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Mean Field Linear Quadratic Control: Uniform Stabilization and Social Optimality

This paper is concerned with uniform stabilization and social optimality for general mean field linear quadratic control systems, where subsystems are coupled via individual dynamics and costs, and the state weight is not assumed with the definiteness condition. For the finite-horizon problem, we first obtain a set of forward-backward stochastic differential equations (FBSDEs) from variational analysis, and construct a feedback-type control by decoupling the FBSDEs. For the infinite-horizon problem, by using solutions to two Riccati equations, we design a set of decentralized control laws, which is further proved to be asymptotically social optimal. Some equivalent conditions are given for uniform stabilization of the systems in different cases, respectively. Finally, the proposed decentralized controls are compared to the asymptotic optimal strategies in previous works.


Introduction
Mean field games have drawn increasing attention in many fields including system control, applied mathematics and economics [6], [8], [14]. The mean field game involves a very large population of small interacting players with the feature that while the influence of each one is negligible, the impact of the overall population is significant. By combining mean field approximations and individual's best response, the dimensionality difficulty is overcome. Mean field games and control have found wide applications, including smart grids [29], [10], [26], finance, economics [15], [9], [34], and social sciences [5], etc. By now, mean field games have been intensively studied in the LQ (linear-quadratic) framework [20], [27], [35], [13], [7], [31]. Huang et al. developed the Nash certainty equivalence (NCE) based on the fixed-point method and designed an ǫ-Nash equilibrium for mean field LQ games with discount costs by the NCE approach [20]. The NCE approach was then applied to the cases with long run average costs [27] and with Markov jump parameters [35], respectively. The works [11], [7] employed the adjoint equation approach and the fixed-point theorem to obtain sufficient conditions for the existence of the equilibrium strategy over a finite horizon. For other aspects of mean field games, readers are referred to [22], [25], [41], [11] for nonlinear mean field games, [38] for oblivious equilibrium in dynamic games, [19], [36] for mean field games with major players, [18], [31] for robust mean field games. Besides noncooperative games, social optima in mean field models have also attracted much interest. The social optimum control refers to that all the players cooperate to optimize the common social cost-the sum of individual costs, which is a type of team decision problem [17]. Huang et al. considered social optima in mean field LQ control, and provided an asymptotic team-optimal solution [21]. Wang and Zhang [37] investigated the mean field social optimal problem where the Markov jump parameter appears as a common source of randomness. For further literature, see [23] for social optima in mixed games, [3] for team-optimal control with finite population and partial information. Most previous results on mean field games and control were given by using the fixed-point method [20], [27], [36], [21], [12], [11], [37]. However, the fixed-point analysis (e.g., from the contraction mapping theorem) is sometimes conservative, particularly for high-dimensional systems. In this paper, we solve the problem by decoupling directly high-dimensional forward-backward stochastic differential equations (FBSDEs). In recent years, some progress has been made for study of the optimal LQ control by tackling the FBSDEs. See [42], [44], [45], [32] for details. This paper investigates uniform stabilization and social optimality for linear quadratic mean field control systems, where subsystems (agents) are coupled via dynamics and individual costs. The state weight Q is not limited to positive semi-definite. This model can be taken as a generation of robust mean field control problems [18], [31], [33]. Since the weight Q in the cost functional is indefinite, the prior boundedness of the state is not implied directly by the finiteness of the cost, which brings about additional difficulty to show the social optimality of decentralized control. For the finite-horizon social control problem, we first obtain a set of FBSDEs by examining the variation of the social cost, and give a centralized feedback-type control laws by decoupling the FBSDEs. With mean field approximations, we design a set of decentralized control laws. By exploiting the uniform convexity property of the problem, the decentralized controls are further shown to have asymptotic social optimality. For the infinitehorizon case, we design a set of decentralized control laws by using solutions of two Riccati equations, which is shown to be asymptotically social optimal. Some equivalent conditions are further given for uniform stabilization of all the subsystems when the state weight Q is positive semi-definite or only symmetric. Furthermore, the explicit expressions of optimal social costs are given in terms of the solutions to two Riccati equations, and the proposed decentralized control laws are compared to the feedback strategies in previous works. Finally, some numerical examples are given to illustrate the effectiveness of the proposed control laws. The main contributions of the paper are summarized as follows.
• We first obtain necessary and sufficient existence conditions of finite-horizon centralized optimal control by variational analysis, and then design a feedback-type decentralized control by tackling FBSDEs with mean field approximations. • In the case Q ≥ 0, the necessary and sufficient conditions are given for uniform stabilization of the systems with the help of the system's observability and detectability. • In the case that Q is indefinite, the necessary and sufficient conditions are given for uniform stabilization of the systems using the Hamiltonian matrices. • The asymptotically optimal decentralized controls are obtained under very basic assumptions (without verifying the fixed-point condition). The corresponding social costs are explicitly given by virtue of the solutions to two Riccati equations. The organization of the paper is as follows. In Section II, the socially optimal control problem is formulated. In Section III, we construct asymptotically optimal decentralized control laws by tackling FBSDEs for the finitehorizon case. In Section IV, for the infinite-horizon case, the asymptotically optimal controls are designed and analyzed, and some equivalent conditions are further given for uniform stabilization in different cases. In Section V, some numerical examples are given to show the effectiveness of the proposed control laws. Section VI concludes the paper. The following notation will be used throughout this paper.
· denotes the Euclidean vector norm or Frobenius matrix norm. For a vector z and a matrix Q, z 2 Q = z T Qz, tr(Q) is the trace of the matrix Q, and Q > 0 (Q ≥ 0) means that Q is positive definite (positive semidefinite). For two vectors x, y, x, y = x T y. C([0, T ], R n ) is the space of all R n -valued continuous functions defined on [0, T ], and For convenience of presentation, we use C, C 1 , C 2 , · · · to denote generic positive constants, which may vary from place to place.

