Geodesic
External estimates of reachable sets for control systems with integral constraints
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 1, Tome 190 (2021), pp. 107-114

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In this paper, we consider the problem of constructing external estimates for reachable sets as level sets of some differentiable Lyapunov–Bellman function for a control system with an integral control constraint from $\mathbb{L}_p$, $p>1$. For an appropriate choice of this function, ellipsoidal and rectangular estimates can be obtained. Constructions proposed are based on integral estimates, maximal solutions, and the comparison principle for systems of differential inequalities. Using time as an argument of the Lyapunov–Bellman function, we obtain more accurate estimates. In the linear nonstationary case, such estimates may coincide with the reachable set. Also we present illustrative examples for nonlinear systems.
Keywords: reachable set, control system, integral constraint, integral inequality, comparison principle, external estimate. 
@article{INTO_2021_190_a9,
     author = {I. V. Zykov},
     title = {External estimates of reachable sets  for control systems with integral constraints},
     journal = {Itogi nauki i tehniki. Sovremenna\^a matematika i e\"e prilo\v{z}eni\^a. Temati\v{c}eskie obzory},
     pages = {107--114},
     publisher = {mathdoc},
     volume = {190},
     year = {2021},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/INTO_2021_190_a9/}
}
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I. V. Zykov. External estimates of reachable sets  for control systems with integral constraints. 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 1, Tome 190 (2021), pp. 107-114. http://geodesic.mathdoc.fr/item/INTO_2021_190_a9/