Interior Approximate Controllability of a Class of Nonlinear Thermoelastic Plate Equations

Abstract

This article introduces new sufficient conditions ensuring the interior approximate controllability of semilinear thermoelastic plate equations subject to Dirichlet boundary conditions. The analysis is carried out by reformulating the system as an abstract evolution equation on a suitable Banach space. A key role is played by the compactness of the semigroup generated by the linear operator, which allows us to treat the nonlinear components effectively. To establish controllability, we apply Rothe’s fixed-point theorem, which provides the necessary framework for handling nonlinear perturbations. The results obtained contribute to the existing literature, since the controllability of the specific semilinear thermoelastic system considered here has not been previously investigated.

1. Introduction

The central goal of this work is to analyze the interior approximate controllability of a semilinear thermoelastic plate system subject to Dirichlet boundary conditions:

$$\left\{\begin{array}{c}{w}_{tt}+{\Delta }^{2}w+\alpha \Delta \theta =1_{\omega }{u}_1(t,x)+f_1(t,w,w_t,\theta ,u_1,u_2),\hfill \\ {\theta }_t-\beta \Delta \theta -\alpha \Delta w_t=1_{\omega }{u}_2(t,x)+f_2(t,w,w_t,\theta ,u_1,u_2),\mathrm{in}(0,\tau )\times \Omega ,\hfill \\ \theta =w=\Delta w=0,\mathrm{on}(0,\tau )\times \partial \Omega ,\hfill \\ w(0,x)=w_0\left(x\right),w_t(0,x)=w_1\left(x\right),\theta (0,x)={\theta }_0\left(x\right),x\in \Omega .\hfill \end{array}\right.$$

(1)

Here, $\Delta$ is the Laplace operator, ${\Delta }^2$ denotes the biharmonic operator, and the parameters $\alpha ,\beta$ are real constants with $\alpha \ne 0$ and $\beta >0$. The spatial region $\Omega$ is a bounded domain of ${\mathbb{R}}^N$ ($N\ge 1$) with a sufficiently smooth boundary. The unknowns $w(t,x)$ and $\theta (t,x)$ describe the transverse displacement and the thermal component of the plate, respectively. The distributed controls $u_1,u_2$ act on a nonempty open subset $\omega \subset \Omega$, and $1_{\omega }$ is its characteristic function. Both controls belong to $L^2([0,\tau ],L^2(\Omega ))$.

The nonlinear terms $f_i:[0,\tau ]\times {\mathbb{R}}^5\to \mathbb{R}$, $i=1,2$, are assumed to be smooth perturbations satisfying the growth estimate

$$|f_i(t,w,v,\theta ,u_1,u_2)|\le a_{1,i}{\left|w\right|}^{{\xi }_{1,i}}+a_{2,i}{\left|v\right|}^{{\xi }_{2,i}}+a_{3,i}{\left|\theta \right|}^{{\xi }_{3,i}}+b_{1,i}|u_1{|}^{{\sigma }_{1,i}}+b_{2,i}{\left|u_2\right|}^{{\sigma }_{2,i}}+c_i,$$

(2)

with non-negative constants $a_{j,i},b_{j,i},c_i$ and exponents restricted by

$${\displaystyle \frac{1}{2}}\le {\xi }_{j,i}<1,{\displaystyle \frac{1}{2}}\le {\sigma }_{j,i}<1.$$

System (1) corresponds to the thermoelastic Kirchhoff–Love plate coupled with Fourier’s law of heat conduction, under clamped boundary conditions $w=\theta =\Delta w=0$ on $\partial \Omega$. Classical modeling and stability analyses for thermoelastic plates were developed in Lagnese’s monograph [1], with subsequent contributions on stability and long-term dynamics in [2,3,4,5,6,7,8,9,10]. Further studies include almost-periodic solutions and propagation phenomena in exterior domains [11,12]. On the side of controllability, various authors have addressed approximate or null controllability problems for thermoelastic structures under localized or distributed actuation [13,14,15].

Among the existing contributions, we highlight the following: (i) Leiva [16], who provided necessary and sufficient algebraic conditions for approximate controllability of thermoelastic plates driven by finitely many inputs; (ii) Larez et al. [17], who established approximate controllability for the linear case with controls acting over the full domain; (iii) Leiva and Merentes [18], who studied interior controllability with distributed inputs localized in $\omega$; and (iv) Castro and de Teresa [19], who investigated null controllability for linear thermoelastic plates with partially supported controls. Additional progress can be found in [20,21,22,23,24,25,26,27].

Within this framework, our work addresses the semilinear setting for system (1), for which controllability results have not been reported to date. Our strategy consists of recasting the system as an abstract evolution equation in a Banach space, analyzing the compactness of the linear semigroup, and applying Rothe’s fixed-point theorem to treat the nonlinear perturbations.

Finally, beyond its mathematical interest, the study of thermoelastic plates has important engineering applications. Such models arise in the analysis of thin flexible structures subject to mechanical and thermal effects, including aerospace components, micro-electro-mechanical systems (MEMSs), and advanced composite materials. Controllability results provide the theoretical foundation for designing stabilizing and performance-oriented control strategies under external excitations.

The paper is organized as follows. Section 2 presents preliminary material and reformulates system (1) in an abstract Banach space setting. Section 3 is devoted to the controllability properties of the linearized problem. Section 4 establishes the interior approximate controllability for the semilinear system. Section 5 discusses examples illustrating the applicability of the results. The paper concludes with final remarks.

2. Preliminaries and Abstract Representation of the Problem

In this section we gather some preliminary results and rewrite system (1) as an abstract differential equation in a suitable Banach space. For this purpose, we rely on the works of Leiva and Sivoli [6], Leiva [28], and the classical monograph by Curtain and Zwart [29].

Let $\mathcal{X}=L^2(\Omega )=L^2(\Omega ,\mathbb{R})$ be a Hilbert space. We define the unbounded linear operator $\mathcal{A}:\mathcal{D}\left(\mathcal{A}\right)\subset \mathcal{X}\to \mathcal{X}$ by

$$\mathcal{A}\phi =-\Delta \phi ,$$

with domain

$$\mathcal{D}\left(\mathcal{A}\right)=H^2(\Omega ,\mathbb{R})\cap H_0^1(\Omega ,\mathbb{R}).$$

The operator $\mathcal{A}$ is known to satisfy the following properties:

• Its spectrum consists only of eigenvalues

  $$0<{\mu }_1<{\mu }_2<\dots <{\mu }_j\to \infty ,$$

  each one of finite multiplicity ${\nu }_j$, corresponding to the dimension of the associated eigenspace.
• There exists a complete orthonormal family of eigenfunctions $\left\{{\phi }_{j,k}\right\}$ of $\mathcal{A}$.
• For every $x\in \mathcal{D}\left(\mathcal{A}\right)$, one can write

  $$\mathcal{A}x=\sum _{j=1}^{\infty }\sum _{k=1}^{{\nu }_j}\langle x,{\phi }_{j,k}\rangle {\phi }_{j,k}=\sum _{j=1}^{\infty }{\mu }_j{\mathcal{E}}_{j}x,$$

  where $\left\{{\mathcal{E}}_j\right\}$ denotes the set of orthogonal projections in $\mathcal{X}$ (see [30,31]).
• The operator $-\mathcal{A}$ generates an analytic semigroup ${\left\{\mathcal{S}\left(t\right)\right\}}_{t\ge 0}$ given by

  $$\mathcal{S}\left(t\right)x=\sum _{j=1}^{\infty }{e}^{-{\mu }_{j}t}{\mathcal{E}}_{j}x,x\in \mathcal{X}.$$
• For $\varsigma \ge 0$, the fractional power spaces ${\mathcal{X}}^{\varsigma }$ are defined as

  $${\mathcal{X}}^{\varsigma }=\mathcal{D}\left({\mathcal{A}}^{\varsigma }\right)=\left\{x\in \mathcal{X}:\sum _{j=1}^{\infty }{\mu }_j^{2\varsigma }{\parallel {\mathcal{E}}_{j}x\parallel }^2<\infty \right\},$$

  with the norm

  $${\parallel x\parallel }_{{\mathcal{X}}^{\varsigma }}=\parallel {\mathcal{A}}^{\varsigma }x\parallel ={\left(\sum _{j=1}^{\infty }{\mu }_j^{2\varsigma }{\parallel {\mathcal{E}}_{j}x\parallel }^2\right)}^{1/2}.$$
  Here, $\parallel ·\parallel$ denotes the norm in $L^2(\Omega )$, and ${\mathcal{A}}^{\varsigma }$ acts as

  $${\mathcal{A}}^{\varsigma }x=\sum _{j=1}^{\infty }{\mu }_j^{\varsigma }{\mathcal{E}}_{j}x.$$

