On the Continuity Properties of the Attainable Sets of Nonlinear Control Systems with Integral Constraint on Controls
Khalik G. Guseinov and Ali S. Nazlipinar
Source: Abstr. Appl. Anal. Volume 2008 (2008), 14 pages.
Abstract
The attainable sets of the nonlinear control systems with integral constraint on the control functions are considered. It is assumed that the behavior of control system is described by differential equation which is nonlinear with respect to phase-state vector and control vector. The admissible control functions are chosen from the closed ball centered at the origin with radius ${{\mu{}}}_{0} $ in ${L}_{p}([{t}_{0},{\theta{}}];{{\mathbb{R}}}^{m})\,(p\in{}(1,+{\infty{}}))$. Precompactness of the solutions set is specified, and dependence of the attainable sets on the initial conditions and on the parameters of the control system is studied.
Full-text: Access denied (no subscription detected)
We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber.
If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text
Links and Identifiers
Permanent link to this document: http://projecteuclid.org/euclid.aaa/1220969149
Digital Object Identifier: doi:10.1155/2008/295817
Mathematical Reviews number (MathSciNet):
MR2393114
Zentralblatt MATH identifier:
05313167
References
V. V. Beletskii, Studies of Motions of Celestial Bodies, Nauka, Moscow, Russia, 1972.
N. N. Krasovskii, Theory of Control of Motion: Linear Systems, Nauka, Moscow, Russia, 1968.
Mathematical Reviews (MathSciNet):
MR0265035
Zentralblatt MATH:
0172.12702
V. I. Ukhobotov, One Dimensional Projection Method in Linear Differential Games with Integral Constraints, Chelyabinsk State University Press, Chelyabinsk, Russia, 2005.
A. G. Chentsov, ``Asymptotic attainability with perturbation of integral constraints in an abstract control problem---I,'' Russian Mathematics, vol. 39, no. 2, pp. 57--68, 1995.
Mathematical Reviews (MathSciNet):
MR1391553
A. G. Chentsov, Asymptotic Attainability, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1997.
Zentralblatt MATH:
0859.93002
Mathematical Reviews (MathSciNet):
MR1427054
R. Conti, Problemi di Controllo e di Controllo Ottimale, UTET, Torino, Italy, 1974.
F. Gozzi and P. Loreti, ``Regularity of the minimum time function and minimum energy problems: the linear case,'' SIAM Journal on Control and Optimization, vol. 37, no. 4, pp. 1195--1221, 1999.
Mathematical Reviews (MathSciNet):
MR1691938
Zentralblatt MATH:
0958.49014
Digital Object Identifier: doi:10.1137/S0363012996312763
H. W. Lou, ``On the attainable sets of control systems with $p$-integrable controls,'' Journal of Optimization Theory and Applications, vol. 123, no. 1, pp. 123--147, 2004.
Mathematical Reviews (MathSciNet):
MR2100266
Zentralblatt MATH:
1070.93007
Digital Object Identifier: doi:10.1023/B:JOTA.0000043994.99474.ff
čommentComment on ref. [15?]: Please update the information of this reference, if possible. M. Motta and C. Sartori, ``Minimum time and minimum energy function for linear systems with controls in $L^p$ for $p\geq1$,'' preprint, 2000.
A. N. Sirotin and A. M. Formalskii, ``Reachability and controllability of discrete-time systems under control actions bounded in magnitude and norm,'' Automation and Remote Control, vol. 64, no. 12, pp. 1844--1857, 2003.
Mathematical Reviews (MathSciNet):
MR2094357
Kh. G. Guseĭnov, A. A. Neznakhin, and V. N. Ushakov, ``Approximate construction of reachable sets of control systems with integral constraints on the controls,'' Journal of Applied Mathematics and Mechanics, vol. 63, no. 4, pp. 557--567, 1999.
Mathematical Reviews (MathSciNet):
MR1729377
Zentralblatt MATH:
0953.93010
Kh. G. Guseinov, O. Ozer, and E. Akyar, ``On the continuity properties of the attainable sets of control systems with integral constraints on control,'' Nonlinear Analysis: Theory, Methods & Applications, vol. 56, no. 3, pp. 433--449, 2004.
Mathematical Reviews (MathSciNet):
MR2032038
Zentralblatt MATH:
1051.93013
Kh. G. Guseinov, O. Ozer, E. Akyar, and V. N. Ushakov, ``The approximation of reachable sets of control systems with integral constraint on controls,'' Nonlinear Differential Equations and Applications, vol. 14, no. 1-2, pp. 57--73, 2007.
Mathematical Reviews (MathSciNet):
MR2346453
Zentralblatt MATH:
1141.93010
Digital Object Identifier: doi:10.1007/s00030-006-4036-6
M. Motta and C. Sartori, ``Minimum time with bounded energy, minimum energy with bounded time,'' SIAM Journal on Control and Optimization, vol. 42, no. 3, pp. 789--809, 2003.
Mathematical Reviews (MathSciNet):
MR2002136
Zentralblatt MATH:
1054.93009
Digital Object Identifier: doi:10.1137/S0363012902385284
Kh. G. Guseinov and A. S. Nazlipinar, ``On the continuity property of $L_p$ balls and an application,'' Journal of Mathematical Analysis and Applications, vol. 335, no. 2, pp. 1347--1359, 2007.
Mathematical Reviews (MathSciNet):
MR2346910
Zentralblatt MATH:
1122.49015
Digital Object Identifier: doi:10.1016/j.jmaa.2007.01.109
M. Motta and F. Rampazzo, ``Multivalued dynamics on a closed domain with absorbing boundary. Applications to optimal control problems with integral constraints,'' Nonlinear Analysis: Theory, Methods & Applications, vol. 41, no. 5-6, pp. 631--647, 2000.
Mathematical Reviews (MathSciNet):
MR1780636
Zentralblatt MATH:
0961.34003
B. T. Polyak, ``Convexity of the reachable set of nonlinear systems under $L_2$ bounded controls,'' Tech. Rep. 02.2002/2003, Institut Mittag-Leffler, Djursholm, Sweden, 2003.
P. Soravia, ``Viscosity solutions and optimal control problems with integral constraints,'' Systems & Control Letters, vol. 40, no. 5, pp. 325--335, 2000.
Mathematical Reviews (MathSciNet):
MR1829774
Zentralblatt MATH:
0985.49016
A. F. Filippov, ``On some questions in the theory of optimal regulation: existence of a solution of the problem of optimal regulation in the class of bounded measurable functions,'' Vestnik Moskovskogo Universiteta, Seriya Matematika, Mekhanika, vol. 1959, no. 2, pp. 25--32, 1959.
Mathematical Reviews (MathSciNet):
MR0122650
A. F. Filippov, Differential Equations with Discontinuous Righthand Sides, vol. 18 of Mathematics and Its Applications, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1988.
Mathematical Reviews (MathSciNet):
MR1028776
Zentralblatt MATH:
0664.34001
Abstract and Applied Analysis
