## Abstract and Applied Analysis

### Generalized Solutions of Functional Differential Inclusions

#### Abstract

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.

#### Article information

Source
Abstr. Appl. Anal., Volume 2008 (2008), Article ID 829701, 35 pages.

Dates
First available in Project Euclid: 9 September 2008

https://projecteuclid.org/euclid.aaa/1220969142

Digital Object Identifier
doi:10.1155/2008/829701

Mathematical Reviews number (MathSciNet)
MR2377427

Zentralblatt MATH identifier
1149.34037

#### Citation

Machina, Anna; Bulgakov, Aleksander; Grigorenko, Anna. Generalized Solutions of Functional Differential Inclusions. Abstr. Appl. Anal. 2008 (2008), Article ID 829701, 35 pages. doi:10.1155/2008/829701. https://projecteuclid.org/euclid.aaa/1220969142

#### References

• 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.
• 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.
• 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).
• A. F. Filippov, Differential Equations with Discontinuous Right-Hand Sides, Nauka, Moscow, Russia, 1985.
• 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.
• 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.
• 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.
• 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.
• T. Ważewski, “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.
• 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.
• A. Bressan, “On a bang-bang principle for nonlinear systems,” Bollettino della Unione Matemàtica Italiana. Supplemento, no. 1, pp. 53–59, 1980.
• 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.
• G. Pianigiani, “On the fundamental theory of multivalued differential equations,” Journal of Differential Equations, vol. 25, no. 1, pp. 30–38, 1977.
• 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).
• 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. skip=.5pt
• 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.
• J. Lygeros, C. Tomlin, and S. Sastry, “Controllers for reachability specifications for hybrid systems,” Automatica, vol. 35, no. 3, pp. 349–370, 1999.
• 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.
• 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.
• 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.
• A. D. Ioffe and V. M. Tikhomirov, Theory of External Problems, Nauka, Moscow, Russia, 1974.
• I. P. Natanson, Theory of Functions of a Real Variable, Nauka, Moscow, Russia, 3rd edition, 1974.
• J.-P. Aubin and A. Cellina, Differential Inclusions: Set-Valued Maps and Viability Theory, vol. 264, Springer, Berlin, Germany, 1984.
• 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.
• A. A. Tolstonogov, Differential Inclusions in a Banach Space, Nauka, Novosibirsk, Russia, 1986.
• 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).
• A. Bressan and G. Colombo, “Extensions and selections of maps with decomposable values,” Studia Mathematica, vol. 90, no. 1, pp. 69–86, 1988.
• A. Fryszkowski, “Continuous selections for a class of nonconvex multivalued maps,” Studia Mathematica, vol. 76, no. 2, pp. 163–174, 1983.
• 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.
• 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.
• 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.
• 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.
• 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.
• A. V. Arutyunov, Optimality Conditions: Abnormal and Degenerate Problems, vol. 526 of Mathematics and Its Applications, Kluwer Academic Publishers, Dordrecht, The Netherlands, 2000.
• 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.
• 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.
• H. Hermes, “The generalized differential equation $\stackrel\dotx\inR(t,x)$,” Advances in Mathematics, vol. 4, pp. 149–169, 1970.
• 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.