Optimal and robust control of the wave equation with Kelvin-Voigt damping and random forcing

In this paper the optimal and robust control of a wave equation with Kelvin-Voigt damping and random forcing is presented. The optimal controller is found using the quadratic performance index, which in-turn is determined from the Hamilton-Jacobi-Bellman (HJB) equation. This equation is reduced to a partial-differential-integral Riccati equation by assuming that the solution of the closed loop Lyapunov function is a non-negative integral function of time and the state variable with a costate vector. A numerical method is then applied to find the solution of the optimal controller for the wave equation. It is shown that the optimal controller reduces the initial sinusoidal wave to a steady state. In robust control, Isaac inequality is established such that the penalty performance from the output to the disturbance is satisfied. Time differential of this Isaac inequality results in the HJB equation. After making an appropriate choice for the closed loop Lyapunov function, the robust controller is obtained, which satisfies the partial-differential-integral form of the Riccati equation instead of the HJB equation. The solution of this Riccati equation for the robust controller stabilizes the wave equation with random forcing. The above formulations are illustrated by considering two cases with Kelvin-Voigt damping of γ= 0.0 and 0.005. Both optimal and robust control for the wave equation are analyzed.