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Abstract and Applied Analysis
Volume 2012 (2012), Article ID 481853, 15 pages
doi:10.1155/2012/481853
Research Article
Forward Euler Solutions and Weakly Invariant Time-Delayed Systems
1Department of Mathematics and Applied Mathematics, VA Commonwealth University, Richmond, Virginia 23284, USA
2Departamento de Matemáticas, Facultad Experimental de Ciencias, Universidad del Zulia, Apartado 526, Maracaibo, Edo Zulia, Venezuela
Received 12 September 2012; Revised 10 December 2012; Accepted 11 December 2012
Academic Editor: Qiji J. Zhu
Copyright © 2012 Norma L. Ortiz-Robinson and Vinicio R. Ríos. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Linked References
- W. C. Chen, “Dynamics and control of a financial system with time-delayed feedbacks,” Chaos, Solitons and Fractals, vol. 37, no. 4, pp. 1198–1207, 2008. View at Publisher · View at Google Scholar · View at Scopus
- H. Gorceki, S. Fuksa, P. Grabowski, and A. Korytowski, Analysis and Synthesis of Time Delay Systems, John Wiley & Sons, New York, NY, USA, 1989.
- Y. Lenbury and P. Pornsawad, “A delay-differential equation model of the feedback-controlled hypothalamus-pituitary-adrenal axis in humans,” Mathematical Medicine and Biology, vol. 22, no. 1, pp. 15–33, 2005. View at Publisher · View at Google Scholar · View at Scopus
- G. Rosen, “Time delays produced by essential nonlinearity in population growth models,” Bulletin of Mathematical Biology, vol. 49, no. 2, pp. 253–255, 1987. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
- A. S. Matveev, “The instability of optimal control problems to time delay,” SIAM Journal on Control and Optimization, vol. 43, no. 5, pp. 1757–1786, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
- J.-P. Aubin, Viability Theory, Birkhäuser, Boston, Mass, USA, 1991. View at Zentralblatt MATH · View at MathSciNet
- G. Haddad, “Monotone trajectories of differential inclusions and functional-differential inclusions with memory,” Israel Journal of Mathematics, vol. 39, no. 1-2, pp. 83–100, 1981. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
- G. Haddad, “Monotone viable trajectories for functional-differential inclusions,” Journal of Differential Equations, vol. 42, no. 1, pp. 1–24, 1981. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
- M. Basin, J. Rodriguez-Gonzalez, and R. Martinez-Zuniga, “Optimal control for linear systems with time delay in control input based on the duality principle,” in Proceedings of the American Control Conference (ACC '03), pp. 2144–2148, Denver, Colo, USA, June 2003. View at Scopus
- R. D. Driver, Ordinary and Delay Differential Equations, Springer-Verlag, New York, NY, USA, 1977. View at MathSciNet
- F. H. Clarke, Yu. S. Ledyaev, R. J. Stern, and P. R. Wolenski, Nonsmooth Analysis and Control Theory, vol. 178, Springer-Verlag, New York, NY, USA, 1998. View at MathSciNet
- F. H. Clarke, Yu. S. Ledyaev, and M. L. Radulescu, “Approximate invariance and differential inclusions in Hilbert spaces,” Journal of Dynamical and Control Systems, vol. 3, no. 4, pp. 493–518, 1997. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
- J.-P. Aubin, “Viability solutions to structured Hamilton-Jacobi equations under constraints,” SIAM Journal on Control and Optimization, vol. 49, no. 5, pp. 1881–1915, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
- F. H. Clarke, Optimization and Nonsmooth Analysis, vol. 5, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, Pa, USA, 1990. View at Publisher · View at Google Scholar · View at MathSciNet
- H. Frankowska, “Lower semicontinuous solutions of Hamilton-Jacobi-Bellman equations,” SIAM Journal on Control and Optimization, vol. 31, no. 1, pp. 257–272, 1993. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
- A. I. Subbotin, “A generalization of the basic equation of the theory of differential games,” Soviet Mathematics, vol. 22, pp. 358–362, 1980.
