Study On The Existence And Generic Stability Of Solutions For Several Types Of Optimal Control Problems

Optimal control theory is to solve a class of functional extremum problems with constraints.So far,it has made great progress and has important applications in real life,such as improving system efficiency in industrial process.At present,part of the research on the optimal control problem focuses on the existence and stability of the optimal solution.In this paper,we mainly study the solvability and generic stability of optimal control problems,where generic stability refers to the stability on a dense residue set of Baire space.Fractional calculus is a generalization of traditional calculus.Different from integer calculus,fractional calculus has the advantages of describing memory and heredity.The system model described by fractional calculus can simulate reality more accurately,so it has practical significance to study it.Non-instantaneous impulse is a special kind of impulses,which reflects the process of keeping this change for a period of time after the system changes suddenly.The research on non-instantaneous impulse system has practical significance in many fields,so it has attracted more and more attention of mathematical workers.Based on these backgrounds,in this paper,we study a class of integer order and two classes of fractional order optimal control problems with non-instantaneous impulses.The following is the specific research content.In the first chapter,it introduces the research significance,research background and three main research issues of this paper.In the second chapter,it introduces some preparatory knowledge related to this paper,including set-valued mapping theory,basic theory of fractional differential equations and stochastic differential equations and other necessary theories.In the third chapter,we study the solvability and generic stability of a class of optimal control problems on integrable function spaces characterized by integer order differential equations with non instantaneous impulses.First,the state equation is transformed into an equivalent integral algebraic equation,and then the uniqueness of the solution of the equation is proved by using the fixed point theorem and Gronwall inequality.Then,by constructing a minimization sequence,we prove the existence of the solution of the optimal control problem.Then,by constructing a set-valued mapping,we obtain the general stability of the optimal solution set by using the related concepts and conclusions of the essential solution.Finally,an example is given to show that the conclusion is valid,and numerical experiments are carried out to simulate the behavior of state function when parameters change.In the forth chapter,it studies the solvability and generic stability of optimal control problems on continuous function spaces characterized by fractional differential equations with non-instantaneous impulses.The proof of the existence and uniqueness of solutions of state equations mainly uses contraction mapping principle,Gronwall inequality of weak singularity,Gamma function and other tools,while the existence and general stability of the solution of the optimal control problem are proved by using the minimization sequence method and the related conclusions of set-valued mapping.Similarly,a specific example is given to show that the conclusion is meaningful.The exact solution of the fractional differential equation in the example is mainly obtained by matlab software,and umerical experiments are carried out to simulate the approximation effect of state function when parameters change.In the fifth chapter,it studies the solvability and generic stability for a class of optimal control problems characterized by fractional stochastic differential equations with non-instantaneous impulses.Firstly,the meaning of the solution of the state equation is given,and the uniqueness of the solution is proved.Then,by constructing Picard iterative sequence of stochastic process,the existence of the solution of the equation is proved by using stochastic analysis tools;Then,the continuous function space with stochastic effect is proved by minimizing the sequence.Finally,the stability of the optimal control problem is discussed.The sixth chapter is the summary of this paper.