Gr. Sarhangi et al., THE BOUNDARY STABILIZATION OF A LINEARIZED SELF-EXCITED WAVE-EQUATION, International Journal of Systems Science, 26(11), 1995, pp. 2125-2137
Citations number
8
Categorie Soggetti
System Science","Computer Science Theory & Methods","Operatione Research & Management Science
We study the linearized self-excited wave equation u(u) - c(2) Delta u
- P(x)u(t) = 0, where P(x) greater than or equal to 0, and P(x) is an
element of L(infinity)(Omega), in a bounded domain Omega subset of R(
n) with smooth boundary Gamma, where boundary damping is present. We o
bserve that the energy is not monotonically non-increasing, owing to n
egative internal damping P which causes self-excitation. Hence, the sy
stem may become unstable. Having considered the partition {Gamma(+), G
amma(-)} of the boundary Gamma on which u = 0 on Gamma(-) and partial
derivative u/partial derivative n + Ku(t) + Lu = 0 on Gamma(+), we fin
d two different bounds for P such that the energy of the system decays
exponentially as t tends to infinity (here, we assume ($) over bar Ga
mma(+) boolean AND Gamma(-) = phi for n > 3). Both bounds depend on Om
ega. Moreover, the second bound depends on the feedback functions K, L
is an element of L(infinity)(Gamma(+)), or more precisely it depends
on a positive function k(x) is an element of L(infinity)(Gamma(+)) whi
ch determines K and L on the partition Gamma(+).