Abstract

In this work, we consider the regional averaged controllability (RAC) problem governed by a class of semilinear hyperbolic systems. We start by giving the definitions of the exact and approximate RAC systems. After that, we state the problem of RAC for semilinear systems. We propose two methods of solution: using a condition of the analytical operator to the nonlinear part of the system to characterize the optimal control via the fixed point theorem and the Hilbert Uniqueness Method (HUM) with an asymptotic condition on the nonlinear part to find the optimal control of the considered problem. Finally, we present a numerical example to show the effectiveness of the main results.

1. Introduction

The study of hyperbolic partial differential equations (PDE) is a historically important subject whose first steps go back to d’Alembert with the equation of waves and to Euler, with equations of the same name describing the evolution of a fluid [1]. An important example of hyperbolic PDEs is provided by the conservation laws of the first order [2], which appear quite naturally in physics, as soon as a balance of energy, mass, quantity of movement, matter… is carried out and that the phenomena of diffusion (thermal or viscosity) are neglected.

Solutions of this kind of problem have undulating characterizations. If a localized perturbation is occurred on the initial input, then points in space far from the support of the perturbation will not instantaneous feel the effects. With respect to a fixed spatiotemporal point, the disturbances have a finite propagation speed and move according to the characteristics of the equation. This property makes it possible to distinguish hyperbolic problems from elliptical or parabolic problems, where the perturbations of the initial conditions will have immediate effects on all points of the domain. Although there are specific requirements that depend on the family of PDEs being investigated, the concept of hyperbolicity is fundamentally qualitative, see [3, 4] for instance.

Hyperbolic systems, it is a part of distributed systems modeling many real-life problems in various areas [5]. Several problems of mechanics are hyperbolic, and therefore the study of hyperbolic problems is of substantial contemporary interest. Furthermore, we primarily use light, sound, and wave phenomena to perceive the outside world through sight and hearing. They are also utilized in metal smelting, medicine, in laser printers and communications technologies, etc. In particular, hyperbolic systems describe optimal control problems with constraints. In the last few years a substantial literature is focused on the study of the control of distributed nonlinear hyperbolic systems and especially bilinear and semilinear hyperbolic systems [6].

The idea of regional controllability is one of controllability’s most significant practical applications, control problems when the objective is not fully characterized as a position have been discussed using this state, but refers only to a subregion of . Exactly, it finds a control which directs the considered system, at the moment , towards a prescribed function defined on a subregion of .

For controllability problems, one considers a control system in the time interval and specifically inquires as to the best way to reach the space of executed instructions (exact controllability) or a dense set in the space of instructions (approximate controllability).

In real-life problems, parameter-dependent system modeling seems to be difficult, since the issues encountered, is that we work with space-temporal systems considered as a nonlinear problem. Others’ difficulty comes from the existence and uniqueness of their solutions. Furthermore, to solve the average control problem associated with a semilinear system a complex formulation of the fixed point method is introduced and applied in infinite dimension.

Averaged control problems for semilinear distributed systems involve the design of control laws for a class of systems described by partial differential equations (PDEs). These systems are characterized by a linear spatial operator and a nonlinear temporal operator. The goal of an averaged control problem is to stabilize the system around a desired equilibrium state or to drive it to a target state while taking into account the effects of averaging.

In this case of an unknown value parameter, it is not possible to control each realization of the system by a single control using an independent control of the parameter. Which motivates this work, is the first time that we consider the averaged time of semilinear distributed systems. Such type of systems is important in theory as in applications and looked as a compromise between linear and nonlinear systems. The average controllability allows us to consider many types of nonautonomous systems. The average controllability introduced by Zuazua [7], purpose is to check the averaged state of a parameterized system instead of the state against the unknown parameter. Moreover, the problem of average controllability has recently been introduced in some papers [8–12].

The paper is organized as follows. In the second section, we begin by stating the problem and giving the definition of the RAC. The third section will be our main result, when we will present two methods to solve the considered problem. First, using a condition of the analytical operator on the nonlinear part of the system to characterize the optimal control via the fixed point theorem. Second, using the HUM with an asymptotic condition on the nonlinear part to find the optimal control. The fourth and last section, we propose a numerical example to show the effectiveness of the main results.

2. Problem Statement

Let an open bounded subset and we denote . Let the semilinear hyperbolic state-space system

The operator is linear elliptic depending on the uncertainty parameter , the nonlinear operator is independent of , is the indicator of such that is a zone of the system domain , is a function in , in is the control and the initial conditions . Let represent the solution of (1) and suppose that . The following definitions give the exact and the approximate averaged controllability of hyperbolic system. First, let a subregion of

Definition 1. We say that (1) is -exactly RAC, if there is independent of such thatwhere is the final target in .

Definition 2. We say that (1) is -approximately RAC, if there is independent of such thatNow, let the RAC problem for (1) and the actuator type zone internal stated:Let , , and .
So, we rewrite (1) asand associated linear system.Consider the operator generating the semigroup on , we define the two operators and Now, we introduce the function.and we denote the inverse of by . The fixed point of (8) where , directly If (6) is -approximately RAC, thenConduct (1) to at time . In the next section, an important situation will be analyzed in the analytical case.

3. Main Results

3.1. First Case Using Analytical Operator

Consider the previous problem (2) of the equation (3). By choosing and let generates the analytic semigroup on .

We consider and we define the real part of the spectrum of ( is a real number providing ). On the dance Banach space , we define for the norm.and ([13]).

Suppose that , , and let , with and that is the map verifying.

This hypothesis is verified by several classes of semilinear hyperbolic systems.

Now, let the functions.

In the next, is equipped with the seminorm:which give us the next theorem.

Theorem 1. We suppose that (6) is -approximately RAC, (11) verified and is a positive constant.

is a positive function that is supposed belong . Then(1)The solution of (2) exists and it is unique, for and such that .(2)The map from is Lipschitz.

Proof. (1)Knowing that , then there exists such that. While (6) is -approximately RAC, then (14) is a norm. Using Schauder fixed-point theorem see [4], The function is a contraction on the nonempty convex closed ball , then admit a unique fixed point solution of (3), for all with Furthermore, for we have We deduce that is contraction while with From (18) we have Therefore, if , then admit a unique fixed point in solution of (2).(2)To prove that the map is Lipshitz, let consider , such that but hence, which concludes the result.

Proposition 1. Let is the solution of the system (3) associate to the control . The solution of the problem (2) is represented by the control sequences

Which converges to .

Proof. The proof is obtained using (20) and (14).

3.1.1. Second Case Using HUM

Here, we address the issue (4) that arises when it is anticipated that the system (1) would verify

The method we will choose is an expansion of the HUM, which has been used to prove controllability in the linear situation (see [3]), as well as the semilinear situation (see [14]).

We consider the set and let

For , the following system admit a unique solution ([3]).

And the solution of (1) can be expressed aswhere and are respectively solutions of systemsthat verify (see [3])and the exist a positive constant verifyingand is solution of the system

The map is Lipschitz continuous, since . So (36) admits a unique solution.

Now, we define the operatorwhere .

Then, the problem of RAC of (1) turns up to solve the equation