On the Properties of the Set of Trajectories of the Control System Described by Urysohn Type Integral Equation

Year 2023, Volume: 11 Issue: 4, 1772 - 1781, 24.10.2023

Nesir Hüseyin

https://doi.org/10.29130/dubited.1186317

Abstract

In this study, the properties of the set of trajectories of the control system described by Urysohn type integral equation are studied. It is assumed that the control functions satisfy integral constraint. It is proved that the set of trajectories is Lipschitz continuous with respect to the parameter which characterizes the bound of control resource. An upper estimation for the diameter of the set of trajectories is given. Robustness of the system’s trajectory with respect to the remaining control resource is discussed.

Keywords

Nonlinear integral equation, Control system, Integral constraint, Set of trajectories, Robustness

Urysohn Tür İntegral Denklem ile Verilen Kontrol Sistemin Yörüngeler Kümesinin Özellikleri Üzerine

Abstract

Bu çalışmada Urysohn tür integral denklem ile verilen control systemin yörüngeler kümesinin özellikleri incelenmektedir. Kontrol fonksiyonların integral kısıtı sağladığı varsayılmaktadır. Yörüngeler kümesinin control kaynağın sınırını belirleyen parametreye göre Lipschitz sürekli olduğu kanıtlanmıştır. Yörüngeler kümesinin çapı için bir üst değerlendirme verilmiştir. Sistemin yörüngesinin, kalan control kaynağına göre robastlığı tartışılmaktadır.

Keywords

Doğrusal olmayan integral denklem, Kontrol sistem, İntegral kısıtlama, Yörüngeler kümesi, Robastlık

References

• [1] W. Heisenberg, Physics and Philosophy. The Revolution in Modern Science, London, Great Britain: George Allen & Unwin, 1958.
• [2] D. Hilbert, Grundzüge Einer Allgemeinen Theorie der Linearen Integralgleichungen, Leipzig und Berlin, Germany: Druck und Verlag von B.G.Teubner,1912.
• [3] N.N. Krasovskii, and A.I. Subbotin, Game-Theoretical Control Problems, New York, USA: Springer, 1988.
• [4] L.S. Pontryagin, V.G. Boltyanskii, R.V. Gamkrelidze, and E.F. Mishchenko, The Mathematical Theory of Optimal Processes, New York, USA: John Wiley & Sons, 1962.
• [5] J. Warga, Optimal Control of Differential and Functional Equations, New York, USA: Academic Press, 1972.
• [6] R. Conti, Problemi di Controllo e di Controllo Ottimale. Torino, Italy: UTET, 1974.
• [7] Kh.G. Guseinov, A.A. Neznakhin, and V.N. Ushakov, “Approximate construction of reachable sets of control systems with integral constraints on the controls,” J. Appl. Math. Mech., vol. 63, no. 4, pp. 557-567, 1999.
• [8] M.I. Gusev, and I.V. Zykov, “On extremal properties of the boundary points of reachable sets for control systems with integral constraints,” Tr. Inst. Math. Mekh. UrO RAN, vol. 23, no. 1, pp. 103-115, 2017.
• [9] G. Ibragimov, I.A. Alias, U. Waziri, and A.B. Jaafaru, “Differential game of optimal pursuit for an infinite system of differential equations,” Bull. Malays. Math. Sci. Soc., vol. 42, pp. 391-403, 2019.
• [10] E.K. Kostousova, “State estimates of bilinear discrete-time systems with integral constraints through polyhedral techniques,” IFAC Papers Online, vol. 51, no. 32, pp. 245-250, 2018.
• [11] N.N. Krasovskii, Theory of Control of Motion: Linear Systems, Moscow, USSR: Nauka, 1968.
• [12] N.N. Subbotina, and A.I. Subbotin, “Alternative for the encounter-evasion differential game with constraints on the momenta of the players' controls,” J. Appl. Math. Mech., vol. 39, no. 3, pp. 376-385, 1975.
• [13] J.P. Aubin, and A. Cellina, Differential Inclusions, Berlin, Germany: Springer, 1984.
• [14] K. Deimling, Multivalued Differential Equations, Berlin, Germany: Walter De Gruyter, 1992.
• [15] A.I. Panasyuk, and V.I Panasyuk, “An equation generated by a differential inclusion,” Mat. Zametki, vol. 27, no. 3, pp. 429-437, 1980.
• [16] E.J. Balder, “On existence problems for the optimal control of certain nonlinear integral equations of Urysohn type,” J. Optim. Theory Appl., vol. 42, no. 3, pp. 447-465, 1984.
• [17] M.L. Bennati, “An existence theorem for optimal controls of systems defined by Uryson integral equations,” Ann. Mat. Pura. Appl., vol. 121, no. 4, pp. 187-197, 1979.
• [18] N. Huseyin, A. Huseyin, and Kh.G. Guseinov, “Approximation of the set of trajectories of the nonlinear control system with limited control resources,” Math. Model. Anal., vol. 23, no. 1, pp. 152-166, 2018.
• [19] N. Huseyin, A. Huseyin, and Kh.G. Guseinov, “Approximation of the set of trajectories of the control system described by a Urysohn type integral equation,” Tr. Inst. Math. Mekh. UrO RAN, vol. 21, no. 2, pp. 59-72, 2015.
• [20] A. Huseyin, N. Huseyin, and K.G. Guseinov, “Approximation of the integral funnel of a nonlinear control system with limited control resources,” Minimax Theory and its Applications, vol. 5, no.2, pp. 327-346, 2020.
• [21] N. Huseyin, “Compactness of the set of trajectories of the control system described by a Urysohn type integral equation,” Int. J. Optim. Control. Theor. Appl. IJOCTA, vol. 7, no.1, pp. 59-65, 2017.
• [22] I.A. Alias, N. Huseyin, and A. Huseyin, “Compactness of the set of trajectories of the control system described by a Urysohn type integral equation with quadratic integral constraints on the control functions,” J. Inequal. Appl., Paper No. 36, 14 pp., 2016.