An Output Stabilization of Bilinear Distributed Systems

Int. Journal of Math. Analysis, Vol. 7, 213, no. 4, 195-211 An Output Stabilization of Bilinear Distributed Systems E. Zerrik, Y. Benslimane MACS Team; Sciences Faculty, Moulay Ismail University zerrik3@yahoo.fr, bensyassine@hotmail.fr A. El Jai IMAGES Lab. LTS. Perpignan University aej@univ-perp.fr Abstract The aim of this paper is to study regional stabilization of the flux of bilinear distributed systems. More precisely it consists in studying the asymptotic behavior of the gradient of such a system not in its whole geometrical evolution domain Ω but only in a subregion ω of Ω. Then we give definitions and under suitable condition we give gradient stabilizing control. We also characterize the control which stabilizes regionally the gradient, and minimizes a given performance cost. Then we develop a numerical approach that is successfully illustrated by simulations. Keywords : Distributed bilinear systems - Stability - Gradient stability - Regional gradient stability 1 Introduction The problem of regional stabilization has been the object of various works [8, 6], and it consists in studying the behavior of a distributed system, not in its whole geometrical evolution domain, but just in a subregion which may be inside or in the boundary of this domain. Many approaches were used to characterize different kinds of stabilization, and mainly characterization of control which achieves the stability and minimizing a given cost criterion. Later the notion of regional stabilization was developed for bilinear system [9, 1] where the authors give sufficient conditions to obtain weak, and strong stabilization. Recently the notion of gradient stabilization was introduced by Zerrik et al [6, 7] in the global and in the regional cases, and it concerns the study of the asymptotic behavior of the gradient for linear distributed system. Necessary and sufficient conditions to stabilize the gradient of the considered system, using the spectrum proprieties for the weak stabilization, the strong stabilization

196 E. Zerrik, Y. Benslimane and A. El Jai were obtained under a dissipation hypothesis. The exponential stabilization is obtained by the resolution of a Riccati equation. Examples of unstable systems with a stabilizable gradient are given. This paper proposes to extend the above results on gradient stabilization to bilinear distributed systems which are close to real application and is organized as follows: In the next section we define different kind of regional gradient stabilization of bilinear distributed systems, and we give sufficient condition to achieve the stabilization of such systems. The third section is focussed on the characterization of control that regionally stabilizes the gradient and minimizes a quadratic performance cost. We also establish an estimation of the stabilization error and the developed approach leads to numerical algorithm. Illustrative simulations are developed for one-dimensional distributed bilinear systems. 2 Regional gradient stabilization 2.1 Preliminaries Let Ω be an open regular domain of IR n, and ω a nonempty subregion of Ω. We consider a bilinear distributed system given by : { ẏ(t) = Ay(t) + v(t)by(t) t y() = y (1) where A is a linear operator with domain D(A) H 1 (Ω), and generates a linear strongly continuous semigroup (T (t)) t on H 1 (Ω), endowed with its usual complex inner product, denoted by.,. and. the associated norm. We define the operator : ω : H 1 (Ω) (L 2 (ω)) n y (χ ω y(x) x 1 y(x) y(x), χ ω,..., χ ω ) x 2 x n (2) where χ ω is the restriction operator to ω, defined by χ ω L 2 (Ω) L 2 (ω) y y /ω (3) (L 2 (Ω)) n is endowed with its usual complex product.,. n and. n is the corresponding norm, and (L 2 (ω)) n is endowed with the restriction of.,. n. B is a linear bounded operator mapping H 1 (Ω) into its self, and let denote G ω = ω ω, where ω is the adjoint operator of ω.

