The study of the stabilizability of the coupled wave equations under various end conditions

Abstract

and are satisfied. M = sup {Il(iu-A)-'\\ I w E R) <ca INTRODUCl'ION(1.1). !" section 4, we show the semigroup generated by the system operator A along with the boundary condition (1.3) satisfies the condition (1.9) of the theorem The of is to bvestigak the st&,iliation of a of which kads to the stabilibility of the system. We obeerve in section 5 that the point s p n t r u m of the system are approaching to the i m a e a r y axis sspptotically d we apply boundary conditioea (1.4-1.5). 1 Eere & @. k1.2 are positive damping coefficients, t E (0,ca) is the time variable and I is tb: apace coordinate along the strings in their equilibrium positions. (For a better observation d e r to more explanation in dynamical caae in section 3.2 of[ll]). We denok the detYection of the string from the equilibrium lines by u(1.1) and u(z,t), reapectively. The coefficient c is the wave speed and a is the spring constant. The natural energy of the above system is given byHaving considered our objectives, we first tried to use multiplier methoda introduced by [2] or [q, and then direct method by [7]. After tremendous work we r e d d that these methoda are not efficient enough to progress our work. However, when we upused to a theorem by Euang [4] which is stated below, we were able to achieve our goal. Euang's TheoremThe system operator generated from (1.1) will he r o oi (2.4) The domain 9 ( A ) is defined with respect to each boundary condition. For instance the domain for operator A correaponding to the system (1.1 -1.3) is Suppose that S(t) is a Co-semigroup generated by A in a Hilbert space satisfying where &=H' x the problem %e state th% following Lemma Lemma 2.1 A is infinifuirnol generator of a contraction Co-8cmigroup. Prooc.disaipativenesa and d m d i t y of A. x 8 ' x H1 is de-in H. To accomplish the wdl-posednae of Ils(t)ll -< 7 t 2 0 7ER+. (1.7) Then S(t) is exponentidly shble iff According to the Lummcr-Phillips theorem [9] it is enough to show the (1.8) {iw I w E R) C p(A), the resolvent set of A,

Ingham type approach for uniform observability inequality of the semi-discrete coupled wave equations

“…The functions u(x, t) and v(x, t) are the displacements of two vibrating strings, measured from their equilibrium positions and the coupling is given by coupling terms αv and αu. This system is motivated by analogous problem in ordinary differential equations for coupled harmonic oscillators (see [13]) and it is well-posed in the energy space H := H 1 0 (0, L)×L 2 (0, L) 2 . That is to say, denoting U = (u, u t , v, v t )…”

Optimal Polynomial Decay to Coupled Wave Equations and Its Numerical Properties

“…The exponential stability of coupled system of wave equations has been studied by Najafi et al [1] in the case of two linear boundary feedback and by Komornik Rao [2] in the case of two nonlinear boundary feedback.…”