Optimal control of nonlinear switched system with mixed constraints and its parallel optimization algorithm

We consider an optimal control problem for a class of nonlinear switched systems with mixed constraints, in which both the mode sequence of active subsystems and the number of switches are not prescribed. By introducing convex combinations of the subsystems and the constraints, an embedded system is formulated. The relationships between the trajectories of the switched system and the corresponding embedded system are discussed. Necessary optimality conditions and the existence condition of optimal solutions are derived. A constructive parallel algorithm is proposed for solving optimal or suboptimal solutions of the considered problem. In order to avoid the case that the switching times may be lost, the time-scaling transformation is applied to transform the variable switching times to pre-fixed switching times. Finally, an optimal control of fed-batch fermentation is solved by applying the proposed algorithm, and the numerical results show that the proposed algorithm is effective.