Geometric bounds on the growth rate of null-controllability cost for the heat equation in small time

Miller, Luc

HAL, hal-00002088 / Harvested from HAL

Résumé

Given a control region $\Omega$ on a compact Riemannian manifold $M$, we consider the heat equation with a source term $g$ localized in $\Omega$. It is known that any initial data in $L^{2}(M)$ can be steered to $0$ in an arbitrarily small time $T$ by applying a suitable control $g$ in $L^{2}([0,T]\times\Omega)$, and, as $T$ tends to $0$, the norm of $g$ grows like $\exp(C/T)$ times the norm of the data. We investigate how $C$ depends on the geometry of $\Omega$. %% 72 words We prove $C\geq d^{2}/4$ where $d$ is the largest distance of a point in $M$ from $\Omega$. When $M$ is a segment of length $L$ controlled at one end, we prove $C\leq \alpha_{*}L^{2}$ for some $\alpha_{*}<2$. Moreover, this bound implies $C\leq\alpha_{*}L_{\Omega}^{2}$ where $L_{\Omega}$ is the length of the longest generalized geodesic in $M$ which does not intersect $\Omega$. The {\em control transmutation method} used in proving this last result is of a broader interest.

Détail

Publié le : 2004-07-05
Classification:  multipliers,  entire functions,  transmutation,  Heat equation,  control cost,  small time asymptotics,  observabillity,  null-controllability,  MSC 35B37, 58J35.,  [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP],  [MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC]