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Mean-Field Stochastic Linear Quadratic Optimal Control Problems: Open-Loop Solvabilities

This paper is concerned with a mean-field linear quadratic (LQ, for short) optimal control problem with deterministic coefficients. It is shown that convexity of the cost functional is necessary for the finiteness of the mean-field LQ problem, whereas uniform convexity of the cost functional is sufficient for the open-loop solvability of the problem. By considering a family of uniformly convex cost functionals, a characterization of the finiteness of the problem is derived and a minimizing sequence, whose convergence is equivalent to the open-loop solvability of the problem, is constructed. Then, it is proved that the uniform convexity of the cost functional is equivalent to the solvability of two coupled differential Riccati equations and the unique open-loop optimal control admits a state feedback representation in the case that the cost functional is uniformly convex. Finally, some examples are presented to illustrate the theory developed.

In [28], two coupled differential Riccati equations are derived by decoupling the optimality system. It is shown that under certain conditions, the two Riccati equations are uniquely solvable and Problem (MF-LQ) admits a unique optimal control which has a state feedback representation. To be precise, if for some δ > 0, then the unique solvability of the two Riccati equations can be obtained from the classical result [29,Theorem 7.2]. However, examples show that the two Riccati equations might still be solvable even if both R(·) andR(·) are negative semi-definite (see Example 6.1). On the other hand, it may happen that Problem (MF-LQ) is open-loop solvable, while the optimal control cannot be obtained by solving the corresponding Riccati equations due to the possible singularities of the terms R + D ⊤ P D and R +R + (D +D) ⊤ P (D +D) (see Example 6.2). Thus, some questions arise naturally: (a) What is the relationship between Problem (MF-LQ) and the solvability of the two Riccati equations? (b) How can we characterize the open-loop solvability of Problem (MF-LQ)? (c) How can we find an optimal control in general? The purpose of this paper is to study Problem (MF-LQ) from an open-loop point of view and to address the above issues. Closed-loop mean-field LQ problems will be investigated in a forthcoming paper.
Our main idea and results of this paper can be informally described as follows. By a representation of the cost functional, we first show that for the open-loop solvability of Problem (MF-LQ), a necessary condition is the convexity of the cost functional and a sufficient condition is the uniform convexity of the cost functional. Under the convexity condition, by adding εE T t |u(s)| 2 ds (ε > 0) to the original cost functional, we get a family of uniformly convex functionals. The corresponding mean-field LQ problems admit unique optimal controls u * ε (·), ε > 0, which form a minimizing sequence of Problem (MF-LQ). Then the open-loop solvability of Problem (MF-LQ) is characterized by the convergence of the sequence, whose limit is an optimal control of Problem (MF-LQ). To construct u * ε (·) explicitly, we further investigate Problem (MF-LQ) with uniformly convex cost functionals. Since the uniform convexity condition is much weaker than (1.4), the result in [28] fails to apply to this case. To overcome this difficulty, we reduce Problem (MF-LQ) to a classical stochastic LQ problem and a deterministic LQ problem. By making use of a result found in [25], we establish the equivalence between the uniform convexity of the cost functional and the solvability of the two Riccati equations. Then by the completion of squares technique, we obtain a state feedback representation of the optimal control via the solutions of the two Riccati equations.
The rest of the paper is organized as follows. Section 2 gives some preliminaries. In Section 3, we study Problem (MF-LQ) from a Hilbert space viewpoint and derive necessary and sufficient conditions for the finiteness and open-loop solvability of the problem by considering a family of uniformly convex cost functionals. Section 4 shows that the solvability of two coupled Riccati equations is necessary for the uniform convexity of the cost functional. In Section 5, we further prove that the solvability of the two coupled Riccati equations is also sufficient for the uniform convexity of the cost functional. Moreover, a state feedback representation is obtained for the optimal control. Some illustrative examples are presented in Section 6.

