Linear-quadratic control with and without stability subject to general implicit continuous-time systems: Coordinate-free interpretations of the optimal costs in terms of dissipation inequality and linear matrix inequality; existence and uniqueness of optimal controls and state trajectories
The linear quadratic control problem for the class of singular, finite- dimensional, time-invariant systems of the form \[ \begin{aligned} E\dot x(t) & = Ax(t)+ Bu(t)\\ y(t) & = Cx(t)+ Du(t),\qquad x(0)= x_ 0\in \mathbb{R}^ n\end{aligned} \] with cost functional \[ J(x_ 0, u(\cdot))= \int^ \infty_ 0 y(t)^ T y(t)dt \] is investigated. Partially, the results are also new for regular systems, i.e., \(\text{det }E\neq 0\). The optimal costs are related to a dissipation inequality and a linear matrix inequality, the latter depending on \(E\), \(A\), \(B\), \(C\), \(D\). Finally, necessary and sufficient conditions for solvability of the optimal control problem and of the optimal controls are presented.
scientific article; zbMATH DE number 614164
scientific article