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

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


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


Zbl 0405.34051
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