Measure differential inclusions - between continuous and discrete
Tóm tắt
The paper is devoted to the study of the measure-driven differential inclusions
,
for arbitrary finite Borel measure μ. This type of results allows one to treat in a similar manner differential and difference inclusions, as well as impulsive problems and therefore to study the evolution of hybrid systems with very complex (including Zeno) behavior. Our method is based on viewing the Borel measures as Lebesgue-Stieltjes measures. We thus obtain, under very general assumptions, the existence of regulated or bounded variation solutions of the considered problem and we indicate some advantages of our approach. MSC:49N25, 34A60, 93C30, 49J53, 37N35, 34A37.
Tài liệu tham khảo
Aubin, J-P: Impulsive Differential Inclusions and Hybrid Systems: A Viability Approach. Lecture Notes, Univ. Paris (2002)
Aubin J-P, Lygeros J, Quincampoix M, Sastry S, Seube N: Impulse differential inclusions: a viability approach to hybrid systems. IEEE Trans. Autom. Control 2002, 47: 2-20. 10.1109/9.981719
Lygeros J, Quincampoix M, Rzeżuchowski T: Impulse differential inclusions driven by discrete measures. Lecture Notes in Computer Science 4416. Hybrid Systems: Computation and Control 2007, 385-398.
Aubin J-P: Impulse and Hybrid Control Systems: A Viability Approach. A Mini-Course. University of California Press, Berkeley; 2002.
Moreau JJ: Bounded variation in time. In Topics in Nonsmooth Mechanics. Edited by: Moreau JJ, Panagiotopoulos PD, Strang G. Birkhäuser, Basel; 1988:1-74.
Code WJ, Loewen PD: Optimal control of non-convex measure-driven differential inclusions. Set-Valued Var. Anal. 2011, 19: 203-235. 10.1007/s11228-010-0138-8
Goebel R, Teel AR: Solutions to hybrid inclusions via set and graphical convergence with stability theory applications. Automatica 2006, 42: 573-587. 10.1016/j.automatica.2005.12.019
Sesekin AN, Fetisova YV: Functional differential equations in the space of functions of bounded variations. Proc. Steklov Inst. Math. 2010, 2: 258-265. suppl.
Silva GN, Vinter RB: Measure driven differential inclusions. J. Math. Anal. Appl. 1996, 202: 727-746. 10.1006/jmaa.1996.0344
Ahmed NU: Measure solutions impulsive evolution differential inclusions and optimal control. Nonlinear Anal. 2001, 47: 13-23. 10.1016/S0362-546X(01)00152-3
Pereira FL, Silva GN: Necessary conditions of optimality for vector-valued impulsive control problems. Syst. Control Lett. 2000, 40: 205-215. 10.1016/S0167-6911(00)00027-X
Pereira FL, Silva GN, Oliveira V: Invariance for impulsive control systems. Autom. Remote Control 2008, 69: 788-800. 10.1134/S0005117908050068
Goncharova E, Staritsyn M: Optimization of measure-driven hybrid systems. J. Optim. Theory Appl. 2012, 153: 139-156. 10.1007/s10957-011-9944-x
Filippova TF: Set-valued solutions to impulsive differential inclusions. Math. Comput. Model. Dyn. Syst. 2005, 11: 149-158. 10.1080/13873950500068542
Leine RI, van de Wouw N: Uniform convergence of monotone measure differential inclusions: with application to the control of mechanical systems with unilateral constraints. Int. J. Bifurc. Chaos 2008, 18: 1435-1457. 10.1142/S0218127408021099
Sesekin AN, Zavalishchin ST: Dynamic Impulse Systems. Kluwer Academic, Dordrecht; 1997.
Kronig R, Penney W: Quantum mechanics in crystal lattices. Proc. R. Soc. Lond. 1931, 130: 499-513. 10.1098/rspa.1931.0019
Atkinson FV: Discrete and Continuous Boundary Problems. Academic Press, New York; 1964.
Sharma RR: An abstract measure differential equation. Proc. Am. Math. Soc. 1972, 32: 503-510. 10.1090/S0002-9939-1972-0291600-3
Shendge GR, Joshi SR: Abstract measure differential inequalities and applications. Acta Math. Hung. 1983, 41: 53-59. 10.1007/BF01994061
Dhage BC, Bellale SS: Existence theorems for perturbed abstract measure differential equations. Nonlinear Anal. 2009, 71: 319-328. 10.1016/j.na.2008.11.057
Schwabik Š, Tvrdý M, Vejvoda O: Differential and Integral Equations. Boundary Problems and Adjoints. Reidel, Dordrecht; 1979.
Wyderka Z: Linear Differential Equations with Measures as Coefficients and Control Theory. Wyd. Uniw. Ślaskiego, Katowice; 1994.
Wyderka Z: Linear differential equations with measures as coefficients and the control theory. Čas. Pěst. Mat. 1989, 114: 13-27.