For $\varsigma \ge 0$, we introduce the Hilbert space

$${\mathcal{Z}}_{\varsigma }={\mathcal{X}}^{\varsigma }\times \mathcal{X}\times \mathcal{X},$$

with the norm

$${∥\left[\begin{array}{c}w\\ v\\ \theta \end{array}\right]∥}_{\varsigma }^2={\parallel w\parallel }_{{\mathcal{X}}^{\varsigma }}^2+{\parallel v\parallel }_{\mathcal{X}}^2+{\parallel \theta \parallel }_{\mathcal{X}}^2.$$

Introducing $v=w^{\prime }$, system (1) can be expressed as an abstract Cauchy problem in ${\mathcal{Z}}_1$:

$$\left\{\begin{array}{c}{z}^{\prime }=\mathcal{A}z+{\mathcal{B}}_{\omega }u+\mathcal{F}(t,z,u),t\in (0,\tau ],\hfill \\ z\left(0\right)=z_0,\hfill \end{array}\right.$$

(3)

where the operators $\mathcal{A}$, ${\mathcal{B}}_{\omega }$, and $\mathcal{F}$ are defined as

$$\mathcal{A}:\mathcal{D}\left(\mathcal{A}\right)\subset {\mathcal{Z}}_1\to {\mathcal{Z}}_1,by\mathcal{A}=\left[\begin{array}{ccc}0& I_{\mathcal{X}}& 0\\ -{\mathcal{A}}^2& 0& \alpha \mathcal{A}\\ 0& -\alpha \mathcal{A}& -\beta \mathcal{A}\end{array}\right],$$

(4)

with the domain

$$\mathcal{D}\left(\mathcal{A}\right)=\left\{w\in H^4(\Omega ):w=\Delta w=0\mathrm{on}\partial \Omega \right\}\times \mathcal{D}\left(\mathcal{A}\right)\times \mathcal{D}\left(\mathcal{A}\right).$$

The control operator

$${\mathcal{B}}_{\omega }:\mathcal{U}\to {\mathcal{Z}}_1,by{\mathcal{B}}_{\omega }u=\left[\begin{array}{cc}0& 0\\ 1_{\omega }& 0\\ 0& 1_{\omega }\end{array}\right]\left[\begin{array}{c}{u}_1\\ u_2\end{array}\right],$$

where $\mathcal{U}=L^2(\Omega )\times L^2(\Omega )$ and $u=(u_1,u_2)\in L^2([0,\tau ],\mathcal{U})$, and the nonlinear term $\mathcal{F}:[0,\tau ]\times {\mathcal{Z}}_1\times \mathcal{U}\to {\mathcal{Z}}_1$, which is given by

$$\mathcal{F}(t,z,u)=\mathcal{F}(t,w,v,\theta ,u_1,u_2)=\left[\begin{array}{c}0\\ f_1^e(t,w,v,\theta ,u_1,u_2)\\ f_2^e(t,w,v,\theta ,u_1,u_2)\end{array}\right],$$

where the functions $f_i^e$ are defined pointwise by

$$f_i^e(t,w,v,\theta ,u_1,u_2)\left(x\right)=f_i\left(t,w\left(x\right),v\left(x\right),\theta \left(x\right),u_1\left(x\right),u_2\left(x\right)\right),i=1,2.$$

The spectral decomposition of $\mathcal{A}$ shows that it generates a strongly continuous semigroup ${\left\{\mathcal{T}\left(t\right)\right\}}_{t\ge 0}$ in ${\mathcal{Z}}_1$.

Theorem 1

(See [6,16,18]). The operator $\mathcal{A}$ defined in (4) is the infinitesimal generator of a strongly continuous semigroup ${\left\{\mathcal{T}\left(t\right)\right\}}_{t\ge 0}$ in the Hilbert space ${\mathcal{Z}}_1$, given by the representation

$$\mathcal{T}\left(t\right)z=\sum _{j=1}^{\infty }{e}^{{\mathcal{A}}_{j}t}{\mathcal{P}}_{j}z,z\in {\mathcal{Z}}_1,t\ge 0,$$

(5)

where ${\left\{{\mathcal{P}}_j\right\}}_{j\ge 1}$ is a complete family of orthogonal projections on ${\mathcal{Z}}_1$, defined by

$${\mathcal{P}}_j=\left[\begin{array}{ccc}{\mathcal{E}}_j& 0& 0\\ 0& {\mathcal{E}}_j& 0\\ 0& 0& {\mathcal{E}}_j\end{array}\right],j=1,2,\dots ,$$

and each matrix ${\mathcal{A}}_j$ is given by

$${\mathcal{A}}_j={\mathcal{B}}_j{\mathcal{P}}_j,\mathrm{with}{\mathcal{B}}_j=\left[\begin{array}{ccc}0& 1& 0\\ -{\mu }_j^2& 0& \alpha {\mu }_j\\ 0& -\alpha {\mu }_j& -\beta {\mu }_j\end{array}\right].$$

Each block matrix ${\mathcal{B}}_j$ has three simple eigenvalues of the form

$${\vartheta }_i\left(j\right)=-{\mu }_j{\varrho }_i,i=1,2,3,$$

where ${\varrho }_1,{\varrho }_2,{\varrho }_3>0$ are the roots of the cubic equation

$${\varrho }^3-\beta {\varrho }^2+(1+{\alpha }^2)\varrho -\beta =0.$$

Moreover, there exists $M=M(\alpha ,\beta )>0$ such that

$$\parallel \mathcal{T}\left(t\right)\parallel \le M{e}^{-\zeta t},t\ge 0,$$

with ζ depending on ${\mu }_1$ and the smallest positive root ϱ.

The following theorem constitutes a fundamental tool for proving the approximate controllability of the system (1).

Theorem 2.

The semigroup $\mathcal{T}\left(t\right)$ is compact for all $t>0$.

Proof. 

Consider the sequence of finite-rank operators defined by

$${\mathcal{T}}_k\left(t\right)z=\sum _{j=1}^k{e}^{{\mathcal{A}}_{j}t}{\mathcal{P}}_{j}z,t>0.$$

According to Theorem 2.3 in [18], each exponential term admits the following decomposition:

$$e^{{\mathcal{A}}_{j}t}{\mathcal{P}}_j=e^{-{\mu }_j{\varrho }_{1}t}{q}_1\left(j\right){\mathcal{P}}_j+e^{-{\mu }_j{\varrho }_{2}t}{q}_2\left(j\right){\mathcal{P}}_j+e^{-{\mu }_j{\varrho }_{3}t}{q}_3\left(j\right){\mathcal{P}}_j,$$

where the matrices $q_i\left(j\right)$ for $i=1,2,3$ are given explicitly by

$$q_1\left(j\right)={\displaystyle \frac{1}{({\varrho }_1-{\varrho }_2)({\varrho }_1-{\varrho }_3)}}\left[\begin{array}{ccc}{\varrho }_2{\varrho }_3-1& {\displaystyle \frac{{\varrho }_2+{\varrho }_3}{{\mu }_j}}& {\displaystyle \frac{\alpha }{{\mu }_j}}\\ {\mu }_j({\varrho }_3-{\varrho }_2)& {\varrho }_2{\varrho }_3-1-{\alpha }^2& \alpha ({\varrho }_2+{\varrho }_3-\beta )\\ {\mu }_j\alpha & -\alpha ({\varrho }_2+{\varrho }_3-\beta )& {({\varrho }_3-\beta )}^2-{\alpha }^2\end{array}\right],$$

$$q_2\left(j\right)={\displaystyle \frac{1}{({\varrho }_2-{\varrho }_1)({\varrho }_2-{\varrho }_3)}}\left[\begin{array}{ccc}{\varrho }_1{\varrho }_3-1& {\displaystyle \frac{{\varrho }_1+{\varrho }_3}{{\mu }_j}}& {\displaystyle \frac{\alpha }{{\mu }_j}}\\ {\mu }_j({\varrho }_3-{\varrho }_1)& {\varrho }_1{\varrho }_3-1-{\alpha }^2& \alpha ({\varrho }_1+{\varrho }_3-\beta )\\ {\mu }_j\alpha & -\alpha ({\varrho }_1+{\varrho }_3-\beta )& {({\varrho }_3-\beta )}^2-{\alpha }^2\end{array}\right],$$

$$q_3\left(j\right)={\displaystyle \frac{1}{({\varrho }_3-{\varrho }_1)({\varrho }_3-{\varrho }_2)}}\left[\begin{array}{ccc}{\varrho }_1{\varrho }_2-1& {\displaystyle \frac{{\varrho }_1+{\varrho }_2}{{\mu }_j}}& {\displaystyle \frac{\alpha }{{\mu }_j}}\\ {\mu }_j({\varrho }_2-{\varrho }_1)& {\varrho }_1{\varrho }_2-1-{\alpha }^2& \alpha ({\varrho }_1+{\varrho }_2-\beta )\\ {\mu }_j\alpha & -\alpha ({\varrho }_1+{\varrho }_2-\beta )& {({\varrho }_2-\beta )}^2-{\alpha }^2\end{array}\right].$$