Preliminaries
Throughout this paper, we denote by R n×m the Euclidean space of all n × m real matrices, and by S n the space of all symmetric n × n real matrices. Recall that the inner product · , · on R n×m is given by M, N → tr (M ⊤ N ), where the superscript ⊤ denotes the transpose of vectors or matrices, and the induced norm is given by |M | = tr (M ⊤ M ). When there is no confusion, we shall use · , · for inner products in possibly different Hilbert spaces, and denote by | · | the norm induced by · , · . For a matrix M ∈ R n×m , we denote by R(M ) the range of M , and if M ∈ S n , we use the notation M > 0 ( 0) to indicate that M is positive (semi-) definite. For a bounded linear operator A form a Banach X space into another Banach space Y , we denote by A * the adjoint operator of A. Let T > 0 be a fixed time horizon. For any t ∈ [0, T ] and Euclidean space H, we let L p (t, T ; H) ( Further, we introduce the following notation: For any S n -valued measurable function F on [t, T ], a.e. s ∈ [t, T ], for some δ > 0.
The following assumptions will be in force throughout this paper.
From the above, we have the following result, which gives a characterization for the optimal controls of Problem (MF-LQ).
T ] and (X * (·), Y * (·), Z * (·)) be the adapted solution to the following (decoupled) mean-field forward-backward stochastic differential equation (MF-FBSDE, for short): Then u * (·) is an optimal control of Problem (MF-LQ) for the initial pair (t, ξ) if and only if and the following stationarity condition holds: Proof. By (2.3), we see that u * (·) is an optimal control of Problem (MF-LQ) for the initial pair (t, ξ) if and only if which is equivalent to (2.6) and the following: Note that the above inequality holds for all u(·) ∈ U[t, T ] if and only if DJ(t, ξ; u * (·))(·) = 0. The result therefore follows.

Finiteness and Open-Loop Solvability of Problem (MF-LQ)
We begin with a representation of the cost functional. For any u(·) ∈ U[t, T ], let X u 0 (·) be the solution of By the linearity of (3.1), we can define bounded linear operators L t : which is a bounded self-adjoint linear operator on U[t, T ]. Then by Proposition 2.2, the cost functional J(t, ξ; u(·)) can be written as Now let us introduce the following conditions.
(H3) The following holds: (H4) There exists a constant δ > 0 such that From (3.3), we see that the map u(·) → J(t, ξ; u(·)) is convex if and only if which is also equivalent to (H3), and u(·) → J(t, ξ; u(·)) is uniformly convex if and only if which is also equivalent to (H4). The following result tells us that (H3) is uniquely open-loop solvable at t, and the unique optimal control for the initial pair (t, ξ) is given by Moreover, Proof. (i) We prove the result by contradiction. Suppose that J 0 (t, 0; u(·)) < 0 for some u(·) ∈ U[t, T ]. By Proposition 2.2, we have Letting λ → ∞, we obtain that which is a contradiction.
(ii) Suppose (H4) holds. Then the operator M t is invertible, and Note that the equality in the above holds if and only if The result therefore follows.
Due to the necessity of (H3) for the finiteness of Problem (MF-LQ), we will assume (H3) holds in the rest of this paper. Now for any ε > 0, consider state equation (1.1) and the following cost functional: Denote the corresponding optimal control problem and value function by Problem (MF-LQ) ε and V ε (· , ·), respectively. By Proposition 3.1, part (ii), for any ξ ∈ L 2 Ft (Ω; R n ), Problem (MF-LQ) ε admits a unique optimal control and the value function is given by Now, we are ready to state the main result of this section.
which implies that Thus, the limit lim ε→0 V ε (t, ξ) exists and On the other hand, for any K, δ > 0, we can find a u δ (·) ∈ U[t, T ], such that Letting ε → 0, we obtain thatV from which we see that Combining (3.15)-(3.16), we obtain the desired result.

Necessary Conditions for the Uniform Convexity and Riccati
Equations Theorem 3.2 tells us that in order to solve Problem (MF-LQ), we need only solve mean-filed LQ problems with uniformly convex cost functionals and then pass to the limit. By Proposition 3.1, under the uniform convexity condition (H4), the unique optimal control u * (·) for the initial pair (t, ξ) is determined by (3.8). However, such a representation is not easy to compute, since M −1 t is in an abstract form and very complicated. Thus, we would like to find some more explicit form of the optimal control. In this section we shall investigate uniform convexity of the cost functional and show the necessity of solvability of two Riccati equations for the uniform convexity of the cost functional.

