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Register a new userExact observability properties of subelliptic wave and Schr{o}dinger equations
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Authors
Cyril Letrouit
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In this survey paper, we report on recent works concerning exact observability (and, by duality, exact controllability) properties of subelliptic wave and Schr{o}dinger-type equations. These results illustrate the slowdown of propagation in directions transverse to the horizontal distribution. The proofs combine sub-Riemannian geometry, semi-classical analysis, spectral theory and non-commutative harmonic analysis.
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We give necessary and sufficient conditions for the controllability of a Schrodinger equation involving the sub-Laplacian of a nilmanifold obtained by taking the quotient of a group of Heisenberg type by one of its discrete sub-groups.This class of nilpotent Lie groups is a major example of stratified Lie groups of step 2. The sub-Laplacian involved in these Schrodinger equations is subelliptic, and, contrarily to what happens for the usual elliptic Schrodinger equation for example on flat tori or on negatively curved manifolds, there exists a minimal time of controllability. The main tools used in the proofs are (operator-valued) semi-classical measures constructed by use of representation theory and a notion of semi-classical wave packets that we introduce here in the context of groups of Heisenberg type.
It is well-known that observability (and, by duality, controllability) of the elliptic wave equation, i.e., with a Riemannian Laplacian, in time $T_0$ is almost equivalent to the Geometric Control Condition (GCC), which stipulates that any geodesic ray meets the control set within time $T_0$. We show that in the subelliptic setting, GCC is never verified, and that subelliptic wave equations are never observable in finite time. More precisely, given any subelliptic Laplacian $Delta=-sum_{i=1}^m X_i^*X_i$ on a manifold $M$ such that $text{Lie}(X_1,ldots,X_m)=TM$ but $text{Span}(X_1,ldots,X_m)subsetneq TM$, we show that for any $T_0>0$ and any measurable subset $omegasubset M$ such that $Mbackslash omega$ has nonempty interior, the wave equation with subelliptic Laplacian $Delta$ is not observable on $omega$ in time $T_0$. The proof is based on the construction of sequences of solutions of the wave equation concentrating on spiraling geodesics (for the associated sub-Riemannian distance) spending a long time in $Mbackslash omega$. As a counterpart, we prove a positive result of observability for the wave equation in the Heisenberg group, where the observation set is a well-chosen part of the phase space.
In this paper, we study the existence and instability of standing waves with a prescribed $L^2$-norm for the fractional Schr{o}dinger equation begin{equation} ipartial_{t}psi=(-Delta)^{s}psi-f(psi), qquad (0.1)end{equation} where $0<s<1$, $f(psi)=|psi|^{p}psi$ with $frac{4s}{N}<p<frac{4s}{N-2s}$ or $f(psi)=(|x|^{-gamma}ast|psi|^2)psi$ with $2s<gamma<min{N,4s}$. To this end, we look for normalized solutions of the associated stationary equation begin{equation} (-Delta)^s u+omega u-f(u)=0. qquad (0.2) end{equation} Firstly, by constructing a suitable submanifold of a $L^2$-sphere, we prove the existence of a normalized solution for (0.2) with least energy in the $L^2$-sphere, which corresponds to a normalized ground state standing wave of(0.1). Then, we show that each normalized ground state of (0.2) coincides a ground state of (0.2) in the usual sense. Finally, we obtain the sharp threshold of global existence and blow-up for (0.1). Moreover, we can use this sharp threshold to show that all normalized ground state standing waves are strongly unstable by blow-up.
In this paper, we consider an optimal bilinear control problem for the nonlinear Schr{o}dinger equations with singular potentials. We show well-posedness of the problem and existence of an optimal control. In addition, the first order optimality system is rigorously derived. Our results generalize the ones in cite{Sp} in several aspects.
In this paper, we study an inverse coefficients problem for two coupled Schr{o}dinger equations with an observation of one component of the solution. The observation is done in a nonempty open subset of the domain where the equations hold. A logarithmic type stability result is obtained. The main method is based on the Carleman estimate for coupled Schr{o}dinger equations and coupled heatn equations, and the Fourier-Bros-Iagolnitzer transform.
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