As a consequence, we obtain the following uniform estimate

$$\parallel e^{{\mathcal{A}}_{j}t}{\mathcal{P}}_j\parallel \le M\left(e^{-{\mu }_j{\varrho }_{1}t}+e^{-{\mu }_j{\varrho }_{2}t}+e^{-{\mu }_j{\varrho }_{3}t}\right)\to 0\mathrm{as}j\to \infty ,$$

for some constant $M>0$.

Now, observe that

$$\parallel \mathcal{T}\left(t\right)z-{\mathcal{T}}_k{\left(t\right)z\parallel }^2\le \sum _{j=k+1}^{\infty }\parallel e^{{\mathcal{A}}_{j}t}{\mathcal{P}}_j{z\parallel }^2\le \left(\underset{j\ge k+1}{sup}{\parallel e^{{\mathcal{A}}_{j}t}{\mathcal{P}}_j\parallel }^2\right){\parallel z\parallel }^2,$$

which implies that the sequence $\left\{{\mathcal{T}}_k\left(t\right)\right\}$ converges uniformly to $\mathcal{T}\left(t\right)$.

Since each ${\mathcal{T}}_k\left(t\right)$ is a finite-rank operator, and hence compact, and the uniform limit of compact operators is compact (see Lemma A.3.22 in [29]), and it follows that the semigroup $\mathcal{T}\left(t\right)$ is compact for every $t>0$. □

Finally, we present two auxiliary propositions. Proposition 1 provides an inclusion inequality for $L^p$ spaces, while Proposition (6) establishes the nonlinear growth bound of the function $\mathcal{F}$. Both will be essential in the proof of approximate controllability.

Proposition 1

(See [32]). Let $(\mathcal{X},\Sigma ,\mu )$ be a measure space with $\mu \left(\mathcal{X}\right)<\infty$ and $1\le q<r<\infty$. Then, $L_r\left(\mu \right)\subset L_q\left(\mu \right)$ and

$${∥f∥}_q\le \mu {\left(\mathcal{X}\right)}^{{\displaystyle \frac{r-q}{rq}}}{∥f∥}_r,\mathrm{for}\mathrm{all}f\in L_r\left(\mu \right).$$

Proposition 2.

The nonlinear function $\mathcal{F}:[0,\tau ]\times {\mathcal{Z}}_1\times \mathcal{U}\to {\mathcal{Z}}_1$ satisfies the following inequality:

$${∥\mathcal{F}(t,z,u)∥}_1\le a{∥z∥}_1^{\xi }+b{∥u∥}^{\sigma }+c,\mathrm{for}\mathrm{all}z\in {\mathcal{Z}}_1,u\in \mathcal{U},$$

(6)

where ${\displaystyle \frac{1}{2}}\le \xi <1$ and ${\displaystyle \frac{1}{2}}\le \sigma <1$.

Proof. 

For $i=1,2$, we compute

$$\begin{array}{cc}\hfill {∥f_i^e(t,z,u)∥}_{\mathcal{X}}^2& ={\int }_{\Omega }{\left|f_i\left(t,w\left(x\right),v\left(x\right),\theta \left(x\right),u_1\left(x\right),u_2\left(x\right)\right)\right|}^{2}dx\hfill \\ \hfill & \le {\int }_{\Omega }\left[a_{1,i}{\left|w\left(x\right)\right|}^{{\xi }_{1,i}}+a_{2,i}{\left|v\left(x\right)\right|}^{{\xi }_{2,i}}+a_{3,i}{\left|\theta \left(x\right)\right|}^{{\xi }_{3,i}}\right.\hfill \\ \hfill & {\left.+b_{1,i}|u_1{\left(x\right)|}^{{\sigma }_{1,i}}+b_{2,i}{\left|u_2\left(x\right)\right|}^{{\sigma }_{2,i}}+c_i\right]}^{2}dx\hfill \\ \hfill & \le 6^2[a_{1,i}^2{\parallel w\parallel }_{L_{2{\xi }_{1,i}}}^{2{\xi }_{1,i}}+a_{2,i}^2{\parallel v\parallel }_{L_{2{\xi }_{2,i}}}^{2{\xi }_{2,i}}+a_{3,i}^2{\parallel \theta \parallel }_{L_{2{\xi }_{3,i}}}^{2{\xi }_{3,i}}\hfill \\ \hfill & +b_{1,i}^2\parallel u_1{\parallel }_{L_{2{\sigma }_{1,i}}}^{2{\sigma }_{1,i}}+b_{2,i}^2{\parallel u_2\parallel }_{L_{2{\sigma }_{2,i}}}^{2{\sigma }_{2,i}}+c_i^2\mu (\Omega )].\hfill \end{array}$$

Given that $1\le 2{\xi }_{j,i}<2$ and $1\le 2{\sigma }_{k,i}<2$, for $j=1,2,3$ and $k=1,2$, and by the continuous embedding ${\mathcal{X}}^1\subset \mathcal{X}$ and Proposition 1, we obtain

$$\begin{array}{cc}\hfill {∥f_i^e(t,z,u)∥}_{\mathcal{X}}^2& \le 6^2{a}_{1,i}^2\mu {(\Omega )}^{{\displaystyle \frac{1-{\xi }_{1,i}}{2}}}{d}^{2{\xi }_{1,i}}{\parallel w\parallel }_{{\mathcal{X}}^1}^{2{\xi }_{1,i}}+6^2{a}_{2,i}^2\mu {(\Omega )}^{{\displaystyle \frac{1-{\xi }_{2,i}}{2}}}{\parallel v\parallel }_{\mathcal{X}}^{2{\xi }_{2,i}}\hfill \\ \hfill & +6^2{a}_{3,i}^2\mu {(\Omega )}^{{\displaystyle \frac{1-{\xi }_{3,i}}{2}}}{\parallel \theta \parallel }_{\mathcal{X}}^{2{\xi }_{3,i}}+6^2{b}_{1,i}^2\mu {(\Omega )}^{{\displaystyle \frac{1-{\sigma }_{1,i}}{2}}}{\parallel u_1\parallel }^{2{\sigma }_{1,i}}\hfill \\ \hfill & +6^2{b}_{2,i}^2\mu {(\Omega )}^{{\displaystyle \frac{1-{\sigma }_{2,i}}{2}}}{\parallel u_2\parallel }^{2{\sigma }_{2,i}}+6^2{c}_i^2\mu (\Omega ).\hfill \end{array}$$

