Controllability cost of conservative systems: resolvent condition and transmutation

This article concerns the exact controllability of unitary groups on Hilbert spaces with unbounded control operator. It provides a necessary and sufficient condition not involving time which blends a resolvent estimate and an observability inequality. By the transmutation of controls in some time L for the corresponding second-order conservative system, it is proved that the cost of controls in time T for the unitary group grows at most like exp(αL²/T) as T tends to 0. In the application to the cost of fast controls for the Schrödinger equation, L is the length of the longest ray of geometric optics which does not intersect the control region. This article also provides observability resolvent estimates implying fast smoothing effect controllability at low cost, and underscores that the controllability cost of a system is not changed by taking its tensor product with a conservative system.     

A spectral approach for the exact observability of infinite-dimensional systems with skew-adjoint generator

Let A be a possibly unbounded skew-adjoint operator on the Hilbert space X with compact resolvent. Let C be a bounded operator from D(A) to another Hilbert space Y. We consider the system governed by the state equation with the output y(t)=Cz(t). We characterize the exact observability of this system only in terms of C and of the spectral elements of the operator A. The starting point in the proof of this result is a Hautus-type test, recently obtained in Burq and Zworski (J. Amer. Soc. 17 (2004) 443-471) and Miller (J. Funct. Anal. 218 (2) (2005) 425-444). We then apply this result to various systems governed by partial differential equations with observation on the boundary of the domain. The Schrödinger equation, the Bernoulli-Euler plate equation and the wave equation in a square are considered. For the plate and Schrödinger equations, the main novelty brought in by our results is that we prove the exact boundary observability for an arbitrarily small observed part of the boundary. This is done by combining our spectral observability test to a theorem of Beurling on nonharmonic Fourier series and to a new number theoretic result on shifted squares.     

New blow-up rates for fast controls of Schrödinger and heat equations

We consider the null-controllability problem for the Schrödinger and heat equations with boundary control. We concentrate on short-time, or fast, controls. We improve recent estimates (see [L. Miller, Geometric bounds on the growth rate of null-controllability cost for the heat equation in small time, J. Differential Equations 204 (2004) 202-226; L. Miller, How violent are fast controls for Schrödinger and plate vibrations?, Arch. Ration. Mech. Anal. 172 (2004) 429-456; L. Miller, Controllability cost of conservative systems: Resolvent condition and transmutation, J. Funct. Anal. 218 (2005) 425-444; L. Miller, The control transmutation method and the cost of fast controls, SIAM J. Control Optim. 45 (2006) 762-772]) on the norm of the operator associating to any initial state the minimal norm control driving the system to zero. Our main results concern the Schrödinger and heat equations in one space dimension. They yield new estimates concerning window problems for series of exponentials as described in [T.I. Seidman, The coefficient map for certain exponential sums, Nederl. Akad. Wetensch. Indag. Math. 48 (1986) 463-478] and in [T.I. Seidman, S.A. Avdonin, S.A. Ivanov, The “window problem” for series of complex exponentials, J. Fourier Anal. Appl. 6 (2000) 233-254]. These results are used, following [L. Miller, The control transmutation method and the cost of fast controls, SIAM J. Control Optim. 45 (2006) 762-772], to deal with the case of several space dimensions.     

Criticality of viscous Hamilton–Jacobi equations and stochastic ergodic control

This paper deals with nonlinear additive eigenvalue problems for viscous Hamilton–Jacobi equations which appear in stochastic ergodic control. Certain qualitative properties of principal eigenvalues and associated eigenfunctions are studied. Such analysis plays a key role in studying the recurrence and transience of feedback diffusions for the corresponding stochastic control problems. Our results can be regarded as a nonlinear extension of the criticality theory for Schrödinger operators with decaying potentials.     

Observability estimate for stochastic Schrödinger equations

In this Note, we present an observability estimate for stochastic Schrödinger equations with nonsmooth lower order terms. The desired inequality is derived by a global Carleman estimate which is based on a fundamental weighted identity for stochastic Schrödinger-like operator. As an interesting byproduct, starting from this identity, one can deduce all the known controllability/observability results for several stochastic and deterministic partial differential equations that are derived before via Carleman estimate in the literature.     

Boundary controllability for the semilinear Schrödinger equations on Riemannian manifolds

We study the boundary exact controllability for the semilinear Schrödinger equation defined on an open, bounded, connected set Ω of a complete, n-dimensional, Riemannian manifold M with metric g. We prove the locally exact controllability around the equilibria under some checkable geometrical conditions. Our results show that exact controllability is geometrical characters of a Riemannian metric, given by the coefficients and equilibria of the semilinear Schrödinger equation. We then establish the globally exact controllability in such a way that the state of the semilinear Schrödinger equation moves from an equilibrium in one location to an equilibrium in another location.     

