Output stabilization of boundary-controlled parabolic PDEs via gradient-based dynamic optimization

Abstract

This paper proposes a new control synthesis approach for the stabilization of boundary-controlled parabolic partial differential equations (PDEs). In the proposed approach, the optimal boundary control is expressed in integral state feedback form with quadratic kernel function, where the quadratic's coefficients are decision variables to be optimized. We introduce a system cost functional to penalize both state and kernel magnitude, and then derive the cost functional's gradient in terms of the solution of an auxiliary “costate” PDE. On this basis, the output stabilization problem can be solved using gradient-based optimization techniques such as sequential quadratic programming. The resulting optimal boundary control is guaranteed to yield closed-loop stability under mild conditions. The primary advantage of our new approach is that the costate PDE is in standard form and can be solved easily using the finite difference method. In contrast, the traditional control synthesis approaches for boundary-controlled parabolic PDEs (i.e., the LQ control and backstepping approaches) require solving non-standard Riccati-type and Klein-Gorden-type PDEs.