The First Order Sufficient Conditions For Mayer Type Optimal Control Problems And Their Applications

In this thesis,the first order sufficient optimality conditions for optimal control problems with linear control systems and Mayer-type cost functions are established.First,under the condition that the function in the performance index is pseudo-convex,the first order sufficient optimality condition of optimal controls for continuous control problems is proved.Second,the discrete approximation of the continuous control problem is obtained by the Euler difference scheme.The first order necessary and sufficient conditions for the discrete optimal control problem are proved.Finally,The discrete optimal control problem is transformed equivalently into a finite dimensional variational inequality problem.A dual projection algorithm is introduced to establish a discrete approximation algorithm for optimal controls.Compared with known results,the pseudo-convex condition is much weaker than the usual convexity conditions,and therefore,the results in this thesis have a wider range of applications.This thesis is divided into four chapters.Chapter 1 gives a brief introduction to the background of this thesis.Chapter 2 proves the sufficiency of the first order necessary optimality condition for optimal controls.Using this condition,the optimal control problem is reformulated into an infinite-dimensional variational inequality problem.In Chapter 3,Euler difference scheme is used to obtain the discrete approximation of the optimal control problem,and the first order necessary and sufficient conditions of the discrete optimal control problem are proved.In Chapter 4,a numerical method for finite-dimensional variational inequalities is introduced for solving the discrete optimal control problems.A specific example is given to illustrate the effectiveness of the algorithm in the discrete approximation of optimal control.