Problem Description
Consider a large population systems with N agents. Agent i evolves by the following stochastic differential equation: where x i ∈ R n and u i ∈ R r are the state and input of the ith agent.
are a sequence of independent 1-dimensional Brownian motions on a complete filtered probability space (Ω, F , {F t } 0≤t≤T , P). The cost function of agent i is given by where ρ > 0 and Q, R are symmetric matrices with appropriate dimensions. Q is allowed to be indefinite. R > 0, and η ∈ C ρ/2 ([0, ∞), R n ). Denote u = {u 1 , . . . , u N }.
The decentralized control set is given by For comparison, define the centralized control sets as , i = 1, · · · , N . In this paper, we mainly study the following problem. (P). Seek a set of decentralized control laws to optimize social cost for the system (1)- The related results can be extended to the case of multidimensional Brownian motions trivially. Here we consider that σ(t) is time-varying and satisfies some growth rate. For convenience of the statement, we assume W i is scalar and σ ∈ C ρ/2 ([0, ∞), R n ). For the finite-horizon problem, our results still hold for the case that the matrices A, B, G, · · · depend on t. Assume A1) The initial states of agents x i (0), i = 1, ..., N are mutually independent and have the same mathematical expectation.
3 The finite-horizon problem For the convenience of design, we first consider the following finite-horizon problem.
We first give equivalent conditions for the convexity of (P1).
where y (N ) = N j=1 y j /N and y i satisfies (ii) Problem (P1) is uniformly convex in u if and only if for any u i ∈ L 2 F (0, T ; R r ), there exists γ > 0 such that Proof. Let x i andx i be the state processes of agent i with the control v andv, respectively. Take any λ 1 ∈ [0, 1] and let λ 2 = 1 − λ 1 . Then Denote u = v −v, and y i = x i −x i . Thus, y i satisfies (4). By the definition of (uniform) convexity, the lemma follows. ✷ By examining the variation of J F soc , we obtain the necessary and sufficient conditions for the existence of centralized optimal control of (P1). To simplify the presentation later, we denote by Theorem 3.1 Suppose R > 0. Then (P1) has a set of optimal control laws if and only if Problem (P1) is convex in u and the following equation system admits a set of solutions (x i , p i , β j i , i, j = 1, · · · , N ): , and furthermore the optimal control is given byǔ i , i, j = 1, · · · , N ) is a set of solutions to the equation system Here α i , i = 1, · · · , N are to be determined. Denote by x i the state of agent i under the controlǔ i . For any u i ∈ L 2 F (0, T ; R r ) and θ ∈ R (θ = 0), let u θ i =ǔ i + θu i . Denote by x θ i the solution of the following perturbed state equation x θ i (0) =x i0 , i = 1, 2, · · · , N.
✷ As an approximation to x (N ) in (10), we obtain Then, by Theorem 3.2, the decentralized control law for agent i may be taken aŝ where P, K, and s are determined by (11)-(13), andx andx i respectively satisfy (15) and  (11)-(12) admit a solution, respectively. The set of decentralized control laws {û 1 , · · · ,û N } in (16) has asymptotic social optimality, i.e., and the corresponding social cost is given by where The infinite-horizon problem Based on the analysis in Section 3, we may design the following decentralized control laws for Problem (P): where P and Π are maximal solutions 1 to the equations and s,x ∈ C ρ/2 ([0, ∞), R n ) are determined by Here s(0) is to be determined, and the existence conditions of P, Π, s andx need to be investigated further.

