## Generalized solutions of functional differential inclusions.(English)Zbl 1149.34037

Summary: We consider the initial value problem for a functional differential inclusion with a Volterra multivalued mapping that is not necessarily decomposable in $$L_{1}^{n}[a,b]$$. The concept of the decomposable hull of a set is introduced. Using this concept, we define a generalized solution of such a problem and study its properties. We have proven that standard results on local existence and continuation of a generalized solution remain true. The question on the estimation of a generalized solution with respect to a given absolutely continuous function is studied. The density principle is proven for the generalized solutions. Asymptotic properties of the set of generalized approximate solutions are studied.

### MSC:

 34K05 General theory of functional-differential equations
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### References:

 [1] A. I. Bulgakov and L. I. Tkach, “Perturbation of a convex-valued operator by a Hammerstein-type multivalued mapping with nonconvex images, and boundary value problems for functional-differential inclusions,” Matematicheskiĭ Sbornik, vol. 189, no. 6, pp. 3-32, 1998, English translation in Sbornik. Mathematics, vol. 189, no. 5-6, pp. 821-848, 1998. · Zbl 0920.34017 [2] A. I. Bulgakov and L. I. Tkach, “Perturbation of a single-valued operator by a multi-valued mapping of Hammerstein type with nonconvex images,” Izvestiya Vysshikh Uchebnykh Zavedeniĭ. Matematika, no. 3, pp. 3-16, 1999, English translation in Russian Mathematics, vol. 43, no. 3, pp. 1-13, 1999. · Zbl 0999.47035 [3] A. F. Filippov, “Classical solutions of differential equations with the right-hand side multi-valued,” Vestnik Moskovskogo Universiteta. Serija I. Matematika, Mehanika, vol. 22, no. 3, pp. 16-26, 1967 (Russian). [4] A. F. Filippov, Differential Equations with Discontinuous Right-Hand Sides, Nauka, Moscow, Russia, 1985. · Zbl 0571.34001 [5] A. I. Bulgakov, “Asymptotic representation of sets of \delta -solutions of a differential inclusion,” Matematicheskie Zametki, vol. 65, no. 5, pp. 775-779, 1999, English translation in Mathematical Notes, vol. 65, no. 5-6, pp. 649-653, 1999. · Zbl 0961.34052 [6] A. I. Bulgakov, O. P. Belyaeva, and A. A. Grigorenko, “On the theory of perturbed inclusions and its applications,” Matematicheskiĭ Sbornik, vol. 196, no. 10, pp. 21-78, 2005, English translation in Sbornik. Mathematics, vol. 196, no. 9-10, pp. 1421-1472, 2005. · Zbl 1144.47044 [7] A. I. Bulgakov, A. A. Efremov, and E. A. Panasenko, “Ordinary differential inclusions with internal and external perturbations,” Differentsial’nye Uravneniya, vol. 36, no. 12, pp. 1587-1598, 2000, English translation in Differential Equations, vol. 36, no. 12, pp. 1741-1753, 2000. · Zbl 0997.34009 [8] A. I. Bulgakov and V. V. Skomorokhov, “Approximation of differential inclusions,” Matematicheskiĭ Sbornik, vol. 193, no. 2, pp. 35-52, 2002, English translation in Sbornik. Mathematics, vol. 193, no. 1-2, pp. 187-203, 2002. · Zbl 1034.34015 [9] T. Wa\Dzewski, “Sur une généralisation de la notion des solutions d’une équation au contingent,” Bulletin de l’Académie Polonaise des Sciences. Série des Sciences Mathématiques, Astronomiques et Physiques, vol. 10, pp. 11-15, 1962. · Zbl 0104.30404 [10] V. I. Blagodatskikh and A. F. Filippov, “Differential inclusions and optimal control,” Trudy Matematicheskogo Instituta Imeni V. A. Steklova, vol. 169, pp. 194-252, 1985, English