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In this paper, we shall study the stabilization and the robustness of a constrained feedback control for bilinear parabolic systems defined on a Hilbert state space. Then, we shall show that stabilizing such a system reduces stabilization only in its projection on a suitable subspace. For this purpose, a new constrained stabilizing feedback control that allows a polynomial decay estimate of the stabilized state is given. Also, the robustness of the considered control is discussed. An illustrating example and simulations are presented.

Bilinear systems represent a small, but important subset of nonlinear systems within which linear systems coexist as a special subclass. Adopting a bilinear model retains a well structured framework, which contains the well- known notional concepts such as time constants and steady-state behaviour. When adopting a bilinear approach, these concepts become operation-dependent quantities which can be appropriately modelled. Bilinear system models represent an important class of nonlinear models that are defined to be linear in both state and control when considered independently, with the nonlinearity (or bilinearity) arising from coupled terms involving products of system state and control (see [

d y ( t ) d t = A y ( t ) + p ( t ) B y ( t ) , y ( 0 ) = y 0 (1)

on a real Hilbert space H with inner product 〈 . , . 〉 ; and corresponding norm ‖ . ‖ , where the linear operator A generates a contraction semigroup ( S ( t ) ) t ≥ 0 on H and B ∈ L ( H ) . While the real valued function p ( . ) ∈ L 2 ( 0 , + ∞ ; ℝ ) represents a control. A function y ( . ) ∈ C ( 0 , t 0 ; H ) , t 0 > 0 , is a mild solution of the system (1) if and only if the solution y ( t ) of the system (1) satisfies the variation of parameters formula:

y ( t ) = S ( t ) y 0 + ∫ 0 t S ( t − τ ) p ( τ ) B y ( τ ) d τ , t ≥ 0 (2)

(see [

span { A z , a d 0 ( A , B ) z , a d 1 ( A , B ) z , ⋯ , a d k ( A , B ) z , ⋯ } = ℝ n , ∀ z ∈ ℝ n − { 0 } (3)

where a d k ( A , B ) is defined recursively as a d 0 ( A , B ) = B , a d 1 ( A , B ) = A B − B A and a d k + 1 ( A , B ) = a d 1 ( A , a d k ( A , B ) ) , ∀ k ∈ ℕ .

Using the following assumptions:

〈 B exp ( t A ) y , exp ( t A ) y 〉 = 0 , , ∀ t ≥ 0 ⇒ y = 0 (4)

the problem of stabilization has been studied in [

p 0 ( t ) = − 〈 y ( t ) , B y ( t ) 〉 (5)

a weak stabilization result is obtained under the weak observability condition:

〈 B S ( t ) y , S ( t ) y 〉 = 0 , ∀ t ≥ 0 ⇒ y = 0 (6)

In the case where B is sequentially continuous from H w ( H endowed with the weak topology) to H , the quadratic feedback control (2) weakly stabilizes the system (1), provided that the following weak observability assumption (4) holds (see [

∫ 0 T | 〈 B S ( t ) y , S ( t ) y 〉 | d t ≥ δ ‖ y ‖ 2 , ∀ y ∈ H , ( T , δ > 0 ) (7)

The strong stabilization result with the following decay estimate

‖ y ( t ) ‖ = O ( 1 t ) , as t → + ∞ (8)

i.e. ‖ y ( t ) ‖ ≤ M t , M > 0 for t large enough, has been obtained using the quadratic feedback control (5) (see [

p r ( t ) = − 〈 y ( t ) , B y ( t ) 〉 ‖ y ( t ) ‖ r , r ∈ ( − ∞ , 2 )

It has been shown in [

p ( t ) = − 〈 B y ( t ) , y ( t ) 〉 1 + | 〈 B y ( t ) , y ( t ) 〉 | (9)

strongly stabilizes the system (1), provided that the assumption (6) holds. It has been established in [

the linear operator A generates a contraction semigroup ( S ( t ) ) t ≥ 0 in H , then the system (1) is strongly

stable with the explicit decay estimate (8), using the control (9), provided that the estimate (7) holds. Here, we will establish an explicit decay estimate of the stabilized state and the robustness of the control (9) for a large class of bilinear systems as considered in [

Let us now recall the following definition concerning the asymptotic behavior of the system (1).

