Studies On Optimal Control Problems Of Some Stochastic Differential-Algebraic Systems

Since the early 1970s,Rosenbrock proposed the concept of differential-algebraic systems,due to its applications in fields of science and engineering,has received wide attention from some academic fields.In recent years,with the development of control theory and the introduction of the differential-algebraic equations,many scholars have devoted themselves to the optimal control problems of differential-algebraic system-s.Based on this,optimal control problems of some stochastic differential-algebraic systems and solvability of the correlated generalized Riccati equations are mainly in-vestigated.The present Ph.D.thesis is divided into eight chapters.In Chapter 1,we introduce the background as well as development status of opti-mal control theory for differential-algebraic systems,and then present the main contents of our thesis.In Chapter 2,we intend to linear quadratic optimal control problem for stochastic differential-algebraic systems.By virtue of Schur's lemma,square completion tech-nique and Moore-Penrose generalized inverse,sufficient conditions for well-posedness of the optimal control problem are obtained.Meanwhile,a relatively comprehensive solvability analysis for the introduced generalized Riccati equations is given.Our re-sults improve and generalize some of the known conclusions.In Chapter 3,an optimal control problem for nonlinear stochastic differential-algebraic systems with index 1 is considered.Based on the existing conclusions for optimal control of standard stochastic differential equations,necessary and sufficient conditions for optimal control are established by means of some proper transforma-tions.In the meantime,new kinds of Riccati equations are given for the application of linear quadratic optimal control.The results of this chapter extend some of the known conclusions.In Chapter 4,we are concerned with the optimal control for a class of discrete stochastic differential-algebraic systems,where the singular matrix is non-square and the system has time-delay.With the aid of augmented matrix techniques and the equiv-alent principle for optimal control problems,solvability of the optimal control problem is obtained through some transformations and dynamic programming.The results gen-eralize and improve some of the known conclusions.In Chapter 5,we take into account of the optimal control problem for affine stochastic differential-algebraic systems with Markovian jumps.Firstly,the exis-tence and uniqueness of stochastic singular affine equations with Markovian jumps are proved.Then,via square completion technique and the generalized It???'s formu-la,we derive sufficient conditions for the optimal control in finite horizon and infinite horizon.At the same time,we consider the solvability of the introduced generalized s-tochastic Riccati equations.Finally,a leader-follower differential game problem arising from game background is presented.The results of this chapter generalize and improve some of the known conclusions.In Chapter 6,we deal with the Nash differential game of an infinite horizon system determined by stochastic differential-algebraic systems with Markovian jumps.Utiliz-ing the generalized It???'s formula as well as the coupled generalized Riccati equations,existence of Nash strategies for a system of linear stochastic differential-algebraic e-quations with Markovian jumps is established.As an application,the stochastic H₂/H_?control with state,control and external disturbance-dependent noise is discussed.And the results generalize some of the known conclusions.In Chapter 7,an optimal control problem for general two-steps differential-algebraic systems is investigated.We establish the first-order necessary optimality con-ditions by using nonsmooth analysis and variational techniques firstly.Then based on the concepts of Drazin inverse and index for matrix,the generalized second-order nec-essary conditions are derived for linear differential-algebraic equations.Also,we give a simple discussion for unfixed switching point case.It is the first time that the second-order necessary optimality conditions are promoted to differential-algebraic systems.In the final Chapter 8,a summary on the main content of this dissertation and future research prospects are presented.