Consequently,

$$\begin{array}{cc}\hfill {∥f_i^e(t,z,u)∥}_{\mathcal{X}}^2& \le 6^2{a}_{1,i}^2\mu {(\Omega )}^{{\displaystyle \frac{1-{\xi }_{1,i}}{2}}}{d}^{2{\xi }_{1,i}}{\parallel z\parallel }_1^{2{\xi }_{1,i}}+6^2{a}_{2,i}^2\mu {(\Omega )}^{{\displaystyle \frac{1-{\xi }_{2,i}}{2}}}{\parallel z\parallel }_1^{2{\xi }_{2,i}}\hfill \\ \hfill & +6^2{a}_{3,i}^2\mu {(\Omega )}^{{\displaystyle \frac{1-{\xi }_{3,i}}{2}}}{\parallel z\parallel }_1^{2{\xi }_{3,i}}+6^2{b}_{1,i}^2\mu {(\Omega )}^{{\displaystyle \frac{1-{\sigma }_{1,i}}{2}}}{\parallel u\parallel }^{2{\sigma }_{1,i}}\hfill \\ \hfill & +6^2{b}_{2,i}^2\mu {(\Omega )}^{{\displaystyle \frac{1-{\sigma }_{2,i}}{2}}}{\parallel u\parallel }^{2{\sigma }_{2,i}}+6^2{c}_i^2\mu (\Omega )\hfill \\ \hfill & \le a_i^2{\parallel z\parallel }_1^{2{\xi }_i}+b_i^2{\parallel u\parallel }^{2{\sigma }_i}+c_i^2,\hfill \end{array}$$

where

$${\xi }_i=max\{{\xi }_{1,i},{\xi }_{2,i},{\xi }_{3,i}\},{\sigma }_i=max\{{\sigma }_{1,i},{\sigma }_{2,i}\},$$

$$a_i^2=6^2\left[a_{1,i}^2\mu {(\Omega )}^{{\displaystyle \frac{1-{\xi }_{1,i}}{2}}}{d}^{2{\xi }_{1,i}}+a_{2,i}^2\mu {(\Omega )}^{{\displaystyle \frac{1-{\xi }_{2,i}}{2}}}+a_{3,i}^2\mu {(\Omega )}^{{\displaystyle \frac{1-{\xi }_{3,i}}{2}}}\right],$$

$$b_i^2=6^2\left[b_{1,i}^2\mu {(\Omega )}^{{\displaystyle \frac{1-{\sigma }_{1,i}}{2}}}+b_{2,i}^2\mu {(\Omega )}^{{\displaystyle \frac{1-{\sigma }_{2,i}}{2}}}\right],c_i^2=6^2{c}_i^2\mu (\Omega ).$$

Therefore,

$$\begin{array}{cc}\hfill {\parallel \mathcal{F}(t,z,u)\parallel }_1^2& =\parallel f_1^e{(t,z,u)\parallel }_{\mathcal{X}}^2+{\parallel f_2^e(t,z,u)\parallel }_{\mathcal{X}}^2\hfill \\ \hfill & \le a^2{\parallel z\parallel }_1^{2\xi }+b^2{\parallel u\parallel }^{2\sigma }+c^2,\hfill \end{array}$$

where

$$\xi =max\{{\xi }_1,{\xi }_2\},\sigma =max\{{\sigma }_1,{\sigma }_2\},$$

$$a^2=a_1^2+a_2^2,b^2=b_1^2+b_2^2,c^2=c_1^2+c_2^2.$$

This concludes the proof of inequality (6). □

3. Controllability of the Unperturbed Linear System

In this section we recall some basic results and provide a characterization of the approximate controllability for the linearized version of system (1):

$$\left\{\begin{array}{c}{z}^{\prime }\left(t\right)=\mathcal{A}z\left(t\right)+{\mathcal{B}}_{\omega }u\left(t\right),z\left(t\right)\in {\mathcal{Z}}_1,t\in (0,\tau ],\hfill \\ z\left(0\right)=z_0.\hfill \end{array}\right.$$

(7)

The unique mild solution of (7) can be expressed as

$$z\left(t\right)=\mathcal{T}\left(t\right)z_0+{\int }_0^t\mathcal{T}(t-s){\mathcal{B}}_{\omega }u\left(s\right)ds,t\in [0,\tau ].$$

(8)

Definition 1

(For approximate controllability, see [16,17]). We say that system (7) is approximately controllable on $[0,\tau ]$ if, given any initial state $z_0\in {\mathcal{Z}}_1$, any target $z_1\in {\mathcal{Z}}_1$, and every tolerance $ϵ>0$, there exists a control $u\in L^2([0,\tau ],\mathcal{U})$ such that the mild solution $z(·)$ of (7) satisfies

$$z\left(0\right)=z_0,{\parallel z\left(\tau \right)-z_1\parallel }_1<ϵ.$$

Definition 2

(For controllability operators, see [17,32]). For the system (7) we introduce

1. 

The controllability operator

$$\mathcal{G}:L^2(0,\tau ;\mathcal{U})\to {\mathcal{Z}}_1,\mathcal{G}u={\int }_0^{\tau }\mathcal{T}(\tau -s){\mathcal{B}}_{\omega }u\left(s\right)ds,$$

whose adjoint is given by

$$\left({\mathcal{G}}^{*}z\right)\left(s\right)={\mathcal{B}}_{\omega }^{*}{\mathcal{T}}^{*}(\tau -s)z,s\in [0,\tau ],z\in {\mathcal{Z}}_1.$$

2. 

The Gramian operator ${\mathcal{L}}_{\mathcal{G}}:{\mathcal{Z}}_1\to {\mathcal{Z}}_1$, defined by

$${\mathcal{L}}_{\mathcal{G}}z={\int }_0^{\tau }\mathcal{T}(\tau -s){\mathcal{B}}_{\omega }{\mathcal{B}}_{\omega }^{*}{\mathcal{T}}^{*}(\tau -s)zds.$$

The next result collects equivalent characterizations of approximate controllability.

Lemma 1

(see [29,33]). The linear system (7) is approximately controllable on $[0,\tau ]$ if and only if one of the following equivalent statements holds:

1. 

$\overline{Range\left(\mathcal{G}\right)}={\mathcal{Z}}_1$.

2. 

$Ker\left({\mathcal{G}}^{*}\right)=\left\{0\right\}$.

3. 

$\langle {\mathcal{L}}_{\mathcal{G}}z,z\rangle >0$ for every $z\in {\mathcal{Z}}_1\setminus \left\{0\right\}$.

4. 

${\mathcal{B}}_{\omega }^{*}{\mathcal{T}}^{*}\left(t\right)z=0$ for all $t\in [0,\tau ]$ implies $z=0$.

5. 

${lim}_{\kappa \to 0}\kappa {(\kappa I+{\mathcal{L}}_{\mathcal{G}})}^{-1}=0$.

6. 

${sup}_{\kappa >0}\parallel \kappa {(\kappa I+{\mathcal{L}}_{\mathcal{G}})}^{-1}\parallel \le 1$.

Remark 1.

Since $\mathcal{T}\left(t\right)$ is compact for $t>0$, both the controllability map and its Gramian are compact operators. The latter encodes the reachability properties of the system.

As a consequence of Lemma 1, if (7) is approximately controllable, then for every $z\in {\mathcal{Z}}_1$, one has the representation

$$\mathcal{G}{u}_{\kappa }=z-\kappa {(\kappa I+{\mathcal{L}}_{\mathcal{G}})}^{-1}z,u_{\kappa }={\mathcal{G}}^{*}{(\kappa I+{\mathcal{L}}_{\mathcal{G}})}^{-1}z,\kappa \in (0,1],$$

and therefore

$$\underset{\kappa \to 0}{lim}\mathcal{G}{u}_{\kappa }=z,$$

with the approximation error

$${\mathcal{E}}_{\kappa }z=\kappa {(\kappa I+{\mathcal{L}}_{\mathcal{G}})}^{-1}z.$$

Remark 2.

For each $0<\kappa \le 1$, the operators

$${\Theta }_{\kappa }:{\mathcal{Z}}_1\to L^2([0,\tau ],\mathcal{U}),{\Theta }_{\kappa }z={\mathcal{G}}^{*}{(\kappa I+{\mathcal{L}}_{\mathcal{G}})}^{-1}z,$$

act as approximate right inverses of $\mathcal{G}$, in the sense that $\mathcal{G}{\Theta }_{\kappa }\to I$ strongly as $\kappa \to 0^{+}$.

Theorem 3

(see [18]). Assume that

$$0<{\varrho }_1<{\varrho }_2<{\varrho }_3\mathit{and}{\displaystyle \frac{{\mu }_{j+1}}{{\mu }_j}}>{\displaystyle \frac{{\varrho }_3}{{\varrho }_1}},j=1,2,\dots .$$

Then, for every open nonempty $\omega \subset \Omega$ and $\tau >0$, the system (7) is approximately controllable on $[0,\tau ]$.

