Insensitizing controls for a semilinear heat equation with a superlinear nonlinearity

C. R. Acad. Sci. Paris, t. 000, S´erie I, p. 1–??, 2002 - PXMA????.TEX - Equations aux d´eriv´ees partielles/Partial differential equations (Contrˆole Optimal/Optimal Control) Insensitizing controls for a semilinear heat equation with a superlinear nonlinearity Olivier BODARTa, Manuel GONZ ´ ALEZ-BURGOSb, Rosario P´ EREZ-GARC´ IAb aLaboratoire de Math´ematiques Appliqu´ees, UMR CNRS 6620, Universit´e Blaise-Pascal (Clermont-Ferrand 2), 63177 AUBIERE, France Phone: (33) 4 73 40 79 65. Fax: (33) 473 40 70 64 E-mail: Olivier.Bo[email protected]clermont.fr bDpto. EDAN, Universidad de Sevilla, Aptdo. 1160, 41080 Sevilla, Spain Phone: (34) 954 55 79 99, (34) 954 55 79 96. Fax: (34) 954 55 28 98 E-mail: [email protected], [email protected] (Re¸cu le jour mois ann´ee, accept´e apr`es r´evision le jour mois ann´ee) Abstract. This Note is concerned with the existence of insensitizing controls for a semilinear heat equation when we consider nonlinearities that behave superlinearly at infinity. We prove the existence of a control insensitizing the L2−norm of the observation of the solution in an open subset Oof the considered domain under appropriate assumptions on the nonlinear term f(y) and the second member ξof the equation. The proof uses global Carleman estimates, parabolic regularity and a fixed point argument. c 2002 Acad´emie des sciences/´ Editions scientifiques et m´edicales Elsevier SAS Contrˆoles insensibilisants pour une ´equation de la chaleur semilin´eaire avec non-lin´earit´e superlin´eaire R´esum´e. Dans cette Note, on ´etudie l’existence de contrˆoles insensibilisants pour une ´equation de la chaleur semi-lin´eaire, avec des non-lin´earit´es superlin´eaires `a l’infini. On d´emontre l’existence de contrˆoles insensibilisant la norme L2de la solution observ´ee dans un ouvert Oinclus dans le domaine consid´er´e, sous des hypoth`eses convenables sur la non-lin´earit´e et le second membre de l’´equation. La d´emonstration fait appel `a des in´egalit´es de Carleman globales, la r´egularisation parabolique et un argument de point fixe. c 2002 Acad´emie des sciences/´ Editions scientifiques et m´edicales Elsevier SAS Version fran¸caise abr´eg´ee Soit Ω ⊂IRN,N≥1, un ouvert born´e et r´egulier. Soit fune fonction de classe C1d´efinie sur IR. Soient ωet Odeux ouverts inclus dans Ω. Pour T > 0 on pose Q= Ω ×(0, T ) et Note pr´esent´ee par First name NAME S0764-4442(00)0????-?/FLA c 2002 Acad´emie des sciences/´ Editions scientifiques et m´edicales Elsevier SAS. Tous droits r´eserv´es. 1 1 O. Bodart, M. Gonz´alez-Burgos, R. P´erez-Garc´ıa Σ = ∂Ω×(0, T ). On consid`ere une solution y=y(x, t;v, τ) de l’´equation (1), o`u ξ∈L2(Q) et y0∈L2(Ω) sont donn´ees, ˆy0∈L2(Ω) est inconnu avec kˆy0kL2(Ω) = 1, τest un petit nombre r´eel inconnu et v∈L2(Q) est un contrˆole `a d´eterminer. On d´esigne par 1ωla fonction caract´eristique de ω. Soit la fonctionnelle φd´efinie par (2), on dit qu’un contrˆole vinsensibilise φsi on a (3). Ce probl`eme, pos´e