Feedback stabilization for a bilinear control system under weak observability inequalities

TL;DR

This paper develops a feedback stabilization method for bilinear control systems with weak observability, providing explicit decay rates and applications to Schrödinger and wave equations.

Contribution

It introduces a novel stabilization approach under weak observability conditions and derives explicit decay rates for regular initial data.

Findings

Explicit weak decay rates for bilinear systems

Application to Schrödinger and wave equations

Stability analysis under weak observability

Abstract

In this paper, we discuss the feedback stabilization of bilinear systems under weak observation properties. In this case, the uniform stability is not guaranteed. Thus we provide an explicit weak decay rate for all regular initial data. Applications to Schr\"odinger and wave equations are provided.

Equations

$$y^{\prime }(t)=Ay(t)+p(t)By(t),y(0)=y_0,$$

$$E^{\prime }(t)\le \mathcal{R}e\left(p(t)⟨By(t),y(t)⟩\right)\cdot$$

$$\mathcal{R}e\{f(y(t))⟨By(t),y(t)⟩\}<0,\forall t\ge 0\cdot$$

$$p_r(t)=-\frac{⟨y(t),By(t)⟩}{\Vert y(t){\Vert }^r}{1}_{\{t\ge 0:y(t)\ne 0\}},r\in \mathbb{R}\cdot$$

$$⟨BS(t)y,S(t)y⟩=0,\forall t\ge 0⟹y=0\cdot$$

$$\exists T,\delta >0;{\int }_0^T|⟨BS(t)y,S(t)y⟩|dt\ge \delta \Vert y{\Vert }^2,\forall y\in H,$$

$$\Vert y(t)\Vert =O\left(\frac{1}{\sqrt{t}}\right)\cdot$$

$$\Vert y\Vert \le \Vert y{\Vert }_L^{\theta }\Vert y{\Vert }_K^{1-\theta },\forall y\in D(A)\cdot$$

$$a_{k+1}\le a_k-C{a}_{k+1}^{\alpha +2},\forall k\ge 0,$$

$$a_k\le \frac{M}{(k+1{)}^{\frac{1}{\alpha +1}}},\forall k\ge 0.$$

$$\displaystyle\int_{0}^{T}|\langle BS(s)y(t),S(s)y(t)\rangle|ds\leq\bigg{(}2T^{\frac{3}{2}}\|B\|^{2}\|y_{0}\|^{2-r}+T^{\frac{1}{2}}\bigg{)}\|y(t)\|^{\frac{r}{2}}\bigg{(}\displaystyle\int_{t}^{t+T}\displaystyle\frac{|\langle y(s),By(s)\rangle|^{2}}{\|y(s)\|^{r}}ds\bigg{)}^{\frac{1}{2}}\cdot$$

$$y^{\prime }(t)=Ay(t)+F_0(y(t)),y(0)=y_0,$$

$${\int }_0^T|⟨BS(t)z,S(t)z⟩|dt\ge \delta \Vert z{\Vert }_L^2,\forall z\in D(A),$$

$$p_0(t)=-⟨y(t),By(t)⟩,$$

$$\Vert y(t){\Vert }^2=O\left(t^{-\frac{\theta }{2-\theta }}\right)ast\to +\infty .$$

$${\int }_0^T|⟨BS(t)z,S(t)z⟩|dt\ge \delta \Vert z{\Vert }_K^2\mathcal{H}\left(\frac{\Vert z{\Vert }^2}{\Vert z{\Vert }_K^2}\right),\forall z\in K\setminus \left\{0\right\},$$

$$\Vert y(t){\Vert }^2=O\left({\mathcal{K}}^{-1}\left(\frac{1}{t}\right)\right)ast\to +\infty \cdot$$

$$S(t)y_0-y(t)={\int }_0^t⟨y(s),By(s)⟩S(t-s)By(s)ds,$$

$$\displaystyle\int_{0}^{T}|\langle BS(s)y(t),S(s)y(t)\rangle|ds\leq\bigg{(}2T^{\frac{3}{2}}\|B\|\|y_{0}\|^{2}+T^{\frac{1}{2}}\bigg{)}\bigg{(}\displaystyle\int_{t}^{t+T}\displaystyle|\langle y(s),By(s)\rangle|^{2}ds\bigg{)}^{\frac{1}{2}}\cdot$$

