Study On Indefinite LQ Optimal Control Of Discrete Markov Jump With Mean Field

In the process of industrial production,there are often Markov jumping phenomena,such as power system,aeronautics and space system,manufacturing system and network control system,which often have sudden failure or environment change,so the optimal control problem of Markov system has been deeply studied and widely used.In recent years,the theory of mean field has been received.It is a mathematical processing method which is widely applied to the lower order approximation of the real physical system under the condition of small fluctuation.It is widely used in the study of complex systems such as mechanics,magnetics,condensed state systems and so on.This paper discusses the indefinite linear two order optimal control problem of the mean field discrete Markov jump system.The main research work is as follows:First,the definition of several stability of the system and the equivalence relation of each stable are given.Then the relevant definitions and theorems of the generalized inverse matrix are introduced,and the formula formula is given.Two,we discuss the indefinite linear quadratic optimal control problem of an average field discrete Markov jump system with finite time domain.First,the performance index and optimal control problem(MF-LQ)N of the system are defined.Then,the sufficient conditions for the existence of the optimal control are obtained,and the optimal control is obtained by the solution of the generalized Riccati equation in the finite time domain of the Markov system.Finally,a numerical example is given to verify the correctness of the theory.Three,we discuss the indefinite linear quadratic optimal control problem of an infinite time domain discrete Markov jump system.First,the performance index and optimal control problem(MF-LQ)? of the system are defined.Then,the Riccati equation in the infinite time domain and the sufficient conditions for the optimal control to be stored in the infinite time domain are derived,and the Riccati equation is used.The minimum nonnegative definite solution is derived,and the optimal control and the optimal performance index are derived.A numerical example is given to illustrate the effectiveness of the method.