Numerical null controllability of semi-linear 1-D heat equations: fixed point, least squares and Newton methods

Numerical null controllability of a semi-linear heat equation via a least squares method Enrique Fern´andez-Cara a, Arnaud M¨unch b aDpto. EDAN, University of Sevilla, Aptdo. 1160, 41080 Sevilla. bLaboratoire de Math´ematiques, Universit´e Blaise Pascal (Clermont-Ferrand 2), UMR CNRS 6620, Campus des C´ezeaux, 63177 Aubi`ere, France Received *****; accepted after revision +++++ Presented by Abstract This note deals with the computation of distributed null controls for a semi-linear 1D heat equation, in the sublinear and slightly superlinear cases. Under sharp growth assumptions, the existence of controls has been obtained in [Fern´andez-Cara & Zuazua, Null and approximate controllability for weakly blowing up semi-linear heat equation, 2000] via a fixed point reformulation; see also [Barbu, Exact controllability of the superlinear heat equation, 2000]. More precisely, Carleman estimates and Kakutani’s theorem together ensure the existence of fixed points for a corresponding linearized control mapping. In practice, the difficulty is to extract from the Picard iterates a convergent (sub)sequence. We introduce and analyze a least squares reformulation of the problem; we show that this strategy leads to an effective and constructive way to compute fixed points. To cite this article: E. Fern´andez-Cara, A. M¨unch, C. R. Acad. Sci. Paris, Ser. I 340 (2005). R´esum´e Contrˆolabilit´e exacte `a z´ero d’une equation de la chaleur semi-lin´eaire par une m´ethode des moindres carr´es. Cette note concerne la d´etermination effective de contrˆoles `a z´ero pour une ´equation de la chaleur semi-lin´eaire, dans le cas l´eg`erement surlin´eaire. Sous des conditions de croissances optimales, l’existence de contrˆoles a ´et´e obtenue dans [Fern´andez-Cara & Zuazua, Null and approximate controllability for weakly blowing up semi-linear heat equation, 2000] par un argument de point fixe ; voir aussi [Barbu, Exact controllability of the superlinear heat equation, 2000]. Pr´ecis´ement, des in´egalit´es de Carleman et le th´eor`eme de Kakutani impliquent l’existence de points fixes pour un op´erateur de contrˆole lin´earis´e associ´e. En pratique, la difficult´e est d’extraire des it´er´es de Picard une sous-suite convergente. Cette note propose et analyse une reformulation du probl`eme par une approche de type moindres carr´es : on montre que celle-ci garantit une construction explicite de points fixes. Pour citer cet article : E. Fern´andez-Cara, A. M¨unch, C. R. Acad. Sci. Paris, Ser. I 340 (2005). Email addresses: [email protected] (Enrique Fern´andez-Cara), [email protected] (Arnaud M¨unch). Preprint submitted to the Acad´emie des sciences 5 mai 2011 Version fran¸caise abr´eg´ee Cette note concerne la nulle contrˆolabilit´e de l’´equation de la chaleur semi-lin´eaire. Plus pr´ecisement, on consid`ere le syst`eme (2), o`u ω⊂(0,1) d´esigne un ouvert non vide de (0,1), T > 0, a∈C1([0,1]) avec a(x)≥a0>0, y0∈L2(0,1) ; la fonction v∈L∞(ω×(0, T)) est le contrˆole et yest l’ ´etat associ´e. On suppose que