An output stabilization of bilinear distributed systems 197 Definition 2.1 The system (1) is said to be regionally weakly gradient stabilizable (r.w.g.s) on ω respectively ( strongly gradient stabilizable (r.s.g.s), exponentially gradient stabilizable (r.e.g.s)), if for any initial condition y H 1 (Ω) the corresponding solution y(t) of (1) is global and ω y(t) converges to weakly respectively (strongly,exponentially) as t +. 2.2 Stabilization control We are concerned with the problem of regional gradient stabilization which consists in finding an appropriate quadratic feedback control that stabilizes regionally the gradient of the system (1). We consider controls of the form v(t) = y(t), Dy(t) (4) where D is a linear bounded operator mapping H 1 (Ω) into its self. The following result gives sufficient conditions for regional weak gradient stabilization of system (1) using the control (4) with D = DB where D L(H 1 (Ω)) Proposition 2.2 Assume that : 1. (T (t)) is a semigroup of contractions 2. B is compact 3. Re( DBz, z z, Bz ) z H 1 (Ω) 4. DBT (t)z, T (t)z T (t)z, BT (t)z = t ω z = then system (1) is regionally weakly gradient stabilizable using the control (4). Proof : Using 1 and 3 system (1) has a unique global mild solution, using the control (4) (see [9]). Let F (z) = z, DBz Bz, since F is locally Lipschitz (B maps bounded sets into bounded sets), and B compact, then there exists z H 1 (Ω) such that y(t) z weakly as t and F (y(t)), y(t) =, t > (see ([2])). So DBT (t)z, T (t)z T (t)z, BT (t)z =, t >, using 4. we obtain ω z =. Let z (L 2 (Ω)) n, ω is a continuous operator then ω y(t), z n ω z, z n as t, we deduce that system (1) is regionally weakly gradient stabilizable. We consider the following illustrative example.

198 E. Zerrik, Y. Benslimane and A. El Jai Example 2.3 Let Ω =], 1[ 2, we consider the following system: y t (x, z, t) =.1 y(x, z, t) + y(t), φ 3 φ 3 + v(t)by(x, z, t) Ω ], + [ y ν (ξ, ξ, t) = Ω ], + [ y() = y H 1 (Ω) (5) where is the Laplace operator. The operator A =.1 + y(t), φ 3 φ 3, has an orthonormal basis of eigenfunctions φ ij (x, z) = 2a ij cos(iπx) cos(jπz), with a ij = (1+(i 2 +j 2 )π 2 )) 1 2, and the corresponding eigenvalues are given by : {.1(i λ ij = 2 + j 2 )π 2 if (i, j) Γ.9π 2 + 1 else where Γ = {(i, j) IN 2 (i, j) (3, )}, A generates a strongly continuous semigroup given by T (t)y = e (.9π2 +1)t y, φ 3 φ 3 + e λijt y, φ ij φ ij. Let consider the operator By = (i,j) Γ (i,j) Γ 1 λ ij y, φ ij φ ij. For (i, j) Γ, ( 1 λ ij, φ ij ), are the eigencouples of B, and since then B is a compact operator. Let ω = { 1 6 } [, 1] be the target subregion. We have BT (t)y, T (t)y = (i,j) Γ 1 λ ij e 2λ ijt y, φ ij 2. Then lim i,j + 1 λ ij = BT (t)y, T (t)y =, t y, φ ij = (i, j) Γ (6) Since ω φ 3 = and from (6) we obtain ω y =, then system (5) is r.w.g.s. We remark also that system (5) isn t weakly stabilizable (for y = φ 3, the solution of (5) y(t) = e (.9π2 +1)t φ 3 don t converge to as t + ). 2.3 Decomposition approach Along this section we consider system (1) with state space Z := H 1 (Ω), and for δ > we decompose of the spectrum of A in two parts of the complex plan σ δ + (A) = {λ : Re(λ) δ} and σ δ (A) = {λ : Re(λ) < δ} If σ δ + (A) is bounded and separated from σ δ (A) in such a way that a rectifiable, simple, closed curve Γ, enclosing an open set containing σ δ + (A) in its interior

An output stabilization of bilinear distributed systems 199 and σ δ + (A) in its exterior, then the state space may be decomposed as Z = Z u Z s where Z u = P Z, Z s = (I P )Z and P is the projection operator mapping Z into itself given by P = 1 (λi A) 1 dλ. 2πi Γ Then the operator A satisfies the spectrum decomposition assumption: A = A u A s where A u = P A, A s = (I P )A. (7) Assume that there exist B u L(Z u ) and B s (Z s ) such that B = B u B s, then system (1) can be decomposed as The solutions of (8) and (9) are given by : and ẏ u (t) = A u y u (t) + v(t)b u y u (t) y u = P y (8) y u = P y Z u ẏ s (t) = A s y s (t) + v(t)b s y s (t) y s = (I P )y (9) y s = (I P )y Z s y u (t) = T u (t)y u + y s (t) = T s (t)y s + v(τ)t u (t τ)b u y u (τ)dτ (1) v(τ)t s (t τ)b s y s (τ)dτ (11) where T u (t) and T s (t) denote the restriction of T (t) on Z u and Z s which are respectively the strongly continuous semigroups generated by A u and A s. By considering the following control : v(t) = G ω D u G ω y u (t), y u (t) (12) where D u L(Z u ) and G ω = ω ω, we obtain the following result: Proposition 2.4 Assume that A satisfies (7), and A s satisfies the spectrum determined growth assumption: ln T s (t) lim t + t = supre(σ(a s )) (13) If system (8) is r.e.g.s on ω using the control (12), then system (1) is r.e.g.s on ω using the same control (12), and if in addition the state of system (8) is bounded, then the state of system (1) remains bounded on Ω.

