On sufficient conditions for stability independent of delay. (English) Zbl 0834.93045

Simple delay systems of the form \[ \dot x(t)= Ax(t)+ Bx(t- h),\qquad h\geq 0 \] are considered and conditions under which the system is stable for all \(h\geq 0\) are found. It is first shown that the system is asymptotically stable, independently of the delay, if \[ |(j\omega I- A)^{- 1} B|< 1,\qquad \forall \omega\geq 0. \] This is then shown to be less conservative than some known results. A sufficient condition which is easy to check is \[ A\text{ stable},\;|PB|_2< 1\quad\text{and}\quad A^T P+ PA= -2I, \] for some positive definite symmetric \(P\). Instead of \(|PB|_2< 1\) one can have \[ \lambda_{\max} \Biggl({|PB|^T+ |PB|\over 2}\Biggr)< 1. \] The results are also generalized to the case of multiple delays.


93D20 Asymptotic stability in control theory
34K20 Stability theory of functional-differential equations
93C05 Linear systems in control theory
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