par J.L. Lions dans [9] fut d’abord ´etudi´e dans [1], o`u les auteurs prouvent l’existence de contrˆoles approximativement insensibilisants. L. de Teresa a prouv´e par ailleurs dans [10], d’une part, qu’il ne peut pas exister de contrˆole insensibilisant pour tout y0∈L2(Ω) quand Ω\ω6=∅. D’autre part, lorsque y0= 0 et sous des hypoth`eses convenables sur ξ, elle prouve l’existence de contrˆoles assurant (3). Dans cette Note, nous ´etendons ce travail `a des non-lin´earit´es plus g´en´erales. L’existence d’un contrˆole vsatisfaisant `a (3) est ´equivalente `a l’existence d’un contrˆole vtel que la solution (y, q) de la cascade de syst`emes (4) v´erifie (5), cf. [9] et [1] pour la d´emonstration. Le r´esultat principal de cette Note est le suivant : Th´ eor` eme 0.1. – Supposons que ω∩ O 6=∅et y0= 0. Supposons en outre que fest de classe C2telle que f(0) = 0 et lim |s|→∞ f′(s) log(1 + |s|)= 0. Soit r∈(N+ 2,∞)donn´e. Alors, pour tout ξ∈Lr(Q)tel que ZZQ exp 1 t3|ξ|2dx dt < ∞, il existe un contrˆole v∈Lr(Q)qui insensibilise la fonctionnelle d´efinie en (2). La structure de la preuve est bas´ee sur un argument de lin´earisation et point fixe, l’´etude fine du cas lin´eaire en ´etant le point cl´e comme dans la plupart des d´emonstrations de ce type (voir [6], [11],...). On consid`ere le probl`eme de contrˆole lin´eaire (8)-(9). Soient B0et Bdeux ouverts tels que B0⊂⊂ B⊂ω∩ O. Dans un premier temps, on construit un contrˆole ˆv∈L2(Q), `a support dans B0×[0, T ], grˆace `a une in´egalit´e d’observabilit´e mixte. Cette in´egalit´e fut d’abord donn´ee dans [10], ainsi que la construction du contrˆole qui en d´ecoule. Dans [2], elle est ´etendue `a des syst`emes plus g´en´eraux, et la d´ependance des constantes vis `a vis des potentiels est explicit´ee. On obtient ainsi une estimation de la norme de ˆven fonction des param`etres aet b. On note (ˆy, ˆq) la solution de (8)-(9) associ´ee au contrˆole ˆv. Ensuite, on consid`ere une fonction telle que θ≡1 dans un voisinage de B0. On pose alors q= (1 −θ) ˆqand y= (1 −θ) ˆy+ 2∇θ· ∇ˆq+ (∆θ)ˆq. Le couple (y, q) v´erifie (8), (9) avec le contrˆole v=−θξ + 2∇θ· ∇ˆy+ (∆θ)ˆy+ (∂t−∆ + a) [2∇θ· ∇ˆq+ (∆θ)ˆq]. Ce contrˆole est `a support dans B×[0, T ]. En utilisant les propri´et´es de r´egularisation de l’´equation de la chaleur, sous des hypoth`eses convenables sur aet b, on montre que v∈Lr(Q) et satisfait l’estimation (10). La r´egularit´e de vpermet d’obtenir des estimations sur ydans un espace de H¨older dont l’injection est compacte dans Z=C0(Q)∩Lr(0, T ;W1,r(Ω)). On pose g(s) =    f(s) sif s6= 0 f′(0) if s= 0. 2 Insensitizing controls for a superlinear heat equation Pour z∈Z, on consid`ere le syst`eme lin´eaire (8) avec les potentiels a=az=g(z)∈L∞(Q), b=bz=f′(z)∈L∞(Q)∩Lr(0, T ;W1,r(Ω)). D’apr`es ce qui pr´ec`ede, il existe vz∈Lr(Q) tel que la solution correspondante (yz, qz) de (8) v´erifie (9) et yz, qz∈Z. On conclut la preuve en appliquant le th´eor`eme du point fixe de Kakutani `a l’application z7→ yz. Les d´etails sont donn´es dans [3]. 