Anar Huseyin     (Sivas Cumhuriyet University, 58140 Merkez, Sivas, Turkey)
Nesir Huseyin     (Sivas Cumhuriyet University, 58140 Merkez, Sivas, Turkey)


In this paper the control system described by a nonlinear differential equation is studied. It is assumed that the control functions have a  quadratic integral constraint, more precisely, the admissible control functions are chosen from the ellipsoid of the space \(L_2([t_0,\theta];\mathbb{R}^m)\). Different properties of the set of trajectories are investigated. It is proved that a small perturbation of the set of control functions causes also appropriate small perturbation of the set of trajectories. It is also shown that the set of trajectories has a small change if along with the integral constraint on the control functions, a sufficiently large norm type geometric constraint on the control functions is introduced. It is established that every trajectory is robust with respect to the fast consumption of the remaining control resource, and hence every trajectory of the system can be approximated by a trajectory generated by full consumption of the total control resource.


Nonlinear control system, Quadratic integral constraint, Set of trajectories, Robustness

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  1. Beletskii V.V. Notes on the Motion of Celestial Bodies. Moscow: Nauka, 1972. 360 p.
  2. Conti R. Problemi di Controllo e di Controllo Ottimale. Torino: UTET, 1974. 239 p. (in Italian)
  3. Filippov A.F. Differential Equations with Discontinuous Right-Hand Sides. Dordrecht: Kluwer, 1988. 304 p.
  4. Gusev M.I., Zykov I.V. On the geometry of the reachable sets of control systems with isoperimetric constraints. Tr. Inst. Mat. Mekh. UrO RAN, 2018. Vol. 24, No. 1. P. 63–75. DOI: 10.21538/0134-4889-2018-24-1-63-75 (in Russian)
  5. Guseinov K.G., Ozer O., Akyar E., Ushakov V.N. The approximation of reachable sets of control systems with integral constraint on controls. Nonlin. Dif. Equat. Appl. (NoDEA), 2007. Vol. 14, No. 1–2. P. 57–73. DOI: 10.1007/s00030-006-4036-6
  6. Guseinov Kh.G., Nazlipinar A.S. On the continuity properties of the attainable sets of nonlinear control systems with integral constraint on controls. Abstr. Appl. Anal., 2008. Art ID 295817, 14 pp. DOI: 10.1155/2008/295817
  7. Huseyin N., Huseyin A., Guseinov Kh.G. Approximations of the set of trajectories and integral funnel of the non-linear control systems with Lp norm constraints on the control functions. IMA J. Math. Control Inform., 2022. Vol. 39, No. 4. P. 1213–1231. DOI: 10.1093/imamci/dnac028
  8. Huseyin A., Huseyin N., Guseinov Kh.G. Approximations of the images and integral funnels of the \(L_p\) balls under a Urysohn-type integral operator. Funktsionalnyi Analiz i ego Prilozheniya, 2022. Vol. 56, No. 4. P. 43–58. DOI: 10.4213/faa3974
  9. Ibragimov G., Ferrara M., Kuchkarov A., Pansera B.A. Simple motion evasion differential game of many pursuers and evaders with integral constraints. Dynamic Games Appl., 2018. Vol. 8, No. 2. P. 352–378. DOI: 10.1007/s13235-017-0226-6
  10. Kolmogorov A.N., Fomin S.V. Introductory Real Analysis. New York: Dover Publications, Inc., 1975. 403 p.
  11. Kostousova E.K. On the polyhedral estimation of reachable sets in the “extended” space for discrete-time systems with uncertain matrices and integral constraints. Tr. Inst. Mat. Mekh. UrO RAN, 2020.Vol. 26, No. 1. P. 141–155. DOI: 10.21538/0134-4889-2020-26-1-141-155 (in Russian)
  12. KrasovskiiN.N. Theory of Control of Motion: Linear Systems. Moscow: Nauka, 1968. 475 p. (in Russian)
  13. Motta M., Sartori C. Minimum time with bounded energy, minimum energy with bounded time. SIAM J. Contr. Optimiz., 2003. Vol. 42, No. 3. P. 789–809. DOI: 10.1137/S0363012902385284
  14. Rousse R., Garoche P.-L., Henrion D. Parabolic set simulation for reachability analysis of linear time-invariant systems with integral quadratic constraint. European J. Contr., 2021. Vol. 58. P. 152–167. DOI: 10.1016/j.ejcon.2020.08.002
  15. Subbotin A.I., Ushakov V.N. Alternative for an encounter-evasion differential game with integral constraints on the players controls. J. Appl. Math. Mech., 1975. Vol. 39, No. 3. P. 387–396. DOI: 10.1016/0021-8928(75)90001-5
  16. Subbotina N.N., Subbotin A.I. Alternative for the encounter-evasion differential game with constraints on the momenta of the players’ controls. J. Appl. Math. Mech., 1975. Vol. 39, No. 3. P. 397–406. DOI: 10.1016/0021-8928(75)90002-7

DOI: http://dx.doi.org/10.15826/umj.2023.1.007

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