Dirichlet Boundary Control Problem for Parabolic Equations with Quadratic Cost: Analyticity and Riccati’s Feedback Synthesis
Irena Lasiecka, Roberto Triggiani · SIAM Journal on Control and Optimization · 1983
For a parabolic equation in y defined on a bounded open domain $\Omega $ with boundary $\Gamma $ and with control function u acting in the Dirichlet boundary condition, we study the optimal quadratic cost problem, which penalizes over an assigned time interval $[0,T]$ the $L_2 (0,T;L_2 ( \cdot ))$-norm of the solution y and of the control u, as well as the $L_2 (\Omega )$-norm of the final state $y(T)$. Feedback synthesis (pointwise in time) of Riccati type: $u^0 (t) = CP(t)y^0 (t)$ of the optimal solution $u^0 $, $y^0 $ is established through a semigroup approach. Moreover, in contrast with the indirect approach of much of the literature, which relies on a Riccati equation to establish existence and numerical computability of the operator $P(t)$, the present approach is instead direct: i.e., the operator $P(t)$ is first defined by an explicit formula in terms of the system data, and only subsequently shown to satisfy, in an appropriate sense, a Riccati-type operator equation. Solution to the Riccati feedback synthesis (§ 3) required some regularity results of the optimal solution $u^0 $, $y^0 $. Accordingly, the regularity question is taken up preliminarily (in § 2) and is carried out, in its own right and in full generality, much beyond the need of the Riccati synthesis.