_{0}^{t} S (t - σ) g_{n} (σ) d σ, for all t \in [0, T] .

(3.10)

Then,

y_{n} (t) \in D (A)

{y_{n}}_{t} (t) \in V

, and

y_{n} \in C ([0, T], D (A)) \cap C^{1} ([0, T]), V) \cap C^{2} ([0, T], H)

. Moreover, we have

{\begin{cases} {y_{n}}_{t t} (t) = A y_{n} (t) + g_{n} (t), \\ y_{n} (0) = ξ_{n}, {y_{n}}_{t} (0) = ξ^{'} . \end{cases}

(3.11)

Through straightforward formal computations, we obtain

\begin{aligned} \frac{d}{d t} E_{y_{n}} (t) & = (⟨ {y_{n}}_{t t} (t), {y_{n}}_{t} (t) ⟩_{H} + ⟨ {y_{n}}_{t} (t), y_{n} (t) ⟩_{V}) \\ = - ⟨ A y_{n} (t), {y_{n}}_{t} (t) ⟩ + ⟨ G_{γ} (({y_{n}}^{t}, {y_{n}}_{t}^{t})), {y_{t}}_{n} ⟩_{H} + ⟨ y_{n} (t), {y_{n}}_{t} ⟩_{V} \\ = ⟨ G_{γ} (({y_{n}}^{t}, {y_{n}}_{t}^{t})), {y_{t}}_{n} (t) ⟩_{H} = - \frac{⟨ B {y_{t}}_{n} (t - δ), {y_{n}}_{t} (t) ⟩_{H}^{2}}{r_{γ} (({y_{n}}^{t}, {y_{n}}_{t}^{t})} . \end{aligned}

Integrating the above equality over the interval $[δ, t]$ , we obtain

E_{y_{n}} (t) - E_{y_{n}} (τ) = - \int_{τ}^{t} \frac{⟨ B {y_{n}}_{σ} (σ - δ), {y_{n}}_{σ} (σ) ⟩_{H}^{2}}{r_{γ} (({y_{n}}^{σ}, {y_{n}}_{σ}^{σ})} d σ, for all 0 \leq τ \leq t \leq T .

By invoking Lebesgue dominated convergence theorem, we deduce that

E_{y} (t) - E_{y} (τ) = - \int_{τ}^{t} \frac{⟨ B y_{σ} (σ - δ), y_{σ} (σ) ⟩_{H}^{2}}{r_{γ} (y^{σ}, y_{σ}^{σ})} d σ, for all 0 \leq τ \leq t \leq T .

(3.12)

This relation holds for all

t \in [0, T]

, demonstrating that

E_{y} (t)

is nonincreasing on

[0, T]

. Furthermore, we establish the inequality:

\begin{aligned} E_{y} (t) \leq E_{y} (τ) \leq E_{y} (0), for all 0 \leq τ \leq t \leq T < T^{*} . \end{aligned}

(3.13)

As a result, the local solution cannot blow-up in finite time, and hence

T^{*} = \infty

, see Travis and Webb (1978).

Remark 1

Since the function $t \to E_{y} (t)$ is nonnegative and nonincreasing, the following properties can be obtained:

There exists some constant $0 \leq σ \leq E_{y} (0)$ such that $E_{y} (t) \to σ$ as $t \to \infty$ .

For all $t \geq δ$ , we have

r_{γ} (y^{t}, y_{t}^{t}) = E_{y} (t - δ)^{γ},

(3.14)

and

r_{γ} (y^{t}, y_{t}^{t}) = sup_{θ \in [t - δ, 0]} {(‖ ψ (θ) ‖_{V}^{2} + ‖ ψ_{θ} (θ) ‖_{H}^{2})}^{γ} \forall t \in [0, δ] .

(3.15)

It is worth noting that $E_{y} (t^{*}) = 0$ , for some $t^{*} > 0$ , if and only if $y (t) = y_{t} (t) = 0$ , $\forall t \geq t^{*}$

We can obviously deduce, from (3.12) that

\int_{0}^{\infty} \frac{⟨ B y_{σ} (σ - δ), y_{σ} (σ) ⟩_{H}^{2}}{r_{γ} (y^{σ}, y_{σ}^{σ})} d σ < \infty .