$$\Vert y(t){\Vert }^2-\Vert y(s){\Vert }^2\le -2{\int }_s^t|⟨y(\tau ),By(\tau )⟩{|}^{2}d\tau ,0\le s\le t\cdot$$

$$\Vert y(t)\Vert \le \Vert y_0\Vert ,\forall t\ge 0\cdot$$

$$s_{k}-s_{k+1}\geq\frac{2\big{(}\displaystyle\int_{0}^{T}|\langle BS(s)y(kT),S(s)y(kT)\rangle|ds\big{)}^{2}}{\big{(}2T^{\frac{3}{2}}\|B\|\|y_{0}\|^{2}+T^{\frac{1}{2}}\big{)}^{2}}$$

$$\|y(kT)\|_{K}^{\frac{4(1-\theta)}{\theta}}\big{(}s_{k}-s_{k+1}\big{)}\geq\frac{2\delta^{2}}{\big{(}2T^{\frac{3}{2}}\|B\|\|y_{0}\|^{2}+T^{\frac{1}{2}}\big{)}^{2}}s_{k}^{2/\theta}.$$

$$\Vert Ay(t)\Vert \le \Vert A{y}_0+F_0(y_0)\Vert +\Vert F_0(y_0)\Vert$$

$$\le \Vert A{y}_0\Vert +2\Vert B{\Vert }^2\Vert y_0{\Vert }^3,$$

$$\Vert y(t){\Vert }_K\le C(\Vert y_0\Vert )\Vert y_0{\Vert }_{D(A)},\forall t\ge 0,$$

$$s_k-s_{k+1}\ge C^{\prime }(\Vert y_0\Vert )s_k^{2/\theta },$$

$$C^{\prime}(\|y_{0}\|)=\frac{2\delta^{2}}{\big{(}2T^{\frac{3}{2}}\|B\|\|y_{0}\|^{2}+T^{\frac{1}{2}}\big{)}^{2}\big{(}C(\|y_{0}\|)\|y_{0}\|_{D(A)}\big{)}^{\frac{4(1-\theta)}{\theta}}}\cdot$$

$$s_k-s_{k+1}\ge C^{\prime }(\Vert y_0\Vert )s_{k+1}^{2/\theta },$$

$$s_k=O\left(\frac{1}{k^{\frac{\theta }{2-\theta }}}\right)\cdot$$

Full text

1. 1 Introduction

2. 2 Stabilization results and decay estimates

3. 2.1 Preliminary

4. 2.2 The case of quadratic control

5. 2.3 The case of normalized control

6. 2.4 Further result

7. 3 Some applications

8. 3.1 Polynomial stabilization of bilinear coupled wave equations

9. 3.2 Weak stabilization of bilinear wave equation

10. 3.3 Weak stabilization of bilinear Schrödinger equation

1. Introduction

Most of control systems which are used to describe processes in physics, engineering, economics.. are generally nonlinear and relatively complex, which makes the identification of mathematical models extremely difficult. Consequently, the first investigations of different concepts in control theory were confined mainly to simple models, namely linear difference equations and linear ordinary differential equations (see [7, 31, 32]). Then there has been several work on generalizing the well known systems theory concepts to systems described by partial differential equations, where the adequate state space is an infinite dimensional functional space (see [11, 10, 14, 15, 21, 23, 32]). In recent years, bilinear systems have been widely used in the modeling of various dynamical systems, since many real physical processes may be appropriately modeled as bilinear systems when linear models are inadequate. Also, bilinear systems provide a better approximation to a nonlinear system than linear ones [16, 17, 19, 20].

In this paper, we are concerned with the question of feedback stabilization of the following homogeneous bilinear system:

[TABLE]

where the state space is an Hilbert $H$ with inner product $\langle\cdot,\cdot\rangle$ and corresponding norm $\|.\|$, the dynamic $A$ is an unbounded operator with domain $D(A)\subset H$ and generates a semigroup of contractions $S(t)$ on $H$, $B$ is a linear bounded operator from $H$ into $H$ and $p(.)$ is a scalar function and represents the control.

