Direct Verification of Parametric Solution for Vibration Reduction Control Problems

Abstract

For the optimal control problems formulated with limited actuator authority, the optimal solution is given by Pontryagin's minimum principle that provides the necessary and sufficient conditions of optimality for systems that are normal. If the resulting solution is a switched control, an alternate computation methodology exists where the problem is formulated as a parametric optimization problem with the switching times as the variables. For vibration reduction problems with constrained endpoints, the parametric formulation is not convex and the Karush-Kuhn-Tucker conditions can only guarantee the local optimality of the solution. In this paper an approach is presented to verify the optimality of the parametric solution for optimal control problems with terminal state inequalities. The verification conditions are derived using the switching function, Karush-Kuhn-Tucker, and the transversality equations. The resulting problem is formulated as a linear program that provides a very efficient test of optimality. Example problems are given to demonstrate the application of this algorithm.