Optimal design of sensors for a damped wave equation
نویسندگان
چکیده
In this paper we model and solve the problem of shaping and placing in an optimal way sensors for a wave equation with constant damping in a bounded open connected subset Ω of IRn. Sensors are modeled by subdomains of Ω of a given measure L|Ω|, with 0 < L < 1. We prove that, if L is close enough to 1, then the optimal design problem has a unique solution, which is characterized by a finite number of low frequency modes. In particular the maximizing sequence built from spectral approximations is stationary.
منابع مشابه
Finding the Optimal Place of Sensors for a 3-D Damped Wave Equation by using Measure Approach
In this paper, we model and solve the problem of optimal shaping and placing to put sensors for a 3-D wave equation with constant damping in a bounded open connected subset of 3-dimensional space. The place of sensor is modeled by a subdomain of this region of a given measure. By using an approach based on the embedding process, first, the system is formulated in variational form;...
متن کاملDecay of Solutions of the Wave Equation with Localized Nonlinear Damping and Trapped Rays
We prove some decay estimates of the energy of the wave equation governed by localized nonlinear dissipations in a bounded domain in which trapped rays may occur. The approach is based on a comparison with the linear damped wave equation and an interpolation argument. Our result extends to the nonlinear damped wave equation the well-known optimal logarithmic decay rate for the linear damped wav...
متن کاملA Spatio-temporal Design Problem for a Damped Wave Equation
We analyze in this work a spatio-temporal optimal design problem governed by a linear damped 1-D wave equation. The problem consists in seeking simultaneously the spatiotemporal layout of two isotropic materials and the static position of the damping set in order to minimize a functional depending quadratically on the gradient of the state. The lack of classical solutions for this kind of nonli...
متن کاملOptimal order finite element approximation for a hyperbolic integro-differential equation
Semidiscrete finite element approximation of a hyperbolic type integro-differential equation is studied. The model problem is treated as the wave equation which is perturbed with a memory term. Stability estimates are obtained for a slightly more general problem. These, based on energy method, are used to prove optimal order a priori error estimates.
متن کاملNonlinear damped partial differential equations and their uniform discretizations
We establish sharp energy decay rates for a large class of nonlinearly first-order damped systems, and we design discretization schemes that inherit of the same energy decay rates, uniformly with respect to the space and/or time discretization parameters, by adding appropriate numerical viscosity terms. Our main arguments use the optimal-weight convexity method and uniform observability inequal...
متن کامل