Mean-field Stochastic Systems Research Articles

Finite-time annular domain [formula omitted] filtering for mean-field stochastic systems with Wiener and Poisson noises

This paper investigates the finite-time annular domain H2/H∞ filtering for mean-field stochastic systems with external disturbance, Wiener and Poisson noises. Initially, the novel concept of finite-time annular domain (FTAD) H2/H∞ filtering is introduced, which ensures the system’s mean square finite-time annular domain boundedness (MSFTADB) and minimizes the H2 and H∞ filtering performance indices. Next, using the Itô-Levy formula and the reverse differential Gronwall inequality, a less conservative sufficient condition for the existence of finite-time annular domain H2/H∞ filter is derived. Furthermore, a new algorithm is devised to establish the relationship between the ranges of adjustable parameters and the minimum value of H2 performance index. Finally, a practical example is provided to verify the effectiveness of the proposed method.

H∞ Filtering of Mean Field Stochastic Differential Systems

This paper addresses the H∞ filtering problem for mean field stochastic differential systems that involve both state-dependent and disturbance-dependent noise. We assume that the state as well as the measurement output is distracted by an uncertain exogenous disturbance. Firstly, a sufficient condition for the stochastic-bounded real lemma is given. Next, H∞ filtering, which is built upon a stochastic-bounded real lemma, is put forward by two linear matrix inequalities. Furthermore, the validation of the theoretical analysis is demonstrated with two examples.

Online Pareto optimal control of mean-field stochastic multi-player systems using policy iteration

Online Pareto optimal control of mean-field stochastic multi-player systems using policy iteration

Infinite Horizon H2/H∞ Control for Discrete-Time Mean-Field Stochastic Systems

In this paper, we deal with the H2/H∞ control problem in the infinite horizon for discrete-time mean-field stochastic systems with (x,u,v)-dependent noise. First of all, a stochastic-bounded real lemma, which is the core of H∞ analysis, is derived. Secondly, a sufficient condition in terms of the solution of coupled difference Riccati equations (CDREs) is obtained for solving the H2/H∞ control problem above. In addition, an iterative algorithm for solving CDREs is proposed and a numerical example is given for verification of the feasibility of the developed results.

Relationship between Nash Equilibrium Strategies and H2/H∞ Control of Mean-Field Stochastic Differential Equations with Multiplicative Noise

The relationship between finite-horizon mean-field stochastic H2/H∞ control and Nash equilibrium strategies is investigated in this technical note. First, the finite-horizon mean-field stochastic bounded real lemma (SBRL) is established, which is key to developing the H∞ theory. Second, for mean-field stochastic differential equations (MF-SDEs) with control- and state-dependent noises, it is revealed that the existence of Nash equilibrium strategies is equivalent to the solvability of generalized differential Riccati equations (GDREs). Furthermore, the existence of Nash equilibrium strategies is equivalent to the solvability of H2/H∞ control for MF-SDEs with control- and state-dependent noises. However, for mean-field stochastic systems with disturbance-dependent noises, these two problems are not equivalent. Finally, a sufficient and necessary condition is presented via coupled matrix-valued equations for the finite-horizon H2/H∞ control of mean-field stochastic differential equations with disturbance-dependent noises.

Nonfragile Finite-Time Stabilization for Discrete Mean-Field Stochastic Systems

In this paper, the problem of non-fragile finite-time stabilization for linear discrete mean-field stochastic systems is studied. The uncertain characteristics in control parameters are assumed to be random satisfying the Bernoulli distribution. A new approach called the “state transition matrix method” is introduced and some necessary and sufficient conditions are derived to solve the underlying stabilization problem. The Lyapunov theorem based on the state transition matrix also makes a contribution to the discrete finite-time control theory. One practical example is provided to validate the effectiveness of the newly proposed control strategy.

Rate of Convergence in the Smoluchowski-Kramers Approximation for Mean-field Stochastic Differential Equations

In this paper we study a second-order mean-field stochastic differential systems describing the movement of a particle under the influence of a time-dependent force, a friction, a mean-field interaction and a space and time-dependent stochastic noise. Using techniques from Malliavin calculus, we establish explicit rates of convergence in the zero-mass limit (Smoluchowski-Kramers approximation) in the Lp\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^p$$\\end{document}-distances and in the total variation distance for the position process, the velocity process and a re-scaled velocity process to their corresponding limiting processes.