In order to construct a stabilizing control $p(t)$ for the system (1), a natural approach is to formally compute the time rate of change of the energy $E(t):=\displaystyle\frac{1}{2}||y(t)||^{2}$, obtaining thus

[TABLE]

Thus, in order to make the energy nonincreasing, one may consider feedback controls $p(t)=f(y(t))$ such that the following ”dissipating energy inequality” holds

[TABLE]

As a class of feedback controls that satisfy the last inequality, one can consider the following family of controls :

[TABLE]

The case $r=0$ has been considered in many works [6, 12, 13, 25]. In [6], a weak stabilization result has been established using the control $p_{0}(t)$ provided that the following assumption is verified:

[TABLE]

Under the assumption

[TABLE]

a strong stabilization result has been obtained using the control $p_{0}(t)$, and the following estimate (see [12, 25]) was given:

[TABLE]

In addition, others polynomial estimates was provided in [27] using the control $p_{r}(t)$ with $r<2.$ The case $r=2$ has been considered in [26], where exponential stabilization results have been established using $p_{2}(t)$ under the observation assumption (3).

Note that the inequality (3) is necessary for uniform stabilization of conservative systems, so we can not expect such a degree of stability under a weaker observability assumption. Accordingly, we will look for weak stabilization when dealing with weak observation assumptions. Such a question was investigated in the case of unbounded linear feedback control (see [3, 4]) and bounded nonlinear feedback (see [2] and references therein).

In this paper, we study the weak stabilization of the bilinear system (1) under weaker observation assumptions than (3) using the controls $p_{r}(t),\;r=0,2.$ In the next section we first present our stabilization results under observation inequalities that extend the classical one (3), and we provide the asymptotic estimate of the resulting state. Then we give a stabilization result under a null-controllability like assumption. In the third section we present applications to Schrödinger and wave equations.

2. Stabilization results and decay estimates

For a fixed $\theta\in(0,1),$ we consider a couple $(K,L)$ of Banach spaces (see [24]) such that:

$\bullet$ $H\subset L,\;D(A)\subset K\subset H,$

$\bullet$ for all $y\in D(A);\;\|y\|_{D(A)}\sim\|y\|_{K},$

$\bullet$ the following interpolation holds: $[K,L]_{\theta}=H$ with;

[TABLE]

Let ${\mathcal{H}}:\mathbb{R}_{+}\rightarrow\mathbb{R}_{+}$ be a continuous and increasing function on $\mathbb{R}_{+}$.

2.1. Preliminary

Let us recall the following technical lemmas which gives useful estimates for the problem of stabilization.

Lemma 2.1**.**

([3], p. 26 and [4, 5]). Let $(a_{k})$ be a sequence of positive real numbers satisfying

[TABLE]

where $C>0$ and $\alpha>-1$ are constants. Then there exists a positive constant $M$ such that:

[TABLE]

Lemma 2.2**.**

[27]**. Let $A$ generate a semigroup of contractions $S(t)$ and let $B:H\to H$ be linear and bounded. Then for all $r\in]-\infty,2]$, the system (1) controlled with $p_{r}$ possesses a unique global mild solution, which satisfies the following decay estimate:

[TABLE]

2.2. The case of quadratic control

Let (1) be as given in the introduction. With the control $p_{0}(t)$, the system (1) becomes

[TABLE]

where $F_{0}(y)=-\langle y,By\rangle By,\;y\in H$.

The next result discusses the asymptotic behaviour of the solution $y(t)$ of (6).

Theorem 2.1**.**

Suppose that:

1. (i)

$A$* generates a semigroup of contractions $S(t)$,* 2. (ii)

$B$* is linear and bounded.*

1. (1)

If there exist $\delta,T>0$ such that

[TABLE]

then for all $y_{0}\in D(A),$ the feedback law

[TABLE]

guarantees the following decay estimate for the respective solution to (1),

[TABLE] 2. (2)

Suppose that the function ${\mathcal{K}}:x\rightarrow x{\mathcal{H}}(x)$ is invertible on $\mathbb{R}_{+}$. If there exist $\delta,T>0$ such that

[TABLE]

then the feedback law (8) guarantees the following decay estimate for the respective solution to (1):

[TABLE]

Remark 2.1**.**

In (9), if we take $\mathcal{H}(x)=1,\,\forall x\geq 0$ then $\|y(t)\|^{2}=O\left(\frac{1}{t}\right),\ as\ t\rightarrow+\infty.$ So, the estimate (10) generalizes the stability result obtained in [12, 25] as an implication of exact observability inequality (3).

Proof.