translation in Proceedings of the Steklov Institute of Mathematics, vol. 169, 1986. · Zbl 0595.49026 [11] A. Bressan, “On a bang-bang principle for nonlinear systems,” Bollettino della Unione Matemàtica Italiana. Supplemento, no. 1, pp. 53-59, 1980. · Zbl 0445.49012 [12] A. E. Irisov and E. L. Tonkov, “On the closure of the set of periodic solutions of a differential inclusion,” in Differential and Integral Equations, pp. 32-38, Gor’ kov. Gos. Univ., Gorki, Russia, 1983. [13] G. Pianigiani, “On the fundamental theory of multivalued differential equations,” Journal of Differential Equations, vol. 25, no. 1, pp. 30-38, 1977. · Zbl 0398.34017 [14] L. N. Lyapin and Yu. L. Muromtsev, “Guaranteed optimal control on a set of operative states,” Automation and Remote Control, vol. 54, no. 3, part 1, pp. 421-429, 1993 (Russian). · Zbl 0817.93016 [15] M. S. Branicky, V. S. Borkar, and S. K. Mitter, “A unified framework for hybrid control: model and optimal control theory,” IEEE Transactions on Automatic Control, vol. 43, no. 1, pp. 31-45, 1998. · Zbl 0951.93002 [16] R. W. Brockett, “Hybrid models for motion control systems,” in Essays on Control: Perspectives in the Theory and Its Applications (Groningen, 1993), H. Trentelman and J. C. Willems, Eds., vol. 14 of Progress in Systems Control Theory, pp. 29-53, Birkhäuser, Boston, Mass, USA, 1993. · Zbl 0844.93011 [17] J. Lygeros, C. Tomlin, and S. Sastry, “Controllers for reachability specifications for hybrid systems,” Automatica, vol. 35, no. 3, pp. 349-370, 1999. · Zbl 0943.93043 [18] A. Puri and P. Varaiya, “Decidability of hybrid systems with rectangular differential inclusions,” in Computer Aided Verification (Stanford, CA, 1994), D. Dill, Ed., vol. 1066 of Lecture Notes in Computer Science, pp. 95-104, Springer, Berlin, Germany, 1994. [19] A. J. Van der Schaft and J. M. Schumacher, An Introduction to Hybrid Dynamical Systems, vol. 251 of Springer Lecture Notes in Control and Information Sciences, Springer, London, UK, 2000. · Zbl 0940.93004 [20] P. Varaiya and A. Kurzhanski, “On problems of dynamics and control for hybrid systems,” in Control Theory and Theory of Generalized Solutions of Hamilton Jacobi Equations. Proceedings of International Seminars, vol. 1, pp. 21-37, Ural University, Ekaterinburg, Russia, 2006. [21] A. D. Ioffe and V. M. Tikhomirov, Theory of External Problems, Nauka, Moscow, Russia, 1974. [22] I. P. Natanson, Theory of Functions of a Real Variable, Nauka, Moscow, Russia, 3rd edition, 1974. [23] J.-P. Aubin and A. Cellina, Differential Inclusions: Set-Valued Maps and Viability Theory, vol. 264, Springer, Berlin, Germany, 1984. · Zbl 0538.34007 [24] M. Kamenskii, V. Obukhovskii, and P. Zecca, Condensing Multivalued Maps and Semilinear Differential Inclusions in Banach Spaces, vol. 7 of de Gruyter Series in Nonlinear Analysis and Applications, Walter de Gruyter, Berlin, Germany, 2001. · Zbl 0988.34001 [25] A. A. Tolstonogov, Differential Inclusions in a Banach Space, Nauka, Novosibirsk, Russia, 1986. · Zbl 0689.34014 [26] A. N. Tikhonov, “On Volterra type functional equations and their applications in some problems of mathematical physics,” Bulletin of Moscow University, Section A, vol. 1, no. 8, pp. 1-25, 1938 (Russian). [27] A. Bressan and G. Colombo, “Extensions and selections of maps with decomposable values,” Studia Mathematica, vol. 90, no. 1, pp. 69-86, 1988. · Zbl 0677.54013 [28] A. Fryszkowski, “Continuous selections for a class of nonconvex multivalued maps,” Studia