Exponential stabilization of unstable diffusion-reaction PDEs with mixed boundary condition

In this paper, both the optimal control problems and the stability for the state-feedback of an unstable linear parabolic diffusion-reaction partical differential equations (PDEs) system with an actuator at the right-hand-side boundary are considered. Using the weak variation approach, we firstly derive the first order necessary conditions for the open loop finite-time horizon problem. Then, we discuss the state-feedback problem of the open loop PDEs system, and get the result of the feedback law through solving a Riccati partial differential equation. Finally, with the feedback law, we prove out the exponential stability for the state-feedback problem that is very innovative and instructive by using the lyapunov method.