Introduction In this work, we study the stabilization of the system described by the equation: dy(t) dt = Ay(t

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C om mun. Fac. Sci. U niv. A nk. Ser. A 1 M ath. Stat.

Volum e 68, N umb er 1, Pages 1114–1122 (2019) D O I: 10.31801/cfsuasm as.508205

ISSN 1303–5991 E-ISSN 2618-6470

http://com munications.science.ankara.edu.tr/index.php?series= A 1

OUTPUT STABILIZATION OF SEMILINEAR PARABOLIC SYSTEMS WITH BOUNDED FEEDBACK

ABDESSAMAD EL ALAMI, ALI BOUTOULOUT, AND RADOUANE YAFIA

Abstract. In this paper, we will study the output feedback stabilization of in…nite-semilinear parabolic systems evolving on a spatial domain and in a subregion ! of (interior to or on its boundary @ ). We consider the condition of admissibility and the decomposition methods technique of the state space via the spectral properties of the system. Then we apply this approach to a regional exponential stabilization problem using bounded feedback. Ap- plications are presented.

1. Introduction

In this work, we study the stabilization of the system described by the equation:

dy(t)

dt = Ay(t) + N y(t) + v(t)By(t); y(0) = y₀; (1) where is an open bounded subset of Rⁿ with smooth boundary @ . The state space H endowed with the inner product h:; :i, and the corresponding norm k:k.

A is the dynamic unbounded operator with domain D(A) H and generates a semigroup of contractions (S(t))t 0on H, and N is a nonlinear operator such that N (0) = 0: B is a linear operator from H to H. The valued function t 7! v(t) represents the control. Let ! be an open and positive Lebesgue measurable subset of , and @ . The operator is de…ne by

: H ! H ;

y ! y = y₌ ; (2)

where the adjoint operator of and set i = :

Taking H = L²(@ ) then H = L²( ). We de…ne the stability of the semilinear system (1) on as:

Received by the editors: May 30, 2018, Accepted: August 30, 2018.

2010 Mathematics Subject Classi…cation. 35K58, 35B35.

Key words and phrases. Output stabilzation, semilinear systems, bounded feedback.

Submitted via ICCDS2018.

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1114

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De…nition 1.1. The system (1) is weakly (resp. strongly, exponentially) stabilizable if there exists a feedback control v(t) = f (y(t)); t 0; f : H ! K :=

R; C such that the corresponding mild solution y(t) of the system (1) satis…es the properties:

(1)

(2) For each initial state y₀ of the system (1) there exists a unique mild solution de…ned for all t 2 R⁺ of the system (1).

(3) f0g is an equilibrium state of the system (1).

(4) y(t) ! 0; weakly (resp. strongly, exponentially), as t ! +1; for all y₀ 2 H:

1.1. Motivation. The following system gives the motivation behind our study 8>

><

>>

:

dy(x₁; x₂; t)

dt = y(x1; x2; t) + y²(x₁; x₂; t)

1 + y²(x1; x2; t)+ u(t); on ]0; 2[² ]0; +1[;

y(0) = y₀ 2 Z;

(3) where u(t) = e⁽¹² ^x²^)t(¹₂ x₂ t² ^e^{( 1}² ^x2)t

1+e^{2( 1}² ^x2)t).

and the state space Z := fy 2 H¹( )=y = 0 on 0 = f0g [0; 1]g (Z is a Hilbert space), that is a closed subspace of H¹( ) endowed with its natural inner product.

Proposition 1.2. The system (3) is not stable on any subregion ! ; but it is stable on @!; where = f0g [0; 2]:

Proof : The system (3) is not stable on any subregion ! , indeed the solution of systems (3) is y(t) = e⁽¹² ^x²^)t which does not tend to zero as t ! +1 on any subregion ! ; but for = f0g [0; 2]; we have

k y(t)kL²( ) e ^tky0kL²( ) 8t > 0:

De…nition 1.3. (see [12]) Let us consider (S(t))t 0linear semigroup, we say that is called an admissible subregion for (S(t))t 0, if:

The mapping : D(A) 7! D( ) is continuous, when D(A) is equipped with the graph norm k:k^A;

there is some > 0 such that Z ₁

0 k S(t)y0k²L²( )dt ky⁰k²; f orally02 D(A): (4) Remark 1.4.

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(2) The …rst condition of admissiblity implies,in particular, that D(A) D( ).

(3)

(3) It follows from 4 that the mapping y₀7! S( )y₀ has a continuous exten- sion from D(A) to L²(0; 1; L²( )) for every t > 0 .

(4) If (S(t))_{t 0} is exponentially stable, then the notion of …nite-time admissibility and in…nite-time admissibility are equivalent.