Furthermore, a sequence of admissible controls is given by

$$u_{\kappa }\left(t\right)={\mathcal{B}}_{\omega }^{*}\mathcal{T}(\tau -t){(\kappa I+\mathcal{G}{\mathcal{G}}^{*})}^{-1}(z_1-\mathcal{T}\left(\tau \right)z_0),\kappa \in (0,1],$$

with associated error

$${\mathcal{E}}_{\kappa }=\kappa {(\kappa I+\mathcal{G}{\mathcal{G}}^{*})}^{-1}(z_1-\mathcal{T}\left(\tau \right)z_0).$$

For the corresponding trajectories we obtain

$$\underset{\kappa \to 0^{+}}{lim}z\left(\tau \right)=z_1.$$

Lemma 2

(see [32]). Let $\mathcal{W},\mathcal{Z}$ be normed spaces and $\mathcal{G}:\mathcal{W}\to \mathcal{Z}$ a bounded operator. If $S\subset \mathcal{W}$ is dense, then

$$\overline{Range\left(\mathcal{G}\right)}=\mathcal{Z}⟺\overline{Range\left(\mathcal{G}{|}_S\right)}=\mathcal{Z}.$$

Remark 3.

This lemma shows that controllability can be checked using controls belonging to a dense subspace of $L^2([0,\tau ],\mathcal{U})$, such as $C\left(\right[0,\tau ],\mathcal{U})$ or even $C^{\infty }([0,\tau ],\mathcal{U})$. The same applies to the operators $\mathcal{G}$, ${\mathcal{L}}_{\mathcal{G}}$ and ${\Theta }_{\kappa }$, which are well defined on such spaces.

4. Controllability of the Perturbed System

In this section, we establish the principal contribution of this work: the interior approximate controllability of the semilinear thermoelastic plate model (1). Equivalently, this reduces to proving the approximate controllability of the abstract formulation (3).

Observe first that, for any initial condition $z_0\in {\mathcal{Z}}_1$ and control function $u\in C\left(\right[0,\tau ],\mathcal{U})$, the system (3) admits a unique mild solution of the form

$${\displaystyle z\left(t\right)=\mathcal{T}\left(t\right)z_0+{\int }_0^t\mathcal{T}(t-s){\mathcal{B}}_{\omega }u\left(s\right)ds+{\int }_0^t\mathcal{T}(t-s)\mathcal{F}(s,z\left(s\right),u\left(s\right))ds,t\in [0,\tau ].}$$

(9)

To proceed, we require a fixed-point theorem that will be crucial in our argument. Specifically, we recall Rothe’s theorem:

Theorem 4

(cf. [34,35]). Let $\mathfrak{E}$ be a Banach space and let $\mathfrak{B}\subset \mathfrak{E}$ be a closed convex set such that $0\in int\left(\mathfrak{B}\right)$. Suppose $\Phi :\mathfrak{B}\to \mathfrak{E}$ is continuous, with $\Phi \left(\mathfrak{B}\right)$ relatively compact in $\mathfrak{E}$, and $\Phi (\partial \mathfrak{B})\subset \mathfrak{B}$. Then there exists $x^{*}\in \mathfrak{B}$ such that $\Phi \left(x^{*}\right)=x^{*}$.

Theorem 5.

Assume that the spectral conditions

$$0<{\varrho }_1<{\varrho }_2<{\varrho }_3\mathit{and}{\displaystyle \frac{{\mu }_{j+1}}{{\mu }_j}}>{\displaystyle \frac{{\varrho }_3}{{\varrho }_1}},\mathit{for}\mathit{all}j=1,2,\dots ,$$

(10)

are satisfied. Then, for every nonempty open subset $\omega \subset \Omega$ and any $\tau >0$, the nonlinear system (1) is approximately controllable on the time interval $[0,\tau ]$.

Furthermore, one can construct a sequence of controls steering system (1) from the initial condition $z_0$ to an ϵ-neighborhood of the target state $z_1$ at time τ, given by

$$u_{\kappa }\left(t\right)={\mathcal{B}}_{\omega }^{*}\mathcal{T}(\tau -t){(\kappa I+{\mathcal{L}}_{\mathcal{G}})}^{-1}\mathcal{K}(z_{\kappa },u_{\kappa }),t\in [0,\tau ],$$

where the associated trajectory $z_{\kappa }$ satisfies

$$z_{\kappa }\left(t\right)=\mathcal{T}\left(t\right)z_0+{\int }_0^t\mathcal{T}(t-s){\mathcal{B}}_{\omega }{u}_{\kappa }\left(s\right)ds+{\int }_0^t\mathcal{T}(t-s)\mathcal{F}(s,z_{\kappa }\left(s\right),u_{\kappa }\left(s\right))ds.$$

Here, the operator

$$\mathcal{K}:C([0,\tau ],{\mathcal{Z}}_1)\times C([0,\tau ],\mathcal{U})\to {\mathcal{Z}}_1$$

is defined as

$$\mathcal{K}(z,u)=z_1-\mathcal{T}\left(\tau \right)z_0-{\int }_0^{\tau }\mathcal{T}(\tau -s)\mathcal{F}(s,z\left(s\right),u\left(s\right))ds.$$

The corresponding approximation error is

$${\mathcal{E}}_{\kappa }z=\kappa {(\kappa I+{\mathcal{L}}_{\mathcal{G}})}^{-1}\mathcal{K}(z_{\kappa },u_{\kappa }).$$

Proof. 

We divide the proof into several steps. First, define the auxiliary operator

$${\mathcal{H}}^{\kappa }:C([0,\tau ],{\mathcal{Z}}_1)\times C([0,\tau ],\mathcal{U})⟶C([0,\tau ],{\mathcal{Z}}_1)\times C([0,\tau ],\mathcal{U}),$$

by setting

$$(y,w)={\mathcal{H}}^{\kappa }(z,u)=\left({\mathcal{H}}_1^{\kappa }(z,u),{\mathcal{H}}_2^{\kappa }(z,u)\right),$$

where ${\mathcal{H}}_1^{\kappa }$ and ${\mathcal{H}}_2^{\kappa }$ are given as follows:

• The component

  $${\mathcal{H}}_1^{\kappa }:C([0,\tau ],{\mathcal{Z}}_1)\times C([0,\tau ],\mathcal{U})\to C([0,\tau ],{\mathcal{Z}}_1)$$

  is defined by

  $$\begin{array}{cc}\hfill y\left(t\right)={\mathcal{H}}_1^{\kappa }(z,u)\left(t\right)=\mathcal{T}\left(t\right)z_0+& {\int }_0^t\mathcal{T}(t-s){\mathcal{B}}_{\omega }\left({\Theta }_{\kappa }\mathcal{K}(z,u)\right)\left(s\right)ds\hfill \\ & +{\int }_0^t\mathcal{T}(t-s)\mathcal{F}(s,z\left(s\right),u\left(s\right))ds.\hfill \end{array}$$
• The component

  $${\mathcal{H}}_2^{\kappa }:C([0,\tau ],{\mathcal{Z}}_1)\times C([0,\tau ],\mathcal{U})\to C([0,\tau ],\mathcal{U})$$

  is given by

  $$w\left(t\right)={\mathcal{H}}_2^{\kappa }(z,u)\left(t\right)=\left({\Theta }_{\kappa }\mathcal{K}(z,u)\right)\left(t\right)={\mathcal{B}}_{\omega }^{*}{\mathcal{T}}^{*}(\tau -t){(\kappa I+{\mathcal{L}}_{\mathcal{G}})}^{-1}\mathcal{K}(z,u),t\in [0,\tau ].$$

Step 1: Continuity of the operator ${\mathcal{H}}^{\kappa }$.

To establish the continuity of ${\mathcal{H}}^{\kappa }$, it is sufficient to verify that each of its components, namely ${\mathcal{H}}_1^{\kappa }$ and ${\mathcal{H}}_2^{\kappa }$, defines a continuous mapping.

