Optimal control of linear systems: a multiresolution approach

This paper deals with the optimal control of continuous-time linear systems with state-space constraints. Recently, for linear systems, with discrete-time constraints, it has been shown that the optimal control can be computed by solving a finite dimensional convex quadratic problem. The optimal trajectory belongs to a class of curves called control theoretic splines. In this paper, we consider the optimal control problem with continuous-time constraints. The method proposed here combines the notion of control theoretic splines with multiresolution approximation theory used for some years in the area of signal processing. This work results in a multiresolution approach to linear control which allows the computation of an approximation of the optimal input with great accuracy. Moreover, the associated output satisfies the state-space constraints at all time.