We present some results showing how to obtain the boundary stabilization. Much work has been done in this direction, both for linear and bilinear problems, see for instance about:

In [3], the quadratic control

v(t) = hy(t); i By(t)i; (5)

was proposed to study the feedback stabilization of (1) with N = 0, and a weak stabilization result was established under the condition

hBS(t)y; S(t)yi = 0; 8t 0 ) y = 0 : (6) In [6], it has been proved that under (6), the same quadratic control( 5) ensures the strong stabilization for a class of semilinear systems.

Recently, the regional exponential stabilization problem of distributed semilinear systems has been resolved (see [2]). Then it has been proved that under the assumption:

ZT

0 jhi!BS(t)y; S(t)yijdt k !yk²; 8 y 2 H; 8T; > 0; (7)

the feedback de…ned by v_j(t) = _R^{B i}^!^y(t)

j(i!y(t))(j = 1; 2);

where R₁(i_!y) = 1 + kB i!ykU;

and; R₂(i_!y) = sup(1; kB i!ykU); guarantees the

regional exponential stabilization. The bilinear …nite-dimensional case has been treated in ([2]).

2. Stabilization results In this paper we need some assumptions:

A1 Assume that i A is dissipative and:

k S(t)y(t)kL²( ) k y(t)kL²( ); A2 The nonlinear operator N is dissipative on !;

(i:e; hi Ny; yiL²( ) 0; 8y 2 H);

such that:

k N y(t)kL²( ) k y(t)kL²( );

A3 A generates a C0-semigroup (S(t))t 0on Hilbert H, and N ( ) : H ! H is a locally Lipschitz, that is, there exists a positive constant K such that

kN(y) N (z)kL²( ) Kky zkL²( ); 8(y; z) 2 H;

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and we can show that K(N ) = sup

y6=z

kN(y) N (z)kL²( )

ky zkL²( )

: Assume that the operator B satis…es:

Z T

0 jhi BS(t)y; S(t)yij^Udt k yk²H!; 8y 2 H; (T; > 0 (8) with (B) = inf

ky0k=1jhi BS(t)y; S(t)yijL²(0;T ;H):

A weak stabilization result was obtained under the weak observability condition:

hBS(t)y; S(t)yi = 0; 8t 0 =) y = 0: (9) 2.1. Regional Boundary exponential stabilization. The …rst result concerns the Regional Boundary exponential stability of the system (1) using properties of the spectrum (A) of A: For this we will assume that the operator A satis…es the following condition:

(H) : A is self-adjoint with compact resolvent, so that A possesses a sequence ( _n)_{n 1}of real eigenvalues, which can be numbered in decreasing order in such away that ! 1. Moreover, there are at most …nitely many nonnegative eigenvalues & = f ¹; 2; :::; Ng(which can be empty) of A, each with …nite-dimensional eigenspace. The eigenvectors ('_n_j); 1 j mn associated with n (mn is the multiplicity of _n) compose a complete system in H (see [3]).

Proposition 2.1. let is an admissible subregion for (S(t))t 0Then the following properties are equivalents:

(1)

(2) The system (1) is Regionally exponentially stable On by the control v(t) = hi By(t); y(t)i

1 + jhi By(t); y(t)ij; ; t > 0: (10) (3) 8 2 (A); Re( ) 0 =) 8 2 N(A I); ' = 0:

Proof: First we show that (1) =) (2) note that from the assumption (H), we have S(t)y02 D(A); 8y⁰2 H and t > 0: we have

S(t)y0=

+1

X

n=1

exp( nt)

mn

X

j=1

hy⁰; '_n_ji:'nj;

Then

k S(t)y0k²L2 ( )=

+1X

n=1

exp(( n+ m)t)

mnX

j=1 mmX

k=1

hy⁰; '_njihy⁰; '_mkih'nj; '_mkiL2 ( );

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taking y₀= '_n₀_j₀ we obtained

k S(t)y₀k²L²( )= exp((2 _n₀)t)k '_n₀_j₀k²L²( )

Furthermore

t!+1lim k S(t)y0k^L²^{( )}= 0 () ⁿ0 < 0; or : k'n0j0k^L²^{( )}= 0;

if k'njkL²( )6= 0; we deduce that

t!+1lim k S(t)y0kL²( )= 0 () ⁿj < 0;

using the variation of constant formula with y₀ as the initial state, we get:

y(t) = S(t s)y0+ Z_t

0

v(s)S(t s)By(s) + S(t s)N y(s)ds;8t 2 [0; T ]; (11)

which give

k S(t)y0kL2 ( ) (1 + ( kBk + Lky0k)T )k y(s)kL2 ( );8 : 0 < s < t:

