Optimal shape and position of the actuators for the stabilization of a string. (English) Zbl 1134.93399

Hébrard, Pascal; Henrott, Antoine

Syst. Control Lett. 48, No. 3-4, 199-209 (2003).

Summary: The energy in a string subject to constant viscous damping \(k\) on a subset \(\omega\) of length \(l>0\) decays exponentially in time; we consider the problem of optimizing the decay rate for the \(\omega\) 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 \(l=\frac 12\), 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.

MSC:

93D15	Stabilization of systems by feedback
74H45	Vibrations in dynamical problems in solid mechanics
74K05	Strings
74M05	Control, switches and devices (“smart materials”) in solid mechanics
93C20	Control/observation systems governed by partial differential equations

Keywords:

Damped wave equation; Optimal location; Spectral abscissa; Genetic algorithms

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