Small-Time Reachable Sets of Linear Systems with Integral Control Constraints: Birth of the Shape of a Reachable Set

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• 作者：Elena Goncharova ; Alexander Ovseevich
• 关键词：Linear control systems ; Reachable sets ; Shapes of convex bodies ; 93B03 ; 93B05 ; 52A23
• 刊名：Journal of Optimization Theory and Applications
• 出版年：2016
• 出版时间：February 2016
• 年：2016
• 卷：168
• 期：2
• 页码：615-624
• 全文大小：398 KB
• 参考文献：1.Schättler, H.: Small-time reachable sets and time-optimal feedback control. In: Mordukhovich, B.S., Sussmann, H.J. (eds.) Nonsmooth Analysis and Geometric Methods in Deterministic Optimal Control. The IMA Volumes in Mathematics and Its Applications, vol. 78, pp. 203–225. Springer, New York (1996)
  2.Krener, A., Schättler, H.: The structure of small-time reachable sets in low dimensions. SIAM J. Control Optim. 27(1), 120–147 (1989)CrossRef MathSciNet MATH
  3.Vershik, A.M., Gerschkovich, V.Y.: Nonholonomic dynamical systems. Geometry of distributions and variational problems. In: Arnold, V.I., Novikov, S.P. (eds.) Dynamical Systems VII, Encyclopaedia of Mathematical Sciences, vol. 16, pp. 1–81. Springer, Berlin (1994). Russian original 1987
  4.Agrachev, A.A., Gamkrelidze, R.V., Sarychev, A.V.: Local invariants of smooth control systems. Acta Appl. Math. 14, 191–237 (1989)CrossRef MathSciNet MATH
  5.Hermes, H.: Nilpotent and high-order approximations of vector field systems. SIAM Rev. 33(2), 238–264 (1991)CrossRef MathSciNet MATH
  6.Kawski, M.: High-order small-time local controllability. In: Sussmann, H.J. (ed.) Nonlinear Controllability and Optimal Control, Monogr. Textbooks Pure Appl. Math., vol. 133, pp. 431–467. Dekker, New York (1990)
  7.Bianchini, R.M., Stefani, G.: Graded approximations and controllability along a trajectory. SIAM J. Control Optim. 28(4), 903–924 (1990)CrossRef MathSciNet MATH
  8.Goncharova, E.V., Ovseevich, A.I.: Birth of the shape of a reachable set. Doklady Math. 88(2), 605–607 (2013)CrossRef MathSciNet MATH
  9.Goncharova, E., Ovseevich, A.: Limit behavior of reachable sets of linear time-invariant systems with integral bounds on control. J. Optim. Theory Appl. 157(2), 400–415 (2013). doi:10.​1007/​s10957-012-0198-z CrossRef MathSciNet MATH
  10.Hörmander, L.V.: Notions of Convexity. In: Oesterlé, J., Weinstein, A. (eds.) Progress in Mathematics, vol. 127. Birkhäuser, Boston (1994)
  11.Schneider, R.: Convex Bodies: The Brunn–Minkowski Theory. Encyclopedia of Mathematics and Its Applications, vol. 44. Cambridge University Press, Cambridge (1993)CrossRef
  12.Goncharova, E.V., Ovseevich, A.I.: Comparative analysis of the asymptotic dynamics of reachable sets of linear systems. Comp. Syst. Sci. Int. 46(4), 505–513 (2007). doi:10.​1134/​S106423070704001​6 CrossRef MathSciNet MATH
  13.Brunovsky, P.: A classification of linear controllable systems. Kybernetika 6, 176–188 (1970)CrossRef MathSciNet
  14.Ananievskii, I.M., Anokhin, N.V., Ovseevich, A.I.: Bounded feedback controls for linear dynamic systems by using common Lyapunov functions. Doklady Math. 82(2), 831–834 (2010)CrossRef
  15.Agrachev, A.A., Sachkov, YuL: Control Theory from the Geometric Viewpoint. Encyclopaedia of Mathematical Sciences, vol. 87. Springer, Berlin (2004)CrossRef
  16.Gromov, M.: Carnot-Carathéodory spaces seen from within. In: Bellaïche, A., Risler, J.J. (eds.) Sub-Riemannian Geometry, Progress in Mathematics, vol. 144, pp. 79–323. Birkhäuser, Basel (1996)CrossRef
  17.Vershik, A.M., Gerschkovich, V.Y.: Bundle of nilpotent Lie algebras over a nonholonomic manifold (nilpotentization). J. Sov. Math. 59(5), 1040–1053 (1992)CrossRef
  
• 作者单位：Elena Goncharova (1)
  Alexander Ovseevich (2)
  
  1. Institute for System Dynamics and Control Theory, Siberian Branch of the Russian Academy of Sciences, Irkutsk, Russia
  2. Institute for Problems in Mechanics of the Russian Academy of Sciences, Moscow, Russia
  
• 刊物主题：Calculus of Variations and Optimal Control; Optimization; Optimization; Theory of Computation; Applications of Mathematics; Engineering, general; Operations Research/Decision Theory;
• 出版者：Springer US
• ISSN：1573-2878

文摘

The paper is concerned with small-time reachable sets of a linear dynamical system under integral constraints on control. The main result is the existence of a limit shape of the reachable sets as time tends to zero. A precise estimate for the rate of convergence is given. Keywords Linear control systems Reachable sets Shapes of convex bodies