| Riccati equation is a differential equation introduced by Riccati when he investigated curvature in 1710s.Since then,it has been one of the most important problems for mathematicians and its early reseaches promoted the development of differential equations.In 1950,when Kalman studied LQ control problem,he introduced Riccati equation to obtain the optimal feedback control of LQ control problem.Moreover,Riccati equation has been widely utilized in the fields of filtering theory,dynamical systems,quantum mechanics and so on.With the development of control theory,the time-inconsistent control problem has been the interdisciplinary frontier of mathematics and finance,but it is still in the exploratory stage.In order to address the time-inconsistent LQ control problem,on the basis of analyzing the essential relationship between time-consistent LQ control problem and Riccati equation,we propose a class of indefinite equilibrium Riccati equation(?)where the involved matrices have the corresponding scales in order to accord with the operation law of matrixes.The indefiniteness of the above equation is mainly caused by the indefiniteness of M,HP+M is not necessarily classically invertible,and its inverse is a generalized inverse.Then,the integral operator contains unknown exponential function.This inevitably leads to the difficulty of obtaining prior estimates.We mainly discuss three special cases of the above indefinite Riccati differential equation,including(1)without any integral term;(2)H=0;(3)H is reversible.For case(1),we discuss the existence and uniqueness of symmetric semi-positive definite continuous solutions of indefinite Riccati differential equations.The result is an extension of the existing results.For case(2),it is a new equilibrium Riccati differential equation.The obtained equilibrium Riccati equation has a unique symmetric semi-positive definite continuous solution.On this base,a linear feedback control is constructed and proved to be the equilibrium control of a class of non-exponential discounted time-inconsistent LQ control problems.This is a new result of the time-inconsistent control problem.For case(3),under appropriate conditions,the symmetric continuous solution of the indefinite equilibrium Riccati equation are obtained.In summary,the above results are of great significance to the investigation of time-inconsistent LQ control problem and lay a foundation for the deeply study of the dynamic behavior of complex dynamical systems. |
