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Robustness with respect to small time-varied delay for linear dynamical systems on Banach spaces. (English) Zbl 1069.34118

The authors consider the time-varied delay problem \[ x'(t)=Ax(t)+Bx(t-\tau(t)),\;t\geq 0,\quad x(\theta)=\xi(\theta),\;-r\leq\theta\leq 0,\tag{P} \] where \(A\) generates a \(C_0\)-semigroup on a Banach space \(X\), \(B\) is a closed densely defined linear operator on \(X\) and \(\tau\) is a continuous function.
Under suitable assumptions, the authors prove that the problem \((P)\) has a unique solution, which is uniformly exponentially stable with small time-varied delay.

MSC:

34K30 Functional-differential equations in abstract spaces
34K20 Stability theory of functional-differential equations
47D06 One-parameter semigroups and linear evolution equations
34K06 Linear functional-differential equations
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