2 E. Zerrik, Y. Benslimane and A. El Jai Proof : Since system (8) is r.e.g.s, then its solution y u is global, and we have: ω y u (t) Me ct y u, for some M, c > (14) Also y s is a mild solution of (9) defined on [, t max [, t max > to conclude that y s (t) is a global solution, we will show that y s (t) is bounded for all t [, t max [. From (13) we have : With (11) we obtain: T s (t) a e bt, t, < b < δ. (15) y s (t) a e bt y s + a B s v u (τ) e b(t τ) y s (τ) dτ. Applying Gronwall s inequality we have : From (12) and (14) we obtain: e bt y s (t) a y s exp( a B s v u (τ) dτ) y s (t) al y s e bt for some L >. (16) Then y s (t) is bounded which implies that y s (t) is a global mild solution, thus y(t) is a global mild solution of (1) (y = y s + y u ). From (16) it follows that y s (t) exponentially as t +, which implies that ω y s (t) n exponentially as t +. Besides, using (1),(14), and the inequality ω y(t) n ω y u (t) n + ω y s (t) n the proof is achieved. The second point is immediate with (16) If system (9) has a global mild solution y s (t), then the condition (13) may be relaxed and we obtain the following result. Proposition 2.5 Let A satisfies (7), A s satisfies the following inequality ln ω T s (t) n lim t + t and for some N >, B s satisfies the following condition: sup Re(σ(A s )) (17) ω B s y n N ω y n for all y Z (18) If system (8) is r.e.g.s on ω, using the control (12) then the system (1) is r.e.g.s on ω using the same control (12). If in addition the state of system (8) is bounded then the state of system (1) remains bounded on Ω.

An output stabilization of bilinear distributed systems 21 Proof : Since system (8) is supposed to be r.e.g.s, then its solution y u (t) is global, also y s (t) is a global mild solution of (9) and since y = y u + y s, then y(t) is a global mild solution of (1). The decomposition of the spectrum of A gives sup Re(σ(A s )) δ. From (17) it s follows that ω T s (t) n a e b t, t, < b < δ, and a >. By (11) and (18) we have : e b t ω y s (t) n a ω y s n + a N v u (τ) e b τ ω y s (τ) n dτ Therefore the proof can be achieved similarly to that of proposition 2.4 3 Stabilization problem with decay estimate Here we characterize the control that stabilizes regionally the gradient of system (1) and minimizes a given performance cost, and we give a decay estimate of the gradient stabilization. Consider the problem + + + min J(v) = P ω By(t), y(t) 2 dt + v(t) 2 dt + Ry(t), y(t) dt v U ad = {v/y(t) is a global solution and J(v) < + } (19) where P ω = G ω P G ω with P and R are positive and self-adjoint operators such that P ω satisfies the following equation : and 1 P ω Ay, y + y, P ω Ay + Ry, y =, y D(A) (2) P ω BT (t)y, T (t)y dt α ω y 2 n, y H 1 (Ω) for some α >. (21) Assume that there exist a, b >, such that t, y H 1 (Ω) ω T (t)y n a ω y n and ω By n b ω y n (22) The main result is based on the following lemma. Lemma 3.1 Denote : µ = 1 P ω By(t), y(t) dt and ν = ( η, < η < 1), ( λ > ) : ν < η ω y 2 n < λ µ Proof : Let ψ(t) = y(t) T (t)y. Firstly we establish the following estimation 1 v(t) 2 dt, then ω ψ(t) n C ω y n ν, t ], 1[, for some C >. (23)