(3.16)

4. Stabilization

In this section, we give sufficient conditions for exponential and strong stabilization of system (1.1).

Let us first consider the following uncontrolled system:

{\begin{cases} z_{t t} (t) + A z (t) = 0, & t \geq 0, \\ z (t) = ψ (t), z_{t} (t) = ψ_{t} (t), & t \in [- δ, 0] . \end{cases}

(4.1)

It’s well known that system (4.1) has a unique solution

z

which satisfies the regularity:

z \in C ([- δ, \infty), V) \cap C^{1} ([- δ, \infty), H)

, if

ψ \in C ([- δ, 0], V) \cap C^{1} ([- δ, 0], H)

The solution $z$ is given by

z (t) = {\begin{cases} C (t) ψ (0) + S (t) ψ_{t} (0), & t \geq 0, \\ ψ (t), & t \in [- δ, 0] . \end{cases}

(4.2)

It follows from (3) that the solution

z

of (4.1) satisfies the following:

‖ z (t) ‖_{V}^{2} + ‖ z_{t} (t) ‖_{H}^{2} = ‖ ψ (0) ‖_{V}^{2} + ‖ ψ_{t} (0) ‖_{H}^{2} = 2 E_{z} (0), for all t \geq 0.

(4.3)

We can now formulate our main result as follows.

Theorem 4.1

Assume that the assumptions of Theorem 3.2 are verified. Suppose also that there exist some positive constants $T$ and $α$ such that the solution z of system (4.1) satisfies

‖ ψ_{t} (0) ‖_{H}^{2} + ‖ ψ (0) ‖_{V}^{2} \leq α \int_{δ}^{T} | ⟨ B z_{t} (t - δ), z_{t} (t) ⟩ | d t .

(4.4)

Then, the control

\begin{aligned} u_{γ} (t) = {\begin{cases} - \frac{⟨ B y_{t} (t - δ), y_{t} (t) ⟩_{H}}{r_{γ} (y^{t}, y_{t}^{t})}, & r_{γ} (y^{t}, y_{t}^{t}) \neq 0, \\ 0, & otherwise, \end{cases} with γ = {0, 1}, \end{aligned}

(4.5)

stabilizes system (1.1) with the following estimate:

E_{y} (t) = {\begin{cases} (\frac{1}{t}) & if γ = 0, \\ (e^{- λ t}) & if γ = 1, \end{cases}

for some

λ > 0

Proof.

To begin with, we decompose the solution of system (1.1) as

y (t) = z (t) + ζ (t)

where

z

is the solution of (4.1) and

ζ

is the solution of the following system:

{\begin{cases} ζ_{t t} (t) + A ζ (t) = u (t) B y_{t} (t - δ), \\ ζ (t) = 0, ζ_{t} (t) = 0, t \in [- δ, 0] . \end{cases}

(4.6)

We clearly have

ζ (t) = \int_{0}^{t} S (t - σ) \frac{⟨ B y_{σ} (σ - δ), y_{σ} (σ) ⟩_{H}}{r_{γ} (y^{σ}, y_{σ}^{σ})} B y_{σ} (σ - δ) d σ .

(4.7)

Moreover, we have

ζ_{t} (t) = \int_{0}^{t} C (t - σ) \frac{⟨ B y_{σ} (σ - δ), y_{σ} (σ) ⟩_{H}}{r_{γ} (y^{σ}, y_{σ}^{σ})} B y_{σ} (σ - δ) d σ .

(4.8)

On the other hand, for

t \geq δ

, define

w (t, s) := S (s) B y_{t} (t - δ), t \geq 0,

satisfying the following system:

\begin{aligned} {\begin{cases} w_{s s} (t, s) + A w (t, s) = 0, s > 0 \\ w (t, 0) = 0, w_{s} (t, 0) = B y_{t} (t - δ) . \end{cases} \end{aligned}

Since

A

is self-adjoint and positive, it follows that for all

t \geq δ

‖ S (s) B y_{t} (t - δ) ‖_{V}^{2} + ‖ C (s) B y_{t} (t - δ) ‖_{H}^{2} = ‖ B y_{t} (t - δ) ‖_{H}^{2} .

