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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 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 direction
We revisit the paper [Mel86] by R. Melrose, providing a full proof of the main theorem on propagation of singularities for subelliptic wave equations, and linking this result with sub-Riemannian geometry. This result asserts that singularities of sub
We present two comparison principles for viscosity sub- and supersolutions of Monge-Ampere-type equations associated to a family of vector fields. In particular, we obtain the uniqueness of a viscosity solution to the Dirichlet problem for the equati
The exact distributed controllability of the semilinear wave equation $partial_{tt}y-Delta y + g(y)=f ,1_{omega}$ posed over multi-dimensional and bounded domains, assuming that $gin C^1(mathbb{R})$ satisfies the growth condition $limsup_{rto infty}
In this article, we present a novel Carleman estimate for ultrahyperbolic operators, in $ R^m_t times R^n_x $. Then, we use a special case of this estimate to obtain improved observability results for wave equations with time-dependent lower order te