Exact Controllability of Linear Stochastic Differential Equations and Related Problems

A notion of $L^p$-exact controllability is introduced for linear controlled (forward) stochastic differential equations, for which several sufficient conditions are established. Further, it is proved that the $L^p$-exact controllability, the validity of an observability inequality for the adjoint equation, the solvability of an optimization problem, and the solvability of an $L^p$-type norm optimal control problem are all equivalent.

In this paper, for any p ∈ [1, ∞), we propose a notion of L p -exact controllability (see Definition 3.1) for FSDE systems. When p = 2 and all the coefficients of the system are bounded, our notion is reduced to the one studied in [18,16]. We point out that since the coefficients B(·) and D k (·) are allowed to be unbounded, the corresponding set of admissible controls is delicate and it makes the controllability problem under consideration more interesting (see below for detailed explanation). Inspired by the results of deterministic systems, for any p ∈ (1, ∞), we introduce a stochastic version of observability inequality (see Theorem 4.2) for the adjoint equation, the validity of which is proved to be equivalent to the L pexact controllability of the linear FSDE (with random coefficients). This provides an approach to study the controllability of linear FSDE systems by establishing an inequality for BSDEs. Moreover, we introduce a family of optimization problems of the adjoint linear BSDE (see Problem (O) and Problem (O) ′ in Section 4), and the solvability of these optimization problems is proved to be equivalent to the L p -exact controllability of the linear FSDE. In other words, we additionally provide an approach to study the exact controllability through infinite-dimensional optimization theory.
As an application, we consider some L p -type norm optimal control problems (see Problem (N) and Problem (N) ′ in Section 5). The norm optimal control problem for deterministic finite or infinite dimensional systems has been investigated by many researchers (see e.g. [7,8,12,22,23]). Recently, Gashi [10] studied a norm optimal control problem (in L 2 sense) for linear FSDE systems with deterministic time-varying coefficients by virtue of the corresponding Hamiltonian system and Riccati equation. Moreover, Wang-Zhang [24] studied a kind of approximately norm optimal control problems for linear FSDEs. In the present paper, with the help of optimization problems for BSDE, we solve the norm optimal control problem for linear FSDE systems with random coefficients (see Theorems 5.1 and 5.3, and Corollary 5.5).
The rest of this paper is organized as follows. In Section 2, we present some preliminaries. Section 3 is devoted to the introduction of the L p -exact controllability for linear FSDE systems. Some sufficient conditions of the L p -exact controllability are established for two types of systems: The diffusion is controlfree and the diffusion is "fully" controlled. In Section 4, we establish the equivalence among the L p -exact controllability, the validity of observability inequality for the adjoint equation and the solvability of optimal control problems. Finally, as an application, a norm optimization problem is considered in Section 5.

Preliminaries
Recall that R n is the n-dimensional (real) Euclidean (vector) space with the standard Euclidean norm | · | induced by the standard Euclidean inner product · , · , and R n×m is the space of all (n × m) (real) matrices, with the inner product A, B = tr [A ⊤ B], ∀A, B ∈ R n×m , so that R n×m is also a Euclidean space. Hereafter, the superscript ⊤ denotes the transpose of a vector or a matrix. We now introduce some spaces, besides L p FT (Ω; R n ) introduced in the previous section. Let H = R n , R n×m , etc., and p, q ∈ [1, ∞).
In the similar manner, one may define This gives the first inclusion in (2.2). Other cases can be proved similarly. Now, we introduce the following definition.
The following result gives a big class of feasible controls for system [A(·), C(·); B(·), D(·)], whose proof is standard.
Proposition 2.2. Let (H1) hold. Let u : [0, T ] × Ω → R n be F-progressively measurable. Suppose the following holds: Then u(·) ∈ U[0, T ], and the solution X(·) ≡ X(· ; x, u(·)) of (1.1) with initial state x under control u(·) satisfies the following: Hereafter, K > 0 will denote a generic constant which could be different from line to line. Further, if then u(·) ∈ U[0, T ], and the following holds: The above result leads us to the following definitions.
In the similar manner, we are able to prove (ii).

Exact Controllability
We now give a precise definition of L p -exact controllability.