O r

exp((2 _n0)t)k '_n0j0k²L2 ( )=k S(t)y0k²L2 ( )=k S(t s)S(s)y0k²L2 ( ):

Then if there exist 2 (A); and 'n0j0 2 N(A ) such that Re( ) 0 and '_n₀_j₀ 6= 0; which give

exp((2 _n0)t)k '_n0j0k²L2 ( ) (1 + ( kBk + Lky0k)T )k y(s)kL2 ( );8 : 0 < s < t:

and we deduce that the system (1) is not regionally stable on : Suppose that (2), and taking y0= y01+ y022 H where

y₀₁= XN n=1

mn

X

j=1

hy0; '_n_ji:'nj;

and

y02=

+1

X

n=N +1 mn

X

j=1

hy⁰; '_n_ji:'nj;

From (2), we have S(t)y₀= S(t)y₀₂, and kS(t)y02k exp((2 _{N +1})t)ky02k: It follows that S(t)y₀₂ ! 0 exponentially as t ! +1: If y022 D(A); then S(t)y022 D(A); : 8t 0 and AS(t)y₀₂ = S(t)Ay₀₂ ! 0 exponentially as t ! +1; which give kS(t)y⁰²k^A = (kS(t)y⁰²k + kS(t)Ay⁰²k)¹² ! 0 exponentially as t ! +1:

Using the continuity of : (D(A); k:k^A) 7! L²( ); we deduce S(t)y02 ! 0 exponentially as t ! +1; or

(1 ( kBk + Lky0k)T )k y(t)kL²( ) k S(t)y₀kL²( ); 8 : t > 0:

Then if 0 < < ¹_kBkT^kBkL we have y(t) ! 0 exponentially as t ! +1;

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2.2. Regional Boundary Strong stabilization. The next result concerns the regional Boundary strong stabilization of (1) on .

Theorem 2.2. Let us consider the following assumptions:

(1)

(2) A generate a linear C0-contraction semigroup (S(t))t 0on H such that (8) holds.

(3) The nonlinear operator N satis…es A₂:

Then the system(1) is strongly stabilizable by the feedback v(t) = hi By(t); y(t)i

1 + jhi By(t); y(t)ij; ; t > 0: (12) Proof : Let y(t) denote the corresponding solution of (1). For t 0 we de…ne the function

! z( ) :=

Z

t

v(s)S( s)By(s) + S( s)N y(s)ds

Applying the variation of constant formula with y(t) as the initial state, we get y( ) = S( t)y(t) + z( ); 8 2 [t; t + T ]; (13) we obtain

k y( )k k S( t)y(t)k + k z( )k Then we have k y( )k k y(t)k + k z( )k:

Furthermore:

k y( )k k y(t)k + ( kBk + Kky⁰k) Z

t k y(s)kds;

where kBk is the norm of B. The Gronwall inequality then yields

k y( )k k y(t)ke⁽ ^kBk+k^ky0k^)T; 8 2 [t; t + T ]; (14) we have the relationhi BS( t)y(t); S( t)y(t)i = hi B(y( ) z( )); S( t)y(t)i

= hi Bz( ); S( t)y(t)i hi By( ); z( )i +hi By( ); y( )i:

It follows that:

jhi BS( t)y(t); S( t)y(t)ij kBkk y(t)kk z( )k

+ kBkk z( )kk y( )k + jhi By( ); y( )ij:

Therefore

jhi BS( t)y(t); S( t)y(t)ij kBkk z( ))k(k y(t)k + k y( )k)

+jhi By( ); y( )ij: ,

Using (14) we deduce that:

jhi BS( t)y(t); S( t)y(t)ij ( kBk²+ Bkkky0k)T k y(t)k²

(1 + e⁽ ^kBk+K^ky0k^)T)e⁽ ^kBk+k^ky0k^)T+ jhi By( ); y( )ij:

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By integrating this inequality over [t; t + T ] we obtain the estimate Z T

0 jhi BS( t)y(t); S( t)y(t)ijd

( kBk²+ kBkkky0k)T²e⁽ ^kBk+k^ky0k^)Tk y(t)k² + ( kBk²k + Bkkky0k)T²(e⁽ ^kBk+k^ky0k^)T)²k y(t)k² +

Z t+T

t jhi By( ); y( )ijd :

We have Z t+T

t jhi By( ); y( )ijd ! 0; as : t ! +1:

Taking > K > 0 such that:

( kBk²+kBkkky0k)T²(1 + e⁽kBk+Kky0k)T)e⁽kBk+kky0k)T < ;

and using (8) we deduce that k y(t)k ! 0; as : t ! +1:

Remark 2.3.