22 E. Zerrik, Y. Benslimane and A. El Jai ψ(t) is written as ψ(t) = and by (22) we obtain : v(s)t (t s)bt (s)y ds + v(s)t (t s)b(y(s) T (s)y )ds, ω ψ(t) n a 2 b ω y n v(s) ds + ab v(s) ω ψ(s) n ds t a 2 b ω y n ν + ab v(s) ω ψ(s) n ds Using Gronwall s inequality, and for all t ], 1[ we have ω ψ(t) n a 2 b ω y n ν + a 3 b 2 ν ω y n v(s) exp( a 2 b ω y n ν + a 3 b 2 ν ω y n exp(ab) (ν < 1) s ab v(τ) dτ)ds Then (23) is satisfied with C = a 2 b ω y n + a 3 b 2 ω y n exp(ab). Now the triangle inequality implies that: 1 P ω BT (t)y, T (t)y dt µ + + 1 1 P ω BT (t)ψ(t), T (t)y dt P ω BT (t)y, ψ(t) dt + and from (22), and (23) there exist ã, b > such that 1 1 P ω Bψ(t), ψ(t) dt P ω BT (t)y, T (t)y dt µ + ã ν ω y 2 n + bν ω y 2 n Since ν < 1 and from (21) we obtain (α (ã + b) ν) ω y 2 n µ, the proof is achieved by choosing η = 1 2 inf(1, α 2 (ã + b) 2 ) Proposition 3.2 If the solution y (t) corresponding to the control: v (t) = P ω By (t), y (t) (24) is global, then v (t) is the unique feedback control solution of (19), and system (1) excited by v (t) is regionally strongly gradient stabilizable. If in addition the condition P ω By, y 2 + 1 2 Ry, y dre( y, By P ωy, y Ay, y ), y D(A) (25) holds for some d >, then the state remains bounded on ω.

An output stabilization of bilinear distributed systems 23 Proof : Let us define the function F (z) = P ω z, z, z H 1 (Ω). For y D(A), we have F (y (t)) = 2 P ω By (t), y (t) 2 Ry (t), y (t) t Integrating, we obtain: P ω By (s), y (s) 2 ds + Ry (s), y (s) ds F (y )ds, t (26) The solution y (t) is continuous with respect to the initial condition (see[4]), and since F and Ry(t), y(t) are continuous then (26) holds for all y H 1 (Ω). It follows that J(v ) is finite for all initial condition y on H 1 (Ω). Now let us show that system (1) controlled by any control v U ad is regionally strongly gradient stabilizable. Indeed: Let ϵ be such that < ϵ < η, since J is finite, Cauchy criterion implies that there exists T > such that for all t > T +1 t P ω By(s), y(s) ds < ϵ and +1 t v(s) 2 ds < ϵ Applying lemma 3.1 with y = y(t), we obtain ω y(t) n < λ ϵ, t > T. Thus ω y(t) n, as t +. Let us show that v (t) is the unique solution of (19): We have F (y(t)) M ω y(t) 2 n, for some M >, then For y D(A) integrating the relation : lim F (y(t)) =. t + F (y(t)) t = P ω By(t), y(t) +v(t) 2 P ω By(t), y(t) 2 v(t) 2 Ry(t), y(t) we obtain J(v) = F (y ) + + P ω By(t), y(t) + v(t) 2 dt. It follows that J(v ) J(v), v U ad. Let y H 1 (Ω), there exists (y n ) D(A) such that y n y as n +. For v U ad, J(v) = F (y n ) + + P ω By n (t), y n (t) + v(t) 2 dt. Thus J(v) F (y n ). By the continuity of F we deduce that J(v) F (y ) = J(v ). The uniqueness of v (t) follows from the uniqueness of y(t) solution of system (1). Now let us show that the state remains bounded on Ω. The inequality (25) gives t F (y(t)) 2 d t y(t) 2 so F (y(t)) d( y(t) 2 y 2 ) y D(A). (27)

