| Multiobjective optimal control problems are to find a control in the set of feasible controls for several object functions,such that in a sense these functions are optimal.The feasible control means the corresponding trajectory with initial value satisfies state equation,and the state-control pairs satisfy the constraints in control systems.Due to the difference of state equation and constraint conditions in control system,multiobjective optimal control problems are also varied.In this paper,we consider a multiobjective optimal control problem in the Euclidean space where the control system involves an ordinary differential equation,the control is point-wisely constrained,and the endpoints of the state satisfy some equality and inequality type constraints.The main work of this paper is as follows:1.Explore the Pontryagin type maximum principle(first-order necessary condition)for the weak Pareto optimal solutions of multiobjective optimal control problems.A convex set related to the first order variation is constructed,and the convex set separation theorem is applied to the set.2.The second-order necessary condition for the weak Pareto optimal solutions of multiobjective optimal control problems is explored.The second order variation method is applied to the control system.The convex set is constructed and convex set separation theorem is used to it.3.Apply the main results to confirm a feasible solution is not weak Pareto optimal. |