Federson M, Mesquita JG, Slavik A: Measure functional differential equations and functional dynamic equations on time scales. J. Differ. Equ. 2012, 252: 3816-3847. 10.1016/j.jde.2011.11.005
Wan X, Sun J: Existence of solutions for perturbed abstract measure functional differential equations. Adv. Differ. Equ. 2011., 2011: Article ID 67
Tvrdý, M: Differential and Integral Equations in the Space of Regulated Functions. Habil. thesis, Praha (2001)
Çamlibel, MK, Heemels, WPMH, van der Schaft, AJ, Schumacher, JM: Solution concepts for hybrid dynamical systems. IFAC (2002)
Heemels WPMH, Camlibel MK, van der Schaft AJ, Schumacher JM: On the existence and uniqueness of solution trajectories to hybrid dynamical systems. Nonlinear and Hybrid Control in Automotive Applications 2002, 391-422.
Bogachev VI: Measure Theory. Springer, Berlin; 2007.
Johnson RA: Atomic and nonatomic measures. Proc. Am. Math. Soc. 1970, 25: 650-655. 10.1090/S0002-9939-1970-0279266-8
Alberti G, Mantegazza C: A note on the theory of SBV functions. Boll. Unione Mat. Ital., B 1997, 11: 375-382.
Aliprantis CD, Border KC: Infinite Dimensional Analysis. Springer, Berlin; 2006.
Bruckner AM, Bruckner JB, Thomson BS: Real Analysis. Prentice Hall, New York; 1997.
Taylor SJ: Introduction to Measure and Integration. Cambridge University Press, Cambridge; 1966.
Fremlin DH 2. In Measure Theory. Torres Fremlin, Colchester; 2003.
Persson J: Fundamental theorems for linear measure differential equations. Math. Scand. 1988, 62: 19-43.
Persson J: Regularization of nonlinear measure differential equations. Matematiche 1989, 44: 113-130.
Aubin J-P, Frankowska H: Set-Valued Analysis. Birkhäuser, Boston; 1990.
Zygmunt W: On superpositionally measurable semi-Carathéodory multifunctions. Comment. Math. Univ. Carol. 1992, 33: 73-77.
Wagner DH: Survey of measurable selection theorems: an update. Lecture Notes in Math. 794. In Measure Theory, Oberwolfach 1979. Springer, Berlin; 1980:176-219. Proc. Conf. Oberwolfach 1979
Castaing C, Valadier M Lecture Notes in Math. 580. In Convex Analysis and Measurable Multifunctions. Springer, Berlin; 1977.
Carter M, van Brunt B: The Lebesgue-Stieltjes Integral - A Practical Introduction. Springer, Berlin; 2000.
Diestel J, Ruess WM, Schachermayer W:Weak compactness in L 1 (μ,X) . Proc. Am. Math. Soc. 1993, 118: 447-453.
Aubin J-P, Cellina A: Differential Inclusions. Springer, Berlin; 1984.
Miller B, Rubinovitch EY: Impulsive Control in Continuous and Discrete-Continuous Systems. Kluwer Academic, Dordrecht; 2003.
Seifert C: Gordon type theorem for measure perturbation. Electron. J. Differ. Equ. 2011., 2011: Article ID 111
Constantin A: Global existence of solutions for perturbed differential equations. Ann. Mat. Pura Appl. (4) 1995, CLXVIII: 237-299.
Cichoń M: Differential inclusions and abstract control problems. Bull. Aust. Math. Soc. 1996, 53: 109-122. 10.1017/S0004972700016774
Or Y, Teel AR: Zeno stability of the set-valued bouncing ball. IEEE Trans. Autom. Control 2011, 56: 447-452.
Dal Maso G, Rampazzo F: On systems of ordinary differential equations with measures as controls. Differ. Integral Equ. 1991, 4: 739-765.
Schechter E: A survey of local existence theories for abstract nonlinear initial value problems. Lecture Notes in Math. 1394. In Nonlinear Semigroups, Partial Differential Equations and Attractors. Springer, Berlin; 1989:136-184.
Miller B: Representation of robust and non-robust solutions of nonlinear discrete-continuous systems. Lecture Notes in Computer Science 1201. Hybrid and Real-Time Systems 1997, 228-239.
Sesekin AN: On sets of discontinuous solutions of nonlinear differential equations. Izv. Vysš. Učebn. Zaved., Mat. 1994, 6: 83-89. (in Russian)
Code, WJ: Measure-Driven Impulsive Systems Stabilization, Optimal Control and Applications. PhD thesis, Vancouver (2009)
Atici FM, Biles DC: First order dynamic inclusions on time scales. J. Math. Anal. Appl. 2004, 292: 222-237. 10.1016/j.jmaa.2003.11.053
Frigon M, Gilbert H: Systems of first order inclusions on time scales. Topol. Methods Nonlinear Anal. 2011, 37: 147-163.
Agarwal RP, O’Regan D, Lakshmikantham V: Discrete second order inclusions. J. Differ. Equ. Appl. 2003, 9: 879-885. 10.1080/1023619031000097044
Smirnov GV: Discrete approximations and optimal solutions of differential inclusions. Cybern. Syst. Anal. 1991, 27: 101-107.
Kloeden PE, Marín-Rubio P: Weak pullback attractors of non-autonomous difference inclusions. J. Differ. Equ. Appl. 2003, 9: 489-502. 10.1080/1023619031000076515
Dontchev A, Lempio F: Difference methods for differential inclusions: a survey. SIAM Rev. 1992, 34: 263-294. 10.1137/1034050
Baier R, Donchev T: Discrete approximation of impulsive differential inclusions. Numer. Funct. Anal. Optim. 2010, 31: 653-678. 10.1080/01630563.2010.483878
