Nonautonomous Control Theory: Linear-Quadratic Dissipative Control Processes

Abstract

This chapter presents an analysis of the normal or strict Willems dissipativity property of a family of linear-quadratic (LQ) control problems, which are described by time-dependent linear control systems and time-dependent quadratic supply rates. These problems give rise in a natural way to a family of nonautonomous linear Hamiltonian systems. An answer is given to a classical question: namely, it is shown that, under a null controllability property, the dissipativity of a LQ control problem is equivalent to the existence of a storage function. Furthermore, when this null controllability property holds, the following statement is valid: if the Hamiltonian family has exponential dichotomy and there exists the Weyl function M ⁻ associated to the solutions bounded at $-\infty$, then the normal or strict dissipativity is characterized by the semidefinite or definite positive character of M ⁻; and the occurrence of uniform weak disconjugacy permits one to establish some weaker equivalences, formulated in terms of one of the principal functions. Some dynamical conditions ensuring the dissipativity of the linear-quadratic problem are also given without assuming the controllability property. The optimality of the results is illustrated by some nontrivial examples, including a Millionščikov–Vinograd type example with highly complex dynamics.