Geodesic
Asymptotic Control Theory for a Closed String. II
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Optimal Control and Dynamical Systems, Tome 321 (2023), pp. 194-214

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We develop an asymptotic control theory for one of the simplest distributed (infinite-dimensional) oscillating systems, namely, for a closed string under a bounded load applied to a single distinguished point. We give a precise description of the classes of string states that admit complete damping, and find an asymptotically exact value of the required time. By using approximate reachable sets instead of the exact ones, we design a feedback control, which turns out to be asymptotically optimal. The main results are exact algebraic formulas for the asymptotic shape of the reachable sets, for the asymptotically optimal time of motion, and for the asymptotically optimal control thus constructed.
Keywords: maximum principle, reachable sets, linear system, string control. 
@article{TM_2023_321_a12,
     author = {Lev V. Lokutsievskiy and Alexander I. Ovseevich},
     title = {Asymptotic {Control} {Theory} for a {Closed} {String.} {II}},
     journal = {Trudy Matematicheskogo Instituta imeni V.A. Steklova},
     pages = {194--214},
     publisher = {mathdoc},
     volume = {321},
     year = {2023},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TM_2023_321_a12/}
}
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Lev V. Lokutsievskiy; Alexander I. Ovseevich. Asymptotic Control Theory for a Closed String. II. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Optimal Control and Dynamical Systems, Tome 321 (2023), pp. 194-214. http://geodesic.mathdoc.fr/item/TM_2023_321_a12/