Geometric control in the presence of a black box. (English) Zbl 1050.35058

Burq, Nicolas; Zworski, Maciej

J. Am. Math. Soc. 17, No. 2, 443-471 (2004).

The purpose of this paper is to show how ideas coming from scattering theory (resolvent estimates) lead to results in control theory and to some closely related eigenfunction estimates. The point of view is the following: a black box in a confined system is replaced by a scattering problem. That permits having isolated dynamical phenomena (such as only one closed orbit) impossible in confined systems. It also permits using some finer results of scattering theory directly. We present two typical applications.
1. In geometric control theory for the Schrödinger equation we are concerned with the following mixed problem: \[ (i\partial_t+ \Delta)u= 0\quad \text{in }\Omega,\qquad u|_{[0,T]\times\Omega}= g^1_{[0,T]\times\Gamma},\;u|_{t=0}= u_0,\tag{1} \] where \(\Omega\) is an open set of \(\mathbb{R}^d\), \(\partial\Omega\) is its boundary and \(\Gamma\) is an open subset of \(\partial\Omega\). The question is to determine a (large) class of functions \(u_0\) for which there exists a control \(g\) such that \(u|_{t=T}= 0\). The following result was established by N. Burg [Mémoire de la S.M.F., 55 (1993):
Theorem 1. Consider \(\Theta= \bigcup_{1\leq j\leq N}\Theta_j\subset\mathbb{R}^d\), a union of mutually disjoint closed sets with strictly convex smooth boundaries and satisfying some geometric assumptions. Let \(\widetilde\Omega\) be a bounded domain with a smooth boundary and containing \(\text{convhull}(\Theta)\). Denote \(\Omega= \widetilde\Omega\setminus\Theta\) and \(\Gamma= \partial\widetilde\Omega\). Then for any \(T\), \(\varepsilon> 0\) and any \(u_0\in H^{1+\varepsilon}_0(\Omega)\) there exists \(g\in L^2([0, T]\times \Gamma)\) such that in (1) we have \(u|_{t> T}= 0\).
The authors show how Theorem 1 can be obtained directly from estimates on the resolvent of the Laplace operator, which in turn can be deduced from semiclassical microlocal analysis or from known results in scattering theory.
2. The second application generalizes a result of Y. Colin de Verdière and B. Parisse [Commun. Partial Differ. Equations 19, 9–10, 1535–1563 (1994; Zbl 0819.35116)], who considered a special case of an isolated trajectory lying on a segment of a constant negative curvature cylinder in dimension two:
Theorem 2: Suppose that \((X,g)\) is a compact Riemannian manifold with a (possibly empty) boundary and \(\gamma\subset X\) is a closed real hyperbolic geodesic not intersecting the boundary. If \(\chi\in{\mathcal C}^\infty(X,[0,1])\) is supported in a sufficiently small neighbourhood of \(\gamma\), then there exists a constant \(C= C(\gamma)\) such that for any eigenfunction \(u\) of the Laplacian \(\Delta_g\) with Dirichlet or Neumann boundary conditions, we have \[ C\int_X| u(x)|^2(1- \chi)(x)\,d\,\text{vol}_g\geq {1\over\log\lambda} \int_X| u(x)|^2\,d\,\text{vol}_g,\;-\Delta_g u=\lambda u.\tag{2} \] The proof is based on putting the closed hyperbolic orbit into a microlocal black box, where that orbit becomes the only trapped orbit in a scattering problem. One can than use scattering estimates to obtain (2).

Reviewer: Viorel Iftimie (Bucureşti)

MSC:

35P25	Scattering theory for PDEs
35B37	PDE in connection with control problems (MSC2000)
35Q40	PDEs in connection with quantum mechanics
35P20	Asymptotic distributions of eigenvalues in context of PDEs
81Q20	Semiclassical techniques, including WKB and Maslov methods applied to problems in quantum theory

Keywords:

pseudodifferential operators; microlocal analysis; scattering theory; Schrödinger equation; compact Riemannian manifold; hyperbolic orbit

Citations:

Zbl 0819.35116

References:

