On one boundary control problem for a pseudo-parabolic equation in a two-dimensional domain

1.   Introduction. Statement of the Problem

In this paper, we consider the following pseudo-parabolic equation in the rectangular domain $\Omega = (0, l_{1})\times(0, l_{2})$:

with Dirichlet boundary conditions

where $\nu(t)$ is the control function and $\varphi(y)$ is a given function, and the initial value condition

It is called that the control function $\nu(t)\in W_{2}^{1}(\mathbb{R_{+}})$ is admissible if it fulfills the conditions $\nu(0) = 0$ and $|\nu(t)| \leq 1$ on the half-line $t\geq 0$.

Suppose that the functions $p(x)\in C^{2}(\Omega)$ and $q(y)\in C^{1}(\Omega)$ satisfy the conditions

It is required that the given functions $\varphi (y)\in W_{2}^{2}(\Omega)$ and $\psi (y)\in L^{2}(\Omega)$ satisfy the following conditions:

where $\varphi_{n}$ and $\psi_{n}$ are the Fourier coefficients of functions $\varphi(y)$ and $\psi(y)$, respectively.

Pseudo-parabolic equations are characterized by the occurrence of a time derivative appearing in the highest order term, which describes various important physical processes. We know that models of the theory of incompressible fluids with memory can be described by equations of pseudo-parabolic type ^[1].

Control Problem. Suppose that the function $\phi(t)$ is given and let $\psi\in L^{2}(\Omega)$. Then we find the control function $\nu(t)$ from the condition

where $u(x, y, t)$ is a solution of the mixed problem (1.1)-(1.4).

In ^[2], the control problem for the parabolic equation with the Neumann–Robin boundary condition was solved using the Laplace operator, and the optimal time for reaching the given temperature in the bounded domain was found. The boundary control problem for the heat transfer equation with the Robin boundary condition was studied in ^[3], and a mathematical model of the heating process of a cylindrical domain was developed.

In ^[4], the control problem related to the nonhomogeneous parabolic equation with Dirichlet boundary condition in a bounded one-dimensional domain was considered, and the optimal estimate of the minimum time required to reach a given temperature of a thin rod was found. In ^[5], the boundary control problem with the Neumann boundary condition for the heat transfer equation in a one-dimensional domain was studied, and an estimate of the minimum time for heating a thin rod was obtained.

The initial-boundary problem for a class of finite degenerate semilinear parabolic equations with a single potential term was studied in ^[6]. Also, the local existence and uniqueness of the weak solution were determined by applying the Galerkin method and the Banach invariance theorem. The initial-boundary value problems for nonlinear parabolic systems with power-type source terms are considered in ^[7].

Control problems for the infinite-dimensional case were studied by Egorov ^[8], who generalized Pontryagin's maximum principle to a class of equations in Banach space, and the proof of a bang-bang principle was shown in the particular conditions. The control problem for a linear parabolic type equation in a one-dimensional domain with a Robin boundary condition was studied by Fattorini and Russell ^[9]. In ^[10], an estimate of Carleman type for the one-dimensional heat equation was proved.

The stability, uniqueness, and existence of solutions of some classical problems for the pseudo-parabolic equation were studied in ^[11]. In ^[12], the point control problems for pseudo-parabolic and parabolic type equations are considered. In ^[13], some problems related to distributed parameter impulse control problems for systems were studied.

In ^[14], the control problem associated with a pseudo-parabolic type equation in a one-dimensional domain was studied, and the existence of an admissible control was proved using the Laplace transform method. Some boundary control problems for the pseudo-parabolic equation can be seen in ^[15]. An initial-boundary value problem for a pseudo-parabolic equation with singular potential was considered by Lian, et al. ^[16], and global existence and blow-up of solutions were studied. In ^[17], a class of semi-linear pseudo-parabolic equations was considered, and the invariance, global existence, non-existence, and asymptotic behavior of some sets with initial energy were proved by introducing a family of potential wells.

In this work, the boundary control problem associated with the pseudo-parabolic type equation is considered. Our main goal is to prove that there is a control function. The boundary control problem studied in this work is reduced to the Volterra integral equation of t the second type using the separation of variables method. The existence of a solution to this integral equation can also be proved using the Laplace transform method. In Section 2, the continuity of the kernel of the integral equation on the half-line $t\geq 0$ is proved. In Section 3, we prove the existence of an admissible control function and derive the required value for it.

2.   Main integral equation

In this section, we consider the reduction of the control problem to the Volterra integral equation of the second kind. For this we first need the following spectral problem:

where $\lambda$ is a constant to be determined later, and with boundary conditions

and

As we know, the above spectral problem is self-adjoint in $L^{2}(\Omega)$ and there exists a sequence of eigenvalues $\{\lambda_{mn}\}$ so that $0 < \lambda_{11}\leq...\leq\lambda_{mn}\rightarrow \infty$, $m, n\rightarrow \infty$. The corresponding eigenfunction $v_{mn}$ forms a complete orthonormal system $\{v_{mn}\}$ in $L^{2}(\Omega)$, and these eigenfunctions belong to $C(\bar{\Omega})$(see ^[18,19]).

Let the eigenfunction $v_{mn}$ is $v_{mn}(x, y) = \vartheta_{m}(x)\, \omega_{n}(y)$, and the eigenfunctions $\vartheta_{m}(x)$, $\omega_{n}(y)$ are solutions of the following spectral problems

and

where $\mu_{m}$ and $\nu_{n}$ are eigenvalues of spectral problems (2.1) and (2.2), respectively. We denote the eigenvalues $\lambda_{mn}$ by $\lambda_{mn} = \mu_m +\nu_n$, $m, n = 1, 2, \cdots$.