(4.9)

Using the boundedness of the operator $B$ and the definition of the energy functional $E_{y} (t)$ as given in (3.5), we obtain

‖ C (s) B y_{t} (t - δ) ‖_{H} \leq ‖ B ‖ \sqrt{E_{y} (t - δ)}, for all t \geq δ .

(4.10)

By combining this estimate with (3.14) and (4.8), we infer that for all

t \geq δ

\begin{aligned} ‖ ζ_{t} (t) ‖_{H} & \leq \int_{0}^{t} {‖ C (t - σ) \frac{⟨ B y_{σ} (σ - δ), y_{σ} (σ) ⟩_{H}}{r_{γ} (y^{σ}, y_{σ}^{σ})} B y_{σ} (σ - δ) ‖}_{H} d σ \\ \leq ‖ B ‖ \int_{0}^{t} \frac{| ⟨ B y_{σ} (σ - δ), y_{σ} (σ) ⟩_{H} |}{\sqrt{r_{γ} (y^{σ}, y_{σ}^{σ})}} d σ . \end{aligned}

(4.11)

However, in view of (3.7) and the boundedness of

B

, we obtain

\begin{aligned} | ⟨ B z_{t} (t - δ), z_{t} (t) ⟩_{H} | & \leq ‖ B ‖ (‖ ζ_{t} (t - δ) ‖ ‖ z_{t} (t) ‖_{H} + ‖ y_{t} (t - δ) ‖_{H} ‖ ζ_{t} (t) ‖_{H}) \\ + | ⟨ B y_{t} (t - δ), y_{t} (t) ⟩_{H} |, for all t \geq δ . \end{aligned}

(4.12)

This together with (4.11), (4.3), and (3.6) imply that

\begin{aligned} | ⟨ B z_{t} (t - δ), z_{t} (t) ⟩_{H} | & \leq ‖ B ‖^{2} \sqrt{2 E_{y} (0)} \int_{0}^{t - δ} \frac{| ⟨ B y_{σ} (σ - δ), y_{σ} (σ) ⟩_{H} |}{\sqrt{r_{γ} (y^{σ}, y_{σ}^{σ})}} d σ \\ + ‖ B ‖^{2} \sqrt{2 E_{y} (0)} \int_{0}^{t} \frac{| ⟨ B y_{σ} (σ - δ), y_{σ} (σ) ⟩_{H} |}{\sqrt{r_{γ} (y^{σ}, y_{σ}^{σ})}} d σ \\ + | ⟨ B y_{t} (t - δ), y_{t} (t) ⟩_{H} |, for all t \geq δ . \end{aligned}

(4.13)

Consequently, for all

t \geq δ

, it can be deduced that

\begin{aligned} | ⟨ B z_{t} (t - δ), z_{t} (t) ⟩_{H} | & \leq 2 ‖ B ‖^{2} \sqrt{2 E_{y} (0)} \int_{0}^{t} \frac{| ⟨ B y_{σ} (σ - δ), y_{σ} (σ) ⟩_{H} |}{\sqrt{r_{γ} (y^{σ}, y_{σ}^{σ})}} d σ \\ + | ⟨ B y_{t} (t - δ), y_{t} (t) ⟩_{H} | . \end{aligned}

(4.14)

Using (3.14) and the fact that

E_{y} (t)

bounded by

E_{y} (0)

, we have

| ⟨ B y_{t} (t - δ), y_{t} (t) ⟩_{H} | \leq (2 E_{y} (0))^{γ / 2} \cdot \frac{| ⟨ B y_{t} (t - δ), y_{t} (t) ⟩_{H} |}{\sqrt{r_{γ} (y^{t}, y_{t}^{t})}}, \forall t \geq δ, with γ \in {0, 1} .

