A localized nonstandard stabilizer for the Timoshenko beam
Article
Tebou, L. (2015). A localized nonstandard stabilizer for the Timoshenko beam
. COMPTES RENDUS MATHEMATIQUE, 353(3), 247-253. 10.1016/j.crma.2015.01.004
Tebou, L. (2015). A localized nonstandard stabilizer for the Timoshenko beam
. COMPTES RENDUS MATHEMATIQUE, 353(3), 247-253. 10.1016/j.crma.2015.01.004
The stabilization of the Timoshenko beam system with localized damping is examined. The damping involves the sum of the bending and shear angle velocities; this work generalizes an earlier result of Haraux, established for a system of ordinary wave equations, to the Timoshenko system. First, we show that strong stability holds if and only if the boundary of the support of the feedback control intersects that of the interval under consideration. Next, we use the frequency domain method combined with the multipliers technique to prove the exponential stability of the associated semigroup when the damping support is a neighborhood of one endpoint of the interval under consideration. When the speed of propagation of the wave generated by the bending and that of the wave generated by the shear angle are distinct, the proof is similar to what is known for two ordinary waves similarly damped. However, when the two speeds are equal, an important identity breaks down, and the proof is carried out by the introduction of an appropriate auxiliary equation whose solution plays a critical role in subsequent estimates.