Mathematica, vol. 76, no. 2, pp. 163-174, 1983. · Zbl 0534.28003 [29] A. I. Bulgakov, “A functional-differential inclusion with an operator that has nonconvex images,” Differentsial’nye Uravneniya, vol. 23, no. 10, pp. 1659-1668, 1987, English translation in Differential Equations, vol. 23, 1987. · Zbl 0715.34023 [30] A. I. Bulgakov, “Continuous branches of multivalued functions, integral inclusions with nonconvex images, and their applications. I,” Differentsial’nye Uravneniya, vol. 28, no. 3, pp. 371-379, 1992. · Zbl 0801.34017 [31] A. I. Bulgakov, “Continuous branches of multivalued functions, integral inclusions with nonconvex images, and their applications. II,” Differentsial’nye Uravneniya, vol. 28, pp. 566-571, 1992. · Zbl 0801.34018 [32] A. I. Bulgakov, “Continuous branches of multivalued mappings, and integral inclusions with nonconvex images and their applications. III,” Differentsial’nye Uravneniya, vol. 28, no. 5, pp. 739-746, 1992. · Zbl 0801.34019 [33] A. Turowicz, “Remarque sur la définition des quasitrajectoires d’un système de commande nonlinéaire,” Bulletin de l’Académie Polonaise des Sciences. Série des Sciences Mathématiques, Astronomiques et Physiques, vol. 11, pp. 367-368, 1963. · Zbl 0121.07002 [34] A. Pliś, “Trajectories and quasitrajectories of an orientor field,” Bulletin de l’Académie Polonaise des Sciences. Série des Sciences Mathématiques, Astronomiques et Physiques, vol. 11, pp. 369-370, 1963. · Zbl 0124.29404 [35] A. I. Bulgakov, “Integral inclusions with nonconvex images and their applications to boundary value problems for differential inclusions,” Matematicheskiĭ Sbornik, vol. 183, no. 10, pp. 63-86, 1992, English translation in Russian Academy of Sciences. Sbornik. Mathematics, vol. 77, no. 1, pp. 193-212, 1994. · Zbl 0789.34020 [36] A. I. Bulgakov and V. P. Maksimov, “Functional and functional-differential inclusions with Volterra operators,” Differential Equations, vol. 17, no. 8, pp. 881-890, 1981. · Zbl 0481.34042 [37] A. V. Arutyunov, Optimality Conditions: Abnormal and Degenerate Problems, vol. 526 of Mathematics and Its Applications, Kluwer Academic Publishers, Dordrecht, The Netherlands, 2000. · Zbl 0987.49001 [38] A. A. Tolstonogov and P. I. Chugunov, “The solution set of a differential inclusion in a Banach space. I,” Sibirskiĭ Matematicheskiĭ Zhurnal, vol. 24, no. 6, pp. 144-159, 1983, English translation in Siberian Mathematical Journal, vol. 24, no. 6, pp. 941-954, 1983. · Zbl 0537.34011 [39] A. A. Tolstonogov and I. A. Finogenko, “Solutions of a differential inclusion with lower semicontinuous nonconvex right-hand side in a Banach space,” Matematicheskiĭ Sbornik, vol. 125(167), no. 2, pp. 199-230, 1984, English translation in Sbornik. Mathematics, vol. 53, no. 1, pp. 203-231, 1986. · Zbl 0588.34012 [40] O. Hájek, “Discontinuous differential equations. I,” Journal of Differential Equations, vol. 32, no. 2, pp. 149-170, 1979. · Zbl 0365.34017 [41] O. Hájek, “Discontinuous differential equations. II,” Journal of Differential Equations, vol. 32, no. 2, pp. 171-185, 1979. · Zbl 0681.34009 [42] H. Hermes, “The generalized differential equation x\?\in R(t,x),” Advances in Mathematics, vol. 4, pp. 149-169, 1970. · Zbl 0191.38803 [43] H. Hermes, “On continuous and measurable selections and the existence of solutions of generalized differential equations,” Proceedings of the American Mathematical Society, vol. 29, pp. 535-542, 1971. · Zbl 0214.09802
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