(4.15)

Integrating (4.14) over the interval

[δ, T]

and taking into account (4.15) we obtain

\begin{aligned} \int_{δ}^{T} | ⟨ B z_{t} (t - δ), z_{t} (t) ⟩_{H} | d t & \leq 2 ‖ B ‖^{2} \sqrt{2 E_{y} (0)} \int_{δ}^{T} \int_{0}^{t} \frac{| ⟨ B y_{σ} (σ - δ), y_{σ} (σ) ⟩_{H} |}{\sqrt{r_{γ} (y^{σ}, y_{σ}^{σ})}} d σ d t \\ + (2 E_{y} (0))^{γ / 2} \int_{δ}^{T} \frac{| ⟨ B y_{t} (t - δ), y_{t} (t) ⟩_{H} |}{\sqrt{r_{γ} (y^{t}, y_{t}^{t})}} d t . \end{aligned}

(4.16)

Therefore

\begin{aligned} \int_{δ}^{T} | ⟨ B z_{t} (t - δ), z_{t} (t) ⟩_{H} | d t & \leq K (2 E_{y} (0))^{γ / 2} \int_{δ}^{T} \frac{| ⟨ B y_{t} (t - δ), y_{t} (t) ⟩_{H} |}{\sqrt{r_{γ} (y^{t}, y_{t}^{t})}} d t, \end{aligned}

(4.17)

where

K = (2 (T - δ) ‖ B ‖^{2} (2 E_{y} (0))^{1 / 2 - γ / 2} + 1) .

Using Cauchy–Schwarz inequality, we obtain

\begin{aligned} \int_{δ}^{T} | ⟨ B z_{t} (t - δ), z_{t} (t) ⟩_{H} | d t & \leq K (2 E_{y} (0))^{γ / 2} \sqrt{T} {(\int_{0}^{T} \frac{| ⟨ B y_{t} (t - δ), y_{t} (t) ⟩_{H} |^{2}}{r_{γ} (y^{t}, y_{t}^{t})} d t)}^{1 / 2} . \end{aligned}

(4.18)

Condition (4.4) yields

‖ ψ (0) ‖_{V}^{2} + ‖ ψ_{t} (0) ‖_{H}^{2} \leq α K \sqrt{T} (2 E_{y} (0))^{γ / 2} {(\int_{0}^{T} \frac{| ⟨ B y_{t} (t - δ), y_{t} (t) ⟩_{H} |^{2}}{r_{γ} (y^{t}, y_{t}^{t})} d t)}^{1 / 2} .

(4.19)

It follows from (3.12) that

(E_{y} (0))^{2 - γ} \leq c (E_{y} (0) - E_{y} (T))

(4.20)

where

c = 4 α^{2} K^{2} T

Since the estimate holds for intervals $[k T, (k + 1) T]$ , it follows that

E_{y} (k (T + 1))^{2 - γ} \leq E_{y} (k T)^{2 - γ} \leq c (E_{y} (k T) - E ((k + 1) T)), k = 0, 1, 2 \dots .

By setting

a_{k} = E_{y} (k T)

, we have

a_{k + 1} \leq a_{k} - c^{- 1} a_{k + 1}^{2 - γ} .

The remainder of the proof will be divided into two cases.

If $γ = 0$ , we apply the result of the following lemma:

Lemma 4.2 Ammari & Tucsnak, 2000, Lemma 5.2

Let $(a_{k})_{k}$ be a sequence of positive real numbers satisfying:

a_{k + 1} \leq a_{k} - c a_{k + 1}^{2 + j} \forall k \geq 0,

where

c > 0

and

j > - 1

are constants. Then, there exists a positive constant

N

such that

a_{k} \leq \frac{N}{(k + 1)^{1 / (1 + j)}} \forall k \geq 0.

One can see from Lemma 4.2 that

a_{k} \leq \frac{M}{(k + 1)} .

Since

E_{y} (t)

is nonincreasing for

t \in [k T, (k + 1) T]

k \in N

, then

E_{y} (t) \leq \frac{M}{t} .

If $γ = 1$ , then

a_{k + 1} \leq \frac{c}{c + 1} a_{k} .

