Null-controllability of the heat equation in unbounded domains

Transcription

1 Chapter 1 Null-controllability of the heat equation in unbounded domains Sorin Micu Facultatea de Matematică-Informatică, Universitatea din Craiova Al. I. Cuza 13, Craiova, 1100 Romania sd micu@yahoo.com Enrique Zuazua Departamento de Matemáticas, Facultad de Ciencias, Universidad Autónoma Cantoblanco, 28049, Madrid Spain enrique.zuazua@uam.es 1.1 Description of the problem Let Ω be a smooth domain of R n with n 1. Given T > 0 and Γ 0 Ω, an open non-empty subset of the boundary of Ω, we consider the linear heat equation: u t u = 0 in Q u = v1 Σ0 on Σ u(x, 0) = u 0 (x) in Ω, (1.1) where Q = Ω (0, T ), Σ = Ω (0, T ) and Σ 0 = Γ 0 (0, T ) and where 1 Σ0 denotes the characteristic function of the subset Σ 0 of Σ. 1

2 2CHAPTER 1. NULL-CONTROLLABILITY OF THE HEAT EQUATION In (1.1), v L 2 (Σ) is a boundary control that acts on the system through the subset Σ 0 of the boundary and u = u(x, t) is the state. System (1.1) is said to be null-controllable at time T if for any u 0 L 2 (Ω) there exists a control v L 2 (Σ 0 ) such that the solution of (1.1) satisfies u(x, T ) = 0 in Ω. (1.2) This article is concerned with the null-controllability problem of (1.1) when the domain Ω is unbounded. 1.2 Motivation and history of the problem We begin with the following well-known result Theorem 1 When Ω is a bounded domain of class C 2 system (1.1) is nullcontrollable for any T > 0. We refer to D.L. Russell [12] for some particular examples treated by means of moment problems and Fourier series and to A. Fursikov and O. Yu. Imanuvilov [3] and G. Lebeau and L. Robbiano [7] for the general result covering any bounded smooth domain Ω and open, non-empty subset Γ 0 of Ω. Both the approaches of [3] and [7] are based on the use of Carleman inequalities. However, in many relevant problems the domain Ω is unbounded. We address the following question: If Ω is an unbounded domain, is system (1.1) null-controllable for some T > 0?. None of the approaches mentioned above apply in this situation. In fact, very particular cases being excepted (see the following section), there exist no results on the null-controllability of the heat equation (1.1) when Ω is unbounded. The approach described in [6] and [9] is also worth mentioning. In this articles it is proved that, for any T > 0, the heat equation has a fundamental solution which is C away from the origin and with support in the strip 0 t T. This fundamental solution, of course, grows very fast as x goes to infinity. As a consequence of this, a boundary controllability result may be immediately obtained in any domain Ω with controls distributed all along its boundary. Note however that, when the domain is unbounded, the solutions and controls obtained in this way grow too fast as x and therefore, these are not solutions in the classical sense. In fact, in the frame of unbounded domains, one has to be very careful in defining

3 1.3. AVAILABLE RESULTS 3 the class of admissible controlled solutions. When imposing, for instance, the classical integrability conditions at infinity, one is imposing additional restrictions that may determine the answer to the controllability problem. This is indeed the case, as we shall explain. There is a weaker notion of controllability property. It is the so called approximate controllability property. System (1.1) is said to be approximately controllable in time T if for any u 0 L 2 (Ω) the set of reachable states, R(T ; u 0 ) = {u(t ) : u solution of (1.1) with v L 2 (Σ 0 )}, is dense in L 2 (Ω). With the aid of classical backward uniqueness results for the heat equation (see, for instance, J.L. Lions and E. Malgrange [8] and J.M. Ghidaglia [4]), it can be seen that null-controllability implies approximate controllability. The approximate control problem for the semi-linear heat equation in general unbounded domains was addressed in [13] where an approximation method was developed. The domain Ω was approximated by bounded domains (essentially by Ω B R, B R being the ball of radius R) and the approximate control in the unbounded domain Ω was obtained as limit of the approximate control on the approximating bounded domain Ω B R. But this approach does not apply in the context of the null-control problem. However, taking into account that approximate controllability holds, it is natural to analyze whether null-controllability holds as well. In [1] it was proved that the null-controllability property holds even in unbounded domains if the control is supported in a subdomain that only leaves a bounded set uncontrolled. Obviously, this result is very close to the case in which the domain Ω is bounded and does not answer to the main issue under consideration of whether heat processes are null-controllable in unbounded domains. 1.3 Available results To our knowledge, in the context of unbounded domains Ω and the boundary control problem, only the particular case of the half-space has been considered: Ω = R n + = {x = (x, x n ) : x R n 1, x n > 0} Γ 0 = Ω = R n 1 = {(x, 0) : x R n 1 } (1.3) (see [10] for n = 1 and [11] for n > 1). According to the results in [10] and [11], the situation is completely different to the case of bounded domains. In fact a simple argument shows that the null controllability result which holds for the case Ω bounded is

