Abstract

We consider a second order linear evolution equation with a dissipative term multiplied by a time-dependent coefficient. Our aim is to design the coefficient in such a way that all solutions decay in time as fast as possible. We discover that constant coefficients do not achieve the goal and neither do time-dependent coefficients, if they are uniformly too big. On the contrary, pulsating coefficients which alternate big and small values in a suitable way prove to be more effective. Our theory applies to ordinary differential equations, systems of ordinary differential equations, and partial differential equations of hyperbolic type.

Keywords

  1. feedback stabilization
  2. time-dependent linear damping
  3. second order evolution equations
  4. decay rates
  5. exponential decay
  6. ultra-exponential decay

MSC codes

  1. 35B40
  2. 93C05
  3. 93D15

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Published In

cover image SIAM Journal on Control and Optimization
SIAM Journal on Control and Optimization
Pages: 1266 - 1294
ISSN (online): 1095-7138

History

Submitted: 6 July 2015
Accepted: 29 February 2016
Published online: 19 May 2016

Keywords

  1. feedback stabilization
  2. time-dependent linear damping
  3. second order evolution equations
  4. decay rates
  5. exponential decay
  6. ultra-exponential decay

MSC codes

  1. 35B40
  2. 93C05
  3. 93D15

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