It follows that

a_{k} \leq {(\frac{c}{c + 1})}^{k + 1} a_{0} .

Let us now set

k = [t / T]

, the integer part of the real number

t / T

Since $E_{y} (t)$ is nonincreasing, we deduce that

E_{y} (t) \leq M e^{- λ t} E_{y} (0),

where

M = (c + 1) / c

and

λ = (1 / T) (\ln (c + 1) - \ln (c))

This achieves the proof.

Remark 2
The stabilization of system (1.1) depends on the value of $γ$
For $γ = 0$ , the system stabilizes with a polynomial rate decay, specifically $O (\frac{1}{t})$ . This indicates that the energy $E_{y} (t)$ of the system decreases as the inverse of time.

For $γ = 1$ , the system stabilizes exponentially, with the energy $E_{y} (t)$ decaying as $O (e^{- λ t})$ for some $λ > 0$ . Exponential stabilization is typically more desirable as it ensures faster decay of energy, leading to quicker stabilization.

5. Applications and Simulation

5.1. Applications

In this section, we discuss two examples to show the applicability of our results.

We initially consider a bilinear wave equation with time delay as a type of second order systems. The importance of this equation is a good description for a wide range of phenomena (mechanical waves, acoustics, electromagnetism, fluid dynamics, etc.).

Example 5.1
On the interval $Ω = (0, 1)$ , we consider the second-order Cauchy problem
${\begin{aligned} y_{t t} (x, t) & = Δ y (x, t) + u (t) y_{t} (x, t - 2), & Ω \times (0, + \infty), \\ y (x, t) & = φ (x, t), y_{t} (x, t) = φ_{t} (x, t), & Ω \times [- 2, 0], \\ y (x, t) & = 0, & {0, 1} \times (0, + \infty), \end{aligned}$
(5.1)
where $u \in L^{\infty} (0, \infty)$ is a real-valued control.

In order to apply our previous result, we rewrite (5.1) as an abstract second-order Cauchy problem (1.1) on $H := L^{2} (0, 1)$ . To this purpose, we introduce the operator
$A = - Δ, with domain D (A) = H^{2} (0, 1) \cap H_{0}^{1} (0, 1) .$
Moreover, there exists an orthonormal basis $(e_{k})_{k \geq 1}$ of $L^{2} (0, 1)$ composed of eigenfunctions of $A$ given by $e_{k} (x) = \sqrt{2} \sin (k π x)$ , $(k \geq 1)$ with corresponding eigenvalues $λ_{k} = (k π)^{2}$ for $k \in N^{}$ . Using the eigenfunctions, we can express the operator $A$ acting on a function $y \in D (A)$ as
$A y = \sum_{k = 1}^{\infty} λ_{k} ⟨ y, e_{n} ⟩ e_{n}, y \in D (A) .$
Furthermore, we introduce the operator $A^{1 / 2}$ defined as
$A^{1 / 2} y = \nabla y = \sum_{k = 1}^{\infty} λ_{k}^{1 / 2} ⟨ y, e_{n} ⟩ e_{n}, y \in V,$
with the space $V = D (A^{1 / 2}) = H_{0}^{1} (0, 1)$ .

It is shown in Travis and Webb (1979) that the operator $- A$ is a generator of strongly continuous cosine family ${C (t)}_{R}$ given by
$C (t) y = \sum_{k \geq 0} ⟨ y, e_{k} ⟩ \cos (k π t) e_{k}, y \in H,$
and the associated sine family ${S (t)}_{t \in R}$ is given by
$S (t) y = \sum_{k \geq 0} \frac{⟨ y, e_{k} ⟩}{k π} \sin (k π t) e_{k}, y \in V .$
The solution of the uncontrolled system,
${\begin{aligned} z_{t t} (x, t) & = Δ z (x, t), & Ω \times (0, + \infty), \\ z (x, t) & = φ (x, t), z_{t} (x, t) = φ_{t} (x, t), & Ω \times [- 2, 0], \\ z (x, t) & = 0, & {0, 1} \times (0, + \infty), \end{aligned}$
is given by
$z (x, t) = {\begin{cases} \sum_{k = 1}^{\infty} (a_{k} \cos (k π t) + b_{k} \sin (k π t)) \sin (k π x), & t \geq 0; \\ φ (x, t), & t \in [- 2, 0], \end{cases}$
(5.2)
and then
$z_{t} (x, t) = {\begin{cases} \sum_{k \geq 0} (a_{k} \sin (k π t) + b_{k} \cos (k π t)) \sin (k π x), & t \geq 0, \\ φ_{t} (x, t) & t \in [- 2, 0] . \end{cases}$
The coefficients $a_{k}$ , $b_{k}$ , are suitable real coefficients determined by the initial data $z (x, 0)$ and $z_{t} (x, 0)$ .

