Necessary Conditions For Optimality Of Discrete Time Control With Time Delay

The idea of optimal control has been accompanying people's life for many years,and has been influencing and guiding people's production and life virtually,but the theory of optimal control has not been formed for a long time.Until the 1940s,Wiener put forward the concept of optimal design for the first time in[42],and the control thought began to be presented to people in the form of theory.In 1960s,the standard form of state space model was put forward for the first time,and the optimal control theory made unprecedented development.Mathematical models are often discrete in the optimal control problems in engineering,biology and economics,and we need to discretize the continuous control problems when solving them with computers.Therefore,the discrete control problem has been paid more and more attention by the theoretical circles at home and abroad.At present,the methods to solve discrete control problems mainly include dynamic programming method and Pontryagin maximum principle.In 1957,Berman[43]put forward the dynamic programming method,which has a wide range of applications.However,when using the dynamic programming method,the following two problems should be considered:one problem is that when the variable dimension is large,the amount of calculation and storage will far exceed the computer's tolerance range,resulting in "dimension disaster";Another problem is that dynamic programming is established on the basis of Berman's optimization principle,which determines the "no aftereffect" of the multi-stage decision-making process,that is,the future decision-making is only determined by the present state,and has nothing to do with the past state.In 1962,Pontryagin and others put forward the maximum principle based on convexity condition in[44],and gave a standardized mathematical description of the optimal control problem,thus promoting the overall development of the optimal control theory research to a great extent.Lu Xianrui and Huang Qingdao gave the maximum principle for solving discrete control problems in[8].Misir J.Mardanov and Samin T.Malik studied the following time-delay discrete control systems in[1]:9(u(·))=?(x(t1))?(?)#12 Where t is the discrete time,to is a given initial time,t1 is a given end time,h>0 is integer,t1-t0>h,and x0 is a given initial state.By assuming that the allowable velocity set satisfies the convexity condition at the moment t+h and does not satisfy it at the moment t,the discrete simulation of Pontryagin maximum principle is obtained under the weak convexity condition,which has strong applicability.In this paper,based on the results obtained by Misir J.Mardanov and Samin T.Malik,the results are extended to discrete optimal control system with time delay in the following form:#12Where t is discrete time,t0 is given start time,t1 is the given end time,h>0is integer,and t1-t0>h.In this paper,a discrete optimization problem with control delay is considered.On the premise that the input data satisfy certain convexity and smoothness assumptions,and considering the special properties of discrete systems,a necessary optimality condition expressed by Hamiltonian function is obtained.On the premise of convexity hypothesis and linearization hypothesis,Euler-type necessary optimality conditions and linearized discrete maximum principle are obtained.This paper is based on the maximum principle,but the requirement of convexity in this paper is weaker than the maximum principle.This paper is divided into seven chapters:In the first chapter,the problems of discrete optimal control with time delay are introduced,and the development of the theory is introduced.In the second chapter,convexity condition,Hamiltonian function and auxiliary matrix function ?0(t),and the following discrete optimal control system model with time delay are introduced:Where xo is the given initial point.In the third chapter,we mainly study the incremental formula of the objective functional when the admissible control u0(t),t ?Ih changes to the following form at the point ? ?Ih:At the same time,it is given that when the right end function of the state equation and the function satisfy the linear condition Admissible control is an optimal necessary condition.In chapter four,some examples are given to illustrate the extensive practicability of the above necessary conditions.The fifth chapter mainly summarizes this paper and gives some expectations for the future development.