The system (1) is weakly (resp. strongly) stabilizable if there exists a feedback control p ( t ) = f ( y ( t ) ) , t ≥ 0 f : H → K : = ℝ , ℂ such that the corresponding mild solution y ( t ) of the system (1) satisfies the properties:

1. For each initial state y 0 of the system (1) there exists a unique mild solution defined for all t ∈ ℝ + of the system (1),

2. { 0 } is an equilibrium state of the system (1),

3. y ( t ) → 0 , weakly (resp. strongly), as t → + ∞ , for all y 0 ∈ H .

In the sequel of this section, we will present an appropriate decomposition of the state space H and the system (1) via the spectral properties of the operator A , and we apply this approach to study the stabilization pro- blem of the system (1). In [

into σ u ( A ) = { λ : R e ( λ ) ≥ − η , η > 0 } and σ s ( A ) = { λ : R e ( λ ) < − η } , then the state space H can be decomposed according to

H = H u ⊕ H s (10)

where H u = P u H = vect { φ j , 1 ≤ j ≤ N } , H s = P s H = vect { φ j , j > N } , P u is given by

P u = 1 2 π i ∫ C ( λ I − A ) − 1 d λ (11)

where C is a curve surrounding σ u ( A ) , P s = I − P u and for all j ≥ 1 , φ j is the eigenvector associated to

the eigenvalue λ j . The projection operators P u and P u commute with A , and we have A = A u + A s with A u = P u A P u and A s = P s A P s . Also, for all y ( t ) ∈ H , we set y u = P u y and y s = P s y . For linear systems, it

has been shown that the initial system can be decomposed into two subsystems on H u and H s . If A s satisfies the spectrum growth assumption:

lim t → + ∞ ln ( ‖ S s ( t ) ‖ ) t = supRe ( σ ( A s ) ) (12)

Which is equivalent to:

‖ S s ( t ) ‖ ≤ M 1 exp ( − η t ) , ∀ t ≥ 0 ( forsome M 1 > 0 ) (13)

where ( S s ( t ) ) t ≥ 0 denotes the semigroup generated by A s in H s , then stabilizing the whole system turns out to stabilizing its projection on H u (see [

B H u ⊂ H u and B H s ⊂ H s (14)

It is easily verified that the condition (14) is equivalent to the fact that the linear operator B commutes with P u . We note that the condition (14) also holds in the special case: H u = H . Let us consider that the system (1) can be decomposed in the following two subsystems:

d y u ( t ) d t = A u y u ( t ) + p ( t ) B u y u ( t ) , y u ( 0 ) = y 0 u ∈ H u (15)

d y s ( t ) d t = A s y s ( t ) + p ( t ) B s y s ( t ) , y s ( 0 ) = y 0 s ∈ H s (16)

in the state spaces H u and H S respectively, and B = B u ⊕ B s . It has been proved that stabilizing a linear system turns out to stabilizing its unstable part (see [

For finite-dimensional systems, the conditions (6) and (7) are equivalent (see [

The following result concerns the strong stabilization of the system (1).

Let

1. A generates a linear C 0 -contraction semigroup ( S ( t ) ) t ≥ 0 on H ,

2. A allows the decomposition (10) of H with dim H u < + ∞ such that (13) holds,

3. B be compact such that

〈 B S ( t ) y , S ( t ) y 〉 = 0 , ∀ t ≥ 0 ⇒ y = 0 (17)

Then, the constrained feedback control law:

p ρ ( t ) = − ρ 〈 y ( t ) , B y ( t ) 〉 1 + | 〈 y ( t ) , B y ( t ) 〉 | , ρ > 0 (18)

strongly stabilizes the system (1).

Proof

The system (1) controlled by (18) possesses a unique mild solution y ( . ) defined on a maximal interval [ 0 , t max [ and given by the variation of constants formula

y ( t ) = S ( t ) y 0 − ρ ∫ 0 t S ( t − τ ) 〈 y ( τ ) , B y ( τ ) 〉 1 + | 〈 y ( τ ) , B y ( τ ) 〉 | B y ( τ ) d τ (19)

corresponds to (18) (see [

Since ( S ( t ) ) t ≥ 0 is a contraction semigroup, we get:

d ‖ y ( t ) ‖ 2 d t ≤ − 2 ρ | 〈 y ( t ) , B y ( t ) 〉 | 2 1 + | 〈 y ( t ) , B y ( t ) 〉 | , ∀ y 0 ∈ D ( A ) (20)

