Optimal control and piecewise parametric programming

This paper deals with the problem of parametric piecewise quadratic programming (in which the cost is a piecewise quadratic function of both the decision variable and a parameter) and the problem of parametric piecewise affine quadratic programming (in which both the cost and the constraint depend on a piecewise affine function of the decision variable and a parameter). Parametric programming seeks a solution for each value of the parameter, and can therefore be used to obtain explicit solutions of some constrained optimal control problems where the state is the parameter. The technique of reverse transformation for parametric programming, introduced in earlier papers, is extended to remove unnecessary overlapping of polytopes on which the solution is defined. The improved technique is then employed for the determination, using dynamic programming, of explicit control for linear systems with a piecewise quadratic cost and explicit control of piecewise affine systems with quadratic cost.