Some results on the controllability of forward stochastic heat equations with control on the drift

In this paper, we establish the null/approximate controllability for forward stochastic heat equations with control on the drift. The null controllability is obtained by a time iteration method and an observability estimate on partial sums of eigenfunctions for elliptic operators. As a consequence of the null controllability, we obtain the observability estimate for backward stochastic heat equations, which leads to a unique continuation property for backward stochastic heat equations, and hence the desired approximate controllability for forward stochastic heat equations. It deserves to point out that one needs to introduce a little stronger assumption on the controller for the approximate controllability of forward stochastic heat equations than that for the null controllability. This is a new phenomenon arising in the study of the controllability problem for stochastic heat equations.     

Null controllability for the parabolic equation with a complex principal part

The paper is devoted to a study of the null controllability for the semilinear parabolic equation with a complex principal part. For this purpose, we establish a key weighted identity for partial differential operators (with real functions α and β), by which we develop a universal approach, based on global Carleman estimate, to deduce not only the desired explicit observability estimate for the linearized complex Ginzburg-Landau equation, but also all the known controllability/observability results for the parabolic, hyperbolic, Schrödinger and plate equations that are derived via Carleman estimates.     

Simultaneous observability and stabilization of some uncoupled wave equations

We consider uncoupled wave equations with different speed of propagation in a bounded domain. Using a combination of the Bardos–Lebeau–Rauch observability result for a single wave equation and a new unique continuation result for uncoupled wave equations, we prove an observability estimate for that system. Applying Lions? Hilbert uniqueness method (HUM), one may derive simultaneous exact controllability results for the uncoupled system; the controls being locally distributed, with their supports satisfying the geometric control condition of Bardos, Lebeau and Rauch. Afterwards, we discuss the related simultaneous stabilization problem; this latter problem is solved by a combination of the new observability inequality, and a result of Haraux establishing an equivalence between observability and stabilization for second order evolution equations with bounded damping operators. Our observability and stabilization results generalize to higher space dimensions some earlier results of Haraux established in the one-dimensional setting.     

Dispersive blow-up for nonlinear Schrödinger equations revisited

The possibility of finite-time, dispersive blow-up for nonlinear equations of Schrödinger type is revisited. This mathematical phenomena is one of the conceivable explanations for oceanic and optical rogue waves. In dimension one, the fact that dispersive blow up does occur for nonlinear Schrödinger equations already appears in [9]. In the present work, the existing results are extended in several ways. In one direction, the theory is broadened to include the Davey–Stewartson and Gross–Pitaevskii equations. In another, dispersive blow up is shown to obtain for nonlinear Schrödinger equations in spatial dimensions larger than one and for more general power-law nonlinearities. As a by-product of our analysis, a sharp global smoothing estimate for the integral term appearing in Duhamel's formula is obtained.     

Ideas in the theory of random media

These lectures discuss the ideas of localization, intermittency, and random fluctuations in the theory of random media. These ideas are compared and contrasted with the older approach based on averaging. Within this framework, the topics discussed include: Anderson localization, turbulent diffusion and flows, periodic Schrödinger operators and averaging theory, longwave oscillations of elastic random media, stochastic differential equations, the spectral theory of Hamiltonians with (an infinite sequence of) wells, random Schrödinger operators, electrons in a random homogeneous field, influence of localization effects on the propagation of elastic waves, the Lyapunov spectrum (Lyapunov exponents), the Furstenberg and Oseledec theorems for ann-tuple of identically distributed unimodular matrices and their relation with the spectral theory of random Schrödinger or string operators, Rossby waves, averaging on random Schrödinger operators, percolation mechanisms, the moments method in the theory of sequences of random variables, the evolution of a magnetic field in the turbulent flow of a conducting fluid or plasma (the so-called kinematical dynamo problem), heat transmission in a randomly flowing fluid.     

Observability Inequalities by Internal Observation and Their Applications

By means of a Carleman estimate, we obtain several observability inequalities by internal observation for hyperbolic equations with lower-order terms. Also, we apply our observability results to an exact controllability problem and an inverse wave source problem.     

