Necessary Conditions Of Convex Constrained Optimal Control Problems On Riemannian Manifolds

The optimal control theory has been applied to the integrated design of the fastest control system,the most fuel saving control system,the least energy consumption control system,linear regulators,and etc.The main theoretical methods to solve the optimal control problem are classical variational method,maximum principle and dynamic programming.In this thesis,we make a supplement to the maximum principle and the second-order necessary conditions in the sense of convex variation.In this thesis,we study the optimal control problem with the state variables on the manifold,the initial state fixed,the end state unconstrained and the control constraint set convex,and derive the second-order necessary conditions of the optimal control problem.It is significant to extend the maximum principle of optimal control problem with state variable in flat space to curved space.Among them,Agrachev and Gamkresize[2]extend the optimal control problem in flat space to curved space.The results given in this paper are difficult to be verified.Because there is no metric on the differential manifold,the resulting second-order necessary conditions are not precise enough.The curvature in our second-order necessary conditions is expressed in a explicit form,which is more accurate and useful for solving some specific problems.Recently,Deng Li and other authors extended the second-order optimality conditions of optimal control problems on flat spaces to manifolds.The control constraint set of the control problem put forward by Deng Li and his colleagues is a general set.What they get is the second-order necessary condition of the optimal control in the sense of needle variation.The control constraint set considered in this paper is a convex set,and what we get is the second-order necessary condition of the optimal control in the sense of convex variation.This paper includes the following points:1?In this paper,the optimal control problem is studied,in which the initial state is fixed and the set of control constraints is convex.The first order asymptotic expansion is obtained by convex variation of the cost functional with respect to the control variable,and then the first order dual equation on the manifold is introduced to obtain the first order necessary condition in the sense of convex variation.2?For the above optimal control problem,the second order asymptotic expansion is obtained by convex variation of the cost functional with respect to the control variable.Under the condition that the first order necessary condition of optimal control problem is satisfied,the second order necessary condition is obtained by introducing an appropriate first order variational equation on a manifold and the second order variational equation related to the curvature tensor.