Asymptotic Behavior of Reachable Setson Small Time Intervals
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ymptotic Behavior of Reachable Sets on Small Time Intervals M. I. Gusev1,2,∗ and I. O. Osipov1,∗∗ Received July 7, 2019; revised July 12, 2019; accepted August 5, 2019
Abstract—The geometric structure of small-time reachable sets plays an important role in control theory, in particular, in solving problems of local synthesis. In this paper, we consider the problem of approximate description of reachable sets on small time intervals for controlaffine systems with integral quadratic constraints on the control. Using a time substitution, we replace such a set by the reachable set on a unit interval of a control system with a small parameter, which is the length of the time interval for the original system. The constraints on the control are given by a ball of small radius in the Hilbert space L2 . Under certain conditions imposed on the controllability Gramian of the linearized system, this reachable set turns out to be convex for sufficiently small values of the parameter. We show that in this case the shape of the reachable set in the state space is asymptotically close to an ellipsoid. The proof of this fact is based on the representation of the reachable set as the image of a Hilbert ball of small radius in L2 under a nonlinear mapping to Rn . In particular, this asymptotic representation holds for a fairly wide class of second-order nonlinear control systems with integral constraints. We give three examples of systems whose reachable sets demonstrate both the presence of the indicated asymptotic behavior and the absence of the latter if the necessary conditions are not satisfied. Keywords: control system, integral constraints, reachable set, convexity, asymptotics.
DOI: 10.1134/S0081543820040070 INTRODUCTION We consider the asymptotic properties of reachable sets of nonlinear control systems with integral quadratic constraints on the control parameters. It is well known (see [1, 2]) that such sets are convex in the case of linear control systems. Moreover, they are ellipsoids in the finitedimensional state space of the system, and their parameters can be described constructively. In the general case of nonlinear systems with integral constraints, the property of convexity of reachable sets is lost, and rather time-consuming approximation algorithms should be used for their construction [3–5]. However, if the time interval is short enough, reachable sets may be convex. The geometric structure of small-time reachable sets plays an important role in control theory, in particular, in solving local synthesis problems. Reachable sets on small time intervals with geometric constraints on the control have been studied by a number of authors (see, for example, [6, 7]). The 1
Krasovskii Institute of Mathematics and Mechanics, Ural Branch of the Russian Academy of Sciences, Yekaterinburg, 620108 Russia 2 Ural Federal University, Yekaterinburg, 620000 Russia e-mail: ∗ [email protected], ∗∗ [email protected]
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ASYMPTOTIC BEHAVIOR OF REACHABLE SETS
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convexity of reachable sets of nonlinear systems is studied in [8, 9].
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