Using the orthogonality of the functions,
$\sin (k π x) and \cos (k π x), k = 1, 2, \dots$
We have
$\begin{aligned} ‖ z_{t} (x, 0) ‖_{L^{2} (0, 1)}^{2} + ‖ z (x, 0) ‖_{H_{0}^{1} (0, 1)}^{2} & = \int_{0}^{1} {| \sum_{k = 0}^{\infty} b_{k} k π \sin (k π x) |}^{2} d x + \int_{0}^{1} {| \sum_{k = 0}^{\infty} a_{k} k π \cos (k π x) |}^{2} d x \\ = \int_{0}^{1} \sum_{k = 0}^{\infty} b_{k}^{2} k^{2} π^{2} \sin^{2} (k π x) d x + \int_{0}^{1} \sum_{k = 0}^{\infty} a_{k}^{2} k^{2} π^{2} \cos^{2} (k π x) d x \\ = \sum_{k = 0}^{\infty} \frac{b_{k}^{2} k^{2} π^{2}}{2} + \sum_{k = 0}^{\infty} \frac{a_{k}^{2} k^{2} π^{2}}{2} . \end{aligned}$
(5.3)

On the other hand, we have
$\begin{aligned} \int_{2}^{4} | ⟨ B z_{t} (x, t - 2), z_{t} (x, t) ⟩_{L^{2} ((0, 1))} | d t & \geq \int_{2}^{4} \int_{0}^{1} \sum_{k = 0}^{\infty} (a_{k} k π)^{2} \sin (k π t) \sin (k π (t - 2)) \\ + (b_{k} k π)^{2} \cos (k π t) \cos (k π (t - 2)) \\ - a_{k} b_{k} (k π)^{2} \cos (k π t) \sin (k π (t - 2)) \\ - a_{k} b_{k} (k π)^{2} \cos (k π (t - 2)) \sin^{2} (k π x) d t d x \\ = \sum_{k = 0}^{\infty} (k π)^{2} \frac{(a_{k}^{2} + b_{k}^{2})}{2} . \end{aligned}$
Then
$‖ z_{t} (x, 0) ‖_{L^{2} ((0, 1))}^{2} + ‖ z (x, 0) ‖_{H_{0}^{1} (0, 1)}^{2} \leq \int_{2}^{4} | ⟨ B z_{t} (t - 2), z_{t} (t) ⟩_{L^{2} (0, 1)} | d t .$
So, condition (4.4) is verified for $α = 1$ .

Theorem 4.1 now provides that system (5.1) is stabilizable by the following control:
$u_{γ} (t) = {\begin{cases} - \frac{\int_{0}^{1} y_{t} (x, t - 2) y_{t} (x, t) d x}{r_{γ} (t)} & r_{γ} (t) \neq 0 \\ 0 & r_{γ} (t) = 0 \end{cases}$
where
$r_{γ} (t) = {(sup_{θ \in [- 2, 0]} \int_{0}^{1} {| \nabla y (x, t + θ) |^{2} + | y_{t} (x, t + θ) |^{2}} d x)}^{γ},$
with the following estimate decay:
$\int_{0}^{1} {| \nabla y (x, t) |^{2} + | y_{t} (x, t) |^{2}} d x = O (e^{- λ t})$
if $γ = 1$ and
$\int_{0}^{1} {| \nabla y (x, t) |^{2} + | y_{t} (x, t) |^{2}} d x = (\frac{1}{t})$
if $γ = 0$ .

