Measure differential inclusions: existence results and minimum problems. (English) Zbl 1479.34008
Summary: We focus on a very general problem in the theory of dynamic systems, namely that of studying measure differential inclusions with varying measures. The multifunction on the right hand side has compact non-necessarily convex values in a real Euclidean space and satisfies bounded variation hypotheses with respect to the Pompeiu excess (and not to the Hausdorff-Pompeiu distance, as usually in the literature). This is possible due to the use of interesting selection principles for excess bounded variation set-valued mappings. Conditions for the minimization of a generic functional with respect to a family of measures generated by equiregulated left-continuous, nondecreasing functions and to associated solutions of the differential inclusion driven by these measures are deduced, under constraints only on the initial point of the trajectory.
MSC:
| 34A06 | Generalized ordinary differential equations (measure-differential equations, set-valued differential equations, etc.) |
| 34A60 | Ordinary differential inclusions |
Keywords:
measure differential inclusion; bounded variation; Pompeiu excess; selection; minimality conditionReferences:
| [1] | Soledad Aronna, M.; Rampazzo, F., L1 limit solutions for control systems, J. Differential Equations, 258, 954-979 (2015) · Zbl 1323.34076 · doi:10.1016/j.jde.2014.10.013 |
| [2] | Aubin, J-P; Frankowska, H., Set-valued Analysis (1990), Boston: Birkhäuser, Boston · Zbl 0713.49021 |
| [3] | Aye, KK; Lee, PY, The dual of the space of bounded variation, Math. Bohem., 131, 1, 1-9 (2006) · Zbl 1112.26008 · doi:10.21136/MB.2006.134078 |
| [4] | Belov, SA; Chistyakov, VV, A selection principle for mappings of bounded variation, J. Math. Anal. Appl., 249, 351-366 (2000) · Zbl 0982.54022 · doi:10.1006/jmaa.2000.6844 |
| [5] | Billingsley, P.: Weak convergence of measures: Applications in probability. In: CBMS-NSF Regional Conference Series in Applied Mathematics (1971) · Zbl 0271.60009 |
| [6] | Boccuto, A.; Candeloro, D.; Sambucini, AR, Stieltjes-type integrals for metric semigroup-valued functions defined on unbounded intervals, PanAm. Math. J., 17, 4, 39-58 (2007) · Zbl 1132.28317 |
| [7] | Çamlibel, M.K., Heemels, W.P.M.H., van der Schaft, A.J., Schumacher, J.M.: On the existence and uniqueness of solution trajectories to hybrid dynamical systems. In: Nonlinear and Hybrid Systems in Automotive Applications, pp 391-422 (2002) |
| [8] | Castaing, C.; Valadier, M., Convex Analysis and Measurable Multifunctions Lecture Notes in Math, vol. 580 (1977), Berlin: Springer, Berlin · Zbl 0346.46038 · doi:10.1007/BFb0087685 |
| [9] | Cichoń, M.; Satco, B., Measure differential inclusions - between continuous and discrete, Adv. Diff. Equations, 56, 18 (2014) · Zbl 1350.49014 · doi:10.1186/1687-1847-2014-56 |
| [10] | Cichoń, M., Satco, B.: On the properties of solutions set for measure driven differential inclusions. Discrete and continuous dynamical systems. Special Issue: SI, pp. 287-296 (2015) · Zbl 1339.34033 |
| [11] | Cichoń, M.; Satco, B.; Sikorska-Nowak, A., Impulsive nonlocal differential equations through differential equations on time scales, Appl. Math. Comp., 218, 2449-245 (2011) · Zbl 1247.34138 · doi:10.1016/j.amc.2011.07.057 |
| [12] | Chistyakov, VV, Asymmetric variations of multifunctions with application to functional inclusions, J. Math. Anal. Appl., 478, 2, 421-444 (2019) · Zbl 1421.49017 · doi:10.1016/j.jmaa.2019.05.035 |
| [13] | Chistyakov, VV; Repovš, D., Selections of bounded variation under the excess restrictions, J. Math. Anal. Appl., 331, 873-885 (2007) · Zbl 1149.26023 · doi:10.1016/j.jmaa.2006.09.004 |
| [14] | Code, WJ; Loewen, PD, Optimal control of non-convex measure-driven differential inclusion, Set-Valued Anal., 19, 203-235 (2010) · Zbl 1214.49036 · doi:10.1007/s11228-010-0138-8 |
| [15] | Dal Maso, G.; Rampazzo, F., On System of ordinary differential equations with measures as controls, Differ. Integral Equ., 4, 4, 739-765 (1991) · Zbl 0731.34087 |
| [16] | Denkowski, Z., Migorski, S., Papageorgiou, N.S.: An introduction to nonlinear analysis: Theory. Kluwer Academic Publishers (2003) · Zbl 1040.46001 |
| [17] | Di Piazza, L.; Marraffa, V.; Satco, B., Closure properties for integral problems driven by regulated functions via convergence results, J. Math. Anal. Appl., 466, 690-710 (2018) · Zbl 1392.26014 · doi:10.1016/j.jmaa.2018.06.012 |
| [18] | Di Piazza, L.; Marraffa, V.; Satco, B., Approximating the solutions of differential inclusions driven by measures, Ann. Mat. Pura Appl., 198, 2123-2140 (2019) · Zbl 1440.34005 · doi:10.1007/s10231-019-00857-6 |