, C(·); B(·), D(·)] is said to be L p -exactly controllable by U[0, T ] on the time interval [0, T ], if for any (x, ξ) ∈ R n × L p FT (Ω; R n ), there exists a u(·) ∈ U[0, T ] such that the solution X(·) ∈ L 1 F (Ω; C([0, T ]; R n )) of (1.1) with X(0) = x satisfies X(T ) = ξ. In the above, U[0, T ] could be U q [0, T ], or U q r [0, T ] for some suitable q 1, and also it could be U p,ρ,σ [0, T ], or U p,ρ,σ r [0, T ]. We emphasize that in defining the system to be L p -exactly controllable (for p 1) by . Depending on the choice of U[0, T ], X(·) might have better integrability/regularity but we do not require any better property than L 1 F (Ω; C([0, T ]; R n )). We will see shortly that this gives us a great reflexibility.

The case D(·) = 0
In this subsection, we consider system [A(·), C(·); B(·), 0], i.e., the state equation reads Thus, the control u(·) does not appear in the diffusion. When all the coefficients in the above are constants, it was shown in [1] that the system is approximately controllable (under some additional conditions) in the following sense: For any (x, ξ) ∈ R n × L 2 FT (Ω; R n ), and any ε > 0, there exists a u ε (·) ∈ L 2 The following is our first result which improves the results of [1] significantly.
Proof. Consider the following system: with v(·) ∈ L q F (Ω; L 1 (0, T ; R n )), q > 1. Then the unique solution X(·) satisfies E sup Let Φ(·) be the solution to the following: Then Φ(·) −1 exists and satisfies the following: Therefore, for any q 1, with the constant K(T, q) depending on T and q (as well as the bound of C k (·)), and we have the following variation of constants formula for X(·): Now, for any q ∈ (1, p), we want to choose some v(·) ∈ L q F (Ω; L 1 (0, T ; R n )) so that X(T ) = ξ which is equivalent to the following: Since ξ ∈ L p FT (Ω; R n ), for anyq ∈ (q, p), we have Thus, Φ(T ) −1 ξ − x ∈ Lq FT (Ω; R n ). Then, by [17, Theorem 3.1], we can find a v(·) ∈ Lq F (Ω; L 1 (0, T ; R n )) such that Since q <q, one has with X(·) defined by (3.3). Then which implies that This means that X(·) is the solution to (3.1), corresponding to (x, u(·)) with Therefore, u(·) ∈ U q [0, T ] which makes X(T ) = ξ. This proves our conclusion.
Let us make some comments on the above result. To this end, let us define Then a result from [17, Theorems 3.1, 3.2] tells us that Thus, In particular, Let us look at an implication of the above. Consider a system of the following form: For terminal state ξ ∈ L p FT (Ω; R n ), with p > 1, if one is only allowed to use the control from L p F (Ω; L 2 (0, T ; R n )), the above system is not controllable. Whereas, by Theorem 3.2, this system is L p -exactly controllable on [0, T ] by U q [0, T ] for any q ∈ (1, p). This is a main reason that we define the L p -exact controllability allowing the control taken from a larger space than L p F (Ω; L 2 (0, T ; R m )), and not restricting the state process X(·) to belong to L p F (Ω; C([0, T ]; R n )). Next, we notice that in Theorem 3.2, condition (3.2) implies that rank B(t) = n m, t ∈ [0, T ], a.s.
This means that the dimension of the control process is no less than that of the state process. Now, if which will always be the case if m < n, then for each t ∈ [0, T ], there exists a θ(t) ∈ R n \ {0} such that The following gives a negative result for the exact controllability, under condition (3.9) with a little more regularity conditions on θ(·) and C(·), which is essentially an extension of (3.6).
Then for any p > 1, Take We claim that the above constructed ξ cannot be hit by the state X(T ) from X(0) = x under any u(·) ∈ U p [0, T ]. We show this by contradiction. Suppose otherwise, then for some Hence, Then (3.13) By a standard estimate for BSDEs and Burkholder-Davis-Gundy inequality, one obtains (3.14) E sup On the other hand, and for any f, g, h ∈ L 2 (t, T ; R), Thus, for each k = 1, 2, · · · , d, The other two negative terms can be estimated similarly. Consequently, making use of (3.14), This leads to which is a contradiction since |θ(T ) ⊤ (ζ 0 − ζ 1 )| = 1. The above implies that the terminal state ξ constructed above cannot be hit by the state under any u(·) ∈ U p [0, T ]. This completes the proof.

