Controllability and observability of some coupled stochastic parabolic systems

This paper is devoted to a study of controllability and observability problems for some stochastic coupled linear parabolic systems only by one control and through an observer, respectively. In order to get a null controllability result, the Lebeau-Robbiano technique is adopted. The key point is to prove an observability inequality for certain stochastic coupled backward parabolic system by an iteration, when terminal values belong to a finite dimensional space. Different from deterministic systems, Kalman-type rank conditions for the controllability of stochastic coupled parabolic systems do not hold any more. Meanwhile, based on the Carleman estimates method, an observability inequality and unique continuation property for general stochastic linear coupled parabolic systems through an observer are derived.

1. Introduction. Let n and m be two positive integers. T > 0 and G is a nonempty bounded domain in R m with a smooth boundary Γ. Assume that G 0 and G 1 are two nonempty open subsets of G satisfying G 1 ⊂ G 0 . Denote by χ G0 the characteristic function of G 0 . Put Q = G × (0, T ) and Σ = Γ × (0, T ).
Let (Ω, F, {F t } t≥0 , P) be a complete filtered probability space, on which a onedimensional standard Brownian motion {W (t)} t≥0 is defined, so that F = {F t } t≥0 is the natural filtration generated by W (·), augmented by all P-null sets in F. Let H be a Banach space. Denote by L 2 F (0, T ; H) the Banach space consisting of all Hvalued F-adapted processes X(·) satisfying E(|X(·)| 2 L 2 (0,T ;H) ) < ∞; by L ∞ F (0, T ; H) the Banach space consisting of all H-valued F-adapted bounded processes; and by L 2 F (Ω; C([0, T ]; H)) the Banach space consisting of all H-valued F-adapted continuous processes X(·) satisfying E(|X(·)| 2 C([0,T ];H) ) < ∞. Also, for a real-valued matrix P , denote by P and |P | its conjugate matrix and determinant, respectively. For a linear operator A, whose domain is dense in a Hilbert space, denote by A * the conjugate operator of it. In the sequel, C is used to denote a generic positive constant and for simplicity, all zero vectors are denoted by 0.
For example, when n = 3, (1) is a system governed by three stochastic parabolic equations. The above assumptions (H 1 )-(H 3 ) become that for t ∈ (t 1 , t 2 ), The first result of this paper is stated as follows.
Theorem 1.1. Suppose that the conditions (H 1 )-(H 3 ) hold. Then for any T > 0, the system (1) is null controllable at the time T , i.e., for any Y 0 ∈ (L 2 (G)) n , there is a control u ∈ L 2 F (0, T ; L 2 (G)) satisfying suppu ⊆ G 0 × [0, T ], so that the corresponding solution Y ∈ L 2 F (Ω; C([0, T ]; L 2 (G))) n of (1) satisfies that Y (T ) = 0 in G, P-a.s. Moreover, If one of the conditions (H 1 )-(H 3 ) does not hold, the null controllability result in Theorem 1.1 may be untrue. In fact, some counterexamples on (H 1 ) and (H 3 ) were given in [11] for the case of n = 2. In section 3, an example on the assumption (H 2 ) will be presented for n = 3.
Based on the result of Theorem 1.1, consider the following stochastic coupled linear parabolic system: where h ∈ R n is a given vector and B is constant. Set N = (h Bh · · · B n−1 h). Then similar to [1], one can get the following controllability result for (3). Corollary 1. If rankN = n and N −1 D(t)N is an upper triangular matrix on a subinterval (t 1 , t 2 ) of (0, T ), P-a.s., the system (3) is null controllable.
Remark 2. Notice that in [1], a Kalman-type rank condition is given for assessing the null controllability of deterministic coupled linear parabolic systems. Especially, when D(t) ≡ 0 and (3) degenerates to a deterministic parabolic system, its null controllability is equivalent to rankN = n. However, by Example 3 in [11] and section 3 of this paper, the null controllability of stochastic coupled parabolic systems is not robust with respect to small perturbations of some coefficients. Therefore, one cannot get an equivalent Kalman rank condition for the controllability of stochastic coupled parabolic systems, different from deterministic systems.
Remark 3. The stochastic coupled linear parabolic system (1) in Theorem 1.1 is very special. In each equation of it, the elliptic operators of principal parts are the same. Also, the coefficients in principal operators depend only on the spatial variable, while those in the lower order terms are independent of it. Moreover, the system has no first order operators with respect to the spatial variable. These restrictions are technical. However, how to remove them is still an open problem and remains to be done.
where D 2 (·) ∈ L ∞ F (0, T ; R n ), the controllability of the associated coupled system seems difficult and remains to be done. This is because we do not know how to prove the following estimate for any solution (Z, Z) of the system (7) on Q:
Also, the coefficients matrices B(·) and D(·) are as follows: Then by [8] and [15], for any Y 0 ∈ (L 2 (G)) n , (4) admits a unique solution in the class of In order to study the observability of general linear coupled stochastic parabolic system (4), assume that there exist a nonempty open subset G * of G 0 and positive constant b 0 , so that the following conditions hold for (x, t) ∈ G * × (0, T ): The other main result of this paper is stated as follows.
Theorem 1.2. If the conditions (P 1 )-(P 2 ) hold, then there exists a positive constant C, so that any solution Y = (y 1 , · · · , y n ) ∈ L 2 F (Ω; C([0, T ]; L 2 (G))) n of (4) satisfies that for any Y 0 ∈ (L 2 (G)) n , In addition, the following unique continuation property holds: The above observability results mean that under the assumptions (P 1 )-(P 2 ), the information on a component y n of Y in a local domain G 0 can determine the whole information of solution vector Y uniquely.
Remark 5. Similar to Example 1 in [11], it is easy to show that if the assumption (P 1 ) does not hold, Theorem 1.2 may be untrue. However, the assumption (P 2 ) seems only technical.
Remark 6. The result of Theorem 1.2 can be generalized to the following more general linear coupled stochastic parabolic systems with first order operators with respect to the spatial variable: . Similar to arguments in [7], M ij are required to satisfy the same condition (P 2 ) asd ij .
Up to now, there have been numerous works addressing controllability and observability problems of deterministic parabolic equations/systems (see, e.g., [2], [4], [5], [6], [17], [18], [19], and the references therein). However, very little is known about the controllability and observability of stochastic parabolic equations/systems. Let us recall some known results in this respect. In [3], [14] and [16], the controllability and observability for some forward and backward stochastic parabolic (single) equations were studied, respectively. In [11], the controllability of some coupled systems governed by two stochastic parabolic equations (the special case of n = 2 in Theorem 1.1) was considered. Hence, only the assumptions (H 1 ) and (H 3 ) in the controllability results of [11] were involved. In this paper, we generalize the null controllability result in [11] to more general case for any integer n ≥ 2.
The key point is to choose suitable weight functions in the proof of an observability inequality for some backward stochastic coupled parabolic systems (Proposition 1) by an iteration method. Also, further explanations on the necessity of (H 2 ) are given in section 3 for n = 3. On the other hand, in [10], a null controllability result for some coupled systems by two backward stochastic parabolic equations by one control was obtained. By a duality technique, this is indeed a special case of Theorem 1.2 for n = 2, since the controllability result in [10] is equivalent to the observability for a coupled system of two forward stochastic parabolic equations through only one observer. Compared to the known results in [10], the coupling appears not only in drift terms, but also in diffusion terms in our paper. Also, the requirement for regularity on coefficients of principal parts may be relaxed to W 1,∞ (G), while the coefficients in diffusion terms are required to be in L ∞ (G), rather than W 1,∞ (G) in [10].
The rest of this paper is organized as follows. In section 2, an observability inequality for some coupled backward stochastic parabolic systems is derived, when terminal values are in a finite dimensional space. Section 3 is devoted to proving the controllability result in Theorem 1.1, based on the Lebeau-Robbiano technique.

LINGYANG LIU AND XU LIU
Also, some examples on the assumptions are given. In section 4, the observability result in Theorem 1.2 is derived by the Carleman estimates method.