[1]	A. Bäcker, R. Schubert, and P. Stifter. On the number of bouncing ball modes in billiards. J. Phys. A: Math. Gen. 30:6783-6795, 1997. · Zbl 0925.81030
[2]	Claude Bardos, Gilles Lebeau, and Jeffrey Rauch, Sharp sufficient conditions for the observation, control, and stabilization of waves from the boundary, SIAM J. Control Optim. 30 (1992), no. 5, 1024 – 1065. · Zbl 0786.93009 · doi:10.1137/0330055
[3]	J.-F. Bony and L. Michel. Microlocalization of resonant states and estimates of the residue of scattering amplitude, Comm. Math. Phys., to appear. · Zbl 1060.35016
[4]	N. Burq. Control for Schrodinger equations on product manifolds. Unpublished, 1992.
[5]	Nicolas Burq, Contrôle de l’équation des plaques en présence d’obstacles strictement convexes, Mém. Soc. Math. France (N.S.) 55 (1993), 126 (French, with English and French summaries). · Zbl 0930.93007
[6]	Nicolas Burq, Semi-classical estimates for the resolvent in nontrapping geometries, Int. Math. Res. Not. 5 (2002), 221 – 241. · Zbl 1161.81368 · doi:10.1155/S1073792802103059
[7]	N. Burq. Smoothing effect for Schrödinger boundary value problems. Preprint, 2002. · Zbl 1061.35024
[8]	Nicolas Burq and Patrick Gérard, Condition nécessaire et suffisante pour la contrôlabilité exacte des ondes, C. R. Acad. Sci. Paris Sér. I Math. 325 (1997), no. 7, 749 – 752 (French, with English and French summaries). · Zbl 0906.93008 · doi:10.1016/S0764-4442(97)80053-5
[9]	Nicolas Burq and Gilles Lebeau, Mesures de défaut de compacité, application au système de Lamé, Ann. Sci. École Norm. Sup. (4) 34 (2001), no. 6, 817 – 870 (French, with English and French summaries). · Zbl 1043.35009 · doi:10.1016/S0012-9593(01)01078-3
[10]	N. Burq and M. Zworski. Eigenfunctions for partially rectangular billiards. math.SP/0312098. · Zbl 1064.35120
[11]	P.A. Chinnery and V.F. Humphrey. Experimental visualization of acoustic resonances within a stadium-shaped cavity. Physical Review E, 53, 1996, 272-276.
[12]	T. Christiansen and M. Zworski, Resonance wave expansions: two hyperbolic examples, Comm. Math. Phys. 212 (2000), no. 2, 323 – 336. · Zbl 0955.58024 · doi:10.1007/s002200000211
[13]	Yves Colin de Verdière and Bernard Parisse, Équilibre instable en régime semi-classique. I. Concentration microlocale, Comm. Partial Differential Equations 19 (1994), no. 9-10, 1535 – 1563 (French, with French summary). · Zbl 0819.35116 · doi:10.1080/03605309408821063
[14]	N. Dencker, J. Sjöstrand, and M. Zworski. Pseudospectra of semiclassical (pseudo)differential operators. Comm. Pure Appl. Math. 57:384-415, 2004. · Zbl 1054.35035
[15]	Mouez Dimassi and Johannes Sjöstrand, Spectral asymptotics in the semi-classical limit, London Mathematical Society Lecture Note Series, vol. 268, Cambridge University Press, Cambridge, 1999. · Zbl 0926.35002
[16]	H. Donnelly. Quantum unique ergodicity. Proc. Amer. Math. Soc. 131, 2945-2951, 2003. · Zbl 1027.58024
[17]	C. Gérard, Asymptotique des pôles de la matrice de scattering pour deux obstacles strictement convexes, Mém. Soc. Math. France (N.S.) 31 (1988), 146 (French, with English summary). · Zbl 0654.35081
[18]	C. Gérard and J. Sjöstrand, Resonances en limite semiclassique et exposants de Lyapunov, Comm. Math. Phys. 116 (1988), no. 2, 193 – 213 (French, with English summary). · Zbl 0698.35118
[19]	Patrick Gérard and Éric Leichtnam, Ergodic properties of eigenfunctions for the Dirichlet problem, Duke Math. J. 71 (1993), no. 2, 559 – 607. · Zbl 0788.35103 · doi:10.1215/S0012-7094-93-07122-0
[20]	Laurent Guillopé, Sur la distribution des longueurs des géodésiques fermées d’une surface compacte à bord totalement géodésique, Duke Math. J. 53 (1986), no. 3, 827 – 848 (French). · Zbl 0611.53042 · doi:10.1215/S0012-7094-86-05345-7
[21]	J.-P. Françoise and V. Guillemin, On the period spectrum of a symplectic mapping, J. Funct. Anal. 100 (1991), no. 2, 317 – 358. · Zbl 0739.58020 · doi:10.1016/0022-1236(91)90114-K
[22]	A. Haraux, Séries lacunaires et contrôle semi-interne des vibrations d’une plaque rectangulaire, J. Math. Pures Appl. (9) 68 (1989), no. 4, 457 – 465 (1990) (French, with English summary). · Zbl 0685.93039
[23]	B. Helffer and J. Sjöstrand, Résonances en limite semi-classique, Mém. Soc. Math. France (N.S.) 24-25 (1986), iv+228 (French, with English summary). · Zbl 0631.35075
