Optimal Control Based On Uncertain Random Dynamical Systems

Probability theory and uncertainty theory are important tools for dealing with indeterminacy.Stochastic optimal control theory is formed on the basis of probability theory,while uncertain optimal control theory is established based on uncertainty theory.Stochastic optimal control theory and uncertain optimal control theory are suitable for dealing with the optimal control problems with a single indeterminate factor(random factor or uncertain factor).The concepts of uncertain random variable and uncertain random process defined in chance theory on the basis of uncertainty theory and probability theory provide the possibility to describe complex systems with both random factors and uncertain factors.The operational laws of chance distribution and expected value offer important theoretical support to deal with such complex systems effectively.Based on the research of chance theory and optimal control theory,this thesis studies the optimal control problem with both random factors and uncertain factors.This thesis focuses on the optimal control problems and the two-person zero-sum game problems of uncertain random dynamical systems,and their applications.The details are as follows:1.An optimal control problem of a discrete-time uncertain random dynamical system is studied.Then recurrence equations for the optimal value function are derived by chance theory and principle of dynamic programming.And the exact solutions for two types of discrete-time uncertain random optimal control problems are obtained.2.An optimal control problem of a discrete-time uncertain random dynamical system with time-delays is studied.It is found that the problem is equivalent to an optimal control problem without time-delay.Subsequently,algorithms for finding the optimal solutions of discrete-time uncertain random optimal control problems with time-delays are designed using recurrence equations.3.An optimal control problem of a continuous-time uncertain random dynamical system is studied.The principle of optimality and equation of optimality for the optimal value function are derived.And the exact solutions for two types of continuous-time uncertain random optimal control problems are obtained.4.An optimal control problem of an uncertain random dynamical system with jump is studied.An equation of optimality for the optimal value function is derived by chance theory and principle of optimality.And the exact solutions for the linear uncertain random optimal control problem with jump and the linear-quadratic uncertain random optimal control problem with jump are obtained.5.The two-person zero-sum game problems of discrete-time and continuous-time uncertain random dynamical systems are respectively studied.Then recurrence equations and equations of optimality for solving the discrete-time uncertain random two-person zero-sum game problem and continuous-time uncertain random two-person zero-sum game problem are derived,respectively.In addition,this thesis investigates the applications of uncertain random optimal control and two-person zero-sum game to advertising investment and economics of counterterrorism,respectively.