Exponential stability for a coupled system of damped–undamped plate equations

Abstract

We consider the transmission problem for a coupled system of undamped and structurally damped plate equations in two sufficiently smooth and bounded subdomains. It is shown that, independently of the size of the damped part, the damping is strong enough to produce uniform exponential decay of the energy of the coupled system.

Stability of the transmission plate equation with a delay term in the moment feedback control

“…Step 3. We apply standard energy method (multipliers h(x).∇u(x, t) and u(x, t)) to problem (1)- (7). We obtain (see Appendix A)…”

“…Stability problems for the undelayed transmission plate equation have been considered by several authors 6‐9 . In particular, Ammari and Vodev 6 considered the system ()–() with α 2 =0 and have shown that if then the solution decays exponentially to zero in the energy space V × L 2 (Ω), where with norm The purpose of this paper is to investigate the stability of problem ()–() in the case where both α 1 and α 2 are different from zero.…”

“…Stability problems for the undelayed transmission plate equation have been considered by several authors. [6][7][8][9] In particular, Ammari and Vodev 6 considered the system (1)-( 6) with 2 = 0 and have shown that if…”

“…The purpose of this paper is to investigate the stability of problem ( 1)- (7) in the case where both 1 and 2 are different from zero. To this end, assume as in Nicaise and Pignotti 4 that…”

Regularity and asymptotic behavior for a damped plate–membrane transmission problem

“…However, by this method we do not obtain optimal polynomial rates. The proof of higher regularity (Section 4) uses arguments similar to [11] where damped plate / undamped plate transmission problems were investigated. In particular, we apply the classical theory of parameter-dependent boundary value problems (see [2]) to obtain sufficiently good estimates in the damped part.…”