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Numerical procedure for optimal control of higher index DAEs
Subdifferentials of convex functions on time scales
1. | Faculty of Computer Science, Białystok University of Technology, ul. Wiejska 45a, 15-351 Bia lystok, Poland |
2. | Faculty of Computer Science, Białystok University of Technology, ul. Wiejska 45a, 15-351 Białystok, Poland, Poland |
References:
[1] |
R. P. Agarwal and M. Bohner, Basic calculus on time scales and some of its applications, Results Math., 35 (1999), 3-22. |
[2] |
D. R. Anderson, Taylor polynomials for nabla dynamic equations on time scales, Panamer. Math. J., 12 (2002), 17-27. |
[3] |
J.-P. Aubin and A. Cellina, "Differential Inclusions. Set-valued Maps and Viability Theory," Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 264, Springer-Verlag, Berlin, 1984. |
[4] |
B. Aulbach and S. Hilger, A unified approach to continuous and discrete dynamics, in "Qualitative Theory of Differential Equations" (eds. B. Sz-Nagy and L. Hatvani), Colloq. Math. Soc. János Bolyai, 53, North Holland, Amsterdam, (1990), 37-56. |
[5] |
B. Aulbach and S. Hilger, Linear dynamic process with inhomogeneous time scale, in "Nonlinear Dynamics and Quantum Dynamical Systems" (eds. G. Leonov, V. Reitmann and W. Timmermann), Math. Res., 59, Akademie-Verlag, Berlin, (1990), 9-20. |
[6] |
N. S. Barnett, P. Cerone and S. S. Dragomir, Majorisation inequalities for Stieltjes integrals, Appl. Math. Lett., 22 (2009), 416-421.
doi: doi:10.1016/j.aml.2008.06.009. |
[7] |
M. Bohner and A. Peterson, "Dynamic Equations on Time Scales. An Introduction with Applications," Birkhäuser Boston, Boston, MA, 2001. |
[8] |
M. Bohner and A. Peterson (eds.), "Advances in Dynamic Equations on Time Scales," Birkhäuser Boston, Boston, MA, 2003. |
[9] |
F. H. Clarke, "Optimization and Nonsmooth Analysis," 2nd edition, Classics in Applied Mathematics, 5, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1990. |
[10] |
C. Dinu, Convex functions on time scales, An. Univ. Craiova Ser. Mat. Inform., 35 (2008), 87-96. |
[11] |
J.-B. Hiriart-Urruty and C. Lemaréchal, "Convex Analysis and Minimization Algorithms. I. Fundamentals," Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 305, Springer-Verlag, Berlin, 1993. |
[12] |
A. B. Malinowska and D. F. M. Torres, On the diamond-alpha Riemann integral and mean value theorems on time scales, Dynam. Systems Appl., 18 (2009), 469-481. |
[13] |
A. B. Malinowska and D. F. M. Torres, Necessary and sufficient conditions for local Pareto optimality on time scales, Journal of Mathematical Sciences, 161 (2009), 803-810, arXiv:0801.2123v1. |
[14] |
D. Mozyrska and D. F. M. Torres, The natural logarithm on time scales, J. Dyn. Syst. Geom. Theor., 7 (2009), 41-48. |
[15] |
K. Murota, "Discrete Convex Analysis," SIAM Monographs on Discrete Mathematics and Applications, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2003. |
[16] |
C. Niculescu and L. E. Persson, "Convex Functions and Their Applications. A Contemporary Approach," CMS Books in Mathematics/Ouvrages de Mathématiques de la SMC, 23, Springer, New York, 2006. |
[17] |
U. M. Özkan and H. Yildirim, Steffensen's integral inequality on time scales, J. Inequal. Appl., (2007), Art. ID 46524, 10 pp. |
[18] |
R. T. Rockafellar, Characterization of the subdifferentials of convex functions, Pacific J. Math., 17 (1966), 497-510. |
[19] |
J. W. Rogers and Q. Sheng, Notes on the diamond-$\alpha$ dynamic derivative on time scales, J. Math. Anal. Appl., 326 (2007), 228-241.
doi: doi:10.1016/j.jmaa.2006.03.004. |
[20] |
Q. Sheng, M. Fadag, J. Henderson and J. M. Davis, An exploration of combined dynamic derivatives on time scales and their applications, Nonlinear Anal. Real World Appl., 7 (2006), 395-413.
