Penalized Weighted Trace Minimization for Optimal Control Device Design and Placement

In this paper, we present a new analytical framework for determining the well-posedness of constrained optimization problems that arise in the study of optimal control device design and placement within the context of infinite dimensional linear quadratic control systems. We first prove the well-posedness of the newly minted "strong form" of the time-independent operator-valued Riccati equation. This form of the equation then enables the use of trace-class operator analysis and the Lagrange multiplier formalism to analyze operator-valued Riccati equation-constrained optimization problems. Using this fundamental result, we then determine the conditions under which there exists unique solutions to two important classes of penalized trace minimization problems for optimal control device placement and design.