Study of exponential stability of coupled wave systems via distributed stabilizer

Abstract

Abstract. Stabilization of the system of wave equations coupled in parallel with coupling distributed springs and viscous dampers are under investigation due to different boundary conditions and wave propagation speeds. Numerical computations are attempted to confirm the theoretical results.2000 Mathematics Subject Classification. 58J45, 93D15, 93D20, 37D99, 35L05. 1.Introduction. Many problems in structural dynamics deal with stabilizing the elastic energy of partial differential equations by boundary or internal energy dissipative controllers for wave equations or the Euler-Bernoulli beam equation. Exponential stability is a very desirable property for such elastic systems. The energy multiplier method [2,6] has been successfully applied to reach to this objective for various partial differential equations and boundary conditions. Stabilization properties of serially connected vibrating strings or beams can be found in several papers [4,5]. There, uniform stabilization can be achieved if we employ dissipative boundary condition at one end. If otherwise, one damper is located at the mid-span joint of two vibrating strings coupled in series, the uniform stabilization property holds if c 1 /c 2 (wave speeds) has certain rational values. Stabilization properties of parallel connected vibrating strings were investigated under various end conditions by [9]. What comes new in this work is, firstly, dealing with the system of wave equations coupled in parallel with distributed viscous damping and springs (suspension system), and secondly, the rate of convergence of the solution when this system goes under the movement by an external disturbance (forcing function) or initial conditions. Having considered this, we are willing to furnish the best possible configuration that guarantees the uniform exponential stability due to different boundary conditions and wave speeds.Let Ω 1 = Ω 2 = Ω = (0, 1) be open sets in R. Also, let ∂Ω 1 ,∂Ω 2 be the boundaries of Ω 1 and Ω 2 , respectively. Throughout, (·) = d()/dt, ( ) = d()/dx, and ∂

Boundary controllability for the quasi-linear wave equations coupled in parallel

“…Controllability for coupled linear wave equations has been investigated by many authors, we refer the readers to [15][16][17][18][19][20], etc. There are few results in higher dimensions for nonlinear cases.…”

Exact boundary controllability of coupled hyperbolic equations

“…These models also arise in a very important applied area-wave propagation in porous media (Boit, 1962;Holger et al, 2010). Stabilizability of coupled wave equations and its applications are discussed by Najafi et al (1997) and Najafi (2001).…”

“…It is known (see, e.g., Avdonin and Ivanov, 1995;2001;Avdonin and Moran, 2001a, Section II.4.5) that the family…”

Energy Decay for the Solutions of a Coupled Wave System

“…A wave system with locally distributed damping coupled in parallel have also been been considered in [8] for the space domain Ω = (0, 1) ⊂ R with friction coefficient α constant. Our result extend the one in [8] in the sense that we allow higher dimension spaces and variable friction coefficient.…”

“…Stability for coupled wave system has been considered in [6], [1], [2] and [8] among others. In [6], both wave equations are damped on the boundary and the coupling is effected by compact operator.…”