ASYMPTOTIC BEHAVIOR OF A SCHR¨ODINGER EQUATION UNDER A CONSTRAINED BOUNDARY FEEDBACK

. Design of controller subject to a constraint for a Schr¨odinger equation is considered based on the energy functional of the system. Thus, the resulting closed-loop system is nonlinear and its well-posedness is proven by the nonlinear monotone operator theory and a complex form of the nonlinear Lax-Milgram theorem. The asymptotic stability and exponential stability of the system are discussed with the LaSalle invariance principle and Riesz ba- sis method, respectively. In the end, a numerical simulation illustrates the feasibility of the suggested feedback control law.


1.
Introduction. The Schrödinger equation is an efficient model to describe optical propagation and dynamics of quantum system, and especially, it can simulate the behavior of micro mechanics precisely ( [9,13,19,32]), so, it has drawn a great deal of attention. There are extensive literatures on Schrödinger equations, involving solvability, controllability, stabilization and so on ( [1,6,7,10,11,14,20,22,23,24,25,28,29,30]). In this paper, we address the following Schrödinger equation    w t (x, t) = −iw xx (x, t), x ∈ (0, 1), t > 0, w(0, t) = 0, w x (1, t) = u(t), w(x, 0) = w 0 (x), where i is the pure imagine unit, u(t) is the control (input) and w 0 (x) is the initial state. In some biological and chemical experiments, the control signals are often limited in an interval due to some constraint conditions, e.g., the restriction of concentration in some types of chemical reactions. Therefore, it is very interesting to study the system (1) with a bounded control, i.e., |u(t)| ≤ β, where β is a positive constant. Based on the L 2 -energy of the system (1) its derivative with respect to t along the locus of the system (1) can be formulated by dE(t) dt = − u(t)w(1, t) , where the notation " " is the imaginary part of a complex number. Thus, the feedback control is taken as follows: Thus, |u(t)| ≤ β and Hence, the closed-loop system (1) with (3) can be formulated by where Since the dynamic system (5) is nonlinear, the premier difficulty is its solvability. A similar problem was discussed in [23]. In this paper, the well-posedness is proven via the maximal monotone operator theory and a complex form of the nonlinear Lax-Milgram theorem. Let y(t) = w(1, t) be the collocated observation (output), y(t) is obviously not continuous with respect to w (refer to [21,Theorem 1]). Thus, the second key point is the asymptotical stability of the output, lim t→∞ y(t) = 0, which is an important step for the exponentially stable analysis. Hence, the novelties in this paper are to provide a complex form of the Lax-Milgram theorem and to prove the exponential stability of the system (5) without the assumption of Theorem 1 in [21] that the collocated output of system is uniformly continuous on its state.
In this paper, the notation " " and " " stand for the real and imaginary part of a complex number, respectively.ẇ(t) is the time derivative of w(t). f (x) and f (x) are the first and second derivative of f (x) with respect to the spatial variable x, respectively.
The remainder of the paper is proceeded as follows. In section 2, the wellposedness of the system (5) is discussed with the maximal monotone operator theory and a complex form of the nonlinear Lax-Milgram theorem, which is proven in Appendix for completeness. In section 3, the asymptotic stability, the asymptotical stability of the output and the exponential stability are proven with the invariant principle in infinite dimensional space [8], an estimation of the upper bound for w(1, t) and Riesz basis method [20,35,36], respectively. In section 4, a numerical simulation is given to verify the theoretical results and the feasibility of the presented feedback law (3). Finally, some concluding remarks are presented in section 5.
2. Well-posedness of the system. This section discusses the well-posedness of the system (5). For this, the state space H is chosen as L 2 (0, 1) with the norm