f:R→Rest localement Lipschitzienne, f(0) = 0 et satisfait (3), de sorte que (2) poss`ede exactement une solution locale en temps. On rappelle ´egalement que sous la condition (4) de croissance sur f, les solutions de (2) sont globalement d´efinies sur [0, T] avec la r´egularit´e usuelle y∈ H =C0([0, T]; L2(0,1)) ∩L2(0, T;H1 0(0,1)) ; voir [2]. Le syst`eme (2) est dit exactement contrˆolable `a z´ero au temps Tsi, pour toute donn´ee y0∈L2(0,1), il existe un contrˆole v∈L∞(qT) et un ´etat associ´e yd´efini globalement, tel que y∈ H et (5) soit v´erifi´e. La r´ef´erence [7] montre que, si la fonction fv´erifie (3) et ne croit pas `a l’infini plus vite que |s|log3/2(1+|s|), alors (2) est contrˆolable `a z´ero ; voir Th´eor`eme 1.1 et ´egalement [1] pour un r´esultat similaire. Ce r´esultat est obtenu par un argument de point fixe, apr`es lin´earisation de (2) : pr´ecis´ement, on consid`ere l’op´erateur Λ0:L2(QT)→L2(QT) qui `a zassocie la solution yzcontrol´ee du syst`eme lin´earis´e (8), o`u la non lin´earit´e f(y) est remplac´ee par le terme y g(z) avec gd´efini par (6). Suivant [8], la solution yzet son contrˆole vzsont choisis de fa¸con `a minimiser la fonctionnelle (9), faisant intervenir une norme L2`a poids du contrˆole et de la solution contrˆol´ee. Les poids ρet ρ0sont d´efinis par (7). Dans un contexte similaire, [7] montre l’existence d’une constante M > 0 telle que l’image de la boule ferm´ee B(0, M) de L2(QT) par Λ0reste dans B(0, M). Le th´eor`eme de Kakutani assure alors l’existence d’au moins un point fixe pour Λ0, mais ne garantit pas la convergence de la suite {yn}n≥0 naturellement d´efinie `a partir de Λ0par yn+1 = Λ0(yn), n≥0, y0donn´e dans L2(QT). Pour cette raison, nous consid´erons le probl`eme de minimisation (12) dont la solution globale est un point fixe de Λ0, ou de fa¸con ´equivalente, une solution control´ee de (2). Dans (12), Λ d´esigne la restriction de Λ0`a Z:= {z∈L1 loc(QT); RRQT(T−t)−1|z|2dx dt < +∞}. On montre tout d’abord que, si g∈C1 b(R), alors la fonction Rde (12) est de classe C1(Z) et sa d´eriv´ee premi`ere est donn´ee par (13), voir Proposition 3.1. Ensuite, on montre que la norme kR0(z)kZ est minor´ee, `a une constante pr`es K:= (1 −K(ω, T, a0,kgkL∞(R))kg0kL∞(R)ky0kL∞), par la norme kz− Λ(z)kZ; voir Proposition 3.2. Remarquablement, il en r´esulte que si la donn´ee initiale y0`a contrˆoler est suffisamment petite de telle fa¸con que K>0, alors les points critiques de Rsont des points fixes de Λ, c’est-`a-dire des solutions control´ees `a z´ero pour le syst`eme non lin´eaire (2) ; voir Corollaire 3.1. Sous ces hypoth`eses, la r´esolution du probl`eme (12) par une m´ethode de gradient g´en`ere la suite {zn}n≥0 d´efinie par zn+1 =zn−ρR0(zn), n ≥0, z0∈Z(1) convergeante vers une solution de (2). Nous renvoyons `a [6] pour des exp´eriences num´eriques mettant en ´evidence des donn´ees (ω, T, a0, y0) pour lesquelles la suite (1) converge tandis que la suite {yn}n≥0 des it´er´es de Picard, bien que restant born´e dans L2(QT), ne converge pas. Mentionnons enfin que cette approche est g´en´erale et s’adapte `a beaucoup d’autres syst`emes contrˆolables rentrant dans le