Studies On Partially Observed Optimal Controls Of Mean-Field Stochastic Systems And A Class Of Fractional-Order Control Systems

Stochastic maximum principle(SMP)is one of the powerful approach to studying optimal control problems.Since the introduction of the mean-field backward stochastic differential equations,the SMP for mean-field models have been springing up.Based upon this,this thesis mainly studies the optimal control problems of mean-field stochas-tic systems.Meanwhile,the exact controllability and continuous dependence are also investigated for a class of fractional neutral integro-differential systems.Our present PH.D.thesis consists of seven chapters.Chapter 1 aims to introduce the historical background,current situations of the optimal control theory,and states the main results of our thesis.Chapter 2 is concerned with the partially observed linear-quadratic optimal control problems.First,based on the classical spike variational method,backward separation approach as well as filtering technique,the necessary and sufficient conditions of the optimal control problems with the non-convex domain is derived.Second,by means of decoupling technique,we obtain two Riccati equations,which are uniquely solvable under certain conditions.Finally,the optimal cost functional is represented by the so-lutions of the Riccati equations for the special case.Our results generalize and improve the partial conclusions of related papers.Chapter 3 considers a class of partially observed risk-sensitive optimal control problems of mean-field type.By virtue of the Girsanov's theorem and the classical spike variational technique,the maximum principle for the partially observed control problems is established.Then,the sufficient condition for the optimality is also ob-tained under some concavity conditions.Finally,we give the linear-quadratic optimal portfolio problems arising from financial background.The results are new.Chapter 4 studies the risk-sensitive control problem of mean-field stochastic dif-ferential delay equation under partial information.We establish the(SMP)under the assumptions that the control domain is not convex and the value function is not smooth.Then based on Ito's formula and inequality technique,some continuous dependence,existence and uniqueness results are proved for the mean-field stochastic differential de-lay equation.Finally,we establish the verification theorem for the mean-field stochastic differential delay equation with a clever construction of the Hamiltonian function.We improve some known results.Chapter 5 deals with an optimal control problem of an infinite horizon system de-scribed by mean-field backward stochastic differential equation with delay and partial information.First,the existence and uniqueness of mean-field backward stochastic d-ifferential equation with average delay is proved.Second,the sufficient and necessary conditions are obtained on infinite horizon.Finally,we study an linear-quadratic opti-mal control problem to derive the optimal control,which is explicitly expressed by the solution of a mean-field forward-backward stochastic filtering equation.The results extend and improve the related conclusions in the previous literatures.Chapter 6 is devoted to the fractional neutral integro-differential equations with state-dependent delay.With the help of the resolvent operator and some analytic meth-ods,the exact controllability and continuous dependence are investigated for a fraction-al differential systems.The results renew the conclusions of related article.In the final Chapter 7,a summary and future research are presented.