A solution to the initial-boundary problem (1.1) - (1.4) is the function $u(x, y, t)$, which is expressed as follows:

where the function $w(x, y, t)$ with the regularity $w(x, y, t) \in C_{x, y, t}^{2, 2, 1}(\Omega_{T})\cap C(\bar{\Omega}_{T})$ is the solution to the mixed problem

with initial-boundary value conditions

We set

where the coefficients $a_{mn}$, $b_{mn}$, $c_{mn}$ and $\gamma_{mn}$ are as follows:

and

Thus, we obtain (see ^[18])

where $\rho_{mn} = \frac{\lambda_{mn}}{1+\lambda_{mn}} < 1$.

Using (2.3) and (2.9), we obtain the solution of the mixed problem (1.1)–(1.4):

From (2.10) and the condition (1.5), we can write

where $\gamma_{mn}$ is defined by (2.8).

According to the properties of the function $\nu(t)$, we have

Note that

Using (2.11), we can write

We set

and

where $\beta_{mn}$ is defined by (2.4), and $\Lambda_{mn}$ is as follows:

Thus, we have the following Volterra integral equation of the second kind:

Lemma 1. The following estimate is valid:

where $C = \text{const} > 0$ and $\beta_{mn}$ is defined by (2.4).

Proof. Step 1. Using (2.1), (2.5), and the formula for integration by parts, we write

where $b_{mn}$ is defined by (2.6).

Then, by (2.2) and (2.5), we have

where $c_{mn}$ is defined by (2.7).

Then we have the following equality:

where $\mu_{m}+\nu_{n} = \lambda_{mn}.$

Step 2. As we know, the following inequality holds for the eigenfunctions of problem (2.1) (see ^[14])

Step 3. By (2.4), (2.16), and (2.17), we obtain

If the function $p(x) \in C^1(\Omega)$, we can write the following estimate (see ^[20])

Therefore,

where $C_{1} = \text{const} > 0$.

Then, by integrating the equation (2.1) from 0 to $l_{1}$, we obtain

According to (2.19) and (2.20), we have the estimate

Thus, we obtain the required estimate

Proposition 1. Assume that $\varphi, \psi\in L^{2}(\Omega)$. Then, the kernel $B(t)$ of the integral equation (2.15) is continuous on the half-line $t\geq 0$.

Proof. According to Lemma 1 and (2.12), we have the estimate

where $C = \text{const} > 0$.

3.   Main result

In this section, we consider the existence of a solution to the Volterra integral equation of the second kind, that is, the existence of an admissible control function.

For any $M > 0$, we denote $W(M)$ the set of functions $\phi\ \in W_{2}^{1}(-\infty, +\infty)$, which satisfying the following conditions

Now, we present the main theorem for proving the existence of admissible control.

Theorem 1. There exists $M > 0$ such that, for any function $\phi \in W(M)$, the equation (2.15) has a solution $\nu(t)$ meeting the condition $|\nu(t)| \leq 1$.

We rewrite the Volterra integral equation (2.15) as follows:

It is known that we can write the Laplace transform of the function $\nu(t)$ as follows:

where $p = \sigma+i\zeta$, $\sigma > 0$, $\zeta\in\mathbb{R}$.

Then, applying the Laplace transform to the integral equation (2.15), we obtain

where $\alpha$ is defined by (2.13).

Then we can write

and

Lemma 2. The following estimate

is valid, where $\alpha = \text{const} > 0$ is defined by (2.13).

Proof. According to Lemma 1 and (2.13), we can write

It is known that we can write the Laplace transform of the function $B(t)$ as follows:

where function $B(t)$ is defined by (2.12). Then we can write

where

and

We can see that the following inequality holds:

As a result, we obtain

Then, using the inequality (3.3), we can obtain the following assumptions:

and

where $C_{1, \sigma}$ and $C_{2, \sigma}$ are defined as follows:

By (3.4) and (3.6), we obtain the required estimate

where $C_{\sigma} = \text{min}(C_{1, \sigma}, C_{2, \sigma})$ is bounded for all $\sigma > 0$.

Let the Laplace transform of function $\phi(t)$ satisfy the condition

If we proceed to the limit as $\sigma\rightarrow 0$ in the equality (3.2), we have

Also, to prove Theorem 1, we need the following lemma.

Lemma 3. Suppose that the function $\phi(t)$ belongs to $W(M)$. Then for the imaginary part of the Laplace transform of function $\phi(t)$ the following estimate holds

Proof. Using the formula for integration by parts in (3.1), we can write

Then, we have

Further, for $\sigma\rightarrow 0$ we obtain

Besides

Thus, we can write the following inequality:

where $C_{2} = \text{const} > 0$.

By using elementary identities and inequalities, we have

Now we present the proof of Theorem 1.

Proof of Theorem 1. Let us show that $\nu\in W_2^1(\mathbb{R}_+)$. Indeed, using (3.7) and (3.8), we can write

Now, we show that the function $\nu(t)$ satisfies the Lipschitz condition. Actually,

Using (3.7), (3.8), and Lemma 3, we have the following estimate

where $\alpha$ is defined by (2.13) and

Theorem 1 is proved.

Use of AI tools declaration

The author declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

Conflict of interest

The authors declare there is no conflict of interest.