A Class Of Generalized Time-inconsistent LQ Control Problem

Time-inconsistent control problems are a hot topic in the forefront of mathematics and finance,which is widely used in finance,management and other fields.This thesis studies a class of time-inconsistent linear-quadratic(LQ)control problems with non-exponential discount,where its controlled system is the following linear ordinary differential equation(?)Our problem is to minimize the following objective functional (?)The matrix and vector here satisfy the appropriate dimensions,the discount factor M is not necessarily positive definite,and Q is not necessarily semi-positive definite.As an optimization problem,we mainly pay attention to three basic problems:the existence,uniqueness and optimality of the solution.This thesis focuses on the existence of the solution.The basic characteristic of the time-inconsistent control problems is not suitable for dynamic programming principle,and often only the closed-loop solution can be introduced.There are two main methods for investigating the optimal control problems:the variational method based on the maximum principle and the dynamic programming method based on the dynamic programming principle.Both of them can not be directly applied to the time-inconsistent control problems,which is the main difficulty to study the time-inconsistent control problems.In order to break through the difficulty,this thesis introduces the linear closed-loop equilibrium control,and reasonably introduces the two-point boundary value problem(not necessarily corresponding to the original problem),the Fredholm integral equation and the Riccati equation.Firstly,the equivalent relationship between the solvability of the original problem and the solvability of these three types of problems is obtained.Further,on the basis of discussing the solvability of the Riccati equation,the closed-loop control is constructed and it is proved that the control is the closed-loop equilibrium control of the original problem.That is,the existence of the closed-loop equilibrium control for a class of generalized time-inconsistent LQ control problems is obtained.The results and research methods of this thesis lay a certain foundation for further study of the time-inconsistent control problems.