Research On Non-cooperative Differential Game For Stochastic Generalized Linear Systems And Its Application

Differential game have been widely studied both in theory and application, and as a scientific and effective tool for decision-making, has been widely used in many aspects, such as national defense and military engineering, production management, economic life, etc.This dissertation investigated a class of dynamic systems:the stochastic Markov jump linear systems and stochastic singular linear systems, which have been used frequently in economic and management. On the basis of the existing literature of optimal control and Stochastic differential game theory, by utilizing the maximum principle, dynamic programming, Riccati equation methods used in dynamic optimization, this dissertation studied the differential game theory of discrete-time stochastic Markov jump linear systems and Continuous time stochastic singular linear systems and its applications in robust control problems and dynamic input-output problems systematically. The main contributions can be concluded as follows:First, in aspect of theory and method research. The non-cooperative differential game theory for discrete-time stochastic Markov jump linear systems and continuous time stochastic singular linear systems were discussed. Firstly, on the basis of the existed stochastic LQ differential game theory, two person zero-sum、nonzero-sum and stackelberg game models of discrete-time stochastic Markov jump linear systems and continuous time stochastic singular linear systems were established. And then by means of the maximum principle, we proved that sufficient conditions for the existence of the equilibrium strategy are equivalent to the solvability of the corresponding differential (algebraic) Riccati equations; morever, we got the explicit solution of the optimal control strategy and the expressions of the optimal value function. Finally, the numerical simulation examples were given to verify the validity of the presented results.These obtained results in this chapter expanded the existing results in stochastic differential game research.Second, in aspect of Application research. The robust control problems of discrete-time stochastic Markov jump linear systems and continuous time stochastic singular linear systems were studied based on game theory approach. By means of the results of stochastic differential game, we viewed the control strategy designer as one player of the game, i.e. P1, the stochastic disturbance as another player of the game, i.e. "nature" P2, respectively, then the robust control problems were transformed into a two person differential game model, in which player P1 faced the problem that how to design his own strategy in the case of various interference strategy implemented by "nature" P2, both balanced with the "nature" and optimized his own objective.Then, based on the result of game model, Corresponding strategies of stochastic H∞, H2/H∞ robust control problems were obtained. Finally, numerical examples were presented to verify the validity of the conclusions. Furtherly, we investigated the application of the non-cooperative differential game theory in dynamic input-output systems. First, the dynamic input-output modes based on stochastic linear systems> discrete-time stochastic Markov jump linear systems and continuous time stochastic singular linear systems Considering random factors are established respectively. Then, we considered the dynamic input-output problem in the context of a two-player, zero-sum stochastic differential game. One of the players in this game is an ’investment’ (PI) and the other is a fictitious player-the market(P2). The ’investment’ has a utility function and selects a strategy, which maximizes the expected utility of the terminal wealth. The market then selects a generalized "scenario", which is represented by a probability measure, to minimize the maximal utility of the’investment’. The closed-form expressions of optimal strategies of the ’investment’ and the optimal value function are derived by solving the associated game. Finally, the explicit expressions of the optimal strategies are constructed, and a numerical simulation is also given.Innovative achievements of the paper concludes to the following two parts.First, in aspect of game theory and research method:the paper obtains saddle-point equilibrium strategy, Nash equilibrium strategy, Stackelberg strategy of discrete-time stochastic Markov jump linear systems and saddle-point equilibrium, Nash equilibrium of continuous time stochastic singular linear systems, replenishing and enriching dynamic non-cooperative differential game theory described by state differential equations.Second, in aspect of the application of game theory:the paper systematically studies dynamic non-cooperative differential game theory of discrete-time stochastic Markov jump linear systems and continuous time stochastic singular linear systems, and applies the theory into robust control problem. Moreover, in consideration of uncertain disturbance factors, the dynamic input-output differential game modes based on stochastic linear systems、 discrete-time stochastic Markov jump linear systems and continuous time stochastic singular linear systems Considering random factors are established discreptly, and takes computer simulation study, providing new quantitative analysis tools and application cases for robust control problems and dynamic input-output models.This thesis was supported by the National Natural Science Fund of China-Noncooperative differential game theory of generalized Markov jump linear systems with application to finance and insurance (71171061), the Natural Science Fund of Guangdong Province-Noncooperative differential game theory of Markov jump linear systems with application to economics (S2011010004970).