1. Statement of the problem Let Ω ⊂IRN,N≥1, be a regular bounded open set. Let fbe a C1function defined on IR. Let ωand Obe two open subsets of Ω. For T > 0 we denote Q= Ω ×(0, T ) and Σ = ∂Ω×(0, T ). We consider the semilinear heat equation: (∂ty−∆y+f(y) = ξ+v1ωin Q, y= 0 on Σ, y(0) = y0+τˆy0in Ω,(1) where ξ∈L2(Q) and y0∈L2(Ω) are given, ˆy0∈L2(Ω) is unknown with kˆy0kL2(Ω) = 1, τis a small unknown real number and v∈L2(Q) is a control function to be determined. Here 1ωis the characteristic function of ω. Let us define φ(y) = 1 2ZZO×(0,T ) |y(x, t;τ, v)|2dx dt, (2) y=y(·,·;τ, v) being a solution of (1) associated to τand v. A control function vis said to insensitize φif ∂φ(y(·,·;τ, v)) ∂τ τ=0 = 0,∀ˆy0∈L2(Ω) with kˆy0kL2(Ω) = 1.(3) This problem, addressed by J.L. Lions in [9], has been studied for globally Lipschitz-continuous nonlinearities. First, in [1] the authors proved the existence of so-called approximately insensitizing controls for unknown data both in the initial and boundary conditions. In [10] it was proved, on the one hand, that we cannot expect the existence of insensitizing controls for every y0∈L2(Ω) when Ω\ω6=∅. On the other hand, for y0= 0 and suitable assumptions on ξ, L. de Teresa proved the existence of controls such that (3) holds. In this note we extend this last result to more general nonlinearities. It is shown in [9] and [1] that the existence of a control vsatisfying (3) is equivalent to the existence of a control vsuch that the solution (y, q) of (∂ty−∆y+f(y) = ξ+v1ω,−∂tq−∆q+f′(y)q=y1Oin Q, y= 0, q = 0 on Σ, y(x, 0) = y0(x), q(x, T ) = 0 in Ω (4) verifies q(x, 0) = 0 in Ω.(5) The main result in this note is the following Theorem 1.1. – Assume that ω∩O 6=∅and y0= 0. Let fbe a C2function verifying f(0) = 0 and lim |s|→∞ f′(s) log(1 + |s|)= 0.(6) 3 O. Bodart, M. Gonz´alez-Burgos, R. P´erez-Garc´ıa Let r∈(N+ 2,∞)be given. Then, for any ξ∈Lr(Q)such that ZZQ exp 1 t3|ξ|2dx dt < ∞,(7) there exists a control function v∈Lr(Q)insensitizing the functional given by (2). As usual in controllability problems for nonlinear equations, we first prove a controllability result for a linearized version of (4). We then apply a fixed point argument to treat the general case. The structure of the proof is quite general (for other controllability results proved in a similar way, see [6], [11],...). We sketch here these two main steps and the details will be given in [3]. Remark 1. – Hypothesis (6) is fulfilled by certain superlinear nonlinearities fsuch as |f(s)|=|p1(s)|logα(1 + |p2(s)|)for all |s| ≥ s0>0, with α∈[0,1), where p1and p2are first order real polynomial functions. In view of known null-controllability results for a semilinear heat equation with homogeneous Dirichlet boundary conditions (see Theorem 1.2 in [6]), one could think of extending Theorem 1.1 to nonlinearities fwith growth at infinity of order |s|logα(1 + |s|)with 1≤α < 3/2. Notice that, in this case, in the absence of control, blow-up phenomena occur. The idea in [6] is to control the system to zero in a short time to prevent the solution from blowing-up and, the only