ASYMPTOTIC BEHAVIOR OF A SCHRÖDINGER EQUATION 385
The system operator A in H is defined by with its domain where and H 2 (0, 1) is the usual Sobolev space [2, Chapter 3]. Then the system (5) can be rewritten as a nonlinear evolutionary equation where w(t) = w(·, t) and w 0 is an initial state.
In addition, if a new operator A 0 is defined by the system (1) can be formulated as where B 0 = δ(· − 1) is the Dirac Delta distribution. Obviously, input operator B 0 is unbounded and the nonhomogeneous term B 0 u(t) with u(t) defined by (3) does not satisfy the Lipschitz continuous condition on w(t). Therefore, the linear nonhomogeneous theory of the Cauchy problem is not suitable for (9). Thus, it is necessary to apply the theory of nonlinear monotone operators to prove the existence and uniqueness of the solution of the system (8) (i.e., (5)). According to the definition of f β (1) in (7), it can be guessed that the operator −A defined by (6) and (7) is monotone. Furthermore, A satisfies the following proposition. Proposition 1. −A is a maximal monotone operator.
Lemma 2.2. Let Z be a complex Hilbert space and L a bounded linear functional on it. Assume that B(x, z) = B 1 (x, z) + iB 2 (x, z) is a complex-valued functional defined on the product space Z × Z, and satisfies the following inequality: ∀z ∈ Z, where B 1 (x, z) and B 2 (x, z) are bounded sesqui-linear forms. And assume that N (x, z) is a complex-valued nonlinear functional defined on Z × Z, and satisfies the following conditions: (1): For every x, N (x, z) is a bounded linear functional with respect to z, and N (x, z) is sequentially continuous with respect to x, that is, Then, there exists a unique x ∈ Z such that In what follows, the proof of Proposition 1 is divided into two steps.
∀f ∈ H, we want to establish the solvability of the resolvent equation We introduce a Hilbert space Z = {ϕ ∈ H 1 (0, 1)|ϕ(0) = 0}, whose inner product and norm are defined by respectively, where H 1 (0, 1) is the usual Sobolev space. Thus for φ ∈ Z, multiplying both sides of the resolvent equation (12) by φ(x) and integrating from 0 to 1, we obtain that then, from (13), we can obtain the following variational equation and (B(w, w)) = w 2 Z . Similar to the proof of the monotonicity of −A, we can deduce that From the following inequalities: it can be deduced that B 1 (w, φ) and B 2 (w, φ) are bounded sesqui-linear forms on Z, and that N (w, φ) is bounded linear functional with respect to φ and is sequentially continuous with respect to w. Thus, according to Lemma 2.2, there exists w ∈ Z such that (13) holds. Further restricting φ so that φ ∈ C ∞ 0 (0, 1), we can derive from (13) that

HAOYUE CUI, DONGYI LIU AND GENQI XU
which implies that (12) holds in the sense of distributions. Thus, w ∈ Z ⊂ H and f ∈ H leads to w ∈ H. So, w ∈ Z ∩ H 2 (0, 1), according to the Sobolev spaces theory [2].
To argue w (1) = −iw β (1), we return to (13) which is true for φ ∈ Z. Integrating by parts in the third term and using the above equation, we get Thus, w (1) = −iw β (1), which shows that there exists a w ∈ dom(A) such that (12) holds. So R(I − A) = H.
3. Stability of the system. In this section, we first study the asymptotical stability, using the LaSalle invariance principle in infinite dimensional space, which is very useful tool to have been applying to the asymptotical stability of nonlinear systems extensively ( [3], [4], [34], etc.). Then, we discuss the asymptotical stability of the output and the exponential stability. For this, we need the following proposition.
Proof. Since −A is monotone, it can be easily obtained that which implies that I −A is an injective (one-to-one) map. Thus, it can be concluded from R(I − A) = H that (I − A) −1 exists. It follows from H 2 (0, 1) ⊂ H = L 2 (0, 1) with compact embedding [2, Chapter 6] that (I − A) −1 is compact .
3.1. Asymptotical stability. It follows from Proposition 2 that the positive trajectory through w of the system (5) (i.e., (8)) defined by is relative compact for any w ∈ dom(A), according to Theorem 3 in [12]. So, we can apply LaSalle invariance principle in the infinite dimensional space to the system (5) (see Theorem 9.2.7 and Corollary 9.2.9 in [8], also [12,15,34]). We denote the set of equilibrium points of {S(t)} t≥0 by E, then where w(x, t) satisfies the following system Obviously, (14) only has the trivial solution, which, together with the LaSalle invariance principle, shows that the system (5) is asymptotically stable. Thus, we can draw the following conclusion: Theorem 3.1. Let A be defined by (6) and (7), and the initial state w 0 ∈ dom(A). Then, the L 2 -energy function (2) decays to zero as t → ∞, i.e., lim t→∞ E(w(·, t)) = 0, where w(x, t) is a solution of (5).
3.3. Exponential stability. It follows from Theorem 3.2 that |w(1, t)| ≤ β always holds for t > 0 large enough. So, the nonlinear system (5) reduces to a linear one for t > t 0 with t 0 > 0 large enough, that is, For the system (17), we can define its system operator L by with domain In what follows, we investigate the asymptotic spectra of L. Assume that λ is an eigenvalue for L, and f ∈ dom(L) is the corresponding eigenfunction, then By a simple calculation, we can obtain from (20) that f (x) = sinh √ iλx and Denote by 2 √ iλ = i(2n + 1)π + ε n (22) where ε n → 0 as n → ∞. Substituting (22) into (21), we get Then, the eigenvalues of operator L are where n is a sufficiently large integer. Correspondingly, the asymptotic expression of eigenfunction is f n (x) = i sin (n + 1/2)πx. Similar to Theorem 2.2 in [20], we can deduce the following theorem.  (18) and (19), there exists a sequence of (generalized) eigenfunctions of L, which forms a Riesz basis for H. Additionally, the following assertions are true.
1. The eigenvalues with sufficiently large module are algebraically simple. So, using Theorem 3.2 and Theorem 3.3, we can obtain the following corollary.
Corollary 1. Let A be defined by (6) and (7), and the initial state w 0 ∈ dom(A). Then, the system (5) is exponentially stable after sufficiently long time.
4. Numerical simulation. In this section, we simulate the dynamical behavior of the system (5) using the finite-difference method. In the numerical experiment, the parameter of boundary restriction on the control u(t), β = 1, and the initial function in the system (5) is taken as w 0 (x) = 8 sin(5πx) + 2ix 3 . The stepsizes of time and space are taken as τ = 0.01 and h = 0.01, respectively. Figure 1 and 2 show the dynamic behaviors of real and imaginary parts of w(x, t), respectively. Figure 3 displays the asymptotic behaviors of w(x, t) at the right end x = 1. For comparison, we also draw the asymptotic curve of w(1, t), the system at the right end x = 1 without restriction on the boundary control u(t), i.e., u(t) = −iw (1, t). The profiles in Figure 3 indicate that the simple modified feedback law (3) is feasible and practical, although the restriction on u(t) reduces the performance of the feedback law, u(t) = −iw(1, t).