doi: doi:10.1016/j.nonrwa.2005.03.008. |
[21] |
Q. Sheng, Hybrid approximations via second order combined dynamic derivatives on time scales, Electron. J. Qual. Theory Differ. Equ., 17 (2007), 13 pp. |
show all references
References:
[1] |
R. P. Agarwal and M. Bohner, Basic calculus on time scales and some of its applications, Results Math., 35 (1999), 3-22. |
[2] |
D. R. Anderson, Taylor polynomials for nabla dynamic equations on time scales, Panamer. Math. J., 12 (2002), 17-27. |
[3] |
J.-P. Aubin and A. Cellina, "Differential Inclusions. Set-valued Maps and Viability Theory," Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 264, Springer-Verlag, Berlin, 1984. |
[4] |
B. Aulbach and S. Hilger, A unified approach to continuous and discrete dynamics, in "Qualitative Theory of Differential Equations" (eds. B. Sz-Nagy and L. Hatvani), Colloq. Math. Soc. János Bolyai, 53, North Holland, Amsterdam, (1990), 37-56. |
[5] |
B. Aulbach and S. Hilger, Linear dynamic process with inhomogeneous time scale, in "Nonlinear Dynamics and Quantum Dynamical Systems" (eds. G. Leonov, V. Reitmann and W. Timmermann), Math. Res., 59, Akademie-Verlag, Berlin, (1990), 9-20. |
[6] |
N. S. Barnett, P. Cerone and S. S. Dragomir, Majorisation inequalities for Stieltjes integrals, Appl. Math. Lett., 22 (2009), 416-421.
doi: doi:10.1016/j.aml.2008.06.009. |
[7] |
M. Bohner and A. Peterson, "Dynamic Equations on Time Scales. An Introduction with Applications," Birkhäuser Boston, Boston, MA, 2001. |
[8] |
M. Bohner and A. Peterson (eds.), "Advances in Dynamic Equations on Time Scales," Birkhäuser Boston, Boston, MA, 2003. |
[9] |
F. H. Clarke, "Optimization and Nonsmooth Analysis," 2nd edition, Classics in Applied Mathematics, 5, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1990. |
[10] |
C. Dinu, Convex functions on time scales, An. Univ. Craiova Ser. Mat. Inform., 35 (2008), 87-96. |
[11] |
J.-B. Hiriart-Urruty and C. Lemaréchal, "Convex Analysis and Minimization Algorithms. I. Fundamentals," Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 305, Springer-Verlag, Berlin, 1993. |
[12] |
A. B. Malinowska and D. F. M. Torres, On the diamond-alpha Riemann integral and mean value theorems on time scales, Dynam. Systems Appl., 18 (2009), 469-481. |
[13] |
A. B. Malinowska and D. F. M. Torres, Necessary and sufficient conditions for local Pareto optimality on time scales, Journal of Mathematical Sciences, 161 (2009), 803-810, arXiv:0801.2123v1. |
[14] |
D. Mozyrska and D. F. M. Torres, The natural logarithm on time scales, J. Dyn. Syst. Geom. Theor., 7 (2009), 41-48. |
[15] |
K. Murota, "Discrete Convex Analysis," SIAM Monographs on Discrete Mathematics and Applications, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2003. |
[16] |
C. Niculescu and L. E. Persson, "Convex Functions and Their Applications. A Contemporary Approach," CMS Books in Mathematics/Ouvrages de Mathématiques de la SMC, 23, Springer, New York, 2006. |
[17] |
U. M. Özkan and H. Yildirim, Steffensen's integral inequality on time scales, J. Inequal. Appl., (2007), Art. ID 46524, 10 pp. |
[18] |
R. T. Rockafellar, Characterization of the subdifferentials of convex functions, Pacific J. Math., 17 (1966), 497-510. |
[19] |
J. W. Rogers and Q. Sheng, Notes on the diamond-$\alpha$ dynamic derivative on time scales, J. Math. Anal. Appl., 326 (2007), 228-241.
doi: doi:10.1016/j.jmaa.2006.03.004. |
[20] |
Q. Sheng, M. Fadag, J. Henderson and J. M. Davis, An exploration of combined dynamic derivatives on time scales and their applications, Nonlinear Anal. Real World Appl., 7 (2006), 395-413.
doi: doi:10.1016/j.nonrwa.2005.03.008. |
[21] |
Q. Sheng, Hybrid approximations via second order combined dynamic derivatives on time scales, Electron. J. Qual. Theory Differ. Equ., 17 (2007), 13 pp. |
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