General Time-Symmetric Mean-Field Forward-Backward Doubly Stochastic Differential Equations

In this paper, a general class of time-symmetric mean-field stochastic systems, namely the so-called mean-field forward-backward doubly stochastic differential equations (mean-field FBDSDEs, in short) are studied, where coefficients depend not only on the solution processes but also on their law. We first verify the existence and uniqueness of solutions for the forward equation of general mean-field FBDSDEs under Lipschitz conditions, and we obtain the associated comparison theorem; similarly, we also verify those results about the backward equation. As the above two comparison theorems’ application, we prove the existence of the maximal solution for general mean-field FBDSDEs under some much weaker monotone continuity conditions. Furthermore, under appropriate assumptions we prove the uniqueness of the solution for the equations. Finally, we also obtain a comparison theorem for coupled general mean-field FBDSDEs.

Mean-Field Control for Stochastic Delay Systems via Static Output Feedback Strategy

In this paper, we consider mean-field control based on the static output feedback (SOF) strategy for stochastic delay systems. First, we define a stabilization problem via SOF gains in block-diagonal forms for systems with a single player, and then solve the problem of minimizing the upper bound of the cost function by cost-guaranteed cost control theory. For this problem, the necessary conditions for the sub-optimality are established using stochastic large-scale matrix equations. The obtained preliminary results are then used to study Pareto optimal strategies in cooperative games, for mean-field stochastic systems involving a large number of players. The primary contribution of this study is the derivation of a design method for decentralized strategies. Furthermore, a new low-order computational algorithm based on Newton's method is developed to obtain the decentralized strategy set. The cost degradation of the proposed decentralized SOF strategy set is then estimated. Finally, a simple numerical example is presented to demonstrate the usefulness and effectiveness of the proposed method. As a result, it is determined that the decentralized SOF strategy works well even when the number of players goes to infinity.

A general maximum principle for partially observed mean-field stochastic system with random jumps in progressive structure

<p style='text-indent:20px;'>We study the progressive optimal control for partially observed stochastic system of mean-field type with random jumps. The cost function and the observation are also of mean-field type. The control is allowed to enter the diffusion, jump coefficient and the observation. The control domain need not be convex. We obtain the maximum principle for the partially observable progressive optimal control by a special spike variation. The maximum principle in the progressive structure is different from the classical case.</p>

A general maximum principle for progressive optimal control of partially observed mean-field stochastic system with Markov chain

A general maximum principle for progressive optimal control of partially observed mean-field stochastic system with Markov chain

Spectral criteria to stability and observability of mean-field stochastic periodic systems

This paper addresses stability and observability of discrete-time mean-field stochastic systems with periodic coefficients and multiplicative noise. By constructing a monodromy operator associated to the periodic coefficients of the considered dynamics, spectral criterion and Lyapunov-type criterion are presented for asymptotic mean square stability and weak stability, respectively. Based on the Lyapunov criterion, a necessary and sufficient condition is supplied for regional stabilizability. Furthermore, Popov–Belevitch–Hautus (PBH) criterion is obtained for observability of mean-field stochastic periodic systems. By the proposed spectral criterion of stability/observability, the intrinsic relationships are clarified for stability/observability among deterministic periodic systems, stochastic periodic systems and mean-field stochastic periodic systems. Finally, as an application of PBH criterion, a Barbashin–Krasovskii stability theorem is established to indicate the connection between observability and stability of the considered systems.

A maximum principle for mean-field stochastic control system with noisy observation

This paper is concerned with an optimal control problem driven by mean-field stochastic differential equation, where the state is partially observed via a noisy process. A new feature of the control problem is that the drift coefficient of observation equation is allowed to be a linear growth function, which immediately causes a difficulty in studying the control problem. A backward separation method with a dominated growth rate idea is used to overcome the difficulty. A maximum principle for optimality of the control problem is derived by an approximate technique. An asset–liability management problem for an insurance firm is studied. An optimal premium strategy in a feedback form is explicitly obtained.

Dynamic optimization problems for mean-field stochastic large-population systems

This paper considers dynamic optimization problems for a class of control average meanfield stochastic large-population systems. For each agent, the state system is governed by a linear mean-field stochastic differential equation with individual noise and common noise, and the weight coefficients in the corresponding cost functional can be indefinite. The decentralized optimal strategies are characterized by stochastic Hamiltonian system, which turns out to be an algebra equation and a mean-field forward-backward stochastic differential equation. Applying the decoupling method, the feedback representation of decentralized optimal strategies is further obtained through two Riccati equations. The solvability of stochastic Hamiltonian system and Riccati equations under indefinite condition is also derived. The explicit structure of the control average limit and the related mean-field Nash certainty equivalence equation systems are discussed by some separation techniques. Moreover, the decentralized optimal strategies are proved to satisfy the approximate Nash equilibrium property. The good performance of the proposed theoretical results is illustrated by a practical example from the engineering field.

