Indefinite Linear Quadratic Control Problem Of Stochastic System And Its Applications

This dissertation is devoted to the study of indefinite LQ control problem of stochastic system.It is mainly on the indefinite LQ control problem of mean-field stochastic system with jump,the indefinite LQ control problem of mean-field stochastic system with infinite horizon,the indefinite LQ control problem of mean-field backward stochastic system,the generalized indefinite LQ control problem with asymmetric information,and the LQ control problem of backward stochastic system with partial information.The main contents,results and innovations are listed as follows in the order of chapters:1.An indefinite LQ control problem of mean-field stochastic system with jump is studied by using the equivalent cost functional method.By introducing an invertible linear transformation,a feedback optimal control of mean-field LQ control problem under standard conditions is obtained.Different from classical techniques of solving LQ control problems,the linear transformation is more convenient for calculations.When the weighting matrices are allowed to be indefinite,we construct a family of equivalent cost functionals.An optimal control for indefinite LQ control problem is thus obtained by seeking an equivalent cost functional which satisfies standard conditions.Notably,our theoretical results provide a new and effective way to investigate the unique solvability of mean-field FBSDE.Specifically,in some cases,a meanfield FBSDE can be regarded as the stochastic Hamiltonian system of a mean-field LQ control problem.We can solve the corresponding LQ control problem by the equivalent cost functional method,and thus obtain the unique solvability of original mean-field FBSDE.Finally,some examples are provided to illustrate our results.2.We study an indefinite LQ control and stabilization problem of mean-field stochastic system with infinite horizon.Inspired by the equivalent cost functional,we discuss the asymptotic properties of Riccati equations arising in finite horizon mean-field LQ control problem,which leads to generalized algebraic Riccati equations.We further consider the stabilizing solution and maximal solution of generalized algebraic Riccati equations.A relationship for algebraic Riccati equations between indefinite and positive-definite mean-field LQ control problem with infinite horizon is established.With this relationship,necessary and sufficient conditions for mean-square stabilization of mean-field stochastic system are first given when the weighting matrices are allowed to be indefinite.Due to the appearance of mean-field terms,the generalized algebraic Riccati equations are partially coupled.On the other hand,we have to decompose the state process in proving the mean-square stabilization of mean-field system.3.An indefinite LQ control problem of mean-field backward stochastic system is investigated.The connection between LQ control problems of mean-field backward stochastic system and forward stochastic system is established under the uniform convexity of cost functional.An optimal control and optimal cost are further obtained via two Riccati equations with given terminal values,an adjoint process and a mean-field BSDE.To derive a feedback representation of optimal control,we further introduce two Riccati equations with given initial values and a mean-field SDE.By using the equivalent cost functional method,we get the unique solvabilities of Riccati equations with given initial values.With these results,a feedback representation of optimal control is first established.On the other hand,sufficient conditions for the uniform convexity of cost functional are also given in terms of Riccati equations,for the first time.4.A generalized indefinite LQ control problem is studied,in which two cooperative controllers are posed with two types of constraints both:admissibility constraints and information constraints.By the complete square method and the variational technique,we derive a generalized stochastic maximum principle.Different from existing literature,both projection and conditional expectation operators are involved in stochastic Hamiltonian system.Focusing on information constraints,the stochastic Hamiltonian system contains two different filtering,which brings new difficulty in decoupling.To overcome this difficulty,we introduce three different conditional expectation terms.By decoupling the stochastic Hamiltonian system,we obtain a Lyapounov-Riccati system,which includes a Lyapounov equation and three Riccati equations.An equivalent relationship between the solvability of Lyapounov-Riccati system and uniform convexity of cost functional is further obtained with asymmetric information.Explicit formulas of optimal control and optimal cost are established via solution of Lyapounov-Riccati system.The information redundancy and monotonicity in LQ control problem are first addressed.Finally,an asset management problem is investigated by applying the theoretical results.5.An LQ control problem of backward stochastic system with partial information is solved.Using the complete square method and the stochastic maximum principle,we give sufficient and necessary conditions for optimal control.Different from control problems of forward stochastic systems,we obtain a stochastic Hamiltonian system with filtering,which is coupled in the initial value.To derive a feedback representation of optimal control,we decouple the stochastic Hamiltonian system twice.A Lyapounov equation,two Riccati equations,a BSDE with filtering and an SDE with filtering are obtained.A feedback representation of optimal control in terms of Lyapounov equation,Riccati equations,BSDE and SDE are established,as well as an explicit formula of optimal cost.Finally,two examples are provided to illustrate our theoretical results.Compared with previous research results,we need only to impose some usual conditions on system equation and cost functional.Further,we give the unique solvability of BSDE with filtering,which has not been investigated.