It follows from (20) that

‖ y ( t ) ‖ ≤ ‖ y 0 ‖ , ∀ t ≥ 0 (21)

From (19) and using the fact that ( S ( t ) ) t ≥ 0 is a contraction semigroup and the Gronwall inequality, we deduce that the map y 0 → y ( t ) is continuous from H to H . Then (21) holds for all y 0 ∈ H by density argument, and hence t max = + ∞ (see [

such that y ( t n ) weakly converges in H and let z ∈ H such that y ( t n ) → z , weakly as n → ∞ . (The ex- istence of the sequence ( t n ) n ∈ ℕ and z are ensured by (21) and by the fact that space H is reflexive.) Taking y ( t n ) as initial state in (19) and using superposition property of the solution, and via the dominated conver-

gence theorem, we obtain 〈 B S ( t ) z , S ( t ) z 〉 = 0 , ∀ t ≥ 0 . It follows from (17) that z = 0 . Hence y ( t ) → 0 , weakly as t → + ∞ , and since dim H u < + ∞ , we have y u ( t ) → 0 , as t → + ∞ . For the component y S ( t ) of y ( t ) we have

y s ( t ) = S s ( t ) y 0 s − ρ ∫ 0 t S s ( t − τ ) 〈 y ( τ ) , B y ( τ ) 〉 1 + | 〈 y ( τ ) , B y ( τ ) 〉 | B s y s ( τ ) d τ (22)

Then for all 0 ≤ t 0 ≤ t , we have:

y s ( t ) = S s ( t − t 0 ) y s ( t 0 ) − ρ ∫ t 0 t S s ( t − τ ) 〈 y ( τ ) , B y ( τ ) 〉 1 + | 〈 y ( τ ) , B y ( τ ) 〉 | B s y s ( τ ) d τ (23)

It follows from (13) that

‖ y s ( t ) ‖ ≤ M 1 e − η ( t − t 0 ) ‖ y s ( t 0 ) ‖ + ρ M 1 ‖ B ‖ ∫ t 0 t e − η ( t − τ ) ‖ y s ( τ ) ‖ d τ (24)

From Gronwall inequality, we obtain:

‖ y s ( t ) ‖ ≤ M 1 e ( ρ M 1 ‖ B ‖ − η ) ( t − t 0 ) ‖ y s ( t 0 ) ‖ , ∀ t ≥ t 0

Taking ρ < η M 1 ‖ B ‖ , we deduce that y s ( t ) → 0 , as t → + ∞ .

Hence y ( t ) = y u ( t ) + y s ( t ) → 0 , as t → + ∞ .

In what follows, we will study the strong stabilizability of the system (1) with the decay estimate (8).

Before we state our main result, the following lemmas will be needed (see [

Let ( s k ) k ≥ 0 be a sequence of positive real numbers satisfying

s k + 1 + C s k + 1 2 + α ≤ s k , ∀ k ≥ 0 (24)

where C > 0 and α > − 1 are constants. Then there exists a positive constant M 2 (depending on α and C ) such that

s k ≤ M 2 ( k + 1 ) α + 1 , k ≥ 0 (25)

Let us now recall the following existing result (see [

Let A generate a contraction semigroup S ( t ) on H and let B be linear operator from H into itself. Then the system (1), controlled by (18) possesses a unique mild solution y ( t ) ∈ H for each y 0 ∈ H which satisfies

∫ 0 T | 〈 S ( s ) y ( t ) , B S ( s ) y ( t ) 〉 | d s = O ( ∫ t t + T | 〈 y ( s ) , B y ( s ) 〉 | 2 1 + | 〈 B y ( s ) , y ( s ) 〉 | d s ) , as t → + ∞ (26)

For almost all T > 0 .

Our main result in this section is stated as follows:

Let

1. A generates a linear C 0 -semigroup ( S ( t ) ) t ≥ 0 such that ( S u ( t ) ) t ≥ 0 is a semigroup of isometries and (13) holds,

2. A allows the decomposition (10) of H with dim H u < + ∞ ,

3. B ∈ L ( H ) such that for all y u ∈ H u , we have

〈 B u S u ( t ) y u , S u ( t ) y u 〉 = 0 , ∀ t ≥ 0 ⇒ y u = 0 (27)

Then the constrained feedback control law:

p ρ , u ( t ) = − ρ 〈 y u ( t ) , B u y u ( t ) 〉 1 + | 〈 y u ( t ) , B u y u ( t ) 〉 | , ρ > 0 , (28)

strongly stabilizes the system (1) with the explicit decay estimate (8).