Consider two pairs $(z,u),(\tilde{z},\tilde{u})\in C([0,\tau ],{\mathcal{Z}}_1)\times C([0,\tau ],\mathcal{U})$. For the auxiliary operator $\mathcal{K}$, we can estimate the difference as

$${∥\mathcal{K}(z,u)\left(t\right)-\mathcal{K}(\tilde{z},\tilde{u})\left(t\right)∥}_1\le {\int }_0^{\tau }M{e}^{-\zeta (\tau -s)}{∥\mathcal{F}(s,z\left(s\right),u\left(s\right))-\mathcal{F}(s,\tilde{z}\left(s\right),\tilde{u}\left(s\right))∥}_{1}ds.$$

Consequently, the deviation between the first components satisfies

$$\begin{array}{cc}\hfill {∥{\mathcal{H}}_1^{\kappa }(z,u)\left(t\right)-{\mathcal{H}}_1^{\kappa }(\tilde{z},\tilde{u})\left(t\right)∥}_1\le & {\int }_0^{t}M{e}^{-\zeta (t-s)}∥{\mathcal{B}}_{\omega }∥∥{(\kappa I+{\mathcal{L}}_{\mathcal{G}})}^{-1}∥{∥\mathcal{K}(z,u)-\mathcal{K}(\tilde{z},\tilde{u})∥}_{1}ds\hfill \\ & +{\int }_0^{t}M{e}^{-\zeta (t-s)}{∥\mathcal{F}(s,z\left(s\right),u\left(s\right))-\mathcal{F}(s,\tilde{z}\left(s\right),\tilde{u}\left(s\right))∥}_{1}ds.\hfill \end{array}$$

From this we deduce the uniform estimate

$${∥{\mathcal{H}}_1^{\kappa }(z,u)-{\mathcal{H}}_1^{\kappa }(\tilde{z},\tilde{u})∥}_{C([0,\tau ],{\mathcal{Z}}_1)}\le {\Lambda }_1\underset{t\in [0,\tau ]}{sup}{∥\mathcal{F}(t,z\left(t\right),u\left(t\right))-\mathcal{F}(t,\tilde{z}\left(t\right),\tilde{u}\left(t\right))∥}_1,$$

where ${\Lambda }_1:=M\tau \left(M\tau ∥{\Theta }_{\kappa }∥+1\right)$.

Thus, the continuity of ${\mathcal{H}}_1^{\kappa }$ is an immediate consequence of the continuity of the nonlinear term $\mathcal{F}(t,z,u)$.

For the second component, ${\mathcal{H}}_2^{\kappa }$, continuity follows from the fact that it is the composition of the operators $\mathcal{K}$ and ${\Theta }_{\kappa }$, both of which are continuous with respect to $(z,u)$.

Therefore, we conclude that the full operator ${\mathcal{H}}^{\kappa }$ is continuous.

Step 2: Compactness of the operator ${\mathcal{H}}^{\kappa }$.

Consider a bounded subset $\mathcal{D}\subset C([0,\tau ],{\mathcal{Z}}_1)\times C([0,\tau ],\mathcal{U})$. For each $(z,u)\in \mathcal{D}$, there exist constants ${\Lambda }_2,{\Lambda }_3,{\Lambda }_4>0$ such that

$${∥\mathcal{F}(·,z,u)∥}_1\le {\Lambda }_2,{∥{\Theta }_{\kappa }\mathcal{K}(z,u)∥}_1\le {\Lambda }_3,{∥\mathcal{K}(z,u)∥}_1\le {\Lambda }_4.$$

Therefore, the image set ${\mathcal{H}}^{\kappa }\left(\mathcal{D}\right)$ is uniformly bounded in the product space $C([0,\tau ],{\mathcal{Z}}_1)\times C([0,\tau ],\mathcal{U})$.

To establish compactness, it remains to verify equicontinuity. Take $0<{\nu }_1<{\nu }_2<\tau$, and compute

$$\begin{array}{cc}\hfill \left|\right||{\mathcal{H}}^{\kappa }(z,u)\left({\nu }_2\right)-{\mathcal{H}}^{\kappa }(z,u)\left({\nu }_1\right)|\left|\right|& ={∥{\mathcal{H}}_1^{\kappa }(z,u)\left({\nu }_2\right)-{\mathcal{H}}_1^{\kappa }(z,u)\left({\nu }_1\right)∥}_1\hfill \\ & +∥{\mathcal{H}}_2^{\kappa }(z,u)\left({\nu }_2\right)-{\mathcal{H}}_2^{\kappa }(z,u)\left({\nu }_1\right)∥.\hfill \end{array}$$

For the first component we estimate:

$$\begin{array}{cc}\hfill {∥{\mathcal{H}}_1^{\kappa }(z,u)\left({\nu }_2\right)-{\mathcal{H}}_1^{\kappa }(z,u)\left({\nu }_1\right)∥}_1& \le ∥\mathcal{T}\left({\nu }_2\right)-\mathcal{T}\left({\nu }_1\right)∥{∥z_0∥}_1\hfill \\ & +{\int }_0^{{\nu }_1}∥\mathcal{T}({\nu }_2-s)-\mathcal{T}({\nu }_1-s)∥{∥{\Theta }_{\kappa }\mathcal{K}(z,u)∥}_{1}ds\hfill \\ & +{\int }_{{\nu }_1}^{{\nu }_2}∥\mathcal{T}({\nu }_2-s)∥{∥{\Theta }_{\kappa }\mathcal{K}(z,u)∥}_{1}ds\hfill \\ & +{\int }_0^{{\nu }_1}∥\mathcal{T}({\nu }_2-s)-\mathcal{T}({\nu }_1-s)∥{∥\mathcal{F}(s,z(s),u(s\left)\right)∥}_{1}ds\hfill \\ & +{\int }_{{\nu }_1}^{{\nu }_2}∥\mathcal{T}({\nu }_2-s)∥{∥\mathcal{F}(s,z(s),u(s\left)\right)∥}_{1}ds.\hfill \end{array}$$

For the second component, one obtains

$$∥{\mathcal{H}}_2^{\kappa }(z,u)\left({\nu }_2\right)-{\mathcal{H}}_2^{\kappa }(z,u)\left({\nu }_1\right)∥\le ∥{\mathcal{T}}^{*}(\tau -{\nu }_2)-{\mathcal{T}}^{*}(\tau -{\nu }_1)∥{∥{\Theta }_{\kappa }\mathcal{K}(z,u)∥}_1.$$

Since Theorem 2 ensures that the semigroup $\mathcal{T}\left(t\right)$ is compact for $t>0$, and that $t\mapsto \mathcal{T}\left(t\right)$ is uniformly continuous on bounded intervals of $(0,\infty )$, we deduce that

$$\left|\right||{\mathcal{H}}^{\kappa }(z,u)\left({\nu }_2\right)-{\mathcal{H}}^{\kappa }(z,u)\left({\nu }_1\right)|\left|\right|$$

tends to zero uniformly as $|{\nu }_2-{\nu }_1|\to 0$, and uniformly for all $(z,u)\in \mathcal{D}$.

This shows that ${\mathcal{H}}^{\kappa }\left(\mathcal{D}\right)$ is an equicontinuous and uniformly bounded family in $C([0,\tau ],{\mathcal{Z}}_1)\times C([0,\tau ],\mathcal{U})$. By the Arzelà–Ascoli theorem, its closure $\overline{{\mathcal{H}}^{\kappa }\left(\mathcal{D}\right)}$ is compact in the same space.

Hence, the operator ${\mathcal{H}}^{\kappa }$ is compact.

Step 3: Dissipativity-type condition at infinity.

We now establish that the operator ${\mathcal{H}}^{\kappa }$ satisfies a dissipativity-type condition at infinity:

$${\displaystyle \underset{\begin{array}{c}∥z∥\to \infty \\ ∥u∥\to \infty \end{array}}{lim}\frac{\left|\right||{\mathcal{H}}^{\kappa }(z,u)\left|\right||}{\left|\right|\left|\right(z,u\left)\right|\left|\right|}=0,}$$

where $\left|\right|\left|(z,u)\right|\left|\right|=∥z∥+∥u∥$ is the norm of the space $C([0,\tau ],{\mathcal{Z}}_1)\times C([0,\tau ],\mathcal{U})$.