| [19] | Federson, M.; Mesquita, JG; Slavík, A., Measure functional differential equations and functional dynamic equations on time scales, J. Diff. Equations, 252, 3816-3847 (2012) · Zbl 1239.34076 · doi:10.1016/j.jde.2011.11.005 |
| [20] | Fraňková, D., Regulated functions, Math. Bohem., 116, 1, 20-59 (1991) · Zbl 0724.26009 · doi:10.21136/MB.1991.126195 |
| [21] | Frigon, M.; López Pouso, R., Theory and applications of first-order systems of Stieltjes differential equations, Adv. Nonlinear Anal., 6, 13-36 (2017) · Zbl 1361.34010 · doi:10.1515/anona-2015-0158 |
| [22] | Goebel, R.; Teel, AR, Solutions to hybrid inclusions via set and graphical convergence with stability theory applications, Automatica, 42, 573-587 (2006) · Zbl 1106.93042 · doi:10.1016/j.automatica.2005.12.019 |
| [23] | Karamzin, DY; De Oliveira, VA; Pereira, FL; Silva, GN, On the properness of an impulsive control extension of dynamic optimization problems, ESAIM: COCV, 21, 857-875 (2015) · Zbl 1318.49068 |
| [24] | Kurzweil, J., Generalized ordinary differential equations and continuous dependence on a parameter, Czechoslovak Math. J., 7, 82, 418-449 (1957) · Zbl 0090.30002 · doi:10.21136/CMJ.1957.100258 |
| [25] | Miller, B.; Rubinovitch, EY, Impulsive Control in Continuous and Discrete-Continuous Systems (2003), Dordrecht: Kluwer Academic Publishers, Dordrecht · Zbl 1065.49022 · doi:10.1007/978-1-4615-0095-7 |
| [26] | Monteiro, GA; Slavik, A., Extremal solutions of measure differential equations, J. Math. Anal. Appl., 444, 568-597 (2016) · Zbl 1356.34094 · doi:10.1016/j.jmaa.2016.06.035 |
| [27] | Monteiro, G.A., Slavik, A., Tvrdy, M.: Kurzweil-Stieltjes integral. Theory and its Applications, World Scientific Series in real analysis. Vol. 15 (2018) · Zbl 1437.28001 |
| [28] | Motta, M.; Sartori, C., On l1 limit solutions in impulsive control, DCDIS Series S, 11, 6, 1201-1218 (2018) · Zbl 1407.49055 |
| [29] | Murray, JM, Existence theorems for optimal control and calculus of variations problems where the states can jump, Siam J. Control Optim., 24, 412-438 (1986) · Zbl 0587.49004 · doi:10.1137/0324024 |
| [30] | Phillips, GM, Interpolation and Approximation by Polynomials (2003), Berlin: Springer, Berlin · Zbl 1023.41002 · doi:10.1007/b97417 |
| [31] | Raymond, JP, Optimal control problems in spaces of functions of bounded variation, Differential Integral Equations, 10, 1, 105-136 (1997) · Zbl 0879.49003 |
| [32] | Raymond, JP; Seghir, D., Existence and characterization of BV-curves for problems of calculus of variations, Nonlinear Anal. T.M.A., 28, 109-1132 (1997) · Zbl 0892.49005 · doi:10.1016/S0362-546X(97)82863-5 |
| [33] | Saks, S., Theory of the Integral (1937), Warszawa: Monografie Matematyczne, Warszawa · Zbl 0017.30004 |
| [34] | Satco, B., Continuous dependence results for set-valued measure differential problems, Electr. Jour. Qualit. Th. Diff. Equat., 79, 1-15 (2015) · Zbl 1349.34051 |
| [35] | Satco, B.: Nonlinear Volterra integral equations in Henstock integrability setting. Electr. J. Diff. Equ., 39 (2008) · Zbl 1169.45300 |
| [36] | Schwabik, Š.: Generalized ordinary differential equations. World Scientific (1992) · Zbl 0781.34003 |
| [37] | Schwabik, Š.; Tvrdý, M.; Vejvoda, O., Differential and Integral Equations (1979), Dordrecht, Praha: Boundary Problems and Adjoints, Dordrecht, Praha · Zbl 0417.45001 |
| [38] | Silva, GN; Vinter, RB, Measure driven differential inclusions, J. Math. Anal. Appl., 202, 727-746 (1996) · Zbl 0877.49004 · doi:10.1006/jmaa.1996.0344 |
| [39] | Silva, GN; Vinter, RB, Necessary conditions for optimal impulsive control problems, SIAM J. Control Optim., 35, 6, 1829-1846 (1997) · Zbl 0886.49028 · doi:10.1137/S0363012995281857 |
| [40] | Slavík, A., Well-posedness results for abstract generalized differential equations and measure functional differential equations, J. Differential Equations, 259, 666-707 (2015) · Zbl 1319.34116 · doi:10.1016/j.jde.2015.02.013 |
| [41] | Taylor, A.E.: General Theory of Functions and Integration. Dover Books on Mathematics Series (2012) |
| [42] | Toth, G., Measures of Symmetry for Convex Sets and Stability (2015), New York: Springer, New York · Zbl 1335.52002 · doi:10.1007/978-3-319-23733-6 |
| [43] | Tvrdý, M., Differential and Integral Equations in the Space of Regulated Functions (2001), Praha: Habil, Thesis, Praha · Zbl 1081.34504 |
| [44] | Vinter, R.: Optimal control. Birkhäuser (2000) · Zbl 0952.49001 |
| [45] | Wiweger, A., Linear spaces with mixed topology, Studia Math., 20, 1, 47-68 (1961) · Zbl 0097.31301 · doi:10.4064/sm-20-1-47-68 |
| [46] | Zavalishchin, ST; Sesekin, AN, Dynamic Impulse Systems (1997), Dordrecht: Kluwer Academic, Dordrecht · Zbl 0880.46031 · doi:10.1007/978-94-015-8893-5 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
