The regulator problem for parabolic equations with dirichlet boundary control - Part I: Riccati\u27s Feedback Synthesis and Regularity of Optimal Solution

This paper considers the regulator problem for a parabolic equation (generally unstable), defined on an open, bounded domain Ω, with control function u acting in the Dirichlet boundary condition: minimize the quadratic functional which penalizes the L2(0, ∞; L2(Ω))-norm of the solution y and the L2(0, ∞; L2(Γ))-norm of the Dirichlet control u. 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 pair u0, 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. © 1987 Springer-Verlag New York Inc