cadre de [8]. English version 1. Problem statement Let ω⊂(0,1) denote a (small) non-empty open interval, let 1ωbe the associated characteristic function and let us assume that T > 0 and a∈C1([0,1]), with a(x)≥a0>0, y0∈L2(0,1). We will consider the null controllability problem for the semi-linear 1-D heat system 2      yt−(a(x)yx)x+f(y) = v1ω,(x, t)∈QT= (0,1) ×(0, T) y(x, t) = 0,(x, t)∈ΣT={0,1} × (0, T) y(x, 0) = y0(x), x ∈(0,1). (2) Here, v∈L∞(ω×(0, T)) is the control and yis the associated state. We assume that f:R→Ris, at least, locally Lipschitz-continuous, f(0) = 0 and |f0(s)| ≤ C(1 + |s|5),p.p. s∈R,(3) so that (2) possesses exactly one local in time solution. We recall (see [2]) that, under the growth condition |f(s)| ≤ C(1 + |s|log(1 + |s|)) ∀s∈R,(4) any solution is globally defined in [0, T] and satisfies y∈ H =C0([0, T ]; L2(0,1)) ∩L2(0, T;H1 0(0,1)). Without such a condition, the solutions to (2) can blow up before t=T; the blow-up time depends on the sizes of ky0kL2(0,1) and kakL∞. The system (2) is said to be null-controllable at time Tif, for any y0∈L2(0,1), there exist controls v∈L∞(qT) and associated states ythat are again globally defined in [0, T], belong to Hand satisfy y(x, T) = 0, x ∈(0,1).(5) Reference [7] provides conditions on funder which (2) is null-controllable (see also [1]): Theorem 1.1 ([7]) Let T > 0be given. Assume that f:R→Ris locally Lipschitz-continuous, f(0) = 0, (3) is satisfied and f(s)(|s|log3/2(1 + |s|))−1→0as |s|→∞. Then (2) is null-controllable at time T. Thus, if fdoes not grow at infinity faster than |s|logp(1+|s|) for some p < 3/2, the action of the control von ω×(0, T) can compensate the blow up of the solution. The same result was obtained before in [4] for p < 1. Let us also mention [3], where a positive positive boundary controllability result is obtained for a specific class of initial and final data and Tlarge enough. 2. Linearization - A fixed point operator For simplicity, let us assume that f∈C1(R). Let us introduce the function g, with g(s) = s−1f(s) if s6= 0, g(0) = f0(0) otherwise. (6) Then g∈C0(R) and f(s) = g(s)sfor all s(recall that f(0) = 0). We will use the weights ρand ρ0 introduced by Fursikov and Imanuvilov [8]:          ρ(x, t) = exp β(x)(T−t)−1, ρ0(x, t)=(T−t)3/2ρ(x, t), β(x) = K1eK2−eβ0(x) the Kiare large positive constants (depending on T,a0,kakC1and kAk∞) and β0∈C∞([0,1]), β0>0 in (0,1), β0(0) = β0(1) = 0, |β0 0|>0 outside ω. (7) For any z∈L1(QT), we also introduce the notation Lg(z)y=yt−(a(x)yx)x+g(z)y , L∗ g(z)q=−qt−(a(x)qx)x+g(z)q and we set P0={q∈C2(QT) : q= 0 on ΣT}. In this linear space, the bilinear form m(z;p, q) := ZZQT ρ−2L∗ g(z)p L∗ g(z)q dx dt +ZZqT ρ−2 0p q dx dt 3 is a scalar product, as a consequence of the unique continuation property. Let Pbe the completion of P0 for this scalar product. It can be shown that Pdoes not depend on z. Let us consider the linear control systems Lg(z)y=v1ω,(x, t)∈QT;y(x, t)=0,(x, t)∈ΣT;y(x, 0) = y0(x), x ∈(0,1).