Article
Full entry |
PDF
(0.2 MB)
Feedback
PDF
(0.2 MB)
Feedback
Keywords:
nonlinear Volterra integral equation; control system; integral constraint
nonlinear Volterra integral equation; control system; integral constraint
Summary:
In this paper the control system with limited control resources is studied, where the behavior of the system is described by a nonlinear Volterra integral equation. The admissible control functions are chosen from the closed ball centered at the origin with radius $\mu $ in $L_p$ $(p>1)$. It is proved that the set of trajectories generated by all admissible control functions is Lipschitz continuous with respect to $\mu $ for each fixed $p$, and is continuous with respect to $p$ for each fixed $\mu $. An upper estimate for the diameter of the set of trajectories is given.
References:
[1] Bock, I., Lovíšek, J.: On a reliable solution of a Volterra integral equation in a Hilbert space. Appl. Math., Praha 48 (2003), 469-486. DOI 10.1023/B:APOM.0000024487.48855.d9 | MR 2025957 | Zbl 1099.45001
[2] Conti, R.: Problemi di Controllo e di Controllo Ottimale. UTET Torino (1974).
[3] Guseinov, K. G.: Approximation of the attainable sets of the nonlinear control systems with integral constraint on controls. Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 71 (2009), 622-645. DOI 10.1016/j.na.2008.10.097 | MR 2518068 | Zbl 1162.93005
[4] Guseinov, K. 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 (2008), Article ID 295817. MR 2393114
[5] Guseinov, K. G., Nazlipinar, A. S.: On the continuity property of $L_p$ balls and an application. J. Math. Anal. Appl. 335 (2007), 1347-1359. DOI 10.1016/j.jmaa.2007.01.109 | MR 2346910 | Zbl 1122.49015
[6] Hlaváček, I.: Reliable solutions of problems in the deformation theory of plasticity with respect to uncertain material function. Appl. Math., Praha 41 (1996), 447-466. MR 1415251
[7] Huseyin, A., Huseyin, N.: Precompactness of the set of trajectories of the controllable system described by a nonlinear Volterra integral equation. Math. Model. Anal. 17 (2012), 686-695. DOI 10.3846/13926292.2012.736088 | MR 3001166 | Zbl 1255.93070
[8] Krasnosel'skij, M. A., Krejn, S. G.: On the principle of averaging in nonlinear mechanics. Usp. Mat. Nauk. 10 (1955), 147-152 Russian. MR 0071596
[9] Krasovskij, N. N.: Theory of Motion Control. Linear Systems. Nauka Moskva (1968), Russian. Zbl 0172.12702
[10] Lakshmikantham, V.: Existence and comparison results for Volterra integral equations in a Banach space. Volterra Equations. Proc. Helsinki Symp., Otaniemi/Finland 1978 Lecture Notes in Mathematics 737 Springer, Berlin (1979), 120-126. DOI 10.1007/BFb0064502 | MR 0551035 | Zbl 0418.45015
[11] Miller, R. K.: Nonlinear Volterra Integral Equations. Mathematics Lecture Note Series W. A. Benjamin, Menlo Park (1971). MR 0511193 | Zbl 0448.45004
[12] Minorsky, N.: Introduction to Non-Linear Mechanics. J. W. Edwards Ann Arbor (1947). MR 0020689
[13] Orlicz, W., Szufla, S.: On some classes of nonlinear Volterra integral equations in Banach spaces. Bull. Acad. Pol. Sci., Sér. Sci. Math. 30 (1982), 239-250. MR 0673260 | Zbl 0501.45013
[14] Polyanin, A. D., Manzhirov, A. V.: Handbook of Integral Equations. CRC Press Boca Raton (1998). MR 1790925 | Zbl 0896.45001
[15] Väth, M.: Volterra and Integral Equations of Vector Functions. Monographs and Textbooks in Pure and Applied Mathematics 224 Marcel Dekker, New York (2000). MR 1738341 | Zbl 0940.45002
Search
Partner of
