Chapters authored
On the Stabilization of Infinite Dimensional Semilinear Systems
This chapter considers the question of the output stabilization for a class of infinite dimensional semilinear system evolving on a spatial domain Ω by controls depending on the output operator. First we study the case of bilinear systems, so we give sufficient conditions for exponential, strong and weak stabilization of the output of such systems. Then, we extend the obtained results for bilinear systems to the semilinear ones. Under sufficient conditions, we obtain controls that exponentially, strongly, and weakly stabilize the output of such systems. The method is based essentially on the decay of the energy and the semigroup approach. Illustrations by examples and simulations are also given.
Part of the book: Nonlinear Systems
Gradient Optimal Control of the Bilinear Reaction–Diffusion Equation
In this chapter, we study a problem of gradient optimal control for a bilinear reaction–diffusion equation evolving in a spatial domain Ω⊂Rn using distributed and bounded controls. Then, we minimize a functional constituted of the deviation between the desired gradient and the reached one and the energy term. We prove the existence of an optimal control solution of the minimization problem. Then this control is characterized as solution to an optimality system. Moreover, we discuss two special cases of controls: the ones are time dependent, and the others are space dependent. A numerical approach is given and successfully illustrated by simulations.
Part of the book: Recent Developments in the Solution of Nonlinear Differential Equations
Fractional Optimal Control Problem of Parabolic Bilinear Systems with Bounded Controls
The purpose of this paper is to study a fractional distributed optimal control for a class of infinite-dimensional parabolic bilinear systems evolving on a spatial domain Ω by distributed controls depending on the control operator. Using the Fréchet differentiability, we prove the existence of an optimal control depending on both time and space, that minimizes a quadratic functional which leads into account, the deviation between the desired state and the reached one. Then, we show characterizations of an optimal distributed control for different admissible controls set. Moreover, we developed an algorithm and give simulations that successfully illustrate the theoretically obtained results.
Part of the book: Vibration Control of Structures