Uniform stabilization of subsystems
We now list some basic assumptions for reference: ) is observable. Assumptions A2) and A3) are basic in the study of the LQ optimal control problem. We will show that under some conditions, A2) is also necessary for uniform stabilization of multiagent systems. In many cases, A3) may be weakened to the following assumption. (22) and (23) ✷ It is shown that the decentralized control laws (16) uniformly stabilize the systems (1) . Theorem 4.1 Let A1)-A3) hold. Then for any N , Proof. See Appendix B.

Asymptotic social optimality
Now we are in a position to state the asymptotic optimality of the decentralized control. Theorem 4.5 Let A1)-A3) hold. For Problem (P), the set of decentralized control laws {û 1 , · · · ,û N } given by (21) has asymptotic social optimality, i.e., Proof. We first prove that for u ∈ U c , J soc (u) which further implies that By (1) which leads to for any r ∈ [0, 1], (36) By J soc (u) < C 1 and basic SDE estimates, we can find a constant C such that

From (35) and (36) we obtain
which implies that for any r ∈ [0, 1], By taking integration with respect to r ∈ [0, 1], we obtain This together with A3) lead to which with (34) further gives By (1), we have From (38) and (39), we obtain that This together with A3) implies that which gives (33). From this with Theorem 4.1, By a similar argument to the proof of Theorem 3.3 combined with Lemma 4.2, the conclusion follows. ✷ If A3) is replaced by A3 ′ ), the decentralized control (21) still has asymptotic social optimality.
Corollary 2 Assume that A1)-A2), A3 ′ ) hold. The decentralized control (21) is asymptotically social optimal. Proof. Without loss of generality, we simply assume A + G = diag{A 1 , A 2 }, where A 1 − (ρ/2)I is Hurwitz, and −(A 2 − (ρ/2)I) is Hurwitz (If necessary, we may apply a nonsingular linear transformation as in the proof of Theorem 4.3). Write This and the observability of ( The other parts of the proof are similar to that of Theorem 4.5. ✷ For the case that Q are indefinite, we have the following result of asymptotic optimality. Theorem 4.6 Let A1)-A2), A3 ′′ ) hold. Assume (22)-(23) admit negative definite solutions P − < 0 and Π − < 0, respectively. Then, the set of decentralized control in (21) is asymptotically socially optimal. Furthermore, if {x i0 } have the same variance, then the asymptotic average social optimum is given by Proof. From the above assumptions and Theorem 4.4, the Riccati equation (22) admits a ρ-stabilizing solution P and a negative definite solution P − ; (23) has a ρstabilizing solution Π and a negative definite solution Π − . By a similar argument in the proof of Lemma 3.1, we obtain for any u ∈ U c , By [39,Theorem 8], the centralized optimal control exists and the optimal state is ρ-stable. Thus, we only need to consider the following control set For any u ∈ U ′ c satisfying J soc (u) ≤ N C, we have Denote By [19], there exists C 1 , C 2 > 0 such that This together with (41) gives Similarly, we have From this and (42), The remainder of the proof can follow by that of Theorem 3.3. For the case that {x i0 } have the same variance, from (18), the asymptotic average social optimum The work [21] investigated mean field LQ problem (P) with Q ≥ 0. To obtain asymptotic social optimality, they need Q > 0 and I − Γ is nonsingular. In Corollary 2, we have loosed the assumption to A3 ′ ), i.e., (A − (ρ/2)I, √ Q) and (A − (ρ/2)I, √ Q(I − Γ)) are detectable. In Theorem 4.6, we further give the condition for the case of indefinite Q. Particularly, for the scalar case, the condition is equivalent to (29)- (30). It can be verified that the assumption Q > 0 and I − Γ is nonsingular implies (29)- (30), but the converse is not true.

