Nonlinear boundary control problem of hyperbolic systems

ABSTRACT

The paper is devoted to the study of optimal control of a hyperbolic system, where the control enters the system through the boundary. The hyperbolic system is the wave equation on a smooth and open domain, where the boundary condition involves the normal derivative at the boundary of z, the time derivative of z times a constant k, and a nonlinear term control. Here, z is the state and u is the control, satisfying some boundedness condition depending on k. The functional cost consists of the energy and the difference between the solution of the system at final time, and a desired state in $L^2$-norm. For a closed convex set, we prove the existence of an optimal control that minimises the cost functional using a priori estimates. Then, using the differentiability of the cost functional with respect to the control, we establish the characterisation by deriving necessary conditions that an optimal control must satisfy. A numerical approach is successfully illustrated by simulations.

KEYWORDS:

• Non-linear system
• hyperbolic system
• boundary control
• optimal control
• controllability

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Notes on contributors

A. Bouhamed

Abdelhak Bouhamed received his M.Sc. degree in applied mathematics from Hassan II University in 2017. He is currently pursuing a Ph.D. degree in optimal control at LISTI Laboratory, ENSA, Ibn Zohr. His research interests include optimal control and controllability of infinite-dimensional systems.

A. El Kabouss

Abella El kabouss Ph.D. degree in Applied Mathematics from Faculty of Science, Moulay Ismail University in Meknes, Morocco. Currently, he is a visiting professor at the Higher School of Technology-Guelmim, and the Faculty of Economics and Management-Guelmim, Morocco. He has written numerous papers in the field of systems theory and particularly in optimal control and controllability of infinite dimensional systems.

H. Bouzahir

Hassane Bouzahir is a Full Professor of Mathematics/Statistics with National School of Applied Sciences at Ibn Zohr University in Agadir, Morocco. He has a very rich experience of Head of Departments and Director of Research Laboratories. Hassan obtained his Ph.D. in Mathematics in April 2001. His publications include many research items: 3 research books, 2 book chapters, entries in the mathematics databases MathScinet and/or Zentralblatt MATH, articles on Management, Modeling and Simulation in Engineering, articles in different Science and Engineering journals.