On the global optimal solution for linear quadratic problems of switched system

Abstract

The global optimal solution for the optimal switching problem is considered in discrete time, where these subsystems are linear and the cost functional is quadratic. The optimal switching problem is a discrete optimization problem. Complete enumeration search is always required to find the global optimal solution, which is very expensive. Relaxation method is an effective method to transform the discrete optimization problem into the continuous optimization problem, while the optimal solution is always not the feasible solution of the discrete optimization problem. In this paper, we propose a special class of relaxation method to transform the optimal switching problem into a relaxed optimization problem. We prove that the optimal solution of this modified relaxed optimization problem is exactly that of the optimal switching problem. Then, the global optimal solution can be obtained by solving the continuous optimization problem easily. Numerical examples are demonstrated to show that the modified relaxation method is efficient and effective to obtain the global optimal solution.