right hand side term being the control, setting v≡0for the rest of the time interval. However, this strategy fails here since, in insensitivity problems, both initial and final times are fixed and, in addition, there is a right hand side term ξin the equation. The problem then remains opened for such nonlinearities. When f≡0and O= Ω, a positive result on the existence of insensitizing controls is proved for a small class of initial data (see Lemma 2 in [10]). The proof of this result is strongly based on properties of the semigroup of the classical heat equation. Even in the case of a linear heat equation with an L∞potencial, giving sufficient conditions on the initial data y0in order to insensitize (2) is an open problem. 2. The linear case Let us consider the following linear controllability problem: (∂ty−∆y+ay =ξ+v1ω,−∂tq−∆q+bq =y1Oin Q, y= 0, q = 0 on Σ, y(x, 0) = 0, q(x, T ) = 0 in Ω.(8) q(0) = 0 in L2(Ω), (9) where a, b ∈L∞(Q) and ξ,ω,Oare given as in Theorem 1.1. First, using an adequate observability inequality, we construct controls in L2(Q). Then, due to the regularizing properties of the heat equation, we exhibit more regular controls. This will be essential in the proof of Theorem 1.1 since fbehaves superlinearly at infinity. We actually seek a control supported in ω∩O 6=∅. Let then B0and Bbe two open sets such that B0⊂⊂ B⊂ω∩ O. The following result holds: Proposition 2.1. – Let ξbe as in Theorem 1.1. Then, there exists a control ˆv∈L2(Q)such that supp ˆv⊂B0×[0, T ]and the corresponding solution (ˆy, ˆq)of (8) satisfies (9). Moreover, ˆvcan be chosen in such a way that kˆvk2 L2(Q)≤exp [C H(T, kak∞,kbk∞)] ZZQ exp CM(T, kak∞,kbk∞) t|ξ|2dx dt, 4 Insensitizing controls for a superlinear heat equation with M(T, kak∞,kbk∞) = 1 + T+Tkak2/3 ∞+Tkbk2/3 ∞+Tka−bk1/2 ∞, H(T, kak∞,kbk∞) = 1 + 1 T+T+Tkak∞+kak2/3 ∞+Tkbk∞+kbk2/3 ∞+ka−bk1/2 ∞, Cbeing a positive constant only depending on Ω,ωand O. A similar result was proved in [10] as a consequence of a global Carleman inequality for the linear heat equation (cf. [7]). In [2], the result is extended to the linear heat equation involving first order terms and one gives the explicit dependence of Mand Hon T,kak∞and kbk∞. This is crucial to prove the existence of insensitizing controls in the nonlinear case when fis not a globally Lipschitz-continuous function. From (ˆy, ˆq) we now build a new control v∈Lr(Q) with support in B×[0, T ], rbeing as in Theorem 1.1. For this, let θ∈ D(B) be a function such that θ≡1 in a neighbourhood of B0. We set q= (1 −θ) ˆqand y= (1 −θ) ˆy+ 2∇θ· ∇ˆq+ (∆θ)ˆq. Then, (y, q) solves (8), (9) for the control v=−θξ + 2∇θ· ∇ˆy+ (∆θ)ˆy+ (∂t−∆ + a) [2∇θ· ∇ˆq+ (∆θ)ˆq]. In the sequel Cwill stand for a generic positive constant depending on Ω, ω,Oand T, which value can change from line to line. One has Proposition 2.2. – 1. For a, b ∈L∞(Q)we have y, q ∈C0(Q)∩Lr(0, T ;W1,r(Ω)) and kykC0(Q)∩Lr(W1,r )+kqkC0(Q)∩Lr(W1,r )≤exp [C(1 + kak∞+kbk∞)] kξkLr(Q)+kˆvkL2(Q). 2. Suppose in addition that b∈Lr(0, T ;W1,r(Ω)). Then v∈Lr(Q),supp v⊂B×[0, T ]and kvkLr(Q)≤exp [C(1 + kak∞+kbk∞)] 1 + k∇bkLr(Q)NkξkLr(Q)+kˆvkL2(Q).