24 E. Zerrik, Y. Benslimane and A. El Jai F and y(t) are continuous with respect to the initial condition, then (27) holds y H 1 (Ω). But F (y(t)) as t +, which achieves the proof. The following result gives the decay estimate of ω y (t) n. and Proposition 3.3 Assume that the following assumptions are verified: P ω y, y d ω y 2 n for some d > (28) Ry, y e ω y 2 n for some e > (29) Re( RAy, y ) Re( P ω By, y RBy, y ), y D(A). (3) If in addition the solution y(t) is global then for any initial condition y such that ω y we have the estimation ω y(t) n = O( 1 ), as t +. t Proof : Let us consider the function W (y(t)) = P ω y(t), y(t) +, and denote W m = P ω y (t), y (t) + positive sequence. Integrating the relation we obtain : W m+1 W m = m+1 m W (y(t)) t m Using lemma 3.1 with y = y (t) we have Ry(s), y(s) ds, t Ry (s), y (s) ds, t, which is a = 2 P ω By(t), y(t) 2 between m and m+1 m+1 P ω By (t), y (t) 2 dt = µ = v (t) 2 =: ν. m W m+1 W m η or W m+1 W m < 1 λ 2 ωy 4 n. (31) Let y D(A) be such that ω y, W = 1 2 P ωy, y M ω y 2 n for some M >. The inequality (28) implies that W > and from (31), we have: 1 where f(z ) = min( λ 2 M, η 2 W m+1 W m f(z )W 2 m (32) W 2 ), then W m is a nondecreasing sequence. Let denote V m = 1, then V m+1 V m = W m W m+1 W m W m+1 and W m W m+1 W m Wm 2 from (32) we obtain V m+1 V m f(y ), then V m+1 V +mf(y ) which implies W that W m, m and we obtain the following estimate 1 + W f(y )m W (y (t)) 2W (y ) 1 + W (y )f(y )t, t (33)

An output stabilization of bilinear distributed systems 25 Now let y H 1 (Ω) such that ω y, there exists a sequence y n D(A) and y n y as n +, also ω y n ( ω is continuous), and using similar above technics we show that (33) is satisfied for y n. But both f(y ) and y (t) are continuous with respect to the initial state y, so (33) holds for all y H 1 (Ω). The condition (3) implies that Ry(t), y(t), and from (29) we obtain: t W (y (t)) (d + et) ω y (t) 2 n We conclude that : ω y (t) 2 n 2U(y ) (d + et)(1 + U(y )f(y )t) (34) since f(y ), W (y ) and e are not null then the proof is achieved 3.1 Numerical approach Our goal here is to calculate the control v solution of problem (19). As shown, this control is given by (24) where P ω is solution of (2). This turn up to consider the problem ẏ(t) = Ay(t) + v(t)by(t) t > v(t) = P ω By(t), y(t) (35) y() = y To solve (35), let t m = mh where m IN, h >, and for m 1, and t m 1 t t m. The system (35) is written as ẏ m (t) = Ay m (t) + v m 1 (t m 1 )By m (t) y m (t m 1 ) = y m 1 (t m 1 ) v m (t) = P ω By m (t), y m (t) (36) y() = y v () = P ω By, y Now we give a relation between the gradient of system (35) and the one of system (36). For that let suppose that the global mild solution of (35) is bounded, i.e. there exists α 1 > such that y(t) α 1 which implies that v(t) α 2, where α 2 = α1 P 2 ω B, and we have the following result: Proposition 3.4 Assume that ω T (t) n Me λt for some λ >, and M 1. Also B satisfies (22) and α 2 λ, then there exists c >, such M B that ω y m (t) ω y(t) n < ch, t m 1 t t m (37)

26 E. Zerrik, Y. Benslimane and A. El Jai Proof : We have ω y(t) n M ω y n e (λ Mb B α 2)t. Indeed : ω y(t) n Me λt ω y n + Mb B α 2 e α(t s) ω y(s) n ds Then e λt ω y(t) n M ω y n + Mb B α 2 e αs ω y(s) n ds Using Gronwall s inequality we obtain ω y(t) n M ω y n e (λ Mb B α 2)t. Now let denote z t m = y m (t) y(t), using Gronwall s inequality we obtain e λt ω zm t n Me λt m 1 ω z t m 1 m 1 n + 2α 2 M 2 b ω y n B he Mb B α 2t +α 2 M 2 b B e λt m 1 ω z t m 1 m 1 n e Mb B α2(t s) ds t m 1 +2α2M 2 3 b 2 ω y n B 2 h 2 e Mb B α 2t We denote α = λ + bm B α 2, it follows that : ω z t m n M ω z t m 1 m 1 n e α(t t m 1) + 2λM ω y n he αt + 2M ω y n λ 2 h 2 e αt M ω z t m 1 m 1 n + 2λM ω y n he α(m 1)h + 2M ω y n λ 2 h 2 e α(m 1)h Let denote M 1 = 2λM ω y n [1 + hλ], we obtain: ω z t m n M ω z t m 1 m 1 n + M 1 he α(m 1)h (38) Finally let show the requested estimation (37): We need to calculate the error ω z1 t n, similarly to above we have: ω z1 t n e λt 2λM ω y n he Mb B α2t + Mb B α 2 ω z1 t n e λs ds, Gronwall s inequality implies that : ω z t 1 n e λt 2λM ω y n he Mb B α 2t + 2Mh 2 λ 2 ω y n e Mb B α 2t then ω z t 1 n M 1 h. We have (m + 1) 2 e ( λ+α 2Mb B )mh as m + ( λ + α 2 Mb B < ). So there exists N 1 > such that for all m > N 1, e ( λ+α 2Mb B )mh ) < and from (38) we have for all m > N 1 1 (1 + m) 2 ω z t m n M ω z t m 1 m 1 n + M 1h m 2