Proof of the first assertion: Let $y_{0}\in D(A).$ We can assume that $y_{0}\neq 0.$

According to Lemma 2.2, the system (6) possesses a unique global mild solution, which is continuous with respect to the initial state, and satisfies the following variation of parameters formula:

[TABLE]

and satisfies the decay estimate

[TABLE]

Moreover since $A$ is dissipative, an approximation argument (see [7]) shows that

[TABLE]

It follows that

[TABLE]

Let us consider the sequence $s_{k}=\|y(kT)\|^{2}$, $k\in\mathbb{N}$.

Applying the inequality (13) for $s=kT$ and $t=(k+1)T]$ and using (12), we derive

[TABLE]

which by, (4) and (7), gives:

[TABLE]

Moreover, we can see from nonlinear semigroup properties that for all $t\geq 0$, we have

[TABLE]

[TABLE]

from which it comes (recall that $\|y_{k}\|_{K}\sim\|y\|_{D(A)}$)

[TABLE]

with C(\|y_{0}\|)=:C_{1}\big{(}1+\|B\|^{2}\|y_{0}\|^{2}\big{)}, and where $C_{1}$ is a positive constant which is independent of $y_{0}\cdot$

Thus

[TABLE]

where

[TABLE]

Then, since $\|y(t)\|^{2}$ decreases in time, this implies

[TABLE]

which, by applying Lemma 2.1, gives

[TABLE]

Then using again that $\|y(t)\|^{2}$ decreases, we deduce that

[TABLE]

Proof of the second assertion.

Let $y_{0}\in D(A)\setminus\{0\},$ let $C(\|y_{0}\|)$ be the constant given in (15) and consider the sequence

[TABLE]

Let us observe that under the assumptions on $\mathcal{H}$, the two sequences $e_{k}$ and $\frac{e_{k}}{s_{k}}$ are decreasing.

Using (9), we derive from (14)

[TABLE]

Moreover, it comes from (15) and the increasing of $\mathcal{H}$ that

[TABLE]

Having in mind that $\|y_{k}\|_{K}\sim\|y\|_{D(A)}\geq\|y_{0}\|,$ we deduce that

[TABLE]

for some constant $C_{2}>0$ depending on $\theta$. Thus, using the decreasing of $e_{k}$ and $\frac{e_{k}}{s_{k}}$ it comes

[TABLE]

and

[TABLE]

Applying Lemma 2.1, we deduce that:

[TABLE]

$\Box$

2.3. The case of normalized control

Let (1) be as given in the introduction and consider the control

[TABLE]

Thus the system (1) takes the form

[TABLE]

where $F_{2}(y)=-\frac{\langle y,By\rangle}{\|y\|^{2}}By,$ if $y\neq 0$ and $F_{2}(0)=0.$

In the following result we provide a uniform estimate for the solution of (16).

Theorem 2.2**.**

Suppose that:

1. (i)

$A$* generates a semigroup of contractions $S(t)$,* 2. (ii)

$B$* is linear and bounded.*

1. (1)

If the estimate (7) is satisfied, then for all $y_{0}\in D(A)$ the feedback law

[TABLE]

leads to the following decay estimate for the respective solution to (1)

[TABLE]

for some constant $C>0$ which is independent of $y_{0}$. 2. (2)

Suppose that ${\mathcal{H}}(x)$ is invertible on $\mathbb{R}_{+}$ and that the function $x\mapsto\frac{1}{x}\,{\mathcal{H}}^{2}(x)$ is increasing on $(0,1).$ If the estimate (9) is satisfied, then the feedback law (17) leads to the following decay estimate for the respective solution to (1):

[TABLE]

for some constant $C>0$ which is independent of $y_{0}$.

Proof.

By dissipativeness, we can assume without loss of generality that $y(t)$ does not vanish, so that the control $p_{2}(t)$ takes the form $p_{2}(t)=-\frac{\langle y(t),By(t)\rangle}{\|y(t)\|^{2}}.$

Proof of the first assertion.

Applying Lemma 2.2 for $r=2$, we deduce that system (16) possesses a unique global mild solution which is continuous with respect to initial states, and given by the following variation of parameters formula

[TABLE]

Moreover, we can show that (see [7])

[TABLE]

from which it comes immediately

[TABLE]

Moreover, we can again see from Lemma 2.2 that the solution $y(t)$ of (16) satisfies the decay estimate

[TABLE]

Now, let us consider the sequence $s_{k}=\|y(kT)\|^{2}$, $k\in\mathbb{N}$.