The case D(·) is surjective
In this subsection, we let d = 1. The case d > 1 can be discussed similarly. For system [A(·), C(·); B(·), D(·)], we assume the following: In this case, [D(t)D(t) ⊤ ] −1 exists and uniformly bounded. We define and introduce the following controlled system: with X(·) being the state and (v(·), Z(·)) being the control. Using our notation, the above system can be , the latter has a simpler structure. For system [ A(·), 0; ( B(·), D(·)), (0, I)], we need the following set: The following result is a kind of reduction.
Theorem 3.6. Let (H1) and (3.16) hold. Let A(·), B(·), D(·) be defined by (3.17). Suppose where 2 ∨ p ≡ max{2, p} and ε > 0 is a given constant. Then the following are equivalent: (iii) Matrix G defined below is invertible: where Y(·) is the adapted solution to the following FSDE: ; R n×n )) and the Matrix G is well defined. Now we prove the conclusion by contradiction. Suppose matrix G is not invertible, then there exists a vector 0 = β ∈ R n such that Now, we claim that by choosing x = β ∈ R n , there will be no (v(·), Z(·)) ∈ U p [0, T ] × L p F (Ω; L 2 (0, T ; R n )) such that the corresponding state precess X(·) ≡ X(· ; x, v(·), Z(·)) satisfies In fact, suppose there exists a pair (v(·), Z(·)) ∈ U p [0, T ] × L p F (Ω; L 2 (0, T ; R n )) such that the above is true. Then applying the Itô's formula to Y(t)X(t) on the interval [0, T ], one obtains the following relationship: It is easy to check that Thus, Making use of (3.28), we get (Ω; L 2 (0, T ; R n×m )) leads to the following: ) to be the unique adapted solution of the following BSDE: Then, by linearity, we see that (X(·), v(·), Z(·)) satisfies the following: The above result is essentially due to Liu-Peng [16]. We have re-organized the way presenting the result. It is worthy of pointing out that, unlike in [16], we allow the coefficients to be unbounded and allow p to be different from 2. Combining Theorems 3.4 and 3.6, we have the following result. As a simple corollary of the above, we have the following result for the case of deterministic coefficients. Corollary 3.8. Let (H1), (3.16) and (3.25) hold. Let A(·) and B(·) be deterministic. Let Φ(·) be the solution to the following ODE: Proof. In the current case, we have On the other hand, Hence, G Ψ.
Then our conclusion follows from the above theorem.
The invertibility of matrix G defined by (3.26) gives a nice criterion for the L p -exact controllability of system [A(·), C(·); B(·), D(·)] through the L p -exact controllability of system [ A(·), 0; ( B(·), D(·)), (0, I)]. However, unless n = 1, in the case of random coefficients, the solution Y(·) of FSDE (3.27) does not have a relatively simple (explicit) form. Thus, the applicability of condition (iii) in Theorem 3.6 is somehow limited. In the rest of this subsection, we will present another sufficient condition for the L p -exact controllability of [ A(·), 0; ( B(·), D(·)), (0, I)] which might have a better applicability. Now, with random coefficients, we still let Φ(·) be the solution to (3.31) which is a random ODE. Presumably, Φ(·) is easier to get than Y(·) (the solution of (3.27)). Define and introduce the following mean-field stochastic Fredholm integral equation of first kind: We have the following result.
Thus, in the case that A(·) and B(·) are deterministic, condition (3.35) is automatically true with M = 1, as long as Ψ(t, τ ) is invertible. We will present an example that A(·) is random and (3.35) holds. We also point out that condition (3.16) implies that m n. Further, condition that Ψ(t, τ ) −1 exists implies that m > n. In fact, if (3.16) holds and m = n, then B(·) = 0 which implies that Ψ(t, τ ) = 0. We will say a little bit more about this shortly.
The following simple example is to show that condition (3.35) is possible for random coefficient case.