Connections between three-dimensional inverse scattering and linear least-squares estimation of random fields

The three-dimensional Schrödinger equation inverse scattering problem with a nonspherically-symmetric potential is related to the filtering problem of computing the linear leastsquares estimate of the three-dimensional random field on the surface of a sphere from noisy observations inside the sphere. The relation consists of associating an estimation problem with the inverse scattering problem, and vice-versa. This association allows equations and quantities for one problem to be given interpretations in terms of the other problem. A new fast algorithm is obtained for the estimation of random fields using this association. The present work is an extension of the connections between estimation and inverse scattering already known to exist for stationary random processes and one-dimensional scattering potentials, and isotropic random fields and radial scattering protentials.     

Controllability of a quantum particle in a moving potential well

We consider a nonrelativistic charged particle in a 1D moving potential well. This quantum system is subject to a control, which is the acceleration of the well. It is represented by a wave function solution of a Schrödinger equation, the position of the well together with its velocity. We prove the following controllability result for this bilinear control system: given ψ₀ close enough to an eigenstate and ψ_f close enough to another eigenstate, the wave function can be moved exactly from ψ₀ to ψ_f in finite time. Moreover, we can control the position and the velocity of the well. Our proof uses moment theory, a Nash-Moser implicit function theorem, the return method and expansion to the second order.     

Exact Internal Controllability of Maxwell's Equations

In this paper we obtain two exact internal controllability results of Maxwell's equations in a general region by using multiplier techniques. The first one is exact controllability in a short time, in which we obtain the ``optimal" (observability) estimates when the location and the shape of the controller is fixed. What happens if we allow the controller to change? Under some conditions, we show that by doing that the system can be exactly controllable within any given time duration, which is our second exact controllability result. Accepted 30 September 1998     

Localizing estimates of the support of solutions of some nonlinear Schrödinger equations – The stationary case

The main goal of this paper is to study the nature of the support of the solution of suitable nonlinear Schrödinger equations, mainly the compactness of the support and its spatial localization. This question touches the very foundations underlying the derivation of the Schrödinger equation, since it is well-known a solution of a linear Schrödinger equation perturbed by a regular potential never vanishes on a set of positive measure. A fact, which reflects the impossibility of locating the particle. Here we shall prove that if the perturbation involves suitable singular nonlinear terms then the support of the solution is a compact set, and so any estimate on its spatial localization implies very rich information on places not accessible by the particle. Our results are obtained by the application of certain energy methods which connect the compactness of the support with the local vanishing of a suitable “energy function” which satisfies a nonlinear differential inequality with an exponent less than one. The results improve and extend a previous short presentation by the authors published in 2006.     

Moment problem formulation of the Schrödinger equation

We show that it is possible to project out in an exact manner the lowest eigenstate of Schrödinger equations. Taking into account the nodeless property of the lowest eigenstate one can replace the full Schrödinger equation by a moment problem whose measure is the eigenstate itself. The infinite set of positivity inequalities linked to this moment problem provides a framework which allows to compute sequences of upper and lower bounds to the unknown eigenvalue and eigenfunction.The effective computation is based on deep convexity properties embedded in the set of hierarchical inequalities associated to this moment problem. The convexity allows to get the bounds through linear programming. We illustrate the method with simple one dimensional problems.Laboratoire de la Direction des Sciences de la Matière du Commissariat à l'Energie Atomique.     

Null Controllability for Some Systems of Two Backward Stochastic Heat Equations with One Control Force

The authors establish the null controllability for some systems coupled by two backward stochastic heat equations. The desired controllability result is obtained by means of proving a suitable observability estimate for the dual system of the controlled system.     

On the observability of time-discrete conservative linear systems

We consider various time discretization schemes of abstract conservative evolution equations of the form , where A is a skew-adjoint operator. We analyze the problem of observability through an operator B. More precisely, we assume that the pair (A,B) is exactly observable for the continuous model, and we derive uniform observability inequalities for suitable time-discretization schemes within the class of conveniently filtered initial data. The method we use is mainly based on the resolvent estimate given by Burq and Zworski in [N. Burq, M. Zworski, Geometric control in the presence of a black box, J. Amer. Math. Soc. 17(2) (2004) 443-471 (electronic)]. We present some applications of our results to time-discrete schemes for wave, Schrödinger and KdV equations and fully discrete approximation schemes for wave equations.     

Boundary-value problem for the nonlinear Schrödinger equation

A mixed boundary-value problem for the nonlinear Schrödinger equation and its generalization is studied by the method used for the inverse scattering problem. A connection is established between conservation laws and boundary conditions in integrable boundary-value problems for higher nonlinear Schrödinger equations. It is shown that the generalized boundary-value problem requires a joint consideration of regular and singular solutions for the nonlinear Schrödinger equation with repulsion.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 169, pp. 151–165, 1988.