Applying the inequality (20) for $s=kT$ and $t=(k+1)T]$ and using (7) and (21), we derive

[TABLE]

which by (4) gives

[TABLE]

We have

[TABLE]

It follows that

[TABLE]

where $C_{*}$ is a positive constant which is independent of $y_{0}\cdot$

Thus

[TABLE]

where $C>0$ is independent of $y_{0}\cdot$

In the sequel, $C>0$ will denote a generic constant which is independent of $y_{0}\cdot$

Since $\|y(t)\|$ decreases in time, this implies

[TABLE]

Applying Lemma 2.1, we deduce that

[TABLE]

Then using again that $\|y(t)\|$ decreases, we deduce that

[TABLE]

Proof of the second assertion.

We consider the sequence

[TABLE]

where $C_{*}$ is the constant given in (22).

Here, the two sequences $e_{k}$ and $\frac{e_{k}}{s_{k}}$ decreasing.

We deduce from the inequalities (21) and (20) that

[TABLE]

which by, (9) and the increasing of $\mathcal{H}$, gives

[TABLE]

Then, using the decreasing of $e_{k}$ and $\frac{e_{k}}{s_{k}}$ it comes

[TABLE]

and

[TABLE]

Applying Lemma 2.1, we deduce that

[TABLE]

$\Box$

2.4. Further result

In this part, we will establish a stability result under a null-controllability like assumption. More precisely, we consider the following estimate:

[TABLE]

for some $\delta,T>0$.

Note that inequality (23) can be seen as an estimate is also a weak observability inequality; since only $S(T)y_{0}$ may be recovered, not $y_{0}$.

Theorem 2.3**.**

Suppose that:

1. (i)

$A$* generates a semigroup of contractions $S(t)$,* 2. (ii)

$B$* is linear and bounded,* 3. (iii)

the estimate (23) holds.

Then for all $y_{0}\in D(A),$ the respective solution to controls $p_{0}(t)$ and $p_{2}(t)$ tends to [math] as $t\to+\infty$.

Proof.

It suffices to prove the case of control $p_{2}(t)$ since the two cases can be treated similarly.

Let $y_{0}\in D(A)$. Then using (21) and (23), we obtain

[TABLE]

for some constant $C>0$.

Moreover, the estimate (20) implies the following integral convergence:

[TABLE]

This, together with (24) implies that

[TABLE]

Then taking into account the inequalities (4) and (22), we deduce that

[TABLE]

Now, let $\epsilon>0$ and let $t_{1}>T$ be such that

[TABLE]

Implementing this in the following variation of constants formula

[TABLE]

it comes

[TABLE]

Hence Gronwall yields

[TABLE]

which completes the proof. $\Box$

Remark 2.2**.**

Note that if (23) holds with the norm of $H$, then one gets the convergence in $H$ for all $y_{0}\in H.$

3. Some applications

Here we give some applications of Theorems 2.1 $\&$ 2.2.

3.1. Polynomial stabilization of bilinear coupled wave equations

We consider the two following initial and boundary coupled problems:

[TABLE]

and

[TABLE]

where $a\in L^{\infty}(\Omega),a\geq 0,\beta\in\mathbb{R}^{*}$, $\Omega$ is a bounded open set of $\mathbb{R}^{n}$ of class $\mathcal{C}^{2}$ and where ${\bf 1}_{E_{1}}$ is the characteristic function of the set $E_{1}:=\{t\geq 0:\;(u(t),u_{t}(t),v(t),v_{t}(t))\neq(0,0,0,0)\}.$

Here, we have:

[TABLE]

which is a skew-adjoint operator satisfying the assumptions $(i)-(ii)$ of Theorems 2.1 $\&$ 2.2, and B=\left(\begin{array}[]{llcc}0&0&0&0\\ 0&a&0&0\\ 0&0&0&0\\ 0&0&0&0\end{array}\right)\in\mathcal{L}(H).

Moreover, the corresponding linear equation becomes in this case:

[TABLE]

According to [1, 30] we show that the observability inequality is given by

Proposition 3.1**.**

There exist $T,\beta_{0}>0$ and $c_{T}>0$ such that for all $0<|\beta|<\beta_{0}$, the following observability inequality holds:

[TABLE]

for all initial data $(\phi_{0},\phi_{1},\psi_{0},\psi_{1})\in\left[H^{1}_{0}(\Omega)\times L^{2}(\Omega)\right]^{2}.$

We remark here that we have (7) for $L=\left[L^{2}(\Omega)\times H^{-1}(\Omega)\right]^{2}$ and $\theta=1/2.$

Thus according to Theorems 2.1 $\&$ 2.2, we have the following stabilization result for the bilinear wave equation.