Proof

Let us consider the system:

d y u ( t ) d t = A u y u ( t ) − ρ 〈 y u ( t ) , B u y u ( t ) 〉 1 + | 〈 y u ( t ) , B u y u ( t ) 〉 | B u y u ( t ) , y u ( 0 ) = y 0 u (29)

Multiplying the system (29) by y u ( t ) , integrating over Ω and using the fact that ( S u ( t ) ) t ≥ 0 is a semigroup of isometries, we obtain:

d ‖ y u ( t ) ‖ 2 d t ≤ − 2 ρ | 〈 y u ( t ) , B u y u ( t ) 〉 | 2 1 + | 〈 y u ( t ) , B u y u ( t ) 〉 | (30)

which proves that the real function t → ‖ y u ( t ) ‖ is decreasing on ℝ + , and we have

‖ y u ( t ) ‖ ≤ ‖ y u ( 0 ) ‖ , ∀ t ≥ 0 (31)

Hence, the system (29) admits a unique mild solution defined for almost all t ≥ 0 (see [

Integrating now the inequality (30) over the interval [ k T , ( k + 1 ) T ] , for k ∈ ℕ and T > 0 , we get:

‖ y u ( ( k + 1 ) T ) ‖ 2 − ‖ y u ( k T ) ‖ 2 ≤ − 2 ρ ∫ k T ( k + 1 ) T | 〈 y u ( τ ) , B u y u ( τ ) 〉 | 2 1 + | 〈 y u ( τ ) , B u y u ( τ ) 〉 | d τ

using now the estimate (26), we deduce that

‖ y u ( ( k + 1 ) T ) ‖ 2 − ‖ y u ( k T ) ‖ 2 ≤ − M ( ∫ 0 T | 〈 S u ( τ ) y u , B u S u ( τ ) y u 〉 | d τ ) 2 (32)

for some M > 0 . Using now the fact that dim ( H u ) < + ∞ , then the assumption (27) is equivalent to

∫ 0 T | 〈 B u S u ( t ) y u , S u ( t ) y u 〉 | d t ≥ δ ‖ y u ‖ 2 , ∀ y u ∈ H u , ( T , δ > 0 ) (33)

From (32) and (33) we have

‖ y u ( ( k + 1 ) T ) ‖ 2 − ‖ y u ( k T ) ‖ 2 ≤ − M δ 2 ‖ y u ( k T ) ‖ 4

using the fact that the map t → ‖ y u ( t ) ‖ is decreasing on ℝ + , we obtain:

‖ y u ( ( k + 1 ) T ) ‖ 2 − ‖ y u ( k T ) ‖ 2 ≤ − M δ 2 ‖ y u ( ( k + 1 ) T ) ‖ 4

which implies that

‖ y u ( ( k + 1 ) T ) ‖ 2 + C ‖ y u ( ( k + 1 ) T ) ‖ 4 ≤ ‖ y u ( k T ) ‖ 2 , C = M δ 2

Letting s k = ‖ y u ( k T ) ‖ 2 , the last inequality can be written as

s k + 1 + C s k + 1 2 ≤ s k , ∀ k ≥ 0

From Lemma.3.1 we have

s k ≤ M 2 k + 1

For k = [ t T ] ( [ t T ] designed the integer part of t T ), then we obtain s k ≤ M 3 t , ( M 3 > 0 ) , which gives

‖ y u ( t ) ‖ 2 ≤ M 3 t

Hence

‖ y u ( t ) ‖ = O ( t − 1 2 ) , as t → + ∞ (34)

For the component y s ( t ) , we shall show that y s ( t ) is defined for all t ≥ 0 and exponentially converges to 0, as t → + ∞ . The system (1) excited by the constrained feedback control (28) admits a unique mild solution defined for almost all t in a maximal interval [ 0 , t max [ defined by

y ( t ) = S ( t ) y 0 + ∫ 0 t S ( t − τ ) p ρ , u ( τ ) B y ( τ ) d τ

Thus

y s ( t ) = S s ( t ) y 0 s + ∫ 0 t S s ( t − τ ) p ρ , u ( τ ) B s y s ( τ ) d τ , ∀ t ∈ [ 0 , t max [ (35)