An output stabilization of bilinear distributed systems 27 then ω zm t n M 2 ω z t m 2 m 2 n + M M 1h (m 1) + M 1h 2 m. 2 Thus ω zm t n M m N 1 ω z t N 1 N 1 n +M m N M 1 1 1 h (N 1 + 1) +M m N M 1 2 1 h 2 (N 1 + 2) +...+M 1h 2 m 2 (39) In the other hand, using (38) for m N 1, we have: ω z t m n M ω z t m 1 m 1 n + M 1 h m M i M 1 h i= then m N 1, ω z t m m n hm 1 m M i, and replacing in (39) we obtain i= ω zm t n M m N N 1 m N 1 M i 1 1 M i M 1 M 1 h + i= i= (m i) h 2 (M m N N 1 m 1 M i M m j + )M i= j=n 1 +1 j 2 1 h, The above series converges then the proof is achieved. Now let solve the algebraic Riccati equation (2) for that: Let H N = span{ϕ i i = 1,..., N} a subspace of H 1 (Ω) where {ϕ i i IN } is an orthonormal basis of H 1 (Ω). H N is endowed with H 1 (Ω) restriction inner product. We define the projection operator : Π N : H 1 (Ω) H N y N y, ϕ i ϕ i (x) i=1 To solve (2) we have to solve the following algebraic Riccati equation : (4) P N ω A N y N, y N + y N, P N ω A N y N + R N y N, y N = (41) where A N, P N ω, and R N are respectively the projection of A, P ω, and R. Let y H 1 (Ω), lim N ΠN (y) y = then lim P ω N Π N y P y = N y H 1 (Ω), or equivalently Pω N Π N converges to P ω strongly in H 1 (Ω), ([3]). Now for m 1 and t m 1 t t m, let us consider the system : y m (t) = Ay m (t) + v t m 1(t N m 1 )By m (t) y m (t m 1 ) = y m 1 (t m 1 ) (42) vm(t) N = P N B N ym(t), N ym(t) N v() = P N B N y N, y N

28 E. Zerrik, Y. Benslimane and A. El Jai The linear operator (A + vm 1(t N m 1 )B) generates a linear strongly continuous semigroup given by T m,n = T m,n i where T m,n = T and i T m,n i (t)y = T m,n y + vm 1(t N m 1 ) T m,n (t s)bt m,n i 1 yds. Then lim T m,n T m =, where T m is the semigroup generated by the N + operator (A + v m 1 (t m 1 )B). Thus the mild solution of system (42) converges strongly to the mild solution of system (36) as N +. This leads to the following algorithm: Step 1 : Initial data: Threshold accuracy ε >, subregion ω, N the dimension of space projection, and y N is the projection of initial state. Step 2 : Choose a time sequence (t m ) m IN, t m+1 = t m + h, with h > small enough for numerical consideration Step 3 : Solve (41) using the algorithm given in [1] gives Pω N Step 4 : v() = Pω N B N y N, y N While ω y m (t m ) < ϵ repeat vm(t) N = Pω N B N ym(t), N ym(t) N solving (42) gives y m+1 (t m+1 ) m m + 1 3.2 Simulation results Example 1. Let Ω =], 1[ and consider the following system y (x, t) =.1 y(x, t) +.68y(x, t) + v(t)y(x, t) t y(, t) y(1, t) = = x x y(x, ) = x 2 (1 x 3 ) 2 (43) We consider the subregion ω =],.4[ and we take ϕ i (x) = a i cos(iπx), with a i = 2 i, and we consider the problem (19) with R = 5 1+(iπ) ω 2 ω. Let take N = 6, the projection of the dynamic A is a diagonal matrix with diag(a 6 ) = (.5813,.2852,.283,.8991, 1.7874, 2.8731) Applying the previous algorithm the matrix Pω 6 = (p ij ) 1 i,j 6 is given by: 1.6725.6623.245.1397.1499.89.6623 1.5564.5736.236.1218.444 Pω 6 =.245.5736.8415.582.1496.458.1397.236.582.6683.4124.814.1499.1218.1496.4124.4252.2482.89.444.458.814.2482.348 and we have the following figures:

