Exact controllability for nonlocal and nonlinear hyperbolic PDEs

Abstract

This paper deals with the internal and boundary exact controllability of some nonlinear hyperbolic systems with local and nonlocal nonlinearities in dimension one. Nonlocal terms in the space and time variables are considered and they appear in the coefficient of the spatial derivative of the state. First, we prove the linearized result. For this, our proof method is based on HUM (Hilbert Uniqueness Method) and observability inequality results. Then, we apply a Fixed-Point technique to prove the nonlinear result with internal control. In a similar way, we will prove the nonlinear result regarding the boundary control. Finally, some possible extensions and open problems concerning others nonlocal systems are presented.