The Second-order Necessary Conditions For Finite-dimensional Optimal Control Problems

Since the birth of cybernetics,the necessary condition for optimal control has been one of the core issues of research.The Pontryagin maximum principle reveals the simi-larities and differences between the optimal control problem and the classical variation from the perspective of the first-order invariant.However,many times,only the first-order necessary conditions are not enough,and second-order necessary conditions are often needed(such as solving optimal control).And expect to reveal the similarities and differences between the optimal control problem and the classical variation problem from the perspective of the second-order invariant.Since the problem involves nonlinear func-tional,the relevant results(especially the second-order invariant results)are not many.Pulses are not only a common phenomenon in the natural world(pulse phenomenon),but also a widely used technology in engineering(pulse control technology).The optimal control problem dominated by the pulse system,and the optimal pulse control problem are important research topics.In this paper,we will discuss the second-order necessary conditions for two kinds of optimal control related to impulsive phenomena:(P1)The optimal control problem dominated by pulsed ordinary differential equa-tions(?)where Ji(i = 1,2,(…,n)is a single value nonlinear map,determining the size of the jump at time li.(P2)The optimal pulse control problem dominated by ordinary differential equations (?)where ?(·)is a segmented constant function:(?)For(P1),first of all,we study the nature of the variation of the controlled system solution with respect to control,as well as the nature of the second-order variation of the performance index with respect to control.Then we introduce second-order adjoint equations(other than the first-order adjoint equations),and obtain the second-order in-variant of optimal control(only dependent on the second-order necessary condition of the optimal pair).For(P2),we not only optimize the pulse time,but also the control value.In order to overcome the difficulties caused by the generalized function and introduce variation rationally,based on the perturbation of ?2 about pulse time,perturbation of ?about control value,we obtain the second-order necessary condition of the(P2),which only depends on optimal pair.That is,the second-order invariant.