Averaged Control Problems Governed by a Semilinear Distributed Systems

. In this work, we consider the regional averaged controllability (RAC) problem governed by a class of semilinear hyperbolic systems. We start by giving the defnitions 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 fxed point theorem and the Hilbert Uniqueness Method (HUM) with an asymptotic condition on the nonlinear part to fnd the optimal control of the considered problem. Finally, we present a numerical example to show the efectiveness of the main results.


Introduction
Te study of hyperbolic partial diferential equations (PDE) is a historically important subject whose frst 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 fuid [1].An important example of hyperbolic PDEs is provided by the conservation laws of the frst 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 difusion (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 efects.With respect to a fxed spatiotemporal point, the disturbances have a fnite propagation speed and move according to the characteristics of the equation.Tis property makes it possible to distinguish hyperbolic problems from elliptical or parabolic problems, where the perturbations of the initial conditions will have immediate efects on all points of the domain.Although there are specifc 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.Tey 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].
Te idea of regional controllability is one of controllability's most signifcant 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 ω of Ω. Exactly, it fnds a control which directs the considered system, at the moment T, towards a prescribed function defned on a subregion ω of Ω.
For controllability problems, one considers a control system in the time interval [0, T] and specifcally 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 difcult, since the issues encountered, is that we work with space-temporal systems considered as a nonlinear problem.Others' difculty 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 fxed point method is introduced and applied in infnite dimension.
Averaged control problems for semilinear distributed systems involve the design of control laws for a class of systems described by partial diferential equations (PDEs).Tese systems are characterized by a linear spatial operator and a nonlinear temporal operator.Te 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 efects 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 frst 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.Te average controllability allows us to consider many types of nonautonomous systems.Te 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][9][10][11][12].
Te paper is organized as follows.In the second section, we begin by stating the problem and giving the defnition of the RAC.Te 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 fxed point theorem.Second, using the HUM with an asymptotic condition on the nonlinear part to fnd the optimal control.Te fourth and last section, we propose a numerical example to show the efectiveness of the main results.

Problem Statement
Let an open bounded subset Ω ⊂ IR n and we denote Q � Ω × ]0, T[.Let the semilinear hyperbolic state-space system Te operator A(σ) is linear elliptic depending on the Let Y u � (X u , zX u /zt) represent the solution of (1) and suppose that Y u (T, σ) ∈ (L 2 (Ω)) 2 � E. Te following defnitions give the exact and the approximate averaged controllability of hyperbolic system.First, let ω a subregion of Ω Defnition 1.We say that (1) is ω-exactly RAC, if there is u ∈ C independent of σ such that where Now, let the RAC problem for (1) and the actuator type zone internal (f, D) stated: and Bu � (0, Proj D fu) t .So, we rewrite (1) as and associated linear system.
Consider the operator A(σ) generating the semigroup S(t, σ) (t≥0) on E, we defne the two operators L(., σ) and G ω Now, we introduce the function.
and we denote the inverse of Conduct (1) to Y d at time T. In the next section, an important situation will be analyzed in the analytical case.
Tis hypothesis is verifed by several classes of semilinear hyperbolic systems.Now, let the functions.

Journal of Mathematics
In the next, ImG ω is equipped with the seminorm: which give us the next theorem.
(2) To prove that the map hence, which concludes the result.

□
Proposition 1.Let Y u n is the solution of the system (3) associate to the control u n .Te solution of the problem ( 2) is represented by the control sequences Which converges to u * ∈ C.
Proof.Te 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 [14]).( 27) Te method we will choose is an expansion of the HUM, which has been used to prove controllability in the linear situation (see [3]), well as the semilinear situation (see [14]).
We consider the set and let And the solution of (1) can be expressed as where ϕ 0 and ϕ 1 are respectively solutions of systems that verify (see [3]) Journal of Mathematics and the exist θ 1 a positive constant verifying and ϕ 2 is solution of the system Now, we defne the operator where χ � Proj * ω Proj ω .Ten, the problem of RAC of (1) turns up to solve the equation Te equation ( 39) is equivalent to the equation For a positive constant θ 2 , let Solve the problem (39), became a fxed point of We defne the operator K ω by with  G * is the dual of  G.

Simulations
In this section, we present a numerical example which illustrates the previous results.It shows that there exists a link between the subregion area and the reached state Consider the one dimensional system excited by a zone actuator located in D. Remark 1.In the next simulation, we will apply the previous algorithm to the second case where the optimal control is given by u(t) � − 〈Z(t), f〉 L 2 (D) .Te characterization established in the frst case by Proposition 1 can be tested using the same algorithm.
Tables 1 and 2 show numerically how the cost and the error respectively grow with respect to the subregion area.
Figures 1 and 2 represent the profle of the energy dissipated to command the system (49) from its initial states to the desired ones at the time T � 3 with the cost E � 2.031 × 10 − 4 .
From the reached state solution of the system (49) presented by Figure 1, we can remark that the desired position given by Figure 1 is very close to the reached position.Terefore, for the reached speed of the system (49) presented by Figure 2, we will have that the desired speed is very close to the reached one.

Conclusion
In this study, we describe the regional averaged controllability problem governed by a class of semilinear hyperbolic systems.Te defnitions of the precise and approximate regional averaged controllability systems are provided frst.Te issue of regional averaged controllability for semilinear Step 2: Repeat (i) Solve the system (30) to fnd Z. (ii) Compute the control u * (t) by the formula u(t) � − 〈Z(t), f〉 L 2 (D) .(iii) Solve (49) to obtain X u (T, σ) and zX u /zt(T, σ).(iv) Until E � ‖    8 Journal of Mathematics systems is then raised.We suggest two approaches to the problem: the Hilbert Uniqueness Method with an asymptotic condition on the nonlinear part to fnd the optimal control of the considered problem, and using a condition of the analytical operator to the nonlinear part of the system to characterize the optimal control via the fxed point theorem.Finally, we give a numerical example to illustrate the effectiveness of our approach and to validate our results.

2 )
and the region ω.(ii) Defne the precision ε and the location D.