An output stabilization of bilinear distributed systems 29 1.15 flux.1.2.3.4.5.6.7.8.9 1 x 1 2 3 Flux.1.5.1.2.3.4.5.6.7.8.9 1 x.5.1 4 5 t=1 t=2.5 t=4.15.2 t=5 Figure 1: The flux evolution on ω =],.4[. The above figure shows how the gradient is stabilized on ω. The error flux stabilization is 5.9 1 7, and the stabilization cost is 3.1 1 4. The following table shows that there exists a relation between the area of target subregion ω, the cost and the error of gradient stabilization. ω ],.3[ ],.4[ ],.7[ ],.9[ ],1[ Error 5.956 1 9 3.8522 1 6 4.7438 1 6 2.648 1 5 2.7163 1 5 Cost 5.43 1 2 5.83 1 2 6.65 1 2 6.79 1 2 6.8 1 2 Table 1 More the area of the target region increases, more both the error gradient stabilization and the cost increase. In some situations we are interested by improving the degree of gradient stabilization on ω Ω, as is showing in the following example. Example 2. Let Ω =], 1[ and consider the following system : y (x, t) =.1 y(x, t) +.5y(x, t) +.7v(t)y(x, t) t y(, t) = y(1, t) = (44) y(x, ) =.5x(1 x) Let ω =],.4[ we consider the problem (19) with R = 3 ω ω. We take N = 5. A 5 the projection of A =.1 +.5id, is diagonal matrix with diag(a 5 ) = (.413,.1521,.3882, 1.7913, 1.9674) and Pω 5 is a symmetric matrix with 1.8654.4911.599.1141.113.4911 1.613.6335.1827.85 Pω 5 =.599.633.929.589.623.1141.1827.589.541.3287.11.85.623.3287.4292 Applying the above algorithm we obtain the following figures

21 E. Zerrik, Y. Benslimane and A. El Jai.15.1 t=3 t=4 t=5 12 x 1 3 1 8 t=14 6.5 4 Flux Flux 2.1.2.3.4.5.6.7.8.9 1 x.1.2.3.4.5.6.7.8.9 1 2 x.5 4 6.1 8 Figure 2: The flux evolution on ω =],.4[. The above figures show that the gradient is regionally stabilizable on ω at t = 14, the error is 9.92 1 4 and the cost is 1.23 1 1. Also we remark that as the area of subregion increases as the error and the cost increase. ω ],.1[ ],.3[ ],.5[ ],.8[ ],1[ Error 8.2691 1 4 9.5172 1 4 9.9658 1 4 9.9878 1 4 9.9954 1 4 Cost 11.9 1 2 12.22 1 2 13.1 1 2 13.5 1 2 14.29 1 2 Table 2 4 Conclusion The problem of regional gradient stabilization of bilinear distributed systems is considered, two approaches are used to characterize the gradient stabilization, the first one is based in the decomposition of the state space, and the second one on solving algebraic Riccati eqution and leads to an effective algorithm which is successfully implemented. A naturel extension of this work is the study of the gradient stabilization of semilinear distributed systems. The work is under consideration. References [1] W.F. Arnold and A.J. Laub, Generalized Eigenproblem Algorithms and Software for Algebraic Riccati Equations, Proc. IEEE, 72 (1984), 1746-1754. [2] J.M Ball and M. Slemrod, Feedback stabilization for distributed semilinear control systems, Appl. Math. Optim, 5 (1979), 169-179. [3] H. Banks and K. Kunisch, The linear regulator problem for parabolic systems, SIAM J. control Optim, 22 (1984), 684-696. [4] A. Pazzy, Semigroups of linear operators and applications to partial differential equation, Springer Verlag, New York, 1983.

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