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Erschienen in:
2020 | OriginalPaper | Buchkapitel
Asymptotic Behaviour of Controllability Gramians and Convexity of Small-Time Reachable Sets
verfasst von : Mikhail Gusev
Erschienen in: Large-Scale Scientific Computing
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Abstract
We study the convexity of reachable sets for a nonlinear control-affine system on a small time interval under integral constraints on control variables. The convexity of a reachable set for a nonlinear control system under integral constraints was proved by B. Polyak under assumption that linearization of the system is controllable and \( L_2\) norms of controls are bounded from above by a sufficiently small number. Using this result we propose sufficient conditions for the convexity of reachable sets of a control-affine system on a small time interval. These conditions are based on the estimates for the asymptotics of the minimal eigenvalue of controllability Gramian of the system linearization, which depends on a small parameter (a length of the interval). A procedure for calculating estimates using the expansion of the Gramian into a series with respect to the small parameter degrees is described and some illustrative examples are presented.