Control system defined by some integral operator

• Autorzy: Majewski M.

Identyfikatory

DOI

10.7494/OpMath.2017.37.2.313

• Języki publikacji: EN

Abstrakty

EN

In the paper we consider a nonlinear control system governed by the Volterra integral operator. Using a version of the global implicit function theorem we prove that the control system under consideration is well-posed and robust, i.e. for any admissible control u there exists a uniquely defined trajectory xu which continuously depends on control u and the operator [formula] is continuously differentiable. The novelty of this paper is, among others, the application of the Bielecki norm in the space of solutions which allows us to weaken standard assumptions.

Słowa kluczowe

EN

Volterra equation; implicit function theorem; sensitivity

• Wydawca: AGH University of Science and Technology Press
• Czasopismo: Opuscula Mathematica
• Rocznik: 2017
• Tom: Vol. 37, no. 2
• Strony: 313--325
• Opis fizyczny: Bibliogr. 19 poz.

Twórcy

autor

Majewski M.

Bibliografia

• [1] A. Bielecki, Une remarque sur I’application de la methode de Banach-Cocciopoli-Tichonov dans la theorie de Vequation s = f (x,y, z,p,q), Bull. Pol. Acad. Sci. Math. 4 (1956), 265-268.
• [2] D. Bors, Stability of nonlinear Urysohn integral equations via global diffeomorphisms and implicit function theorems, J. Integral Equations Appl. 27 (2015) 3, 343-366.
• [3] D. Bors, A. Skowron, S. Walczak, Systems described by Volterra type integral operators, Discrete Contin. Dyn. Syst. Ser. B 19 (2014) 8, 2401-2416.
• [4] O. Diekmann, Limiting behaviour in an epidemic model, Nonlinear Anal. 1 (1977) 5, 459-470.
• [5] S. Fukuda (ed.), Emotional Engineering, vol. 3, Springer International Publishing, Cham, 2015.
• [6] M. Galewski, M. Koniorczyk, On a global diffeomorphism between two Banach spaces and some application, Studia Sci. Math. Hungar. 52 (2015) 1, 65-86.
• [7] M. Galewski, M. Koniorczyk, On a global implicit function theorem and some applications to integro-differential initial value problems, Acta Math. Hungar. 148 (2016) 2, 257-278.
• [8] G. Gripenberg, S.O. Londen, O. Staffans, Volterra Integral and Functional Equations, Cambridge University Press, 1990.
• [9] D. Idczak, A global implicit function theorem and its applications to functional equations, Discrete Contin. Dyn. Syst. Ser. B 19 (2014) 8, 2549-2556.
• [10] D. Idczak, A. Skowron, On the diffeomorphisms between Banach and Hilbert spaces, Adv. Nonlinear Stud. 12 (2012) 1, 89-100.
• [11] S.O. Londen, On some nonlinear Volterra integrodifferential equations, J. Differential Equations 11 (1972) 1, 169-179.
• [12] S.O. Londen, On the asymptotic behavior of the solutions of a nonlinear volterra equation, Ann. Mat. Pura Appl. (IV) 93 (1972), 263-269.
• [13] A.G. MacFarlane (ed.), Frequency-Response Methods in Control Systems, John Wiley & Sons Inc, New York, 1979.
• [14] J. Mawhin, Problernes de Dirichlet variationnels non-lineaires, Presses de l’Universite de Montreal, Montreal, Quebec, Canada, 1987; (see also Polish ed. Metody wariacyjne dla nieliniowych problemów Dirichleta, WNT, Warszawa, 1994).
• [15] M. Mousa, R. Miller, A. Michel, Stability analysis of hybrid composite dynamical systems: Descriptions involving operators and differential equations, IEEE Trans. Automat. Control 31 (1986) 3, 216-226.
• [16] A. Piskorek, Równania całkowe. Elementy teorii i zastosowania, WNT, Warszawa, 1997.
• [17] M. Podowski, A study of nuclear reactor models with nonlinear reactivity feedbacks: Stability criteria and power overshoot evaluation, IEEE Trans. Automat. Control 31 (1986), 108-115.
• [18] M. Renardy, W.J. Hrusa, J.A. Nohel, Mathematical Problems in Viscoelasticity, Longman Scientific & Technical and J. Wiley, New York, 1987.
• [19] F.G. Tricomi, Integral Equations, Dover Publications, Inc., 1985.

Uwagi

PL

Opracowanie ze środków MNiSW w ramach umowy 812/P-DUN/2016 na działalność upowszechniającą naukę (zadania 2017).