It follows from (13) that

‖ y s ( t ) ‖ ≤ M 1 e − η t ‖ y 0 s ‖ + ρ M 1 ‖ B ‖ ∫ 0 t e − η ( t − τ ) ‖ y s ( τ ) ‖ d τ

For almost all t ∈ [ 0 , t max [ . The Gronwall inequality then yields:

‖ y s ( t ) ‖ ≤ M 1 ‖ y 0 s ‖ e ( ρ M 1 ‖ B ‖ − η ) t , ∀ t ≥ 0 (36)

Taking ρ < η M 1 ‖ B ‖ , it follows from (36) that y s ( t ) is bounded on [ 0 , t max [ so t max = + ∞ , and therefore

(36) holds for all t ≥ 0 . Hence, from (34) and (36), the solution of (1) satisfies the estimate (8). This completes the proof of Theorem 3.3.

1. Since the function t → ‖ y u ( t ) ‖ decreasing in ℝ + , we have

∃ t 0 ≥ 0 ; y u ( t 0 ) = 0 ⇔ y u ( t ) = 0 , ∀ t ≥ t 0

In this case, we have

p ρ , u ( t ) = 0 , ∀ t ≥ t 0 ⇒ y ( t ) = S s ( t − t 0 ) y s ( t 0 ) , ∀ t ≥ t 0

Hence, using (13) the system (1) is exponentially stable.

2. The constrained feedback control (28) depends only on the unstable part y u ( t ) and we have

| p ρ , u ( t ) | < ρ , ∀ t ≥ 0

3. The constrained feedback control (28) satisfies

| p ρ , u ( t ) | = O ( 1 t ) , as t → + ∞

4. We note that (27) is weaker than (6). The converse is not true as we can see taking an orthonormal basis ( ϕ n ) n ≥ 1 of H , A z = − 〈 z , ϕ 1 〉 ϕ 1 and B z = ∑ n = 1 + ∞ 〈 z , ϕ n 〉 n 2 ϕ n .

5. In the case dim H u = + ∞ and B is nonlinear and locally Lipschitz, such that B ( 0 ) = 0 , then using the same techniques as in [

In this section, we study the robustness of the controls (18) and (28), under a class of perturbations of the system (1).

In this part, we consider the strong robustness of the feedback (18). Then, we will show that the stability property of the system (1) remains invariant under a certain class of bounded perturbations.

Let us consider the following perturbed system

d y ( t ) d t = A y ( t ) + p ( t ) B y ( t ) + ξ ( y ( t ) ) , y ( 0 ) = y 0 (37)

where the linear bounded operator ξ = ξ u + ξ s is such that the system (37) is decomposed into two following subsystems:

d y u ( t ) d t = A u y u ( t ) + p ( t ) B u y u ( t ) + ξ u ( y u ( t ) ) , y u ( 0 ) = y 0 u ∈ H u (38)

d y s ( t ) d t = A s y s ( t ) + p ( t ) B s y s ( t ) + ξ s ( y u ( t ) ) , y s ( 0 ) = y 0 s ∈ H s (39)

The following main result concerns the strong stability of the system (37).

Let

1. A generates a linear C 0 -contraction semigroup ( S ( t ) ) t ≥ 0 on H such that (13) holds,

2. The operator B is compact such that (6) holds,

3. The linear operator ξ is compact and satisfying ‖ ξ s ‖ < η M 1 and 〈 ξ ( y ) , y 〉 ≤ 0 , ∀ y ∈ H .

Then the system (37) is strongly stabilizable.

Proof

First, let us note that 0 remains an equilibrium state of the perturbed system (37), which can be written in the form:

d y ( t ) d t = A y ( t ) + g ( y ( t ) ) , y ( 0 ) = y 0 (40)

where g = f + ξ and f ( y ) = { − ρ 〈 B y , y 〉 1 + | 〈 B y , y 〉 | B y , y ≠ 0 ; 0 , y = 0.

Since f and ξ are locally Lipschitz, so is g . Also g is dissipative: 〈 ξ ( y ) , y 〉 ≤ 0 , ∀ y ∈ H .