Y(T ) = GX(T ).
Note that X(r)X(s) J 0 (s, ξ; u(·)) J 0 (s, Note that M(·) : [t, T ] → S n is continuous. The result therefore follows. Now, let us introduce the following Riccati equation: A solution P (·) of (4.9) is said to be strongly regular if for some δ > 0. The Riccati equation (4.9) is said to be strongly regularly solvable, if it admits a strongly regular solution. By a standard argument using Gronwall's inequality, one can show that if the regular solution of (4.9) exists, it must be unique. Compared with the strongly regular solution, the notion of regular solution, which is closely related to the closed-loop strategy, was introduced in [26]. The interested reader is referred to [25] for further information.
The following result shows that the strongly regular solvability of the Riccati equation (4.9) is necessary for the uniform convexity of the cost functional. To prove the above result, we need the following lemma, whose proof can be found in [25].  If there exists a constant β > 0 such that for all Θ(·) ∈ L 2 (t, T ; R m×n ), Proof of Theorem 4.2. We only need to show that the condition stated in Lemma 4.3 holds. To this end, let Θ(·) ∈ L 2 (t, T ; R m×n ) and P (·) ≡ P Θ (·) be the corresponding solution of (4.11). For any deterministic u(·) ∈ L 2 (t, T ; R m ), let X u (·) be the solution of By the uniqueness of solutions, X u (·) also solves Thus, by applying Itô's formula to s → P (s)X u (s), X u (s) , we have (noting (H4) and (4.14)) Hence, for any u(·) ∈ L 2 (t, T ; R m ), the following holds: where Φ(·) is the solution of the following R n×n -valued ordinary differential equation (ODE, for short): Consequently, (4.16) becomes Similar to the previous argument, by applying Itô's formula to r → P (r)X(r), X(r) , we can derive that By Proposition 4.1, we have which implies that P (s) αI, ∀s ∈ [t, T ]. The proof is completed.
From Theorem 4.2, we see that the Riccati equation (4.9) is strongly regularly solvable under the uniform convexity condition (H4). With the strongly regular solution P (·) of (4.9), we may further introduce the following deterministic LQ optimal control problem.
Consider the state equation We pose the following deterministic LQ problem.

Sufficiency of the Riccati equations
In the previous section, we proved that the solvability of the Riccati equations (4.9) and (4.23) is necessary for the uniform convexity of the cost functional. In this section, we shall show that it is also sufficient. Moreover, under the uniform convexity condition, the optimal control can be represented explicitly as a state feedback form via the solutions of the Riccati equations.
First we need the following lemma.
To prove the second, for any v(·) ∈ L 2 (t, T ; R m×n ), let y v (·) be the solution to the following ODE: y v (t) = 0.

Similar to the previous argument, one can show that B is invertible and
Observe that E[X u 0 (·)] satisfies (5.3) with v(·) = E[u(·)]. The result therefore follows. Now we present the main result of this section, which gives a characterization for the uniform convexity of the cost functional as well as a feedback representation of the optimal control. Theorem 5.2. Let (H1)-(H2) hold. Then the map u(·) → J 0 (t, 0; u(·)) is uniformly convex if and only if the Riccati equation (4.9) admits a strongly regular solution P (·) such that and the corresponding Riccati equation (4.23) admits a solution Π(·). In this case, the unique optimal u * (·) of Problem (MF-LQ) at (t, ξ) is given by with (η(·), ζ(·)) andη(·) being the (adapted) solutions to the following BSDE η(T ) = g, and ordinary differential equation respectively, and X * (·) is the solution of the closed-loop system Moreover, the value V (t, ξ) is given by Proof. The "only if " part has been proved in Section 4. Let us now show the "if " part. For any ξ ∈ L 2 Ft (Ω; R n ) and u(·) ∈ U[t, T ], let X(·) ≡ X(· ; t, ξ, u(·)) be the corresponding solution of (1.1). Set Then (5.11) Now we rewrite the cost functional as follows: ds.

Examples
In this section we present two illustrative examples. In the first example, the condition (1.4) does not hold, but the corresponding Riccati equations are still solvable. Thus, by Theorem 5.2, the cost functional is uniformly convex. This example shows that the uniform convexity condition (H4) is indeed weaker than (1.4). By Theorem 5.2, the cost functional J(t, ξ; u(·)) is uniformly convex in u(·), and for any initial pair (t, ξ) ∈ [0, 1) × L 2 Ft (Ω; R), the problem admits a unique optimal control u * (·) given by u * (s) = − 2 s + 1 X * (s) + 2 s + 1 − 2Π(s) E[X * (s)], s ∈ [t, 1] with X * (·) being the solution to the following closed-loop system: Now we present an example in which the mean-field LQ problem is open-loop solvable, but the cost functional is not uniformly convex. Hence, the optimal control cannot be constructed directly in terms of the Riccati equations. However, an optimal could still be found by making use of Theorem 3.2 and 5.2. It is easy to see that P (·) ≡ 2 is the unique solution of (6.5). However, since J ε (t, ξ; u(·)) = J(t, ξ; u(·)) + εE It is clear that for t ∈ [0, T ), u * ε (s) converges uniformly to which, by Theorem 3.2, is an optimal control of the original problem at (t, ξ).