Stabilization for the semilinear wave equation with geometric control\n condition

2012/05/11 by Romain Joly, Joly, Romain, Camille Laurent +1 · 2 citations
Engineering · Mathematics · #Stability and Controllability of Differential Equations #Advanced Mathematical Physics Problems #Numerical methods for differential equations

paper · pdf · doi:10.48550/arxiv.1205.2459

Abstract

In this article, we prove the exponential stabilization of the semilinear\nwave equation with a damping effective in a zone satisfying the geometric\ncontrol condition only. The nonlinearity is assumed to be subcritical,\ndefocusing and analytic. The main novelty compared to previous results, is the\nproof of a unique continuation result in large time for some undamped equation.\nThe idea is to use an asymptotic smoothing effect proved by Hale and Raugel in\nthe context of dynamical systems. Then, once the analyticity in time is proved,\nwe apply a unique continuation result with partial analyticity due to Robbiano,\nZuily, Tataru and H "ormander. Some other consequences are also given for the\ncontrollability and the existence of a compact attractor.\n

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