2. An observability inequality in finite dimensional spaces. This section is devoted to establishing an observability inequality for some coupled backward stochastic parabolic systems, based on the Lebeau-Robbiano inequality. To this aim, for the operator A defined in (2) eigenvalues and the corresponding eigenfunctions of −A, respectively. Also, |e i | L 2 (G) = 1, for i = 1, 2, · · · . For any positive integer k, set X k = span{e 1 , e 2 , · · · , e k } and denote by P k the orthogonal projection from L 2 (G) to X k . By [9] and [13], one has the following Lebeau-Robbiano inequality.
Then there exists a positive constant C, independent of t 1 and t 2 , such that for any positive integer k and Z t2 ∈ L 2 (Ω, F t2 , P; (X k ) n ), the corresponding solution (Z, Z) = (z 1 , · · · , z n ,z 1 , · · · ,z n ) of (7) satisfies that Proof. The whole proof is divided into five parts.
Step 5. Taking k = n − 1 in (20), one has that In the following, we estimate the last two terms in (24). By Itô's formula, Then by the first equation of (7), it is easy to show that for any ε > 0, This, together with (24), indicates Put (10) and (25) with (9), we get that Hence, we get the desired estimate (8) in Proposition 1.
3. Controllability of coupled stochastic parabolic systems. In this section, similar to [14], by the Lebeau-Robbiano technique, one can get the desired null controllability result for (1) in Theorem 1.1. First, by the duality, the observability estimate (8) in Proposition 1 implies the following controllability result.
Proposition 2. Assume that (H 1 )-(H 3 ) hold. Then for any positive integer k and Y (t 1 ) ∈L 2 (Ω, F t1 , P; (L 2 (G)) n ), one can find a control u k ∈ L 2 F (t 1 , t 2 ; L 2 (G)), such that suppu k ⊆ G 0 ×[t 1 , t 2 ], and the corresponding solution Y (·) of (1) in Q 1 satisfies that P k (Y (t 2 )) = 0 in G, P-a.s. Moreover, there exists a positive constant C, independent of t 1 and t 2 , such that and On the other hand, it is easy to show the following decay property for solutions of (1).
Proposition 3. Assume that u ≡ 0 in Q 1 . Then for any positive integer k and Y (t 1 ) ∈L 2 (Ω, F t1 , P; (L 2 (G)) n ), if P k (Y (t 1 )) = 0, the corresponding solution Y (·) of (1) in Q 1 satisfies that P k (Y (t)) = 0 and By the classical Lebeau-Robbiano technique, by Propositions 2 and 3, one can get the null controllability result of (1) in Theorem 1.1. Now, we give a sketch of its proof.
Sketch of a poof of Theorem 1.1. The whole proof is divided into four parts.
Step 1. First, introduce some notations. For any ∈ N, write ]. Then it is easy to check that Let {σ } ∈N be a sequence of sufficiently large positive constants, which will be specified later. For any Y 0 ∈ L 2 (G) n , take a control u ≡ 0 in G × [0, t 1 ]. For the corresponding Step 2. First, consider the system (1) on the interval I 0 = [T 0 , T 1 ]: By Proposition 2, for σ 1 ∈ N, there exists a control u 1 ∈ L 2 F (I 0 ; L 2 (G 0 )), such that the corresponding solution Y 11 of (26) satisfies P σ1 (Y 11 (T 1 )) = 0 in G, P-a.s. Moreover, On the other hand, consider the system (1) with u ≡ 0 on the interval J 0 = [T 1 , T 2 ]: By Proposition 3, the solution Y 12 of (27) satisfies ) n , and P σ1 (Y 12 (t)) = 0 in G, P-a.s., for any t ∈ J 0 .
Step 4. By (29) and (30), it is easy to show that By Weyl's formula, for sufficiently large σ , it holds that By a similar argument, by (28), it follows that Choose a control function u as follows: s. The proof of Theorem 1.1 is completed.
Next, we give a proof of Corollary 1.

4.
Observability of coupled stochastic parabolic systems. This section is devoted to observability problems for the coupled stochastic parabolic system (4).