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**On the convexity of the reachable set with respect to a part of coordinates at small time intervals.**
*(Russian.
English summary)*
Zbl 1485.93051

Summary: We investigate the convexity of the reachable sets for some of the coordinates of nonlinear systems with integral constraints on the control at small time intervals. We have proved sufficient convexity conditions in the form of constraints on the asymptotics of the eigenvalues of the Gramian of the controllability of a linearized system for some of the coordinates. There are two nonlinear third-order systems under study as examples. The system linearized along a trajectory generated by zero control is uncontrollable, and the system in the other example is completely controllable. We investigate the sufficient conditions for convexity of projection of reachable sets. Numerical modeling has been carried out, demonstrating the non-convexity of some projections even for small time intervals.

### Keywords:

nonlinear control systems; reachable sets; integral constraints; convexity; linearization; small time interval; asympotics
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\textit{I. O. Osipov}, Vestn. Udmurt. Univ., Mat. Mekh. Komp'yut. Nauki 31, No. 2, 210--225 (2021; Zbl 1485.93051)

### References:

[1] | Kurzhanski A. B., Varaiya P., Dynamics and control of trajectory tubes. Theory and computation, Birkh{ä}user, Basel, 2014 · Zbl 1336.93004 |

[2] | Vdovin S. A., Taras’yev A. M., Ushakov V. N., “Construction of the attainability set of a Brockett integrator”, Journal of Applied Mathematics and Mechanics, 68:5 (2004), 631-646 · Zbl 1106.49035 |

[3] | Patsko V. S., Pyatko S. G., Fedotov A. A., “Three-dimensional reachability set for a nonlinear control system”, Journal of Computer and Systems Sciences International, 42:3 (2003), 320-328 · Zbl 1110.93307 |

[4] | Gornov A. Yu., Computational technologies for solving optimal control problems, Nauka, Novosibirsk, 2009 |

[5] | Guseinov K. G., Ozer O., Akyar E., Ushakov V. N., “The approximation of reachable sets of control systems with integral constraint on controls”, Nonlinear Differential Equations and Applications, 14:1-2 (2007), 57-73 · Zbl 1141.93010 |

[6] | Filippova T. F., “Ellipsoidal estimates of reachable sets for control systems with nonlinear terms”, IFAC-PapersOnLine, 50:1 (2017), 15355-15360 |

[7] | Polyak B. T., “Sonvexity of the reachable set of nonlinear systems under L2 bounded controls”, Dynamics of Continuous, Discrete and Impulsive Systems. Ser. A: Mathematical Analysis, 11 (2004), 255-267 · Zbl 1050.93007 |

[8] | Polyak B. T., “Local programming”, Comput. Math. Math. Phys., 41:9 (2001), 1259-1266 · Zbl 1040.90048 |

[9] | Gusev M. I., “On convexity of reachable sets of a nonlinear system under integral constraints”, IFAC-PapersOnLine, 51:32 (2018), 207-212 |

[10] | Gusev M. I., Osipov I. O., “Asymptotic behavior of reachable sets on small time intervals”, Proceedings of the Steklov Institute of Mathematics, 309, suppl. 1 (2020), 52-64 |

[11] | Goncharova E., Ovseevich A., “Small-time reachable sets of linear systems with integral control constraints: birth of the shape of a reachable set”, Journal of Optimization Theory and Applications, 168:2 (2016), 615-624 · Zbl 1333.93045 |

[12] | Gusev M. I., “The limits of applicability of the linearization method in calculating small-time reachable sets”, Ural Mathematical Journal, 6:1 (2020), 71-83 · Zbl 1448.93050 |

[13] | Gusev M. I., “Estimates of the minimal eigenvalue of the controllability Gramian for a system containing a small parameter”, Mathematical Optimization Theory and Operations Research, Springer, Cham, 2019, 461-473 · Zbl 1441.93027 |

[14] | Cockayne E. J., Hall G. W. C., “Plane motion of a particle subject to curvature constraints”, SIAM Journal on Control, 13:1 (1975), 197-220 · Zbl 0305.53004 |

[15] | Patsko V. S., Fedotov A. A., “Attainability set at instant for one-side turning Dubins car”, Trudy Instituta Matematiki i Mekhaniki UrO RAN, 24, no. 1, 2018, 143-155 (in Russian) |

[16] | Zykov I. V., Osipov I. O., A program for constructing the reachable sets of nonlinear systems with integral control constraints by the Monte Carlo method, certificate of state registration of a computer program № 2020661557, 2020 |

[17] | Zykov I. V., “An algorithm for constructing reachable sets for systems with multiple integral constraints”, Mathematical Analysis With Applications, Springer, Cham, 2020, 51-60 · Zbl 1471.93032 |

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