[24]	Lars Hörmander, The analysis of linear partial differential operators. III, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 274, Springer-Verlag, Berlin, 1985. Pseudodifferential operators. Lars Hörmander, The analysis of linear partial differential operators. IV, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 275, Springer-Verlag, Berlin, 1985. Fourier integral operators. Lars Hörmander, The analysis of linear partial differential operators. III, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 274, Springer-Verlag, Berlin, 1985. Pseudodifferential operators. Lars Hörmander, The analysis of linear partial differential operators. IV, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 275, Springer-Verlag, Berlin, 1985. Fourier integral operators.
[25]	A. Iantchenko, J. Sjöstrand, and M. Zworski, Birkhoff normal forms in semi-classical inverse problems, Math. Res. Lett. 9 (2002), no. 2-3, 337 – 362. · Zbl 1258.35208 · doi:10.4310/MRL.2002.v9.n3.a9
[26]	Mitsuru Ikawa, Decay of solutions of the wave equation in the exterior of several convex bodies, Ann. Inst. Fourier (Grenoble) 38 (1988), no. 2, 113 – 146 (English, with French summary). · Zbl 0636.35045
[27]	S. Jaffard, Contrôle interne exact des vibrations d’une plaque rectangulaire, Portugal. Math. 47 (1990), no. 4, 423 – 429 (French, with English summary). · Zbl 0718.49026
[28]	Jean-Pierre Kahane, Pseudo-périodicité et séries de Fourier lacunaires, Ann. Sci. École Norm. Sup. (3) 79 (1962), 93 – 150 (French). · Zbl 0105.28601
[29]	G. Lebeau, Contrôle de l’équation de Schrödinger, J. Math. Pures Appl. (9) 71 (1992), no. 3, 267 – 291 (French, with English summary). · Zbl 0838.35013
[30]	J.-L. Lions, Contrôlabilité exacte, perturbations et stabilisation de systèmes distribués. Tome 1, Recherches en Mathématiques Appliquées [Research in Applied Mathematics], vol. 8, Masson, Paris, 1988 (French). Contrôlabilité exacte. [Exact controllability]; With appendices by E. Zuazua, C. Bardos, G. Lebeau and J. Rauch. · Zbl 0653.93002
[31]	E. Lindenstrauss. Invariant measures and arithmetic quantum unique ergodicity, preprint, 2003.
[32]	R. B. Melrose and J. Sjöstrand, Singularities of boundary value problems. I, Comm. Pure Appl. Math. 31 (1978), no. 5, 593 – 617. , https://doi.org/10.1002/cpa.3160310504 R. B. Melrose and J. Sjöstrand, Singularities of boundary value problems. II, Comm. Pure Appl. Math. 35 (1982), no. 2, 129 – 168. · Zbl 0546.35083 · doi:10.1002/cpa.3160350202
[33]	L. Miller. How violent are fast controls for Schrödinger equation? preprint, 2003.
[34]	Peter Sarnak, Arithmetic quantum chaos, The Schur lectures (1992) (Tel Aviv), Israel Math. Conf. Proc., vol. 8, Bar-Ilan Univ., Ramat Gan, 1995, pp. 183 – 236. · Zbl 0831.58045
[35]	Johannes Sjöstrand, Geometric bounds on the density of resonances for semiclassical problems, Duke Math. J. 60 (1990), no. 1, 1 – 57. · Zbl 0702.35188 · doi:10.1215/S0012-7094-90-06001-6
[36]	J. Sjöstrand, A trace formula and review of some estimates for resonances, Microlocal analysis and spectral theory (Lucca, 1996) NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., vol. 490, Kluwer Acad. Publ., Dordrecht, 1997, pp. 377 – 437. · Zbl 0877.35090
[37]	Johannes Sjöstrand and Maciej Zworski, Complex scaling and the distribution of scattering poles, J. Amer. Math. Soc. 4 (1991), no. 4, 729 – 769. · Zbl 0752.35046
[38]	J. Sjöstrand and M. Zworski. Quantum monodromy and semiclassical trace formulæ. Journal d’Mathématiques Pures et Appl. 81:1-33, 2002. · Zbl 1038.58033
[39]	J.A.K. Suykens and J. Vandewalle . Nonlinear Modeling: advanced black-box techniques, Kluwer Academic Publishers Boston, June 1998.
[40]	Siu-Hung Tang and Maciej Zworski, From quasimodes to resonances, Math. Res. Lett. 5 (1998), no. 3, 261 – 272. · Zbl 0913.35101 · doi:10.4310/MRL.1998.v5.n3.a1
[41]	Jared Wunsch and Maciej Zworski, Distribution of resonances for asymptotically Euclidean manifolds, J. Differential Geom. 55 (2000), no. 1, 43 – 82. · Zbl 1030.58024
[42]	S. Zelditch. Quantum unique ergodicity. math-ph/0301035.
[43]	Steven Zelditch and Maciej Zworski, Ergodicity of eigenfunctions for ergodic billiards, Comm. Math. Phys. 175 (1996), no. 3, 673 – 682. · Zbl 0840.58048
[44]	J.-L. Lions, Contrôlabilité exacte, perturbations et stabilisation de systèmes distribués. Tome 1, Recherches en Mathématiques Appliquées [Research in Applied Mathematics], vol. 8, Masson, Paris, 1988 (French). Contrôlabilité exacte. [Exact controllability]; With appendices by E. Zuazua, C. Bardos, G. Lebeau and J. Rauch. · Zbl 0653.93002