Closed-form boundary State feedbacks for a class of 1-D partial integro-differential equations

In this paper, a problem of boundary stabilization of a class of linear parabolic partial integro-differential equations (P(I)DEs) in one dimension is considered using the method of backstepping, avoiding spatial discretization required in previous efforts. The problem is formulated as a design of an integral operator whose kernel is required to satisfy a hyperbolic P(I)DE. The kernel P(I)DE is then converted into an equivalent integral equation and by applying the method of successive approximations, the equation's well posedness and the kernel's smoothness are established. It is shown how to extend this approach to design optimally stabilizing controllers. An adaptation mechanism is developed to reduce the conservativeness of the inverse optimal controller, and the performance bounds are derived. For a broad range of physically motivated special cases feedback laws are constructed explicitly and the closed-loop solutions are found in closed form. A numerical scheme for the kernel P(I)DE is proposed; its numerical effort compares favorably with that associated with operator Riccati equations.

I. Introduction

After ABOUT three decades of development, partial differential equation (PDE) control theory, and boundary control in particular, consists of a wealth of mathematically impressive results that solve stabilization and optimal control problems. Two of the main driving principles in this development have been generality and extending the existing finite dimensional results. The latter objective has led to extending (at least) two of the basic control theoretic results to PDEs: pole placement and optimal/robust control. While these efforts have been successful, by following the extremely general finite-dimensional path (, where and can be any matrices), they have diverted the attention from structure-specific opportunities that exist in PDEs. Such opportunities have recently started to be capitalized on in the elegant work on distributed control of spatially invariant systems by Bamieh et al. [5].