Optimal shape and position of the actuators for the stabilization of a string

Abstract The energy in a string subject to constant viscous damping k on a subset ω of length l>0 decays exponentially in time; we consider the problem of optimizing the decay rate for the ω which are the unions of at most N intervals. This rate is given by the spectral abscissa of the linear operator associated to the wave equation. We are interested in small values of k; therefore, we consider the derivative of the spectral abscissa at k=0. We prove that, except for the case l= 1 2 , when the number of intervals is not fixed a priori an optimal domain does not exist. We study numerically the case of one or two intervals using a genetic algorithm. These numerical results are not intuitive. In particular, the optimal position of one interval is never at the middle of the string.

[1]  F. J. Narcowich,et al.  Exponential decay of energy of evolution equations with locally distributed damping , 1991 .

[2]  Carlos Castro,et al.  Achieving Arbitrarily Large Decay in the Damped Wave Equation , 2001, SIAM J. Control. Optim..

[3]  P. Freitas.  Optimizing the Rate of Decay of Solutions of the Wave Equation Using Genetic Algorithms: A Counterexample to the Constant Damping Conjecture , 1999 .

[4]  Jeffrey Rauch,et al.  Decay of solutions to nondissipative hyperbolic systems on compact manifolds , 1975 .

[5]  P. Freitas,et al.  ON SOME EIGENVALUE PROBLEMS RELATED TO THE WAVE EQUATION WITH INDEFINITE DAMPING , 1996 .

[6]  C. Bardos,et al.  Sharp sufficient conditions for the observation, control, and stabilization of waves from the boundary , 1992 .

[7]  E. Zuazua,et al.  The rate at which energy decays in a damped String , 1994 .

[8]  E. Zuazua,et al.  Stability Results for the Wave Equation with Indefinite Damping , 1996 .

[9]  Steven J. Cox.  Designing for Optimal Energy Absorption II, The Damped Wave Equation , 1998 .

[10]  G. M.,et al.  Partial Differential Equations I , 2023, Applied Mathematical Sciences.

[11]  Tosio Kato.  Perturbation theory for linear operators , 1966 .

[12]  S. Cox,et al.  Designing for Optimal Energy Absorption, 1: Lumped Parameter Systems , 1998 .

[13]  Michael L. Overton,et al.  Perturbing the Critically Damped Wave Equation , 1996, SIAM J. Appl. Math..

[14]  Bopeng Rao,et al.  Energy Decay Rate of Wave Equations with Indefinite Damping , 2000 .