M.I. Gusev, I.O. Osipov. Asymptotic behavior of reachable sets on small time intervals ... P. 86-99

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.

Keywords: control system, integral constraints, reachable set, convexity, asymptotics

Received July 7, 2019

Revised July 12, 2019

Accepted August 5, 2019

Mikhail Ivanovich Gusev, Dr. Phys.-Math. Sci., Krasovskii Institute of Mathematics and Mechanics, Ural Branch of the Russian Academy of Sciences, Yekaterinburg, 620108 Russia; Prof., Ural Federal University, Yekaterinburg, 620083 Russia, e-mail: gmi@imm.uran.ru

Ivan Olegovich Osipov, doctoral student, Krasovskii Institute of Mathematics and Mechanics, Ural Branch of the Russian Academy of Sciences, Yekaterinburg, 620108 Russia, e-mail: 79193053374@yandex.ru

REFERENCES

1.   Krasovskii N.N. Teoriya upravleniya dvizheniem [Theory of motion control]. Moscow: Nauka Publ., 1968, 476 p.

2.   Kurzhanskii A.B. Upravlenie i nablyudenie v usloviyakh neopredelennosti [Control and Observation Under the Conditions of Uncertainty]. Moscow: Nauka Publ., 1977, 392 p.

3.   Guseinov Kh.G., Nazlipinar A.S. Attainable sets of the control system with limited resources. Trudy Inst. Mat. i Mekh. UrO RAN, 2010, vol. 16, no. 5, pp. 261–268.

4.   Guseinov K.G., Ozer O., Akyar E., Ushakov V.N. The approximation of reachable sets of control systems with integral constraint on controls. Nonlinear Diff. Eq. Appl., 2007, vol. 14, no. 1-2, pp. 57–73. doi: 10.1007/s00030-006-4036-6 

5.   Gusev M.I., Zykov I.V. On Extremal properties of boundary points of reachable sets for a system with integrally constrained control. IFAC-PapersOnLine, 2017, vol. 50, no. 1, pp. 4082–4087. doi: 10.1016/j.ifacol.2017.08.792 

6.   Krener A., Sch$\ddot{\mathrm{a}}$ttler H. The structure of small-time reachable sets in low dimensions. SIAM J. Control Optim., 1989, vol. 27, no. 1, pp. 120–147. doi: 10.1137/0327008 

7.   Sch$\ddot{\mathrm{a}}$ttler, 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, N Y: Springer, 1996, vol. 78, pp. 203–225. doi: 10.1007/978-1-4613-8489-2_9 

8.   Polyak B.T. Сonvexity of the reachable set of nonlinear systems under $L_2$ bounded controls. Dynamics of Continuous, Discrete and Impulsive Systems Series A: Mathematical Analysis, 2004, vol. 11, no. 2-3, pp. 255–267.

9.   Reiпig G. Convexity of reachable sets of nonlinear ordinary differential equations. Autom. Remote Control, 2007, vol. 68, no. 9, pp. 1527–1543. doi: 10.1134/S000511790709007X 

10.   Goncharova E., Ovseevich A. Small-time reachable sets of linear systems with integral control constraints: birth of the shape of a reachable set. J. Optim. Theory. Appl., 2016, vol. 168, no. 2, pp. 615–624. doi: 10.1007/s10957-015-0754-4 

11.   Gusev M.I. On convexity of reachable sets of a nonlinear system under integral constraints. IFAC-PapersOnLine, 2018, vol. 51, no. 32, pp. 207–212. doi: 10.1016/j.ifacol.2018.11.382 

12.   Polyak B.T. Local programming. Comput. Math. Math. Phys., 2001, vol. 41, no. 9, pp. 1259–1266.

13.   Gusev M.I. Estimates of the minimal eigenvalue of the controllability Gramian for a system containing a small parameter. In: Khachay M., Kochetov Y., Pardalos P. (eds) Mathematical Optimization Theory and Operations Research (MOTOR 2019), Ser. Lecture Notes in Computer Science, 2019, vol. 11548, pp. 461–473. doi: 10.1007/978-3-030-22629-9_32 

14.   Zykov I.V. On external estimates of the reachable sets for control systems with integral constraints. Izv. IMI UdGU, 2019, vol. 53, pp. 61–72 (in Russian). doi: 10.20537/2226-3594-2019-53-06 

15.   Lee E.B., Markus L. Foundations of optimal control theory. N Y; London; Sydney: John Wiley & Sons, Inc., 1967, 576 p. Translated to Russian under the title Osnovy teorii optimal’nogo upravleniya, Moscow: Nauka Publ., 1972, 576 p. ISBN: 0471522635 .

16.   Cockayne E.J., Hall G.W.C. Plane motion of a particle subject to curvature constraints. SIAM J. Control, 1975, vol. 13, no. 1, pp. 197–220. doi: 10.1137/0313012 

17.   Berdyshev Yu.I. Nelineynye zadachi posledovatelnogo upravleniya i ih prilozhenie [Nonlinear sequential control problems and their application]. Yekaterinburg, IMM UrO RAN, 2015. 193 с. ISBN: 978-5-8295-0381-9 .

18.   Patsko V.S., Pyatko S.G., Fedotov A.A. Three-dimensional reachability set for a nonlinear control system. J. Computer and Systems Sciences International, 2003, vol. 42, no. 3, pp. 320–328.

Cite this article as: M.I. Gusev, I.O. Osipov. Asymptotic behavior of reachable sets on small time intervals, Trudy Instituta Matematiki i Mekhaniki URO RAN, 2019, vol. 25, no. 3, pp. 86–99.