4 4CHAPTER 1. NULL-CONTROLLABILITY OF THE HEAT EQUATION no longer true. Indeed, the null-controllability of (1.1) with initial data in L 2 ( R n +) and boundary control in L 2 (Σ) is equivalent to an observability inequality for the adjoint system { ϕt + ϕ = 0 on Q (1.4) ϕ = 0 on Σ. More precisely, it is equivalent to the existence of a positive constant C > 0 such that ϕ(0) 2 L 2 (R n ) C ϕ 2 + x n dx dt (1.5) holds for every smooth solution of (1.4). When Ω is bounded, Carleman inequalities provide the estimate (1.5) and, consequently, null-controllability holds (see for instance [3]). In the case of a half-space, by using a translation argument, it is easy to see that (1.5) does not hold (see [11]). In the case of bounded domains, by using Fourier series expansion, the control problem may be reduced to a moment problem. However, Fourier series cannot be used directly in R n +. Nevertheless, it was observed by M. Escobedo and O. Kavian in [2] that, on suitable similarity variables and at the appropriate scale, solutions of the heat equation on conical domains may be indeed developed in Fourier series on a weighted L 2 space. This idea was used in [10] and [11] to study the null-controllability property when Ω is given by (1.3). Firstly, we use similarity variables and weighted Sobolev spaces to develop the solutions in Fourier series. A sequence of one-dimensional controlled systems like those studied in [10] is obtained. Each of these systems is equivalent to a moment problem of the following type: given S > 0 and (a n ) n 1 (depending on the Fourier coefficients of the initial data u 0 ) find f L 2 (0, S) such that S 0 Σ f(s)e ns ds = a n, n 1. (1.6) This moment problem turns out to be critical since it concerns the family of real exponential functions {e λns } n 1 with λ n = n, in which the usual summability condition on the inverses of the exponents, n 1 1 λ n <, does not hold. It was proved that, if the sequence (a n ) n 1 has the property that, for any δ > 0, there exists C δ > 0 such that a n C δ e δn, n 1, (1.7)

5 1.4. OPEN PROBLEMS 5 problem (1.6) has a solution if and only if a n = 0 for all n 1. Since (a n ) n 1 depend on the Fourier coeficients of the initial data, the following negative controllability result for the one-dimensional systems is obtained: Theorem 2 When Ω is the half line, there is no non-trivial initial datum u 0 belonging to a negative Sobolev space which is null-controllable in finite time with L 2 boundary controls. This negative result was complemented by showing that there exist initial data with exponentially growing Fourier coefficients for which nullcontrollability holds in finite time with L 2 controls. We mention that in [10] and [11] we are dealing with solutions defined in the sense of transposition, and therefore, the solutions in [6] and [9] that grow and oscillate very fast at infinity are excluded. 1.4 Open problems As we have already mentioned, the null-controllability property of (1.1) when Ω is unbounded and different from a half-line or half-space is still open. The approach based on the use of the similarity variables may still be used in general conical domains. But, due to the lack of orthogonality of the traces of the normal derivatives of the eigenfunctions, the corresponding moment problem is more complex and remains to be solved. When Ω is a general unbounded domain, the similarity transformation does not seem to be of any help since the domain one gets after transformation depends on time. Therefore, a completely different approach seems to be needed when Ω is not conical. However, one may still expect a bad behaviour of the null-control problem. Indeed, assume for instance that Ω contains R n +. If one is able to control to zero in Ω an initial data u 0 by means of a boundary control acting on Ω (0, T ), then, by restriction, one is able to control the initial data u 0 R n + with the control being the restriction of the solution in the larger domain Ω (0, T ) to R n 1 (0, T ). A careful development of this argument and of the result it may lead to remains to be done. Acknowledgements: The first author was partially supported by Grant PB of DGES (Spain) and Grant A3/2002 of CNCSIS (Romania). The second author was partially supported by Grant PB of DGES (Spain) and the TMR network of the EU Homogenization and Multiple Scales (HMS2000).

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7 Bibliography [1] V. Cabanillas, S. de Menezes and E. Zuazua, Null controllability in unbounded domains for the semilinear heat equation with nonlinearities involving gradient terms, J. Optim. Theory Applications, 110 (2) (2001), [2] M. Escobedo and O. Kavian, Variational problems related to self-similar solutions of the heat equation, Nonlinear Anal. TMA, 11 (1987), [3] A. Fursikov and O. Yu. Imanuvilov, Controllability of evolution equations, Lecture Notes Series #34, Research Institute of Mathematics, Global Analysis Research Center, Seoul National University, [4] J.M. Ghidaglia, Some backward uniqueness results, Nonlinear Anal. TMA, 10 (1986), [5] O. Yu. Imanuvilov and M. Yamamoto, Carleman estimate for a parabolic equation in Sobolev spaces of negative order and its applications, in Control of Nonlinear Distributed Parameter Systems, G. Chen et al. eds., Marcel-Dekker, 2000, pp [6] B.F. Jones, Jr., A fundamental solution of the heat equation which is supported in a strip, J. Math. Anal. Appl., 60 (1977), [7] G. Lebeau and L. Robbiano, Contrôle exact de l équation de la chaleur, Comm. P.D.E., 20 (1995), [8] J.L. Lions and E. Malgrange, Sur l unicité rétrograde dans les problèmes mixtes paraboliques, Math. Scan., 8 (1960), [9] W. Littman, Boundary control theory for hyperbolic and parabolic partial differential equations with constant coefficients, Annali Scuola Norm. Sup. Pisa, Serie IV, 3 (1978), [10] S. Micu and E. Zuazua, On the lack of null-controllability of the heat equation on the half-line, Trans. AMS, 353 (2001), [11] S. Micu and E. Zuazua, On the lack of null-controllability of the heat equation on the half-space, Portugalia Matematica, 58 (2001),

8 8 BIBLIOGRAPHY [12] D.L. Russell, Controllability and stabilizability theory for linear partial differential equations. Recent progress and open questions, SIAM Rev., 20 (1978), [13] L. de Teresa and E. Zuazua, Approximate controllability of the heat equation in unbounded domains, Nonlinear Anal. TMA, 37 (1999),