(10) The proof of this result can be found in [3] and combines the regularizing effect and properties of the semigroup generated by the heat equation with Dirichlet boundary conditions. Remark 2. – In the case of the heat equation, the previous technique provides a new method of construction of more regular controls from controls in L2(Q). This allows to prove both null controllability and insensitizing local results for the heat equation with nonlinear Fourier boundary conditions (see [5,4]). 3. Idea of the proof of Theorem 1.1 Let f,ξbe as in Theorem 1.1. Let us define g(s) =    f(s) sif s6= 0 f′(0) if s= 0. Let us set Z=C0(Q)∩Lr(0, T ;W1,r(Ω)). For any z∈B(0; R)⊂Z,R > 0 being positive, we consider the linear system (8) with potentials a=az=g(z)∈L∞(Q), b=bz=f′(z)∈L∞(Q)∩Lr(0, T ;W1,r(Ω)). In view of Propositions 2.1 5 O. Bodart, M. Gonz´alez-Burgos, R. P´erez-Garc´ıa and 2.2 and the assumptions on ξ, we infer that there exists vz∈Lr(Q) such that the corresponding solution (yz, qz) of (8) satisfies yz, qz∈Z, (9) and ||yz||Z≤exp [C(1 + kg(z)k∞+kf′(z)k∞)] kξkLr(Q)+    exp 1 2t3ξ   L2(Q)!,(11) kvzkLr(Q)≤C(Ω, ω, O, T, R) kξkLr(Q)+    exp 1 2t3ξ   L2(Q)!.(12) Let us define A:B(0; R)⊂Z7−→ Lr(Q) and Λ : B(0; R)⊂Z7−→ Zas A(z) = {v∈Lr(Q) : (y, q) satisfies (8) for az,bz,(9), v verifying (12)}, Λ(z) = {y∈Z: (y, q) solution of (8) for az,bzwith v∈ A(z), y satisfying (11)}. For fixed z∈B(0; R), Λ(z) is a non empty convex closed set in Z. In fact, in view of Theorem 9.1 p. 341 in [8], its Corollary and (12), Λ(z) is uniformly bounded in the H¨older space H1+β,(1+β)/2(Q) with β= 1 −(N+ 2)/r. The imbedding of H1+β,(1+β)/2(Q) being compact in Z, there exists a compact set K⊂Zsuch that Λ(z)⊂Kfor every z∈B(0; R). Due to (6), (11) we infer the existence of R > 0 large enough such that Λ B(0; R)⊂B(0; R). Finally, one can prove that Λ is an upper hemicontinuous multivalued mapping. We can then apply Kakutani’s fixed point theorem and conclude that there exists y∈Zsuch that y∈Λ(y). This ends the proof of Theorem 1.1. References [1] O. Bodart, C. Fabre, Controls insensitizing the norm of the solution of a semilinear heat equation, J. Math. Anal. Appl. 195 (3) (1995) 658–683. [2] O. Bodart, M. Gonz´alez-Burgos, R. P´erez-Garc´ıa, Insensitizing controls for a heat equation with a nonlinear term involving the state and the gradient (in preparation). [3] O. Bodart, M. Gonz´alez-Burgos, R. P´erez-Garc´ıa, Existence of insensitizing controls for a semilinear heat equation with a superlinear nonlinearity (in preparation). [4] O. Bodart, M. Gonz´alez-Burgos, R. P´erez-Garc´ıa, A local result on insensitizing controls for the heat equation with nonlinear boundary Fourier conditions (in preparation) [5] A. Doubova, E. Fern´andez-Cara, M. 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Zuazua, Exact boundary controllability for the semilinear wave equation, in: Brezis H., Lions J.L. (Eds.), NonLinear Partial Differential Equations and their Applications Vol. X, Pitman, 1991, 357–391. 6