(1) Note that the feedback (12) is a bounded function in time and is uniformly bounded with respect to initial states,

jv(t)j kBk; 8t 0; 8y02 H:

(2) Since ky(t)k decreases, then we have 9t0 0; y(t₀) = 0 , y(t) = 0; 8t t0. In this case we have v(t) = 0; 8t t0.

3. Applications

3.1. Example. Let us consider the following semilinear heat equation, and

=]0; 1[²: 8>

>>

<

>>

>:

@y(x1; x2; t)

@t = Ay(x₁; x₂; ; t) + N y(x₁; x₂; t) + v(t)By(x₁; x₂; t); in Q;

y(0) = y0;@y(:; t)

@ = 0; on @Q;

(15) where y(t) is the temperature pro…le at time t; and Q = ]0; +1[: We suppose that the system is controlled via the ‡ow of a liquid v(t). Here we take the state space H = L²(0; 1) and the operator A is de…ned by Ay = @²y

@x²₁ +@²y

@x²₂; 8y 2 D(A) with

D(A) = fy 2 H²(0; 1)j @y(0; t)

@ = 0; on : @Qg:

The domain of A gives the homogeneous Neumann boundary. Here we take the state space H = U = L²(0; 1). The spectrum of A is given by the simple eigenvalues

kj = ²(j + k)², j; k 2 N and eigenfunctions '(k;j)(x1; x2) = ek(x1)ej(x2);

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where e_k(x₁) =p

2 cos(k x₁) if k 6= 0 and e0(x₁) = 1: The operator of control B;

is de…ned by : By = hy; '(1;0)i:'(1;0); B is linear operator, and we have

< By; y >= jhy; '(1;0)ij²; and

N y = hy; '(1;1)i:'(1;1)

1 + jhy; '(1;1)ij

which is nonlinear operator, dissipative and locally Lipschitz. Let us show that the subregion = f¹2g [0; 1]: From the Green formula and using the Schwartz inequality, we have

Z +1

0 k y(s)k²ds < +1;

and thus is admissible for S(t) Then the systems (15) is -exponentially stabilizable by the control:

v(t) = 2 ²jhi y(t); '(1;0)ij²

1 + jhi y(t); '(1;0)ij²; : t > 0:

Conclusion. In this work we have considered the problem of regional boundary exponential stabilization with output of a constrained parabolic semilinear system under the condition of admissibility for semigroup linear. Also the question of the regional boundary strong stabilizing controls is discussed.

References

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[3] Zerrik, EL. and Ouzara, M., Output stabilization for in…nite-dimentionel bilinear systems, Int. J. Appl. Math. Comput. Sci., Vol. 15, No. 2, (2005), 187-195.

[4] Ball, J. and Slemrod, M., Feedback stabilization of distributed semilinear control systems, Appl. Math. Opt., 5, (1979), 169-179.

[5] Kato, T., Perturbation theory for linear operators, New York., Springer, 1980.

[6] Berrahmoune, L., Stabilization and decay estimate for distributed bilinear systems, Systems and Control Letters, 36 (1999), 167-171.

[7] Berrahmoune, L., Stabilization of bilinear control systems in Hilbert space with nonquadratic feedback, Rend. Circ. Mat. Palermo, 58 (2009), 275-282.

[8] Ouzahra, M., Stabilization of in…nite-dimensional bilinear systems using a quadratic feedback control, International Journal of Control, 82 (2009), 1657-1664.

[9] Ouzahra, M., Exponential and weak stabilization of constrained bilinear systems, SIAM J.

Control Optim., 48, Issue 6 (2010), 3962-3974.

[10] Pazy, A., Semi-groups of linear operators and applications to partial di¤erential equations, Springer Verlag, New York, 1983.

[11] Quinn, J. P., Stabilization of bilinear systems by quadratic feedback control, J. Math. Anal.

Appl., 75 (1980), 66-80.

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[12] Weiss,G., Admissibility of unbounded control operators. SIAM J Control Optim., 27 (1989), 527-545.

Current address : Abdessamad El Alami (Corresponding author): Laboratory of MACS, De- partment of Mathematics, Moulay Ismail University, Faculty of Sciences Meknes, Morocco.

E-mail address : elalamiabdessamad@gmail.com

ORCID Address: http://orcid.org/0000-0002-5251-6047

Current address : Laboratory of MACS, Department of Mathematics, Moulay Ismail University, Faculty of Sciences Meknes, Morocco.

E-mail address : boutouloutali@yahoo.fr

Current address : Ibn Zohr University, CST Campus Universitaire Ait Melloul Agadir, Mo- rocco.

E-mail address : r.yafia@uiz.ac.ma