As a second application, we consider an abstract version of the vibrating beam equation with time delay.
Example 5.2
We consider on $Ω = (0, π)$ , the following system:
${\begin{aligned} y_{t t} (x, t) & = - Δ^{2} y (x, t) + u (t) y_{t} (x, t - 2 π), & Ω \times (0, + \infty), \\ y (x, t) & = φ (x, t), y_{t} (x, t) = φ_{t} (x, t), & Ω \times [- 2 π, 0], \\ y (x, t) & = Δ y (x, t) = 0, & {0, π} \times (0, + \infty), \end{aligned}$
(5.4)
where $u \in L^{\infty} (0, + \infty)$ is a real-valued control.

Let $H = L^{2} (Ω)$ and define the operator $A : D (A) \to H$ as follows:
$A = Δ^{2}, D (A) = {y \in H^{4} (Ω), y_{/ \partial Ω}, Δ y_{/ \partial Ω} = 0} .$
It is well known that the operator $A$ is self-adjoint and positive. Then, we can introduce the square root of $A$ defined by
$A^{1 / 2} = - Δ, V = D (A^{1 / 2}) = H^{2} (Ω) \cap H_{0}^{1} (Ω) .$
The spectrum of $A$ is given by $σ (A) = {k^{4}, k \geq 1}$ , corresponding to the eigenfunctions $(e_{k})_{k \geq 1}$ , defined by $e_{k} = \sqrt{2 / π} \sin (k x)$ $\forall k \in N^{}$ .

It is shown in Fattorini (2011) that the operator $- A$ is a generator of strongly continuous cosine family ${C (t)}_{R}$ given by
$C (t) y = \sum_{k \geq 0} ⟨ y, e_{k} ⟩ \cos (k^{2} t) e_{k}, y \in H,$
and the associated sine family ${S (t)}_{t \in R}$ is given by
$S (t) y = \sum_{k \geq 0} \frac{⟨ y, e_{k} ⟩}{k π} \sin (k^{2} t) e_{k}, y \in V .$
Then, the solution of the linear beam equation,
${\begin{aligned} z_{t t} (x, t) & = - Δ^{2} z (x, t), & Ω \times (0, + \infty), \\ z (x, t) & = φ (x, t), z_{t} (x, t) = φ (x, t), & Ω \times [- 2, 0], \\ z (x, t) & = Δ z (x, t) = 0, & {0, π} \times (0, + \infty), \end{aligned}$
takes the form
$z (x, t) = {\begin{cases} \sum_{k \geq 0} (a_{k} \cos (k^{2} t) + b_{k} \sin (k^{2} t)) e_{k}, & t \geq 0, \\ φ (x, t), & t \in [- 2 π, 0], \end{cases}$
and then
$z_{t} (x, t) = {\begin{cases} \sum_{k \geq 0} (- k^{2} a_{k} \sin (k^{2} t) + b_{k} k^{2} \cos (k^{2} t)) e_{k}, & t \geq 0, \\ φ_{t} (x, t) & t \in [- 2 π, 0], \end{cases}$
where the coefficients $a_{k}$ and $b_{k}$ depend on the initial conditions $z (x, 0)$ and $z_{t} (x, 0)$ .