- J.-M. Bony, “Principe du maximum, inégalite de Harnack et unicité du problème de Cauchy pour les opérateurs elliptiques dégénérés,” Université de Grenoble. Annales de l'Institut Fourier, vol. 19, no. 1, pp. 277–304, 1969. View at MathSciNet
- M. G. Crandall, “A generalization of Peano's existence theorem and flow invariance,” Proceedings of the American Mathematical Society, vol. 36, pp. 151–155, 1972. View at Zentralblatt MATH · View at MathSciNet
- M. Nagumo, “Uber die Lage der Integralkurven gewöhnlicher Differentialgleichungen,” Proceedings of the Physico-Mathematical Society of Japan, vol. 24, pp. 551–559, 1942. View at Zentralblatt MATH · View at MathSciNet
- R. M. Redheffer and W. Walter, “Flow-invariant sets and differential inequalities in normed spaces,” Applicable Analysis, vol. 5, no. 2, pp. 149–161, 1975. View at Zentralblatt MATH · View at MathSciNet
- F. H. Clarke, Yu. S. Ledyaev, R. J. Stern, and P. R. Wolenski, “Qualitative properties of trajectories of control systems: a survey,” Journal of Dynamical and Control Systems, vol. 1, no. 1, pp. 1–48, 1995. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
- H. Frankowska, S. Plaskacz, and T. Rzeżuchowski, “Measurable viability theorems and the Hamilton-Jacobi-Bellman equation,” Journal of Differential Equations, vol. 116, no. 2, pp. 265–305, 1995. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
- Yu. S. Ledyaev, “Criteria for viability of trajectories of nonautonomous differential inclusions and their applications,” Journal of Mathematical Analysis and Applications, vol. 182, no. 1, pp. 165–188, 1994. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
- V. M. Veliov, “Sufficient conditions for viability under imperfect measurement,” Set-Valued Analysis, vol. 1, no. 3, pp. 305–317, 1993. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
- T. Donchev, V. Ríos, and P. Wolenski, “Strong invariance and one-sided Lipschitz multifunctions,” Nonlinear Analysis. Theory, Methods & Applications, vol. 60, no. 5, pp. 849–862, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
- V. R. Ríos and P. R. Wolenski, “Proximal characterization of the reachable set for a discontinuous differential inclusion,” in Geometric Control and Nonsmooth Analysis, F. Ancona, A. Bressan, P. Cannarsa, F. H. Clarke, and P. R. Wolenski, Eds., vol. 76, pp. 270–279, Worldscientific, Singapore, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
- A. I. Subbotin, Generalized Solutions of First-Order PDEs: The Dynamical Optimization Perspective, Birkhauser, Boston, Ma, USA, 1995. View at MathSciNet
- N. Yu. Lukoyanov, “On Hamilton-Jacobi formalism in time-delay control systems,” Trudy Instituta Matematiki i Mekhaniki UrO RAN, vol. 16, no. 5, pp. 269–277, 2010.
- A. Bressan, “Singularities of stabilizing feedbacks,” Rendiconti del Seminario Matematico dell Úniversita e del Politecnico di Torino, vol. 56, no. 4, pp. 87–104, 1998. View at Zentralblatt MATH · View at MathSciNet
- J.-M. Coron and L. Rosier, “A relation between continuous time-varying and discontinuous feedback stabilization,” Journal of Mathematical Systems, Estimation, and Control, vol. 4, no. 1, pp. 67–84, 1994. View at Zentralblatt MATH · View at MathSciNet
- E. P. Ryan, “On Brockett's condition for smooth stabilizability and its necessity in a context of nonsmooth feedback,” SIAM Journal on Control and Optimization, vol. 32, no. 6, pp. 1597–1604, 1994. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
- X. Xiang and H. Kuang, “Optimal control of time-delay systems,” Journal of Mathematics and System Science, vol. 13, no. 1, pp. 31–41, 2000.
- E. Hewitt and K. Stromberg, Real and Abstract Analysis, Springer-Verlag, New York, NY, USA, 1975. View at MathSciNet
- D. Bothe, “Multivalued differential equations on graphs,” Nonlinear Analysis. Theory, Methods & Applications, vol. 18, no. 3, pp. 245–252, 1992. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
- K. Deimling, Multivalued Differential Equations, vol. 1, Walter de Gruyter & Co., Berlin, Germany, 1992. View at Publisher · View at Google Scholar · View at MathSciNet