For example, we may choose Then Thus, A direct computation shows that (denotingā(τ ) = E τ a(τ )) Hence, for t − τ > 0 small, Ψ(τ, t) is invertible, and Then Some direct (lengthy) calculations show that Hence, As a result, we obtain The above shows that (3.35) holds.
As we mentioned earlier, condition (3.16) implies that m n, and if m = n and (3.16) holds, then B(·) = 0. Hence, in order Ψ to be invertible, one must have m > n. The following result is concerned with a case that D(·) is surjective and m = n. Proof. In the current case, D(·) −1 is bounded. If for any x ∈ R n and ξ ∈ L p FT (Ω; R n ), one can find a u(·) ∈ U p [0, T ] such that X(0) = x and X(T ) = ξ, then we let which will lead to Hence, (X(·), Z(·)) is an adapted solution to the following BSDE: Then X(0) cannot be arbitrarily specified. Hence, L p -exact controllability is not possible for system [A(·), C(·); B(·), D(·)].
In the above two subsections, we have discussed the two extreme cases: either D(·) = 0, or D(·) is full rank (for the case d = 1). The case in between remains open. Some partial results have been obtained, but they are not at a mature level to be reported. We hope to present them in a forthcoming paper.

Duality and Observability Inequality
As we know that for deterministic linear ODE systems, the controllability of the original systems is equivalent to the observability of the dual equations. We would like to see how such a result will look like for our FSDE system [A(·), C(·); B(·), D(·)]. To this end, let us first look at an abstract result, whose proof should be standard. But for reader's convenience, we present a proof. Proposition 4.1. Let X and Y be Banach spaces, K : X → Y be a bounded linear operator, and K * : Y * → X * be the adjoint operator of K. Then K is surjective if and only if there exists a δ > 0 such that Further, if X and Y are reflexive and the map x * → |x * | 2 X * from X * to R is Fréchet differentiable, then (4.1) is also equivalent to the following: For any y ∈ Y, the functional (4.2) J(y * ; y) = 1 2 |K * y * | 2 X * + y, y * , y * ∈ Y * , admits a minimum over Y * . In addition, if the norm of X * is strictly convex, then for any y ∈ Y, the optimal solution of (4.2) is necessarily unique.
Conversely, suppose (4.1) holds. Then R(K * ) is closed and K * is injective. Thus, by Banach Closed Range Theorem, R(K) is also closed and proving that K is surjective. Now, for any y ∈ Y, consider functional y * → J(y * ; y). Clearly, under condition (4.1), we have that y * → J(y * ; y) is coercive and weakly lower semi-continuous. Hence, if {y * k } k 1 is a minimizing sequence, then it is bounded. By the reflexivity of Y * , we may assume that y * k converges weakly to someȳ * ∈ Y * . Then by the weakly lower semi-continuity of the functional J(· ; y),ȳ * must be a minimum.
Since y ∈ Y is arbitrary, we obtain that R(K) = Y. Then by what we have proved, (4.1) holds. Finally, if we further assume that the norm of X * is strictly convex, then we must have the uniqueness of the optimal solution to y * → J(y * ; y).
For any given 1 < p < ρ ∞ and 2 < σ ∞, we denote q ≡ p p−1 , and Clearly, with the above notations, we have In the rest of this paper, we will keep the above notations. We now present the first main result of this section. (respectively, where (Y (·), Z(·)) (with Z(·) ≡ (Z 1 (·), · · · , Z d (·))) is the unique adapted solution to the following BSDE: Proof. We only prove the equivalence between system's L p -exactly controllability on [0, T ] by U p,ρ,σ [0, T ] and the validity of the observability inequality (4.6). The other part can be proved with the similar procedure.
Theorem 4.2 provides an approach to study the controllability of stochastic linear systems by establishing an inequality for BSDEs. The following example illustrates this approach.