Theorem 3.1**.**

We suppose that $supp\,(a)$ satisfies a Lions Geometric Control Condition (LGCC), as in [30]. Then, we have:

1. (1)

The energy of (26) satisfies the estimate:

[TABLE]

and for all initial data $(u_{0},u_{1},v_{0},v_{1})\in\left(\left[H^{2}(\Omega)\cap H^{1}_{0}(\Omega)\right]\times H^{1}_{0}(\Omega)\right)^{2}$. 2. (2)

The energy of (25) satisfies the estimate:

[TABLE]

3.2. Weak stabilization of bilinear wave equation

We consider the following initial and boundary problem:

[TABLE]

and

[TABLE]

where $a\in L^{\infty}(\Omega),a\geq 0$ and $\Omega$ is a bounded open set of $\mathbb{R}^{n}$ of class $\mathcal{C}^{2}$.

In this case, we have:

[TABLE]

and $A$ is a skew-adjoint operator satisfying $(i)-(ii)$ of Theorems 2.1 and 2.2, and we have

[TABLE]

The corresponding linear equation is:

[TABLE]

According to [18, 29] we show that the observability inequality is given by

Proposition 3.2**.**

There exists $T$ and $c_{T}>0$ such that the following observability inequality holds:

[TABLE]

for all non-identically zero initial data $(\phi_{0},\phi_{1})\in\left[H^{2}(\Omega)\cap H^{1}_{0}(\Omega)\right]\times H^{1}_{0}(\Omega).$

We remark here that we have (9) for ${\mathcal{H}}(x)=\exp(-\frac{c_{T}}{x^{1/2}}),\,\forall\,x>0.$

Thus according to Theorems 2.1 and 2.2 we have the following stabilization result for the bilinear wave equation.

Theorem 3.2**.**

We suppose that $meas(supp\,(a))\neq 0$. We have:

1. (1)

The energy of (31) satisfies the estimate:

[TABLE]

and for all initial data $(u_{0},u_{1})\in\left[H^{2}(\Omega)\cap H^{1}_{0}(\Omega)\right]\times H^{1}_{0}(\Omega)$. 2. (2)

For all initial data $(u_{0},u_{1})\in\left[H^{2}(\Omega)\cap H^{1}_{0}(\Omega)\right]\times H^{1}_{0}(\Omega),$ the energy of (32) satisfies the estimate:

[TABLE]

3.3. Weak stabilization of bilinear Schrödinger equation

We consider the following initial and boundary problem:

[TABLE]

and

[TABLE]

where $a\in L^{\infty}(\Omega),a\geq 0$ and $\Omega$ is a bounded open set of $\mathbb{R}^{n}$ of class $\mathcal{C}^{2}$.

In this case, we have:

[TABLE]

and $A$ is a skew-adjoint operator satisfying assumptions $(i)-(ii)$ of Theorems 2.1 and 2.2, and we have $B=a\,I\!d\in\mathcal{L}(H).$

Moreover the linear equation becomes in this case:

[TABLE]

From [18, 29], we can show that the following observability inequality:

Proposition 3.3**.**

There exists $T$ and $c_{T}>0$ such that the following observability inequality holds:

[TABLE]

for all non-identically zero initial data $\phi_{0}\in H^{2}(\Omega)\cap H^{1}_{0}(\Omega).$

Here, (9) holds for ${\mathcal{H}}(x)=\exp(-\frac{c_{T}}{x^{1/2}}),\,\forall\,x>0.$ We also notice that the constant $c_{T}$ in (3.3) can be taken large enough, so that the function $\frac{1}{x}{\mathcal{H}}^{2}(x)$ is increasing on $(0,1)$.

By applying Theorems 2.1 and 2.2, we obtain the following stabilization result for the bilinear Schrödinger equation.

Theorem 3.3**.**

We suppose that $meas(supp\,(a))\neq 0$. We have:

1. (1)

The energy of (37) satisfies the estimate:

[TABLE]

and for all initial data $u_{0}\in H^{2}(\Omega)\cap H^{1}_{0}(\Omega)$. 2. (2)

The energy of (38) satisfies the estimate:

[TABLE]