Exponential stabilization of an unstable parabolic PDE system using boundary optimal control

In this paper, we consider an unstable linear parabolic diffusion-reaction PDE system with an actuator at the right boundary via the Dirichlet condition. We begin to derive the first order necessary conditions for the open loop finite-time horizon problem using the classical variational approach. Then, we discuss the the state-feedback problem, and obtain the feedback law by formulating the Riccati partial differential equation (PDE). Finally, with the feedback law, we could prove the stability for the state-feedback problem exponential stability with any given decay rate in the time domain without solving the Riccati PDE.