(8) Then, we define the mapping Λ0:L2(QT)7→ L2(QT) where, for any z∈L2(QT), yz= Λ0(z) is, together with vz, the unique solution to the linear extremal problem      Minimize J(z;y, v) := 1 2ZZQT ρ2|y|2dx dt +1 2ZZqT ρ2 0|v|2dx dt Subject to v∈L2(qT),(y, v) satisfies (8). (9) The couple (yz, vz) is characterized as follows : yz= Λ0(z) = ρ−2L∗ g(z)pz, vz=−ρ−2 0pz|qT, where pzis the unique solution to the linear (elliptic) problem m(z;pz, q) = Z1 0 y0(x)q(x, 0) dx ∀q∈P;pz∈P. (10) The well-posedness of (10) is deduced from the estimate kq(,0)k2 H1 0(0,1) ≤C0m(z;q, q)∀q∈P, C0=C0(ω, T, a0,kakC1,kgkL∞(R))>0, implying in particular that the right hand side of (10) is continuous with respect to the norm defined by m; see [6,8]. Therefore, in order to solve the null controllability problem for (2), it ”suffices” to find a solution to the fixed point equation y= Λ0(y), y ∈L2(QT).(11) This fixed point formulation has been used in [7] to prove Theorem 1.1. Precisely, it is shown that there exists M > 0 such that Λ0maps the closed ball B(0; M)⊂L2(QT) into itself. Then, Kakutani’s theorem provides the existence of at least one fixed point for Λ0. However, this property does not guarantee the convergence of the bounded sequence {yn}n≥0naturally defined by yn+1 = Λ0(yn) for all n≥0, with y0 given in L2(QT). 3. Least squares reformulation of the null controllability problem Let us introduce the function ζ(t)≡(T−t)−1/2and the space Z:= L2(ζ2, QT) = {z∈L1 loc(QT) : RRQTζ2|z|2dx dt < +∞ } and let us denote by Λ the restriction to Zof the mapping Λ0. Obviously, Λ(z)∈Zfor all z∈Z. Let us consider the following least squares reformulation of (11):      Minimize R(z) := 1 2kz−Λ(z)k2 Z Subject to z∈Z. (12) Any solution to (11) solves (12). Conversely, if ysolves (12), we necessarily have R(y) = 0 (because (2) is null controllable with control-states (y, v) such that J(z;y, v)<+∞); hence, yalso solves (11). Therefore (11) and (12) are, in the present context, equivalent. The idea is to solve (12) by using gradient techniques. To this purpose, it is crucial to determine conditions under which Ris differentiable and, also, to compute R0(z). 4 Proposition 3.1 Let us assume that g∈C1 b(R). Then R∈C1(Z). Moreover, for any z∈Z, the derivative R0(z)is given by R0(z) = 1−ρ−2g0(z)pz(z−yz) + ζ−2g0(z)yzλz+pzµz,(13) where pzis the unique solution to (10),yz=ρ−2L∗ g(z)pz,λzis the unique solution to the linear (adjoint) problem m(z;q, λz)=(z−yz, ρ−2L∗ g(z)q)Z∀q∈P;λz∈P, and, finally, µz=ρ−2L∗ g(z)λz. The behavior of R0(z) is explained in the following result: Proposition 3.2 Let the assumptions in Proposition 3.1 be satisfied. There exists a constant Kthat depends on ω,T,a0,kakC1and kgkL∞(R)but is independent of zand y0, such that the following holds for all z∈Z: kR0(z)kZ≥1−Kkg0kL∞(R)ky0kL∞kz−Λ(z)kZ.(14) A very relevant consequence of Proposition 3.2 is the following: Corollary 3.1 Let us the assumptions in Proposition 3.1 be satisfied and let Kbe the constant furnished by Proposition 3.2. If Kkg0kL∞(R)ky0kL∞<1, then the critical points of Rare global minima and, consequently, solve (11). For the proof of Proposition 3.2, the following technical lemmas are needed: Lemma 3.2 For any q∈Pone has (ζρ)−1q∈L∞(QT). Furthermore, there exists C > 0, only depending on ω,T,a0,kakC1and kgkL∞(R), such that k(ζρ)−1qk2 