Numerical solution of minimax optimal control problems by multiple shooting technique

This paper presents the application of the multiple shooting technique to minimax optimal control problems (optimal control problems with Chebyshev performance index). A standard transformation is used to convert the minimax problem into an equivalent optimal control problem with state variable inequality constraints. Using this technique, the highly developed theory on the necessary conditions for state-restricted optimal control problems can be applied advantageously. It is shown that, in general, these necessary conditions lead to a boundary-value problem with switching conditions, which can be treated numerically by a special version of the multiple shooting algorithm. The method is tested on the problem of the optimal heating and cooling of a house. This application shows some typical difficulties arising with minimax optimal control problems, i.e., the estimation of the switching structure which is dependent on the parameters of the problem. This difficulty can be overcome by a careful application of a continuity method. Numerical solutions for the example are presented which demonstrate the efficiency of the method proposed.