CONTROLLABILITY AND OBSERVABILITY FOR SOME FORWARD STOCHASTIC COMPLEX DEGENERATE/SINGULAR GINZBURG–LANDAU EQUATIONS

. This paper is addressed to establishing controllability and observability for some forward linear stochastic complex degenerate/singular Ginzburg–Landau equations. It is suﬃcient to establish appropriate observability inequalities for the corresponding backward and forward equations. The key is to prove the Carleman estimates of the forward and backward linear stochastic complex degenerate/singular Ginzburg–Landau operators. Compared with the existing deterministic results, it is necessary to overcome the diﬃculties caused by some complex coeﬃcients and random terms. The results obtained cover those of deterministic cases and generalize those of stochastic degenerate parabolic equations. Moreover, the limit behavior of the coeﬃcients in the equation is discussed


Introduction and main results
The Ginzburg-Landau equation was first given in [16], which is a typical nonlinear equation in the physics community. It may describe a variety of phenomena including light propagation in nonlinear fibers, phenomena related to pulse formation and superconductivity, and plays an important role in the theory of amplitude equations. Real-valued Ginzburg-Landau equations were first derived as long-wave amplitude equations in [22,26]. The complex Ginzburg-Landau equation was established as a standard 1-D model for some fluid flows (see [27]). The deterministic complex Ginzburg-Landau equation is one of the most frequently studied equations in physics and mathematics. For instance, the Cauchy problems, numerical methods to establish approximate solutions and control problems for the deterministic Ginzburg-Landau equations have been extensively researched (see [1,7,10,[23][24][25]).
And, the uniformly parabolic equations without degeneracies or singularities have been developed in various directions. However, more recently, several situations where the equation is not uniformly parabolic have been investigated. Indeed, many problems coming from physics (see [19]), biology (see [4,8]) and mathematical finance (see [17]) are described by parabolic equations which admit some kind of degeneracy. Another inspiring situation is the case of parabolic equations with singular lower order terms. The corresponding cases arise in quantum mechanics (see [2]), or in combustion problems (see [3]).
In practice, due to the interference of random factors, stochastic processes give a natural replacement for deterministic functions in mathematical descriptions. Compared with deterministic case, some substantially difficulties arise in the study of the stochastic partial differential equations. For example, the solution to a stochastic partial differential equation is non-differentiable with respect to noise variable, and the usual compactness embedding result is not valid for solution spaces of the stochastic evolution equations. Further, the "time" in the stochastic setting is not reversible. Indeed, many tools and methods, which are effective in the deterministic case, do not work anymore in the stochastic setting.
Recently, stochastic complex Ginzburg-Landau equations have received more and more attention, see for example [11,12,18]. In this paper, we will study some linear stochastic complex Ginzburg-Landau equations with both degeneracies and singularities. In fact, the linearized complex Ginzburg-Landau equation also models some different phenomena, such as the amplitude equation in pattern formation and the reaction diffusion of two chemicals in one dimension [6]. It is noted that many properties of Ginzburg-Landau equations are between that of parabolic equations and Schrödinger equations. Therefore, this paper is also devoted to considering its limit behavior where degenerate Ginzburg-Landau equations, degenerate parabolic equations and degenerate Schrödinger equations are considered simultaneously. In the following, the problem of this paper is stated in detail.
Let T > 0 and Q = (0, 1) × (0, T ). Assume G 0 = (x 1 , x 2 ) to be a given nonempty open subset of (0, 1), and denote by χ G0 the characteristic function of the set G 0 . Fix a complete filtered probability space (Ω, F, {F t } t≥0 , P), on which a one-dimensional standard Brownian motion {B(t)} t≥0 is defined such that F = {F t } t≥0 is the natural filtration generated by B(·), augmented by all the P-null sets in F. Let H be a Banach space, and let C( Moreover, denote by i the imaginary unit, and for any complex number c, we denote by c, Re c and Im c, its complex conjugate, real part and imaginary part, respectively.
Consider the following forward linear stochastic complex degenerate/singular Ginzburg-Landau equation: x β ydt = (c 1 y + χ G0 u)dt + (c 2 y + v)dB(t) in Q, y(t, 1) = 0 on (0, T ), y(0, x) = y 0 (x) in (0, 1), where complex-valued coefficients c 1 ∈ L ∞ F (0, T ; L ∞ ((0, 1); C)), and c 2 ∈ L ∞ F (0, T ; W 1,∞ ((0, 1); C)). Also, α ∈ [0, 2), a, b, c, d, µ, β ∈ R satisfy some conditions which will be given later. In (1.1), (u, v) is the control variable, y is the state variable, and y 0 ∈ L 2 ((0, 1); C) is a given initial value. Unless otherwise stated, we assume that all functions mentioned in this paper are complex-valued. Next, we assume that exponents α, β and parameter µ satisfy the following conditions: • sub-critical potentials: We separate the case where both the exponent β and the parameter µ are critical. In the case of (1.3), the potential is called critical, and otherwise it is called sub-critical. As we shall show later, the case of a critical potential requires a specific functional setting and a special care in the derivation of Carleman estimates. The Carleman-type estimate was first introduced by Carleman to study the uniqueness for elliptic equations in [5], which has become an important tool in studying controllability for stochastic partial differential equations.
