Geodesic
On the asymptotics of eigenvalues of the controllability gramian of a linear system with a small parameter
Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Proceedings of the Voronezh spring mathematical school “Modern methods of the theory of boundary-value problems. Pontryagin readings – XXX”. Voronezh, May 3-9, 2019. Part 2, Tome 191 (2021), pp. 115-122

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In this paper, the problem on asymptotics of eigenvalues of the controllability Gramian for a linear system with a small parameter is considered. This problem arises in the study of the convexity of reachable sets for control systems with integral constraints on the control on a small time interval. To verify the asymptotics of the eigenvalues, the controllability Gramian is expanded in a series in powers of a small parameter. We show that coefficients of this expansion can be calculated recursively. Also, we present examples demonstrating applications of the approach proposed to second-order systems.
Keywords: control system, integral constraint, reachable set, convexity, asymptotics, controllability Gramian. 
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     author = {I. O. Osipov},
     title = {On the asymptotics of eigenvalues of the controllability gramian of a linear system with a small parameter},
     journal = {Itogi nauki i tehniki. Sovremenna\^a matematika i e\"e prilo\v{z}eni\^a. Temati\v{c}eskie obzory},
     pages = {115--122},
     publisher = {mathdoc},
     volume = {191},
     year = {2021},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/INTO_2021_191_a10/}
}
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I. O. Osipov. On the asymptotics of eigenvalues of the controllability gramian of a linear system with a small parameter. Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Proceedings of the Voronezh spring mathematical school “Modern methods of the theory of boundary-value problems. Pontryagin readings – XXX”. Voronezh, May 3-9, 2019. Part 2, Tome 191 (2021), pp. 115-122. http://geodesic.mathdoc.fr/item/INTO_2021_191_a10/