Comparison to previous solutions
In this section, we compare the proposed decentralized control laws with the feedback decentralized strategies in previous works. We first introduce a definition from [4]. Definition 4.1 For a control problem with an admissible control set U, a control law u ∈ U is said to be a representation of another control u * ∈ U if (i) they both generate the same unique state trajectory, and (ii) they both have the same open-loop value on this trajectory. For Problem (P), let f = 0, and G = 0. In [21,Theorem 4.3], the decentralized control laws are given by u i (t) = −R −1 B T (P x i (t) +s(t)), i = 1, · · · , N, (43) where P is the stabilizing solution of (22), ands =Kx † + φ. HereK satisfies in whichĀ = A − BR −1 B T P and φ(0) is to be determined by φ ∈ C ρ/2 ([0, ∞), R n ). By comparing this with (23)- (25), one can obtain thatK = Π − P ,x =x † and φ = s. From the above discussion, we have the equivalence of the two sets of decentralized control laws.

Proposition 4.1
The set of decentralized control laws {û 1 , · · · ,û N } in (21) is a representation of {ȗ 1 , · · · ,ȗ N } given by (43). Remark 4.4 The work [21] studied the problem (P) with Q ≥ 0 by the fixed-point approach. In Theorem 4.3, they have shown that the fixed-point equation admits a unique solution, when (A−(ρ/2)I, √ Q) is detectable and Ξ = Γ T Q + QΓ − Γ T QΓ ≤ 0. In fact, the above assumption is merely a sufficient condition to ensure A3 ′ ) (A − (ρ/2)I, Q − Ξ) is detectable). Remark 4.5 The work [24] investigated asymptotic solvability of mean field LQ games by the re-scaling method. They considered (1)-(2) with Q ≥ 0 and derived a low-dimensional ordinary differential equation system by dynamic programming. Actually, the method proposed in this paper can be viewed as a type of direct approach. Different from [24], we tackle directly high-dimensional FBSDEs along the line of maximum principle.

Concluding Remarks
In this paper, we have considered uniform stabilization and social optimality for mean field LQ multiagent systems. For finite-and infinite-horizon problems, we design the decentralized control laws by decoupling FBSDEs, respectively, which are further shown to be asymptotically optimal. Some equivalent conditions are further given for uniform stabilization of the systems in different cases. Finally, we compare such decentralized control laws with the asymptotic optimal strategies in previous works. An interesting generalization is to consider mean field LQ control systems with heterogeneous coefficients by the direct approach [16].Also, the variational analysis may be applied to construct decentralized control laws for the nonlinear social control model.