The main purpose of this paper is to study the null controllability and observability for forward linear stochastic complex degenerate/singular Ginzburg-Landau equation (1.1). The null controllability of (1.1) is formulated as follows. For any y 0 ∈ L 2 ((0, 1); C), one can find a pair of control (u, v) ∈ L 2 F (0, T ; L 2 (G 0 ; C)) × L 2 F (0, T ; L 2 ((0, 1); C)), such that the solution y to (1.1) satisfies y(T ) = 0 in (0, 1), P-a.s. On the other hand, the observability for (1.1) is stated as follows. If u = v = 0 in (1.1), find (if possible) a positive generic constant C 1 = C 1 (a, b, c, d) such that for any y 0 ∈ L 2 ((0, 1); C), the solution y to (1.1) satisfies that The observability is one of the most important properties in structural theory. The observability inequality (1.5) means that the terminal value can be dominated by its local information of any solution to (1.1) in G 0 × (0, T ). Such kind of inequalities are closely related to control problems, unique continuation properties and inverse problems.
In the last decades, the theory of controllability and observability for deterministic and stochastic uniformly parabolic equations has been largely developed (see [11,13,14,28] and the references therein). More recently, there are several papers which are concerned with the control problems for deterministic and stochastic degenerate equations (see [4,20,31]). In addition, parabolic equations with singular potentials have also been extensively studied. In this aspect, we refer to [9,29,30] for deterministic system, and [15,32] for stochastic system.
There are also some known controllability and observability results about the deterministic and stochastic complex Ginzburg-Landau equations (see [10-12, 23, 24] and the references therein). However, as far as we know, nothing about the null controllability and observability are known for stochastic complex degenerate/singular Ginzburg-Landau equations. In this paper, we study the controllability and observability problems of the general forward linear stochastic complex degenerate/singular Ginzburg-Landau equations for different critical cases of the exponents α, β and parameter µ.
The controllability result for forward linear stochastic complex degenerate Ginzburg-Landau equation (1.1) can be stated as follows.
The assumption condition bc = ad is technical, which shows up in some cross terms of the weighted identify, and we now do not know how to drop it.
Remark 1.4. The assumption condition a > 2c in (H 2 ) is not needed in deterministic case. The detailed explanations can refer to Remark 3.8.
Remark 1.5. In the case of deterministic degenerate/singular equation, i.e., a = c = 1, b = d = 0, c 2 = v = 0, and choosing all functions to be real-valued in (1.1), then one can get the null controllability result for forward degenerate/singular parabolic equations from Theorem 1.2. This is the main result in [29].
The corresponding observability estimates for backward linear stochastic complex degenerate/singular Ginzburg-Landau equation (1.6) are established. Proposition 1.6. Assume that (H 1 ) or (H 2 ) holds. Then, for any w T ∈ L 2 (Ω, F T , P; L 2 ((0, 1); C)), the solution to (1.6) satisfies Moreover, in the case of (H 1 ), C 2 (a, b, c, d) is given by , (1.8) and the specific form of C 2 (a, b, c, d) in (H 2 ) is where and C 0 will be defined later by (3.4).
In the rest of paper, unless otherwise stated, we shall denote by C a generic positive constant independent of a, b, c, d, which may change from line to line.
Remark 1.7. Choosing α = µ = 0 in (1.1), one can obtain the null controllability and observability for the onedimensional linear stochastic Ginzburg-Landau equation, which is consistent with the results in [11]. Compared with [11], we choose different weighted functions and use the Hardy inequalities to deal with the difficulties caused by degeneracy and singularity. This leads to more complicated assumptions about the coefficients than [11].
On the other hand, the observability estimates for forward equation (1.1) are as follows: Then, the observability estimate (1.5) holds for any solution to equation (1.1). Furthermore, in the case of (H 1 ), C 1 (a, b, c, d) is given by 11) and the specific form of where K 0 is given by (1.10), and C 0 will be defined later by (3.4).
Remark 1.9. Notice that Theorem 1.8 is valid only for the sub-critical case (i.e., (1.2)). The reason why this result is not available for the critical case (i.e., (1.3)) is that the Carleman estimate we established in this case is based on the H * α (0, 1)-norm of the solution instead of H 1 α (0, 1)-norm.
Remark 1.10. When system (1.1) reduces to a real-valued forward stochastic degenerate parabolic equation (a = 1, b = c = d = 0), the observability estimates for forward stochastic degenerate parabolic equations can be obtained from the above results. These forms are the same as the known one given in [20].
From the observability estimate (1.5), the unique continuation property of the general forward stochastic degenerate/singular parabolic equations can be obtained immediately. From the observability constants (1.11) and (1.12) in Theorem 1.8, we have the following limit behavior of coefficients in (1.1). Corollary 1.12. Assume that (H 1 ) or (H 3 ) holds. If b, d → 0, then the observability estimate (1.5) also holds for stochastic parabolic equations with degeneracy and singularity. Remark 1.13. It is obviously that blow-up phenomena for constant C 1 (a, b, c, d) could occur when a → 0, which means that the internal observability estimate cannot be obtained by using our method for stochastic degenerate Schrödinger equations with singularity. However, the corresponding boundary observability estimate can be derived by Theorem 3.2.
The rest of this paper is organized as follows. In Section 2, we give a pointwise weighted identity for linear stochastic complex degenerate/singular Ginzburg-Landau operator. In Section 3, the global Carleman estimates for the forward and backward linear stochastic degenerate/singular Ginzburg-Landau equations are established. Finally, in Section 4, we prove the main results.