Remark 4.8. The notion of adaptability represents a fundamental difference between deterministic and stochastic systems. From the derivation of Fréchet derivative (see the third line of (4.30)), we can obtain naturally a process: which is closely linked to our problem. But unfortunately,Γ(ϕ(·))(·) is not F-adapted when p = 2σρ σρ−2ρ+2σ (equivalentlyq = ν). Hence, in order to meet the requirement of adaptability, we use Γ(ϕ(·))(t) = E t [Γ(ϕ(·))(t)], t ∈ [0, T ] to replaceΓ(ϕ(·))(·). However, this treatment leads to some difficulty. As a matter of fact, through a direct calculation, we can obtain that the following equation holds for any p ∈ (1, ∞). But due to the introduction of conditional expectation, we only get an inequality where K * and K * 0 are given by (4.12) and (4.20), respectively. Equivalently, with (Y (·), Z(·)) being the adapted solution to BSDE (4.8). Note that we have changed from U p,ρ,σ [0, T ] to U p,ρ,σ r [0, T ] in the above. We now pose the following optimization problem.
Problem (O) ′ . Minimize (4.39) subject to BSDE (4.8) over L q FT (Ω; R n ). Suppose ϕ(·), ψ(·) ∈ L ν F (0, T ; Lq(Ω; R m )). A similar procedure as Problem (O) leads to Unlike the case of Problem (O), in the above we do not need conditional expectation. Due to this, the following lemma holds without the constraint p  Proof. It suffices to calculate the following: Hence, our conclusion follows.
Then similar to Theorem 4.7, we have the following result.
For simplicity, we denote the above U p,ρ,σ [0, T ]-norm optimal control problem by Problem (N). Note that the system (1.1) is L p -exactly controllable on [0, T ] by U p,ρ,σ [0, T ] if and only if, for any (x, ξ) ∈ R n × L p FT (Ω; R n ), the U p,ρ,σ [0, T ]-admissible control set U(x, ξ) is not empty. We callū ∈ U(x, ξ) a U p,ρ,σ [0, T ]norm optimal control to Problem (N) if For any given 2 p < ρ ∞ and 2 < σ ∞, we similarly introduce the U p,ρ,σ r [0, T ]-norm optimal control problem reading: In the previous section, we have given some equivalent conditions for the L p -exact controllability of system (1.  • For any (x, ξ) ∈ R n × L p FT (Ω; R n ), Problem (O) admits a unique optimal solutionη ∈ L q FT (Ω; R n ); • For any (x, ξ) ∈ R n × L p FT (Ω; R n ), Problem (N) admits a unique optimal controlū(·) ∈ U p,ρ,σ [0, T ]. Moreover, the unique norm optimal controlū to Problem (N) is given by (4.36), and the minimal norm is given by The minimal value of functional J(·; x, ξ) is given by Proof. (Sufficiency). Since for any (x, ξ) ∈ R n × L p FT (Ω; R n ), Problem (N) admits an optimal control, then the corresponding U p,ρ,σ [0, T ]-admissible control set U(x, ξ) is not empty. Therefore, the system (1.1) is L p -exactly controllable. By Theorem 4.2 and Theorem 4.7, the first statement holds true.
Since conditional expectation is not introduced in Γ ′ (·), we can solve the U p,ρ,σ r [0, T ]-norm optimal control problems for any p 2, whose proof is similar to that of Theorem 5.1. • For any (x, ξ) ∈ R n × L p FT (Ω; R n ), Problem (O) ′ admits a unique optimal solutionη ′ ∈ L q FT (Ω; R n ); • For any (x, ξ) ∈ R n × L p FT (Ω; R n ), Problem (N) ′ admits a unique optimal controlū ′ (·) ∈ U p,ρ,σ r [0, T ]. Moreover, the unique norm optimal controlū ′ (·) to Problem (N) ′ is given by (4.43), and the minimal norm is given by The minimal value of functional J ′ (·; x, ξ) is given by [0, T ] coincide with L µ F (0, T ; R m ), and then both Problem (N) and Problem (N)' imply the L µ F (0, T ; R m )-norm optimal control problem. Precisely, Theorems 5.1 and 5.3 provide the same result for the L µ F (0, T ; R m )-norm optimal control problem when the system is L p -exactly controllable with p 2. However, when 1 < p < 2, only Theorem 5.1 gives a result, and Theorem 5.3 is invalid.
Moreover,ū(·) defined by (5.9) is the unique weighted norm optimal control, and the minimal weighted norm is given by