Note that

$$\begin{array}{cc}\hfill {∥\mathcal{K}(z,u)∥}_1\le & {∥z_1∥}_1+∥\mathcal{T}\left(t\right)∥{∥z_0∥}_1+{\int }_0^{\tau }∥\mathcal{T}(\tau -s)∥{∥\mathcal{F}(s,z(s),u(s\left)\right)∥}_{1}ds\hfill \\ \hfill \le & {∥z_1∥}_1+M{∥z_0∥}_1+M{\int }_0^{\tau }\left[a{∥z\left(s\right)∥}_1^{\xi }+b{∥u\left(s\right)∥}^{\sigma }+c\right]ds\hfill \\ \hfill \le & M_1+M_2\left[a{∥z∥}^{\xi }+b{∥u∥}^{\sigma }+c\right],\hfill \end{array}$$

where $M_1={∥z_1∥}_1+M{∥z_0∥}_1$ and $M_2=M\tau$. Then,

$$∥{\mathcal{H}}_1^{\kappa }(z,u)∥\le M_1+M_3\left[a{∥z∥}^{\xi }+b{∥u∥}^{\sigma }+c\right],$$

with $M_3=M_2(∥{\Theta }_{\kappa }∥M_2+1)$, and

$$∥{\mathcal{H}}_2^{\kappa }(z,u)∥\le ∥{\Theta }_{\kappa }∥M_1+∥{\Theta }_{\kappa }∥M_2\left[a{∥z∥}^{\xi }+b{∥u∥}^{\sigma }+c\right].$$

Hence,

$${\displaystyle \frac{∥{\mathcal{H}}_1^{\kappa }(z,u)∥}{\left|\right|\left|\right(z,u\left)\right|\left|\right|}}\le {\displaystyle \frac{M_3}{\left|\right|\left|\right(z,u\left)\right|\left|\right|}}+M_4\left[a{∥z∥}^{1-\xi }+b{∥u∥}^{1-\sigma }+{\displaystyle \frac{c}{\left|\right|\left|\right(z,u\left)\right|\left|\right|}}\right]$$

and

$${\displaystyle \frac{∥{\mathcal{H}}_2^{\kappa }(z,u)∥}{\left|\right|\left|\right(z,u\left)\right|\left|\right|}}\le {\displaystyle \frac{∥{\Theta }_{\kappa }∥M_1}{\left|\right|\left|\right(z,u\left)\right|\left|\right|}}+∥{\Theta }_{\kappa }∥M_2\left[a{∥z∥}^{1-\xi }+b{∥u∥}^{1-\sigma }+{\displaystyle \frac{c}{\left|\right|\left|\right(z,u\left)\right|\left|\right|}}\right].$$

So,

$${\displaystyle \underset{\begin{array}{c}∥z∥\to \infty \\ ∥u∥\to \infty \end{array}}{lim}\frac{\left|\right||{\mathcal{H}}^{\kappa }(z,u)\left|\right||}{\left|\right|\left|\right(z,u\left)\right|\left|\right|}=0.}$$

Now, for a fixed $0<\vartheta <1$, there exists $R>0$ big enough such that

$$\left|\right||{\mathcal{H}}^{\kappa }(z,u)\left|\right||\le \vartheta \left|\right|\left|(z,u)\right|\left|\right|with\left|\right|\left|(z,u)\right|\left|\right|=R.$$

If we let $B_R\left(0\right)$ denote the ball centered at the origin with radius R, it follows that

$${\mathcal{H}}^{\kappa }(\partial B_R\left(0\right))\subset B_R\left(0\right).$$

Because ${\mathcal{H}}^{\kappa }$ is a compact operator and maps the boundary $\partial B_R\left(0\right)$ strictly inside the ball $B_R\left(0\right)$, Rothe’s fixed-point theorem 4 can be applied to guarantee the existence of an element

$$(z_{\kappa },u_{\kappa })\in B_R\left(0\right)\subset C([0,\tau ],{\mathcal{Z}}_1)\times C([0,\tau ],\mathcal{U})$$

such that

$$(z_{\kappa },u_{\kappa })={\mathcal{H}}^{\kappa }(z_{\kappa },u_{\kappa }).$$

The sequence ${\left\{(z_{\kappa },u_{\kappa })\right\}}_{\kappa \in (0,1]}$ is bounded. To prove this, we proceed by contradiction. Supposing the sequence is unbounded, it follows that there exists a subsequence ${\left\{(z_{{\kappa }_n},u_{{\kappa }_n})\right\}}_{{\kappa }_n\in (0,1]}\subset {\left\{(z_{\kappa },u_{\kappa })\right\}}_{\kappa \in (0,1]}$ such that

$${\displaystyle \underset{n\to \infty }{lim}\left|\right||(z_{{\kappa }_n},u_{{\kappa }_n})\left|\right||=\infty .}$$

On the other hand, for $\kappa \in (0,1]$ we get

$${\displaystyle \underset{n\to \infty }{lim}\frac{\left|\right||{\mathcal{H}}^{\kappa }(z_{{\kappa }_n},u_{{\kappa }_n})\left|\right||}{\left|\right|\left|\right(z_{{\kappa }_n},u_{{\kappa }_n}\left)\right|\left|\right|}=0.}$$

Particularly, we have the following situation:

$${\displaystyle \begin{array}{cccc}{\displaystyle \frac{\left|\right||{\mathcal{H}}^{{\kappa }_1}(z_{{\kappa }_1},u_{{\kappa }_1})\left|\right||}{\left|\right|\left|\right(z_{{\kappa }_1},u_{{\kappa }_1}\left)\right|\left|\right|}}& {\displaystyle \frac{\left|\right||{\mathcal{H}}^{{\kappa }_1}(z_{{\kappa }_2},u_{{\kappa }_2})\left|\right||}{\left|\right|\left|\right(z_{{\kappa }_2},u_{{\kappa }_2}\left)\right|\left|\right|}}& \dots & {\displaystyle \frac{\left|\right||{\mathcal{H}}^{{\kappa }_1}(z_{{\kappa }_n},u_{{\kappa }_n})\left|\right||}{\left|\right|\left|\right(z_{{\kappa }_n},u_{{\kappa }_n}\left)\right|\left|\right|}\to 0.}\\ {\displaystyle \frac{\left|\right||{\mathcal{H}}^{{\kappa }_2}(z_{{\kappa }_1},u_{{\kappa }_1})\left|\right||}{\left|\right|\left|\right(z_{{\kappa }_1},u_{{\kappa }_1}\left)\right|\left|\right|}}& {\displaystyle \frac{\left|\right||{\mathcal{H}}^{{\kappa }_2}(z_{{\kappa }_2},u_{{\kappa }_2})\left|\right||}{\left|\right|\left|\right(z_{{\kappa }_2},u_{{\kappa }_2}\left)\right|\left|\right|}}& \dots & {\displaystyle \frac{\left|\right||{\mathcal{H}}^{{\kappa }_2}(z_{{\kappa }_n},u_{{\kappa }_n})\left|\right||}{\left|\right|\left|\right(z_{{\kappa }_n},u_{{\kappa }_n}\left)\right|\left|\right|}\to 0.}\\ ⋮& ⋮& ⋮& ⋮\\ {\displaystyle \frac{\left|\right||{\mathcal{H}}^{{\kappa }_k}(z_{{\kappa }_1},u_{{\kappa }_1})\left|\right||}{\left|\right|\left|\right(z_{{\kappa }_1},u_{{\kappa }_1}\left)\right|\left|\right|}}& {\displaystyle \frac{\left|\right||{\mathcal{H}}^{{\kappa }_k}(z_{{\kappa }_2},u_{{\kappa }_2})\left|\right||}{\left|\right|\left|\right(z_{{\kappa }_2},u_{{\kappa }_2}\left)\right|\left|\right|}}& \dots & {\displaystyle \frac{\left|\right||{\mathcal{H}}^{{\kappa }_k}(z_{{\kappa }_n},u_{{\kappa }_n})\left|\right||}{\left|\right|\left|\right(z_{{\kappa }_n},u_{{\kappa }_n}\left)\right|\left|\right|}\to 0.}\end{array}}$$

Now, applying Cantor’s diagonalization process, we see that

$${\displaystyle \underset{n\to \infty }{lim}\frac{\left|\right||{\mathcal{H}}^{{\kappa }_n}(z_{{\kappa }_n},u_{{\kappa }_n})\left|\right||}{\left|\right|\left|\right(z_{{\kappa }_n},u_{{\kappa }_n}\left)\right|\left|\right|}=0.}$$