A weighted identity for linear stochastic complex degenerate/singular Ginzburg-Landau operator
In this section, we are devoted to establishing a pointwise weighted identity for the following linear stochastic complex degenerate/singular Ginzburg-Landau operator: which will play a crucial role in the sequel. First, define two unbounded operators where y ∈ H 2 loc ((0, 1]) denotes y xx ∈ L 2 loc ((0, 1]), and In the case of sub-critical i.e., (1.2), In the case of critical i.e., (1.3), for α ∈ [0, 1), and for α ∈ [1, 2), We denote For a fixed weight function ∈ C 3 (Q; R) and auxiliary function Φ ∈ C 1 (Q; C), we set θ = e , z = θp.
Then, by an elementary calculation, we can get that (2.4)

By (2.2) and (2.3), it is easy to check that
where We have the following pointwise weighted identity for the operator L in (2.1).
Proof. In [11], by choosing n = 1, a 11 = x α , a 0 = 1, and b 0 = 0, we can get the result immediately. From this Lemma, we give the proof of Theorem 2.1.
Proof of Theorem 2.1. By (2.5), (2.6), and Re (ic) = Im c, it is easy to see that (2.9) By (2.3), we can obtain that Next, we compute the last two terms in the right side of sign of the equality (2.9). By the definition of I 2 in (2.4), and a simple calculation, we have that Similarly, by the definition of I 1 in (2.4) and a simple calculation, it holds that (2.14) Finally, combining (2.9)-(2.14) with Lemma 2.2, we can get (2.7).

(3.5)
In the sequel, for any n ∈ N, we denote by O(s n ) a function of order s n , for sufficiently large s. Then, we give the following Carleman estimates.
(i) Assume that (H 1 ) holds. Then, there exist two positive constants s 0 = s 0 (α, η, µ, a, b, c, d) and C, such that for all s ≥ s 0 , every solution p to (3.1) satisfies where
(ii) Assume that (H 3 ) holds. Then, there exist two positive constants s 1 = s 1 (α, η, a, b, c, d) and C, such that for all s ≥ s 1 , every solution p to (3.1) satisfies where The condition a ≥ c is not needed in the case of (ii), but it is necessary in the observability inequality. To avoid confusion about the conditions, we relax the conditions here to be consistent with those for the observability estimates.
Step 1. Let us estimate A 1 in (3.11). Recalling z(t, 1) = 0 on (0, T ) and the definition of V in (2.8), we have The reasonableness of the computations may be delicate since we work in non-standard weighted spaces, specially in the critical potentials. For this reason, we make computations that may be justified by the regularization process described in [29]. In order to understand the computations related to A 1 , it helps to replace z by z n := θp n , where p n is the solution to the regularized problem in which the potential µ x β has been replaced by µ (x+ 1 n ) β . Therefore, the quantity that we actually need to compute is the following one: where V n is obtained by replacing z in V by z n .
Therefore, by the definitions of A in (2.4) and Φ in (3.10), we get that Hence, Step 2. Let us estimate A 2 in (3.11). By (3.1) and a > 0, we know that From the definition of A in (2.4) and noting that |γ t | ≤ Cγ 1+ 1 k , one can obtain that (3.14) From z(0, x) = z(T, x) = 0 in (0, 1), one can see that And then we estimate "cE Q µ x β |dz| 2 dx" in two cases: the case of a sub-critical exponent 0 < β < 2 − α and the case of a critical exponent β = 2 − α.

