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 control-affine 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 \(\mathbb{L}_{2}\). 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 \(\mathbb{L}_{2}\) under a nonlinear mapping to \(\mathbb{R}^{n}\). 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.
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REFERENCES
N. N. Krasovskii, Theory of Motion Control (Nauka, Moscow, 1968) [in Russian].
A. B. Kurzhanskii, Control and Observation under Uncertainty (Nauka, Moscow, 1977) [in Russian].
Kh. G. Guseinov and A. S. Nazlipinar, “Attainable sets of the control system with limited resources,” Trudy Inst. Mat. Mekh. UrO RAN 16 (5), 261–268 (2010).
K. G. Guseinov, O. Ozer, E. Akyar, and V. N. Ushakov, “The approximation of reachable sets of control systems with integral constraint on controls,” Nonlinear Diff. Equations Appl. 14 (1–2), 57–73 (2007). doi 10.1007/s00030-006-4036-6
M. I. Gusev and I. V. Zykov, “On extremal properties of boundary points of reachable sets for a system with integrally constrained control,” IFAC-PapersOnLine 50 (1), 4082–4087 (2017). doi 10.1016/j.ifacol.2017.08.792
A. Krener and H. Schättler, “The structure of small-time reachable sets in low dimensions,” SIAM J. Control Optim. 27 (1), 120–147 (1989). doi 10.1137/0327008
H. Schättler, “Small-time reachable sets and time-optimal feedback control,” Nonsmooth Analysis and Geometric Methods in Deterministic Optimal Control, Ed. by B. S. Mordukhovich and H. J. Sussmann (Springer, New York, 1996), pp. 203–225.
B. T. Polyak, “Convexity of the reachable set of nonlinear systems under \(L_{2}\) bounded controls,” Dyn. Contin. Discrete Impuls. Syst., Ser. A: Math. Anal. 11 (2–3), 255–267 (2004).
G. Reißig, “Convexity of reachable sets of nonlinear ordinary differential equations,” Autom. Remote Control 68 (9), 68, 1527–1543 (2007).
E. Goncharova and A. Ovseevich, “Small-time reachable sets of linear systems with integral control constraints: Birth of the shape of a reachable set,” J. Optim. Theory Appl. 168 (2), 615–624 (2016). doi 10.1007/s10957-015-0754-4
M. I. Gusev, “On convexity of reachable sets of a nonlinear system under integral constraints,” IFAC-PapersOnLine 51 (32), 207–212 (2018). doi 10.1016/j.ifacol.2018.11.382
B. T. Polyak, “Local programming,” Comp. Math. Math. Phys. 41 (9), 1259–1266 (2001).
M. I. Gusev, “Estimates of the minimal eigenvalue of the controllability Gramian for a system containing a small parameter,” in Mathematical Optimization Theory and Operations Research: Proceedings of the 18th International Conference, Yekaterinburg, Russia, 2019 (Springer, New York, 2019), Ser. Lecture Notes in Computer Science 11548, pp. 461–473 (2019).
I. V. Zykov, “On external estimates of the reachable sets for control systems with integral constraints,” Izv. Inst. Mat. Inform. UdGU 53, 61–72 (2019). doi 10.20537/2226-3594-2019-53-06
E. B. Lee and L. Markus, Foundations of Optimal Control Theory (Wiley, New York, 1967; Nauka, Moscow, 1972).
E. J. Cockayne and G. W. C. Hall, “Plane motion of a particle subject to curvature constraints,” SIAM J. Control 13 (1), 197–220 (1975). doi 10.1137/0313012
Yu. I. Berdyshev, Nonlinear Sequential Control Problems and Their Application (IMM UrO RAN, Yekaterinburg, 2015) [in Russian].
V. S. Patsko, S. G. Pyatko, and A. A. Fedotov, “Three-dimensional reachability set for a nonlinear control system,” J. Comput. Syst. Sci. Int. 42 (3), 320–328 (2003).
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Translated by E. Vasil’eva
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Gusev, M.I., Osipov, I.O. Asymptotic Behavior of Reachable Setson Small Time Intervals. Proc. Steklov Inst. Math. 309 (Suppl 1), S52–S64 (2020). https://doi.org/10.1134/S0081543820040070
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DOI: https://doi.org/10.1134/S0081543820040070