## Robust stability and stability radius for variational control systems.(English)Zbl 1395.93469

Summary: We consider an integral variational control system on a Banach space $$X$$ and we study the connections between its uniform exponential stability and the $$(I(\mathbb R_{+},X),O(\mathbb R_{+},X))$$ stability, where $$I$$ and $$O$$ are Banach function spaces. We identify the viable classes of input spaces and output spaces related to the exponential stability of systems and provide optimization techniques with respect to the input space. We analyze the robustness of exponential stability in the presence of structured perturbations. We deduce general estimations for the lower bound of the stability radius of a variational control system in terms of input-output operators acting on translation-invariant spaces. We apply the main results at the study of the exponential stability of nonautonomous systems and analyze in the nonautonomous case the robustness of this asymptotic property.

### MSC:

 93D09 Robust stability 34D30 Structural stability and analogous concepts of solutions to ordinary differential equations 93C30 Control/observation systems governed by functional relations other than differential equations (such as hybrid and switching systems)
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