The assumption 〈 ξ ( y ) , y 〉 ≤ 0 , ∀ y ∈ H , together with (6) guarantees the following implication

〈 g ( S ( t ) y ) , S ( t ) y 〉 = 0 ⇒ y = 0

Then the weak stability of the perturbed system (37) follows from Theorem 2.4 of Ball [

y s ( t ) = S s ( t − t 0 ) y s ( t 0 ) − ρ ∫ t 0 t S s ( t − τ ) 〈 y ( τ ) , B y ( τ ) 〉 1 + | 〈 y ( τ ) , B y ( τ ) 〉 | B s y s ( τ ) d τ + ∫ t 0 t S s ( t − τ ) ξ s ( y s ( τ ) ) d τ (41)

It follows from (13) that

‖ y s ( t ) ‖ ≤ M 1 e − η ( t − t 0 ) ‖ y s ( t 0 ) ‖ + ( ρ M 1 ‖ B ‖ + M 1 ‖ ξ s ‖ ) ∫ t 0 t e − η ( t − τ ) ‖ y s ( τ ) ‖ d τ (42)

From Gronwall inequality, we obtain:

‖ y s ( t ) ‖ ≤ M 1 e ( ρ M 1 ‖ B ‖ + M 1 ‖ ξ s ‖ − η ) ( t − t 0 ) ‖ y s ( t 0 ) ‖ , ∀ t ≥ t 0

Taking ρ < η − M 1 ‖ ξ s ‖ M 1 ‖ B ‖ , we obtain y s ( t ) → 0 , as t → + ∞ . Hence, the solution y ( t ) of the system (37) strongly converges to 0 , as t → + ∞ .

Our second main result in this section is stated as follows:

Let

1. A generate a linear C 0 -semigroup ( S ( t ) ) t ≥ 0 such that ( S u ( t ) ) t ≥ 0 is a semigroup of isometries and (13) holds,

2. A allows the decomposition (10) of H with dim H u < + ∞ ,

3. B ∈ L ( H ) satisfies (27),

4. ‖ ξ s ‖ ≤ η M 1 and 〈 ξ u ( y ) , y 〉 ≤ 0 , for all y ∈ H u .

Then the constrained feedback control (28) strongly stabilizes the system (37) with the explicit decay estimate (8).

Proof

Let us consider the system:

d y u ( t ) d t = A u y u ( t ) − ρ 〈 y u ( t ) , B u y u ( t ) 〉 1 + | 〈 y u ( t ) , B u y u ( t ) 〉 | B u y u ( t ) + ξ u ( y u ( t ) ) , y u ( 0 ) = y 0 u (43)

Multiplying the system (43) by y u ( t ) and integrating over Ω and using the fact that ( S u ( t ) ) t ≥ 0 is a semigroup of isometries and the hypothesis: 〈 ξ u ( y u ) , y u 〉 ≤ 0 , ∀ y u ∈ H u , we obtain:

d ‖ y u ( t ) ‖ 2 d t ≤ − 2 ρ | 〈 y u ( t ) , B u y u ( t ) 〉 | 2 1 + | 〈 y u ( t ) , B u y u ( t ) 〉 | (44)

which gives ‖ y u ( t ) ‖ ≤ ‖ y u ( 0 ) ‖ , ∀ t ≥ 0 . Then, the system (43) admits a unique global mild solution y ( t ) de-

fined for almost all t ≥ 0 . By the same argument as in the proof of Theorem 3.3, the solution y ( t ) of the system (37) satisfies:

‖ y ( t ) ‖ = O ( t − 1 2 ) , as t → + ∞

which completes the proof of Proposition 4.2.

In this part, we will give an illustrating example of the established results.

ExampleLet us consider the following 1-d bilinear heat equation:

{ ∂ y ( x , t ) ∂ t = ∂ 2 y ( x , t ) ∂ x 2 + p ( t ) B y ( t ) , x ∈ ( 0 , 1 ) , t > 0 , ∂ y ( 0 , t ) ∂ x = ∂ y ( 1 , t ) ∂ x = 0 , ∀ t > 0 , (45)

where y ( t ) is the temperature profile at time t . We suppose that the system is controlled via the flow of a liquid p ( t ) in an adequate metallic pipeline. Here we take the state space H = L 2 ( 0 , 1 ) and the operator A is defined by A y = ∂ 2 y ∂ x 2 , with D ( A ) = { y ∈ H 2 ( 0 , 1 ) | ∂ y ( 0 , t ) ∂ x = ∂ y ( 1 , t ) ∂ x = 0 } . The domain of A gives the ho-