L∞(QT)≤Ckqk2 P∀q∈P. Lemma 3.3 With the notation of Proposition 3.1, one has: kpzkP≤Cky0kL∞∀z∈Z and kλzkP≤Ckζρ−1k∞kz−yzkZ∀z∈Z, where Cdepends on ω,T,a0,kakC1and kgkL∞(R). We refer to [6] for the proofs of these lemmas, based on global Carleman inequalities. Proof of Proposition 3.2 - Let z∈Zbe given and let us introduce f:= z−yz. In view of Proposition 3.1, one has kR0(z)kZ≥1 kfkZ (R0(z), f) =1 kfkZZZQTζ2(1 + ρ−2g0(z)pz)|f|2+g0(z)(yzλz+pzµz)fdx dt ≥ kfkZ−1 kfkZZZQT ζ2ρ−2|g0(z)| |pz| |f|2dx dt −1 kfkZZZQT |g0(z)|(|yz||λz|+|pz||µz|)|f|dx dt. In view of Lemmas 3.2 and 3.3, ZZQT ζ2ρ−2|g0(z)| |pz| |f|2dx dt ≤ kρ−2g0(z)pzk∞ZZQT ζ2|f|2dx dt ≤ kζρ−1k∞kg0(z)k∞k(ζρ)−1pzk∞kfk2 Z≤C G1kpzkPkfk2 Z≤C G1ky0kL∞kfk2 Z, 5 where G1:= kg0kL∞(R). On the other hand, from Lemma 3.2 we also have ZZQT |g0(z)|(|yz||λz|+|pz||µz|)|f|dx dt ≤ kg0(z)k∞"ZZQT ρ−2|L∗ g(z)pz|2dx dt1/2 k(ζρ)−1λzk∞ +ZZQT ρ−2|L∗ g(z)λz|2dx dt1/2 k(ζρ)−1pzk∞#ZZQT ζ2|f|2dx dt1/2 ≤C G1ky0kL∞kλzkP+kpzkPkζρ−1k∞kfkZkfkZ ≤C G1ky0kL∞kfk2 Z. Consequently, kR0(z)kZ≥ kfkZ−Kkg0kL∞(R)ky0kL∞kfkZand we get (14) for some K.2 It is therefore very appropriate to try to solve the null controllability problem for (2) by applying a gradient method to the extremal problem (12). Indeed, under the hypotheses of Corollary 3.1, the sequence {zn}n≥0defined by (1) converges, as n→+∞, toward a fixed point of Λ. The limit ˜zand its associated control v˜zsolve (2) and satisfy (5). We refer to [6] for details and for some numerical experiments with f(s) = Cs logp(1 + |s|), C > 0, p∈R, exhibiting data for which the sequence {zn}n≥0 given by (1) converges, in contrast with the sequence {yn}n≥0defined by yn+1 = Λ(yn), n≥1. Let us finally mention that this least squares approach can also be used in the context of many other controllable systems for which appropriate Carleman estimates are available: Navier-Stokes systems, wavelike equations, etc.; see [8], [9]. References [1] V. Barbu, Exact controllability of the superlinear heat equation, Appl. Math. Optim. Optimization, Theory and Applications 42(1), (2000) 73–89. [2] T. Cazenave, A. Haraux, Introduction aux probl`emes d’´evolutions semi-lin´eaires, Math´ematiques et Applications, Ellipses, Paris 1989. [3] J-M. Coron, E. Tr´elat, Global steady-state controllability of one dimensional semilinear heat equations, SIAM J. Control Optim. 43(2) 2004, 549-569. [4] E. Fernandez-Cara, Null controllability of the semi-linear heat equation, Esaim:COCV (1997), no. 2, 87-103. [5] E. Fernandez-Cara and A. M¨unch, Numerical null controllability of the 1-d heat equation: primal algorithm, Submitted to SIAM J. Control Optim. [6] E. Fernandez-Cara and A. M¨unch, Numerical null controllability of a semi-linear 1-d heat equation, Preprint. [7] E. Fernandez-Cara and E. Zuazua, Null and approximate controllability for weakly blowing up semilinear, Ann. Inst. Henri Poincar´e, Analyse non lin´eaire. 17 (2000), no. 5, 583–616. [8] A.V. Fursikov and O. Yu. Imanuvilov, Controllability of Evolution Equations, Lecture Notes Series, number 34. Seoul National University, Korea, (1996) 1–163. [9] I. Lasiecka and R. 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