Mean-field Linear-quadratic Optimal Control Research Articles

Infinite Horizon Mean-Field Linear Quadratic Optimal Control Problems with Jumps and the Related Hamiltonian Systems

Infinite Horizon Mean-Field Linear Quadratic Optimal Control Problems with Jumps and the Related Hamiltonian Systems

Turnpike Properties for Mean-Field Linear-Quadratic Optimal Control Problems

Turnpike Properties for Mean-Field Linear-Quadratic Optimal Control Problems

A Nonhomogeneous Mean-Field Linear-Quadratic Optimal Control Problem and Application

A Nonhomogeneous Mean-Field Linear-Quadratic Optimal Control Problem and Application

Optimal control of mean-field jump-diffusion systems with noisy memory

ABSTRACTThis paper is concerned with an optimal control problem under mean-field jump-diffusion systems with delay and noisy memory. First, we derive necessary and sufficient maximum principles using Malliavin calculus technique. Meanwhile, we introduce a new mean-field backward stochastic differential equation as the adjoint equation which involves not only partial derivatives of the Hamiltonian function but also their Malliavin derivatives. Then, applying a reduction of the noisy memory dynamics to a two-dimensional discrete delay optimal control problem, we establish the second set of necessary and sufficient maximum principles under partial information. Moreover, a natural link between the above two approaches is established via the adjoint equations. Finally, we apply our theoretical results to study a mean-field linear-quadratic optimal control problem.

Indefinite Mean-Field Stochastic Linear-Quadratic Optimal Control: From Finite Horizon to Infinite Horizon

In this paper, the finite-horizon and the infinite-horizon indefinite mean-field stochastic linear-quadratic optimal control problems are studied. Firstly, the open-loop optimal control and the closed-loop optimal strategy for the finite-horizon problem are introduced, and their characterizations, difference and relationship are thoroughly investigated. The open-loop optimal control can be defined for a fixed initial state, whose existence is characterized via the solvability of a linear mean-field forward-backward stochastic difference equation with stationary conditions and a convexity condition. On the other hand, the existence of a closed-loop optimal strategy is shown to be equivalent to any one of the following conditions: the solvability of a couple of generalized difference Riccati equations, the finiteness of the value function for all the initial pairs, and the existence of the open-loop optimal control for all the initial pairs. It is then proved that the solution of the generalized difference Riccati equations converges to a solution of a couple of generalized algebraic Riccati equations. By studying another generalized algebraic Riccati equation, the existence of the maximal solution of the original ones is obtained together with the fact that the stabilizing solution is the maximal solution. Finally, we show that the maximal solution is employed to express the optimal value of the infinite-horizon indefinite mean-field linear-quadratic optimal control. Furthermore, for the question whether the maximal solution is the stabilizing solution, the necessary and the sufficient conditions are presented for several cases.

Indefinite Mean-Field Stochastic Linear-Quadratic Optimal Control

This paper is concerned with the discrete-time indefinite mean-field linear-quadratic optimal control problem. The so-called mean-field type stochastic control problems refer to the problem of incorporating the means of the state variables into the state equations and cost functionals, such as the mean-variance portfolio selection problems. A dynamic optimization problem is called to be nonseparable in the sense of dynamic programming if it is not decomposable by a stage-wise backward recursion. The classical dynamic-programming-based optimal stochastic control methods would fail in such nonseparable situations as the principle of optimality no longer applies. In this paper, we show that both the well-posedness and the solvability of the indefinite mean-field linear-quadratic problem are equivalent to the solvability of two coupled constrained generalized difference Riccati equations and a constrained linear recursive equation. We characterize the optimal control set completely, and obtain a set of necessary and sufficient conditions on the mean-variance portfolio selection problem. The results established in this paper offer a more accurate solution scheme in tackling directly the issue of nonseparability and deriving the optimal policies analytically for the mean-variance-type portfolio selection problems.

Discrete-time mean-field Stochastic linear–quadratic optimal control problems, II: Infinite horizon case

This paper first presents results on the equivalence of several notions of L2-stability for linear mean-field stochastic difference equations with random initial value. Then, it is shown that the optimal control of a mean-field linear–quadratic optimal control with an infinite time horizon uniquely exists, and the optimal control can be expressed as a linear state feedback involving the state and its mean, via the minimal nonnegative definite solution of two coupled algebraic Riccati equations. As a byproduct, the open-loop L2-stabilizability is proved to be equivalent to the closed-loop L2-stabilizability. Moreover, the minimal nonnegative definite solution, the maximal solution, the stabilizing solution of the algebraic Riccati equations and their relations are carefully investigated. Specifically, it is shown that the maximal solution is employed to construct the optimal control and value function to another infinite time horizon mean-field linear–quadratic optimal control. In addition, the maximal solution being the stabilizing solution, is completely characterized by properties of the coefficients of the controlled system. This enriches the existing theory about stochastic algebraic Riccati equations. Finally, the notion of exact detectability is introduced with its equivalent characterization of stochastic versions of the Popov–Belevitch–Hautus criteria. It is then shown that the minimal nonnegative definite solution is the stabilizing solution if and only if the uncontrolled system is exactly detectable.

Fully Coupled Mean-Field Forward-Backward Stochastic Differential Equations and Stochastic Maximum Principle

We discuss a new type of fully coupled forward-backward stochastic differential equations (FBSDEs) whose coefficients depend on the states of the solution processes as well as their expected values, and we call them fully coupled mean-field forward-backward stochastic differential equations (mean-field FBSDEs). We first prove the existence and the uniqueness theorem of such mean-field FBSDEs under some certain monotonicity conditions and show the continuity property of the solutions with respect to the parameters. Then we discuss the stochastic optimal control problems of mean-field FBSDEs. The stochastic maximum principles are derived and the related mean-field linear quadratic optimal control problems are also discussed.

Linear-Quadratic Control of Discrete-Time Stochastic Systems with Indefinite Weight Matrices and Mean-Field Terms

In this paper, the linear-quadratic optimal control problem is considered for discrete-time stochastic systems with indefinite weight matrices in the cost function and mean-field terms in both the cost function and system dynamics. A set of generalized difference Riccati equations (GDREs) is introduced in terms of algebraic equality constraints and matrix pseudo-inverse. It is shown that the solvability of the GDRE is not only sufficient but also necessary for the well-posedness of the indefinite mean-field linear-quadratic optimal control problem and the existence of optimal feedback as well as open-loop controls.