An example on the effect of time delays in boundary feedback stabilization of wave equations. (English) Zbl 0592.93047

Datko, R.; Polis, M. P.; Lagnese, J.

SIAM J. Control Optimization 24, 152-156 (1986).

Given the one-dimensional wave equation with viscous damping: \(u_{tt}- u_{xx}+2au_ t+a^ 2u=0\), \(a\geq 0\), the asymptotic behavior of solutions u is studied when u is subjected to the boundary conditions: \(u(0,t)=0\), \(u_ x(1,t)=-ku_ t(1,t-\epsilon)\), \(\epsilon\geq 0\). If \(\epsilon >0\) it is shown by direct methods using a result of R. Datko [Q. Appl. Math. 36, 279-292 (1978; Zbl 0405.34051)] that there exists a stability limit, say \(K_ 1(a)\), such that for \(0<k<K_ 1(a)\), u is uniformly exponentially stable, while for \(k=K_ 1(a)\) the imaginary axis contains a limit point of the spectrum, \(\epsilon\in R\), \(meas(R)=0\), giving marginal stability, and for \(k>K_ 1(a)\) the system admits (a.s. wrt \(\epsilon)\) exponentially unstable solutions.

Reviewer: G.Leugering

MSC:

93D15	Stabilization of systems by feedback
35L05	Wave equation
93C20	Control/observation systems governed by partial differential equations
35R10	Partial functional-differential equations
93C05	Linear systems in control theory
93D20	Asymptotic stability in control theory

Keywords:

boundary stabilization; delay systems; one-dimensional wave equation with viscous damping

Citations:

Zbl 0405.34051