Linear Quadratic Optimization for Systems in the Behavioral Approach

In this paper the following formulation of the linear quadratic optimal control problem for dynamical systems in the behavioral setting is proposed: given a linear, time-invariant, and complete system, find the set of trajectories of the system that optimize a quadratic-type cost function and that satisfy some linear static constraints. This formulation provides a unifying framework, where several generalized versions of the classical LQ optimal control can be stated and solved. The existence of solutions is first discussed. It is shown that a necessary and sufficient condition for the existence of solutions may be obtained as a by-product of a reduction procedure translating the problem into an equivalent one of minimum complexity. Such a procedure is based on the theory of l2-systems in the behavioral setting. Once the complexity is reduced, a parametrization of the set of optimal solutions is obtained by employing a behavioral realization technique and a two-step optimization procedure.