The regulator problem for parabolic equations with Dirichlet boundary control

This paper considers the regulator problem for a parabolic equation (generally unstable), defined on an open, bounded domain , with control functionu acting in the Dirichlet boundary condition: minimize the quadratic functional which penalizes theL ₂(0, ; L₂())-norm of the solutiony and theL ₂(0, ; L₂())-norm of the Dirichlet controlu. The paper is divided in two parts. Part I derives, in a constructive way, the algebraic Riccati equation satisfied by the candidate Riccati operator solution (unique in our case) and, moreover, studies the regularity properties of the optimal pairu ⁰, y⁰. Part II studies a Galerkin approximation of the regulator problem. It shows first the uniform analyticity and the uniform exponential stability of the underlying discrete (approximating) semigroups. Then it establishes the desired convergence properties, in particular, pointwise Riccati operators convergence and, as a final goal, convergence of the original dynamics acted upon by the discrete feedbacks.Research partially supported by the National Science Foundation under Grant DMS-8301668.