Approximate controllability and its well-posedness for the semilinear reaction-diffusion equation with internal lumped controls

Exact Internal Controllability for the Semilinear Heat Equation

Using Browder-Minty’s surjective theorem from the theory of monotone operators, We consider the exact internal controllability for the semilinear heat equation. We show that the system is exactly controllable in L(Ω) if the nonlinearities are globally Lipschitz continuous. Furthermore, we prove that the controls depend Lipschitz continuously on the terminal states, and discuss the behaviour of ...

Approximate controllability of a reaction-diffusion system

An open-loop control for a system coupling a reaction-diffusion system and an ordinary differential equation is proposed in this study. We use a flatness-like property, indeed, the solution can be expressed in terms of an infinite series depending on a flat output, its derivatives and its integrals. This series is shown to be convergent if the flat output is Gevrey of order 1 < a ≤ 2. Approxima...

Exact Controllability for a Semilinear Wave Equation with Both Interior and Boundary Controls

The purpose of this paper is to prove the existence of the exact controllability of a semilinear wave equation with both interior and boundary controls. Let Ω be a bounded open subset of Rn with a smooth boundary, let f (y) be an accretive mapping of L2(0,T ;H−1(Ω)) into L2(0,T ;H 0 (Ω)) with respect to a duality mapping J ,D( f ) = L2(0,T ;L2(Ω)) and having at most a linear growth in y. Consid...

On partial approximate controllability of semilinear systems

Approximate controllability for the semilinear heat equation in R involving gradient terms

We prove the approximate controllability of the semilinear heat equation in RN , when the nonlinear term is globally Lipschitz and depends both on the state u and its spatial gradient u. The approximate controllability is viewed as the limit of a sequence of optimal control problems. In order to avoid the difficulties related to the lack of compactness of the Sobolev embeddings, we work with th...