Stabilization Control for Linear Continuous-Time Mean-Field Systems

Abstract

This paper investigates the stabilization and control problems for linear continuous-time mean-field systems (MFS). Under standard assumptions, necessary and sufficient conditions to stabilize the mean-field systems in the mean square sense are explored for the first time. It is shown that, under the assumption of exact detectability (exact observability), the mean-field system is stabilizable if and only if a coupled algebraic Riccati equation (ARE) admits a unique positive semi-definite solution (positive definite solution), which coincides with the classical stabilization results for standard deterministic systems and stochastic systems.One of the key techniques in the paper is the obtained solution to the forward and backward stochastic differential equation (FBSDE) associated with the maximum principle for an optimal control problem. Actually, with the analytical FBSDE solution, a necessary and sufficient solvability condition of the optimal control, under mild conditions, is derived. Accordingly, the stabilization condition is presented by defining an Lyaponuv functional via the solution to the FBSDE and the optimal cost function.It is worth of pointing out that the presented results are different from the previous works [1] for stabilization and also different from the works [2], [3], [4] on optimal control.

Indefinite Linear Quadratic Mean Field Social Control with Multiplicative Noise

“…Proof. See Appendix C. Remark 4.2: In [33], some similar results were given for the stabilization of mean field systems. However, only the limiting problem was considered in their work and the mean field term in dynamics and costs is Ex(t) instead of x (N ) (t).…”

“…where Φ i satisfies (B.2). By (38) and the arbitrariness of x i0 we obtain that E On the other hand, (33) and (28) gives E x (N ) (t) −x(t) Proof of Theorem 4.3. (iii)⇒(i) has been proved in Theorem 4.1.…”

“…This procedure shares a similar philosophy with the direct method [24], [25]. In recent years, some progress for optimal LQ control has been made by tackling FBSDEs [48], [11], [51], [6], [31], [39], [33].…”

Event-Based Consensus for General Linear Multiagent Systems under Switching Topologies

“…Also its distributed cooperative control [1,2] has attracted more and more researchers' attention. For example, the controllability [3][4][5][6][7][8][9][10][11] and the consensus problem [12][13][14][15][16][17][18][19] are widely studied in multi-agent systems. Among them, as the basic problem of multi-agent cooperative control, consensus is widely used in formation control [20], cluster control, sensor network and other aspects, which is a research hot issue in the control discipline at present.…”

Variance-constrained $H_{\infty}$ filtering for multi-rate time-varying networked systems based on stochastic protocols

“…Filtering or state estimation problems have long been of far-reaching significance in signal processing, communication, control, and other fields and received many attentions in past few decades [1,2,8,18,22,31]. The celebrated Kalman filtering approach is one of the most popular ways to deal with the state estimation of a dynamic system [3,5,9].…”