Condition (4.4) is verified; indeed,
$\begin{aligned} \int_{2 π}^{4 π} | ⟨ B z_{t} (t - 2 π), z_{t} (t) ⟩_{L^{2} (Ω)} | d t & \geq \int_{2 π}^{4 π} \int_{0}^{π} \sum_{k = 0}^{\infty} (a_{k} k)^{2} \sin (k^{2} t) \sin (k^{2} (t - 2 π)) \\ + (b_{k} k)^{2} \cos (k^{2} t) \cos (k^{2} (t - 2 π)) \\ - a_{k} b_{k} k^{2} \cos (k^{2} t) \sin (k^{2} (t - 2 π)) \\ - a_{k} b_{k} k^{2} \cos (k^{2} (t - 2 π) \sin (k^{2} t)) \sin^{2} (k x) d t d x \\ = \sum_{k = 0}^{\infty} k^{2} \frac{a_{k}^{2} + b_{k}^{2}}{2} \\ = ‖ z_{t} (x, 0) ‖_{L^{2} (Ω)}^{2} + ‖ z (x, 0) ‖_{H^{2} (Ω) \cap H_{0}^{1} (Ω)}^{2} . \end{aligned}$
It follows that the control
$u_{γ} (t) = {\begin{aligned} - \frac{\int_{0}^{π} y_{t} (x, t) y_{t} (x, t - 2 π) d x}{r_{γ} (t)}, & r_{γ} (t) \neq 0, \\ 0 & r_{γ} = 0, \end{aligned}$
where
$r_{γ} (t) = {(sup_{θ \in [- 2 π, 0]} \int_{0}^{π} {| Δ y (x, t + θ) |^{2} + | y_{t} (x, t + θ) |^{2}} d x)}^{γ}$
stabilizes system (5.4) with the following estimate decay:
$\int_{0}^{π} {| Δ y (x, t) |^{2} + | y_{t} (x, t) |^{2}} d x = (e^{- λ t})$
if $γ = 1$ and
$\int_{0}^{π} {| Δ y (x, t) |^{2} + | y_{t} (x, t) |^{2}} d x = (\frac{1}{t})$
if $γ = 0$ .
5.2. Simulation Results

In this subsection, numerical simulations have been given to show the effectiveness of the stabilizing control which given by (1.8). In order to illustrate the previous results numerically, we use the finite difference method to perform numerical computations of both displacement and velocity. Let us consider the following system:

{\begin{aligned} y_{t t} (x, t) & = Δ y (x, t) + u (t) y_{t} (x, t - 2), & (0, 1) \times (0, + \infty), \\ y (x, t) & = \sin (π x), y_{t} (x, t) = 0, & (0, 1) \times [- 2, 0], \\ y (0, t) & = 0, y (1, t) = 0, & t \in (0, + \infty), \end{aligned}

(5.5)

where the control

u (t) = - \int_{0}^{1} y_{t} (x, t) y_{t} (x, t - 2) d x .

(5.6)

Figure 1 shows that the displacements $y (x, t)$ and the velocity $y_{t} (x, t)$ in the spatial domain $Ω = (0, 1)$ of length $L$ are discretized into a finite number of $N x = 69$ . The calculation of the spatial step size is performed using $d x = 1 / (N x + 1)$ . Similarly, for temporal discretization, the time domain is discretized using the parameter $N t$ , representing the number of time steps. The temporal step size is computed as $d t = T / N t = 10^{- 4}$ , where the total time is fixed at $T = 9$ .

Figure 1.

The displacement $y (x, t)$ and the velocity of system (5.5).

Figure 1 shows that the displacement $y (x, t)$ and the velocity of system (5.5) is stabilized at time $t = 6$ . Figure 2 shows the control of (5.6).

Figure 2.

Evolution of control (5.6).

Figure 3 shows that the energy of the system,

E_{y} (t) = \frac{1}{2} (‖ \nabla y (x, t) ‖_{L^{2} (0, 1)}^{2} + ‖ y_{t} (x, t) ‖_{L^{2} (0, 1)}^{2}),

decreases to zero at time

t = 6

Figure 3.

Energy of system (5.5).

6. Conclusion

In this paper, the question of stabilization with decay estimates for a class of second-order bilinear systems with time delay is explored. Under sufficient conditions we showed that the chosen controls stabilize the system with estimate decay. Some illustrating examples are given with simulation. Various questions remain open. This is the case of semilinear systems.

Footnotes

Acknowledgments

The authors are highly grateful to the anonymous referees for their valuable comments.

Funding

The authors disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work is supported by the Natural Science Foundation of Sichuan Province (No. 2026NSFSC0003).

Declaration of Conflicting Interests

The authors declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

ORCID iDs

Khalil El Kazoui

Hassan Ezzaki

Baowei Feng

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