4
, one gets that Combining the above inequality with (3.13)-(3.16), we can get (3.18) Step 3. Let us estimate A 3 in (3.11). By the definitions of A and Φ, we can get From (3.10), it is easy to see that By the definitions of E and F in (2.8), and noting that Φ x = 0, we have Therefore, By z(t, 1) = 0 on (0, T ), one can get that It is easy to check that Therefore, it holds that By observing that |γ t | ≤ Cγ 1+ 1 k , |γγ t | ≤ Cγ 3 , and |γ tt | ≤ Cγ 1+ 2 k , one can conclude that (3.23)

By (3.19)-(3.23), we obtain that
(3.24) Step 4. In this part, we compute the last term A 4 in (3.11). By the definition of B 1 in (2.8), we have We produce estimate of term A 4 in two cases: the case of a sub-critical exponent 0 < β < 2 − α and the case of a critical exponent β = 2 − α.
(ii) Assume that (H 2 ) holds. Then, there exist two positive constants s 4 = s 4 (α, η, a, b, c, d) and C, such that for all s ≥ s 4 , every solution (h, H) to (3.29) satisfies where Remark 3.8. It is worth noting that (i) only needs a > 0, but (ii) needs a > 2c. The reason is that in order to remove the term containing |H x | 2 in (i), we can use the improved Hardy-Poincaré inequality (see Lem. 3.2) to choose the desired coefficients. However, the coefficients of the singular terms in (ii) can only be µ(α). If the stochastic equation reduces to the deterministic case, the condition a > 2c is not necessary.
Remark 3.9. Similar to (ii) in Theorem 3.2, we can also get that, for µ < µ(α), And, for µ = µ(α), we have Proof of Theorem 3.7. The proof is similar to the proof of Theorem 3.2. The main difference is that a, c are replaced by −a, −c, and Step 2. We only prove Step 2 here.
Step 2. Let us estimate A 2 , where A 2 is obtained by replacing a, c of A 2 in (3.11) by −a, −c. From (3.29) and a > 0, we know From the definition of A in (2.4) and |γ t | ≤ Cγ 1+ 1 k , it is easy to see that Further, for any ε > 0, one can obtain that On the other hand, similar to (3.16), one can see that (3.37) And then we estimate "−cE Q µ x β |dz| 2 dx" in two cases: the case of a sub-critical exponent 0 < β < 2 − α and the case of a critical exponent β = 2 − α.

Proofs of the main results
In the section, we give proofs of controllability and observability results for forward linear stochastic complex degenerate/singular Ginzburg-Landau equation (1.1), respectively. First, by the standard duality technique ( [28]) and observability estimate (1.7), the null controllability result in Theorem 1.2 can be obtained immediately. Therefore, we only need to prove Proposition 1.6. Proof of Proposition 1.6. In the case of (H 1 ), choose a cut-off function ξ ∈ C ∞ (R; [0, 1]) such that , and it is easy to see that G 1 ⊆ G 0 .
Similar to the proof of (4.4), we can get It follows that On the other hand, notice that d(|w| 2 ) = wdw + wdw + |dw| 2 . Hence, for any 0 ≤ t 1 ≤ t 2 ≤ T ,  By Gronwall's inequality, it follows that Combining the above equality with (4.8), we have Notice thatĈ 3 e C(1+aK0) is C 1 (a, b, c, d) in (1.8). Then, we have completed the proof for the case of (H 1 ). In the case of (H 2 ), similar to (H 1 ), we know x α |w x | 2 dxdt.
Combining the above equality with (4.14), we get Notice that C 1 e C(1+aK0) is C 1 (a, b, c, d) in (1.11). For the case of (H 1 ), we have completed the proof.
Combining the above equality with (4.17), we have that Notice that C 2 is C 1 (a, b, c, d) in (1.12). The proof of Theorem 1.8 is completed.