Computation of Linear Quadratic Regulator using Krotov Sufficient Conditions

Abstract

This paper revisits the problem of optimal control law design for unconstrained linear systems, including time-varying and time-invariant, with quadratic performance index using the global optimal control framework introduced by Vadim Krotov. Krotov framework works on the idea of total decomposition of the original optimal control problem with respect to time, by an ad hoc choice of the so-called Krotov function, and then provides the sufficient conditions for the global optimal control law to exist based on another optimization problem, which is completely equivalent to the original problem. Subsequently, the solution of this equivalent optimal control problem is determined by an iterative procedure. In this paper, we propose suitable Krotov function for linear quadratic regulator design problem and subsequently, show that by imposing convexity conditions on this equivalent optimization problem, there is no need to compute the solution iteratively. As a consequence, the obtained results are not only in agreement with the ones obtained using Calculus of Variations (CoV) approach, which gives the necessary conditions, but also provide a natural way to apply Krotov sufficient conditions for synthesizing control laws to more involved optimal control problems viz. for nonlinear systems, involving non-quadratic performance functional etc.