mogeneous Neumann boundary condition imposed at the ends of the bar which require specifying how the heat flows out of the bar and means that both ends are insulated. The spectrum of A is given by the simple eigen-

values λ j = − π 2 ( j − 1 ) 2 , j ∈ ℕ ∗ and eigenfunctions φ 1 ( x ) = 1 and φ j ( x ) = 2 cos ( ( j − 1 ) π x ) for all j ≥ 2 . Then the subspace H u is the one-dimensional space spanned by the eigenfunction φ 1 , and we have S u ( t ) y u = 〈 y u , φ 1 〉 φ 1 so S u ( t ) = I H u (the identity) and hence ( S u ( t ) ) t ≥ 0 , is a semigroup of isometries. The operator of control B , is defined by: B y = ∑ j = 1 + ∞ α j 〈 y , φ j 〉 φ j , α j ≥ 0 , ∀ j ≥ 1 , such that ∑ j = 1 + ∞ α j 2 < ∞ (see [

d y 1 ( t ) d t = d y u ( t ) d t = − ρ α 1 y u 3 ( t ) 1 + α 1 y u 2 ( t ) , ∀ t ≥ 0 (46)

which implies that

d d t y u 2 ( t ) = − 2 ρ α 1 y u 4 ( t ) 1 + α 1 y u 2 ( t ) (47)

Letting x ( t ) = y u 2 ( t ) , we obtain

x ′ ( t ) x − 2 ( t ) = − 2 ρ α 1 1 + α 1 x ( t ) , ∀ t > 0

Integrating now the last equality from 0 to t , we get

x − 1 ( t ) − x − 1 ( 0 ) = ∫ 0 t 2 ρ α 1 1 + α 1 x ( τ ) d τ

from (47), we deduce that the nonnegative scalar map t → x ( t ) is decreasing for all t ≥ 0 , and we have

x − 1 ( t ) ≥ 2 ρ α 1 t 1 + α 1 x ( 0 ) , ∀ t > 0

which means that

| y u ( t ) | ≤ 1 + α 1 y 0 u 2 2 ρ α 1 t , ∀ t > 0 , y 0 u ≠ 0

Then

| y u ( t ) | = O ( 1 t ) , as t → + ∞

Furthermore, the control in this case is defined by

p ρ , u ( t ) = − ρ α 1 y u 2 ( t ) 1 + α 1 y u 2 ( t ) , ∀ t ≥ 0 (48)

For j ≥ 2 , the functions y j ( t ) are characterized by y j ( 0 ) = 〈 y ( 0 ) , φ j 〉 , ∀ j ≥ 2 and satisfy

∂ ∂ t y j ( t ) = ( λ j − ρ α 1 α j y 1 2 ( t ) 1 + α 1 y 1 2 ( t ) ) y j ( t ) , t ≥ 0

which implies that

| y j ( t ) | ≤ e − π 2 ( j − 1 ) 2 t | y j ( 0 ) | , j ≥ 2

Then

‖ y s ( t ) ‖ ≤ e − π 2 t ‖ y s ( 0 ) ‖ , ∀ t ≥ 0

Hence, the system (45) is strongly stable with the decay rate estimate (8).

Let us reconsider the above example with the perturbation ξ defined by:

ξ ( y ) = − β y , β ∈ ( 0 , π 2 2 ) , ∀ y ∈ H

It is clear that the function ξ satisfies the conditions of Proposition 4.2.1. Then the perturbed closed-loop system remains stable, i.e., the control (48) still stabilizes the perturbed system i.e., the control (48) is strongly robust.

In this part, taking in the system (45), the operator B = I and y 0 ( x ) = x + 0.3 .

Then we obtain the results shown in Figures 1-5.

In this work, we have considered the problem of strong stabilization with polynomial decay rate of the stabilized state for bilinear parabolic systems that can be decomposed in the stable and unstable parts (15) and (16) under a weaker condition (27). We have also considered the problem of using a stabilizing feedback control for the unstable part (15) only that can make the whole system (1) stable. Various questions remain open. This is the case

of stabilization for nonlinear systems. Finally, we have studied the robustness problem of the stabilizing controls with respect to a class of perturbations, but a confrontation to more realistic situations remain done. This leads us to consider the stabilization problem for stochastic bilinear system.