Goncharova, Elena; Ovseevich, Alexander Limit behavior of reachable sets of linear time-invariant systems with integral bounds on control. (English) Zbl 1267.93014 J. Optim. Theory Appl. 157, No. 2, 400-415 (2013). Summary: In this paper, a linear dynamic system is considered under an \(L_p\)-constraint on control. We establish the existence of the limit shape of reachable sets as time goes to infinity. Asymptotic formulas are obtained for reachable sets and their shapes. The results bridge the cases of geometric bounds on control and constraints on the total impulse of control, and create a unified picture of the structure of the limit shapes of reachable sets. Cited in 2 Documents MSC: 93B03 Attainable sets, reachability 93C05 Linear systems in control theory 93C15 Control/observation systems governed by ordinary differential equations Keywords:linear control systems; reachable sets; shapes of convex bodies; asymptotic formulas; limit behavior PDFBibTeX XMLCite \textit{E. Goncharova} and \textit{A. Ovseevich}, J. Optim. Theory Appl. 157, No. 2, 400--415 (2013; Zbl 1267.93014) Full Text: DOI References: [1] Ovseevich, A. I., Asymptotic behaviour of attainable and superattainable sets, Sopron, Hungary, 1990, Basel · Zbl 0734.93015 [2] Goncharova, E.V., Ovseevich, A.I.: Asymptotics of reachable sets of linear dynamical systems with impulsive control. J. Comput. Syst. Sci. Int. 46(1), 46-55 (2007) · Zbl 1272.93025 [3] Figurina, T.Y., Ovseevich, A.I.: Asymptotic behavior of attainable sets of linear periodic control systems. J. Optim. Theory Appl. 100(2), 349-364 (1999) · Zbl 0914.93008 [4] Goncharova, E., Ovseevich, A.: Asymptotics for shapes of singularly perturbed reachable sets. SIAM J. Control Optim. 49(2), 403-419 (2011) · Zbl 1217.93025 [5] Protasov, V.Y.: The generalized joint spectral radius. A geometric approach. Izv. Math. 61(5), 995-1030 (1997) · Zbl 0893.15002 [6] Schneider, R.: Convex Bodies: the Brunn-Minkowski Theory. Encyclopedia of Mathematics and Its Applications, vol. 44. Cambridge University Press, Cambridge (1993) · Zbl 0798.52001 [7] Lutwak, E.: The Brunn-Minkowski-Firey theory I: mixed volumes and the Minkowski problem. J. Differ. Geom. 38, 131-150 (1993) · Zbl 0788.52007 [8] Firey, W.J.: Some applications of means of convex bodies. Pac. J. Math. 14(1), 53-60 (1964) · Zbl 0126.38405 [9] Dontchev, A.L., Veliov, V.M.: On the behaviour of solutions of linear autonomous differential inclusions at infinity. C. R. Acad. Bulg. Sci. 36, 1021-1024 (1983) · Zbl 0551.34009 [10] Dontchev, A.L., Veliov, V.M.: Singular perturbation in Mayer’s problem for linear systems. SIAM J. Control Optim. 21, 566-581 (1983) · Zbl 0519.49002 [11] Arnold, V.I.: Mathematical Methods of Classical Mechanics. Springer, Berlin (1989) [12] Weyl, H.: Mean motion. Am. J. Math. 60, 889-896 (1938) · JFM 64.0256.02 [13] Weyl, H.: Mean motion. Am. J. Math. 1, 143-148 (1939) · JFM 65.0274.01 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.