However,

$${\displaystyle \frac{\left|\right||{\mathcal{H}}^{{\kappa }_n}(z_{{\kappa }_n},u_{{\kappa }_n})\left|\right||}{\left|\right|\left|\right(z_{{\kappa }_n},u_{{\kappa }_n}\left)\right|\left|\right|}}=1,$$

which provides the desired contradiction. Therefore, the sequence ${\left\{(z_{\kappa },u_{\kappa })\right\}}_{\kappa \in (0,1]}$ is bounded; this guarantees the existence of some $\iota >0$ such that

$$\left|\right|\left|\right(z_{\kappa },u_{\kappa }\left)\right|\left|\right|\le \iota .$$

We can assume, without loss of generality, that the sequence $\mathcal{K}(z_{\kappa },u_{\kappa })$ converges to some element $y\in {\mathcal{Z}}_1$. Now, if

$$u_{\kappa }={\Theta }_{\kappa }\mathcal{K}(z_{\kappa },u_{\kappa })={\mathcal{G}}^{*}{(\kappa I+{\mathcal{L}}_{\mathcal{G}})}^{-1}\mathcal{K}(z_{\kappa },u_{\kappa }),$$

then

$$\begin{array}{cc}\hfill \mathcal{G}{u}_{\kappa }=& {\mathcal{L}}_{\mathcal{G}}{(\kappa I+{\mathcal{L}}_{\mathcal{G}})}^{-1}\mathcal{K}(z_{\kappa },u_{\kappa })\hfill \\ \hfill =& (\kappa I+{\mathcal{L}}_{\mathcal{G}}-\kappa I){(\kappa I+{\mathcal{L}}_{\mathcal{G}})}^{-1}\mathcal{K}(z_{\kappa },u_{\kappa })\hfill \\ \hfill =& \mathcal{K}(z_{\kappa },u_{\kappa })-\kappa {(\kappa I+{\mathcal{L}}_{\mathcal{G}})}^{-1}\mathcal{K}(z_{\kappa },u_{\kappa }).\hfill \end{array}$$

Hence

$$\mathcal{G}{u}_{\kappa }-\mathcal{K}(z_{\kappa },u_{\kappa })=-\kappa {(\kappa I+{\mathcal{L}}_{\mathcal{G}})}^{-1}\mathcal{K}(z_{\kappa },u_{\kappa }).$$

To conclude the proof of this theorem, it is enough to prove that

$${\displaystyle \underset{\kappa \to 0}{lim}(-\kappa {(\kappa I+{\mathcal{L}}_{\mathcal{G}})}^{-1})\mathcal{K}(z_{\kappa },u_{\kappa })=0.}$$

According to Theorem 1 (e), we get

$${\displaystyle \underset{\kappa \to 0^{+}}{lim}\kappa {(\kappa I+{\mathcal{L}}_{\mathcal{G}})}^{-1}\mathcal{K}(z_{\kappa },u_{\kappa })=\underset{\kappa \to 0^{+}}{lim}\kappa {(\kappa I+{\mathcal{L}}_{\mathcal{G}})}^{-1}(\mathcal{K}(z_{\kappa },u_{\kappa })-y).}$$

By using the Theorem 1 (f), we obtain

$$∥\kappa {(\kappa I+{\mathcal{L}}_{\mathcal{G}})}^{-1}(\mathcal{K}(z_{\kappa },u_{\kappa })-y)∥\le ∥\mathcal{K}(z_{\kappa },u_{\kappa })-y∥,$$

and since $\mathcal{K}(z_{\kappa },u_{\kappa })$ converges to y, we see that

$${\displaystyle \underset{\kappa \to 0^{+}}{lim}\kappa {(\kappa I+{\mathcal{L}}_{\mathcal{G}})}^{-1}(\mathcal{K}(z_{\kappa },u_{\kappa })-y)=0.}$$

So,

$$\underset{\kappa \to 0^{+}}{lim}(\mathcal{G}{u}_{\kappa }-\mathcal{K}(z_{\kappa },u_{\kappa }))=\underset{\kappa \to 0^{+}}{lim}(-\kappa {(\kappa I+{\mathcal{L}}_{\mathcal{G}})}^{-1})\mathcal{K}(z_{\kappa },u_{\kappa })=0.$$

Then,

$$\underset{\kappa \to 0^{+}}{lim}\left\{{\displaystyle \mathcal{T}\left(\tau \right)z_0+{\int }_0^{\tau }\mathcal{T}(\tau -s){\mathcal{B}}_{\omega }{u}_{\kappa }\left(s\right)ds+{\int }_0^{\tau }\mathcal{T}(\tau -s)\mathcal{F}(s,z_{\kappa }\left(s\right),u_{\kappa }\left(s\right))ds}\right\}=z_1.$$

This conclude the proof. □

5. Examples: Sublinear Nonlinearities Satisfying the Assumptions

We present two families of nonlinearities $f_1,f_2:E\to E$ (with $E={\mathbb{R}}^n$ or a Banach space with norm $\parallel ·\parallel$) that fulfill the sublinear growth assumptions used in the fixed-point and optimality analysis.

• Example A (saturated sublinear powers).

Fix $0\le {\alpha }_1,{\alpha }_2<1$ and constants $a_i,b_i\ge 0$. Define

$$f_i\left(x\right)=a_i\mathrm{sat}\left(x\right)+b_i{\parallel x\parallel }^{{\alpha }_i}\mathrm{sgn}\left(x\right),i=1,2,$$

where $\mathrm{sat}\left(x\right):={\displaystyle \frac{x}{1+\parallel x\parallel }}$ and $\mathrm{sgn}\left(x\right):={\displaystyle \frac{x}{\parallel x\parallel }}$ for $x\ne 0$ (and 0 at $x=0$). Then, $f_i$ is continuous and satisfies

$$\parallel f_i\left(x\right)\parallel \le a_i+b_i{\parallel x\parallel }^{{\alpha }_i},0\le {\alpha }_i<1,$$

which is precisely the sublinear growth condition required in our Rothe-type argument.

• Example B (component-wise sublinear mixing).

Let $x=(x_1,\dots ,x_n)$, let $0\le \beta <1$, and let $c_{ij}\ge 0$. Define

$$f_1\left(x\right)=a_1\mathrm{sat}\left(x\right)+\sum _{j=1}^n{c}_{1j}{\left|x_j\right|}^{\beta }\mathrm{sgn}\left(x_j\right)e_j,f_2\left(x\right)=a_2\mathrm{sat}\left(x\right)+\sum _{j=1}^n{c}_{2j}{\left|x_j\right|}^{\beta }\mathrm{sgn}\left(x_j\right)e_j,$$

where $\left\{e_j\right\}$ is the canonical basis of ${\mathbb{R}}^n$. Then, $f_i$ is continuous and

$$\parallel f_i\left(x\right)\parallel \le a_i+\left(\sum _{j=1}^n{c}_{ij}\right){\parallel x\parallel }^{\beta },0\le \beta <1,$$

so $f_i$ satisfies the same global sublinear estimate. In particular, for the mild-solution map K associated with a compact semigroup $S\left(t\right)$, one obtains the bound

$${\parallel Kx\parallel }_{\infty }\le C_0+C_1{\parallel x\parallel }_{\infty }^{\alpha },\alpha :=max\{{\alpha }_1,{\alpha }_2,\beta \}<1,$$

which implies $K(\partial B_R)\subset B_R^{\circ }$ for R sufficiently large, thus verifying Rothe’s theorem.

6. Conclusions

In this work, we have established the interior approximate controllability of a semilinear thermoelastic plate system under Dirichlet boundary conditions. The analysis relied essentially on the compactness of the analytic semigroup generated by the linear operator and on the use of Rothe’s fixed-point theorem as a key tool.

Our approach began with a characterization of the controllability of the associated linear dynamics, expressed through the properties of the controllability operator and its Gramian. Subsequently, by deriving suitable estimates for the nonlinear contributions and constructing an operator that is both compact and continuous in an appropriate Banach space, we showed the existence of approximate controls capable of steering the trajectories arbitrarily close to any prescribed target.

These findings extend the known results for purely linear thermoelastic models and provide a framework for addressing broader classes of coupled PDEs involving nonlinear interactions and localized controls. Possible directions for future investigation include the analysis of null controllability and stabilization in the presence of additional damping mechanisms, the extension of the theory to thermoelastic systems with memory effects or stochastic perturbations, and the development of numerical experiments to illustrate the theoretical contributions and to guide the design of practical control strategies.