Coordinate-Free Interpretations of the Optimal Costs for LQ-Problems Subject to Implicit Systems

摘要

We consider linear-quadratic control problems with and without stability, subject to an arbitrary implicit continuous-time system, in a simple distributional framework, and it will be shown that the associated optimal costs, if existent, are solutions of our Dissipation Inequality for implicit systems. This concept is related to the Linear Matrix Inequality, which is expressed in original system coefficients only, and the above-mentioned optimal costs turn out to be characterizable uniquely by certain solutions of this inequality. However, these solutions need not be rank minimizing if the underlying system is not standard, and we will specify why this is the case. Our statements are valid for regular as well as for singular problems, and the possible significance of the algebraic Riccati equation will be illustrated for both regular and singular problems. Furthermore, we will present necessary and sufficient conditions for solvability of our problems and for existence of optimal controls and associated optimal state trajectories. Finally, we will elaborate on the uniqueness of these controls and state trajectories.