## Journal of Applied Mathematics

### Convexity of the Set of Fixed Points Generated by Some Control Systems

#### Abstract

We deal with an application of the fixed point theorem for nonexpansive mappings to a class of control systems. We study closed-loop and open-loop controllable dynamical systems governed by ordinary differential equations (ODEs) and establish convexity of the set of trajectories. Solutions to the above ODEs are considered as fixed points of the associated system-operator. If convexity of the set of trajectories is established, this can be used to estimate and approximate the reachable set of dynamical systems under consideration. The estimations/approximations of the above type are important in various engineering applications as, for example, the verification of safety properties.

#### Article information

Source
J. Appl. Math., Volume 2009 (2009), Article ID 291849, 14 pages.

Dates
First available in Project Euclid: 2 March 2010

https://projecteuclid.org/euclid.jam/1267538751

Digital Object Identifier
doi:10.1155/2009/291849

Mathematical Reviews number (MathSciNet)
MR2556825

Zentralblatt MATH identifier
1175.93102

#### Citation

Azhmyakov, Vadim. Convexity of the Set of Fixed Points Generated by Some Control Systems. J. Appl. Math. 2009 (2009), Article ID 291849, 14 pages. doi:10.1155/2009/291849. https://projecteuclid.org/euclid.jam/1267538751

#### References

• E. Zeidler, Nonlinear Functional Analysis and Its Applications. I: Fixed-Point Theorems, Springer, New York, NY, USA, 1990.
• W. Takahashi, Nonlinear Functional Analysis, Yokohama Publishers, Yokohama, Japan, 2000.
• L. D. Berkovitz, Optimal Control Theory, vol. 1, Springer, New York, NY, USA, 1974.
• H. O. Fattorini, Infinite-Dimensional Optimization and Control Theory, vol. 62 of Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, UK, 1999.
• R. A. Adams, Sobolev Spaces, vol. 65 of Pure and Applied Mathematics, Academic Press, New York, NY, USA, 1975.
• K. Atkinson and W. Han, Theoretical Numerical Analysis, vol. 39 of Texts in Applied Mathematics, Springer, New York, NY, USA, 2nd edition, 2005.
• E. D. Sontag, Mathematical Control Theory, vol. 6 of Texts in Applied Mathematics, Springer, New York, NY, USA, 2nd edition, 1998.
• A. B. Kurzhanski and P. Varaiya, Ellipsoidal techniques for reachability analysis,'' in Hybrid Systems: Computation and Control, vol. 1790 of Lecture Notes in Computer Science, pp. 202--214, Springer, New York, NY, USA, 2000.
• B. T. Polyak, S. A. Nazin, C. Durieu, and E. Walter, Ellipsoidal parameter or state estimation under model uncertainty,'' Automatica, vol. 40, no. 7, pp. 1171--1179, 2004.
• B. T. Polyak, Convexity of the reachable set of nonlinear systems under ${L}_{2}$ bounded controls,'' Dynamics of Continuous, Discrete & Impulsive Systems. Series A, vol. 11, no. 2-3, pp. 255--267, 2004.
• V. Azhmyakov, A numerically stable method for convex optimal control problems,'' Journal of Nonlinear and Convex Analysis, vol. 5, no. 1, pp. 1--18, 2004.
• E. H. Zarantonello, Projections on convex sets in Hilbert space and spectral theory. I. Projections on convex sets,'' in Contributions to Nonlinear Functional Analysis, pp. 237--341, Academic Press, New York, NY, USA, 1971.
• C. D. Aliprantis and K. C. Border, Infinite-Dimensional Analysis, Springer, Berlin, Germany, 2nd edition, 1999.
• A. F. Filippov, Differential Equations with Discontinuous Righthand Sides, vol. 18 of Mathematics and Its Applications (Soviet Series), Kluwer Academic Publishers, Dordrecht, The Netherlands, 1988.
• J.-B. Hiriart-Urruty and C. Lemaréchal, Fundamentals of Convex Analysis, Grundlehren Text Editions, Springer, Berlin, Germany, 2001.
• I. Ekeland and R. Temam, Convex Analysis and Variational Problems, Studies in Mathematics and Its Applications, North-Holland, Amsterdam, The Netherlands, 1976.
• F. H. Clarke, Optimization and Nonsmooth Analysis, vol. 5 of Classics in Applied Mathematics, SIAM, Philadelphia, Pa, USA, 2nd edition, 1990.
• A. D. Ioffe and V. M. Tihomirov, Theory of Extremal Problems, vol. 6 of Studies in Mathematics and Its Applications, North-Holland, Amsterdam, The Netherlands, 1979.
• V. Azhmyakov, V. G. Boltyanski, and A. Poznyak, Optimal control of impulsive hybrid systems,'' Nonlinear Analysis: Hybrid Systems, vol. 2, no. 4, pp. 1089--1097, 2008.