5.
Conclusions. The nonlinearity of dynamic system (5) only results from the external restriction, and after a period of time, the motion of (5) restores the linear one, again. This conclusion seems to be commonplace but the constraint causes a certain trouble to analyze the system. In engineering, the control strategy (3) is called the so-called "saturating control law" [21,34]. The method of stability analysis is slightly different with the ones used in [21,34]. In fact, the assumption in Theorem 1 is simplified and weakened, that is to say, we generalize Theorem 1 in [21].  When the boundary conditions in the system (1) are replaced by w x (0, t) = 0, w x (1, t) = u(t), it can easily be proven that the control law (3) make the energy E(t) of the corresponding system decay. If the boundary conditions are w(0, t) = 0, w(1, t) = u(t) and the control law u(t) is chosen as follows:  then, it can be deduced that |u(t)| ≤ β and Thus, similar to (16), there exists c > 0 such that Therefore, for the boundary conditions w x (0, t) = 0, w x (1, t) = u(t) or w(0, t) = 0, w(1, t) = u(t), the similar results may be obtained. Contrasting with the Timoshenko beam with the saturating control [27], we discussed a first order complex system here. For this, the complex form of the Lax-Milgram theorem was introduced, titled Lemma 2.2, to prove the well-posedness of the closed-loop system. It, a generalization form of the nonlinear Lax-Milgram theorem, is also true for the real Hilbert space only if the imaginary part B 2 (x, z) is deleted, and it can be applied to other nonlinear systems. Moreover, we further improved and perfected the theoretical analysis of [27] here. Recently, the constrained systems begun to absorb attentions in engineering [16,26]. So, the general theory for constrained systems is very interesting and meaningful, which will be discussed in our future work.
Appendix. In this appendix, we give a proof of Lemma 2.2, which is divided into four steps. Proof.
According to the assumption conditions of B(x, z), there exists a bounded linear operator B defined on Z such that Bx, z = B(x, z). Since N (x, z) is a bounded linear functional with respective to z, there exists an operator N : Z → Z such that N (x, z) = N (x), z for each z ∈ Z. In addition, the continuity of N (x, z) with respective to x implies that N is weak* sequentially continuous. Moreover, N (z, x)) ≥ 0, ∀x, z ∈ Z, which shows that N (x) is monotone. The Riesz' representation theorem means that there is a f ∈ Z such that L(z) = (f, z), ∀z ∈ Z. Thus (11) can be rewritten as follows Bx, z + N (x), z = f, z , ∀z ∈ Z.
Step 2. The range of B + N , Range(B + N ), is close in Z.
Let y ∈ Range(B + N ), then there are sequences {x n }, {y n } ⊂ Z such that y n = B(x n ) + N (x n ) and lim n→∞ y n = y. So, the monotonicity of N (z) and (10)  Hence, y ∈ Range(B + N ). So, f = 0, which is a contradiction. Thus, Range(B + N ) = Z, that is to say, there exists x ∈ Z such that (11) holds.
Step 4. Uniqueness. Assume that there are two elements x and y such that (11)  which implies that x = y. That is to say, the element of Z that satisfies (11) is unique.