Noncooperative and Cooperative Multiplayer Minmax H∞ Mean-Field Target Tracking Game Strategy of Nonlinear Mean Field Stochastic Systems With Application to Cyber-Financial Systems

Noncooperative and Cooperative Multiplayer Minmax H_∞ Mean-Field Target Tracking Game Strategy of Nonlinear Mean Field Stochastic Systems With Application to Cyber-Financial Systems

Pareto Suboptimal Strategy for Uncertain Mean-Field Nonlinear Stochastic Systems

In this paper, a static output feedback (SOF) Pareto suboptimal strategy for an uncertain mean-field nonlinear stochastic system is discussed. It is assumed that the nonlinear function in the stochastic system for each player is unknown and constrained by a norm-bounded function. As a preliminary result, the single-player case problem is solved in terms of the guaranteed cost control technique. As a result, it is shown that the necessary conditions satisfying the suboptimality of the upper bound of the cost are established by stochastic coupled-matrix equations (SCMEs). Next, the aforementioned results are applied to a mean-field stochastic system with a large population. Notably, a new necessary condition is described by the solvability condition related to large-scale SCMEs. To avoid the treatment of high-order computation for solving SCMEs, a new reduced-order numerical technique based on the fixed point iteration method is introduced. Consequently, the centralized strategy set is computed, although a large population is considered. Finally, to demonstrate the effectiveness of the proposed scheme, an academic example is solved.

Robust Incentive Stackelberg Games With a Large Population for Stochastic Mean-Field Systems

A static output feedback (SOF) strategy for robust incentive Stackelberg games with a large population for mean-field stochastic systems is investigated. First, the saddle point equilibrium condition of external disturbance and control strategy is derived based on stochastic algebraic matrix equations (SAMEs). Then, a centralized SOF incentive Stackelberg strategy is derived through restructuring the follower’s strategies and the leader’s incentive strategy. Moreover, to avoid the high dimension of design procedure, a new designing algorithm of low-dimensional approximation SOF incentive Stackelberg strategy is proposed. It is shown that the difference in the equilibrium values between using the centralized SOF incentive Stackelberg strategy and using the low-dimensional approximation SOF incentive Stackelberg strategy is <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$O(\sqrt {\varepsilon })=O(1/\sqrt {N})$ </tex-math></inline-formula> , where <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$N$ </tex-math></inline-formula> denotes the population size. Finally, a numerical example with a large population size demonstrates the effectiveness of the proposed approximation SOF incentive Stackelberg strategy.

Optimal control for unknown mean-field discrete-time system based on Q-Learning

Solving the optimal mean-field control problem usually requires complete system information. In this paper, a Q-learning algorithm is discussed to solve the optimal control problem of the unknown mean-field discrete-time stochastic system. First, through the corresponding transformation, we turn the stochastic mean-field control problem into a deterministic problem. Second, the H matrix is obtained through Q-function, and the control strategy relies only on the H matrix. Therefore, solving H matrix is equivalent to solving the mean-field optimal control. The proposed Q-learning method iteratively solves H matrix and gain matrix according to input system state information, without the need for system parameter knowledge. Next, it is proved that the control matrix sequence obtained by Q-learning converge to the optimal control, which shows theoretical feasibility of the Q-learning. Finally, two simulation cases verify the effectiveness of Q-learning algorithm.

Linear-quadratic non-zero sum differential game for mean-field stochastic systems with asymmetric information

In this study, we consider a class of asymmetric information linear-quadratic non-zero sum stochastic differential game problems. The system dynamics are governed by a forward linear mean-field stochastic differential equation and the cost functional is quadratic. In order to facilitate some practical applications, the mean-field terms for the state process and control process are considered in both the system dynamics and cost functional. Using the classical calculus of variation and dual methods, the open-loop Nash equilibrium point can be expressed by introducing an auxiliary mean-field forward backward stochastic differential equation, which comprises one forward and two backward components. Due to some Riccati equations and ordinary differential equations that possess unique solutions, the Nash equilibrium point can also be represented in feedback form for several special cases under asymmetric information. The corresponding filtering equations are derived, and the existence and uniqueness of the solutions are proved. An investment problem in finance is discussed to demonstrate the good performance of the theoretical results.

Controllability Gramian and Kalman rank condition for mean-field control systems

This paper is concerned with the exact controllability of linear mean-field stochastic systems with deterministic coefficients. With the help of the theory of mean-field backward stochastic differential equations (MF-BSDEs, for short) and some delicate analysis, we obtain a mean-field version of the Gramian matrix criterion for the general time-variant case, and a mean-field version of the Kalman rank condition for the special time-invariant case.