Sobolev Type Fractional Dynamic Equations and Optimal Multi-Integral Controls with Fractional Nonlocal Conditions

References

[1] B. Ahmad, J. J. Nieto, Existence of solutions for anti-periodic boundary value problems involving fractional differential equations via Leray-Schauder degree theory. Topol. Methods Nonlinear Anal. 35, No 2 (2010), 295-304.Suche in Google Scholar

[2] D. Baleanu, K. Diethelm, E. Scalas, J. J. Trujillo, Fractional Calculus: Models and Numerical Methods. Ser. on Complexity, Nonlinearity and Chaos, 3, World Scientific Publishing Co. Pte. Ltd., Singapore-Hackensack, NJ (2012).Suche in Google Scholar

[3] E. Bazhlekova, Existence and uniqueness results for a fractional evolution equation in Hilbert space. Fract. Calc. Appl. Anal. 15, No 2 (2012), 232-243; DOI: 10.2478/s13540-012-0017-0; http://link.springer.com/article/10.2478/s13540-012-0017-0.Suche in Google Scholar

[4] M. Benchohra, E. P. Gatsori, S. K. Ntouyas, Controllability results for semilinear evolution inclusions with nonlocal conditions. J. Optim. Theory Appl. 118, No 3 (2003), 493-513.Suche in Google Scholar

[5] A. Debbouche, D. Baleanu, Controllability of fractional evolution nonlocal impulsive quasilinear delay integro-differential systems. Comput. Math. Appl. 62, No 3 (2011), 1442-1450.Suche in Google Scholar

[6] A. Debbouche, D. Baleanu, R. P. Agarwal, Nonlocal nonlinear integrodifferential equations of fractional orders. Bound. Value Probl. 2012, No 78 (2012), 10 pp.10.1186/1687-2770-2012-78Suche in Google Scholar

[7] A. Debbouche, D. F.M. Torres, Approximate controllability of fractional nonlocal delay semilinear systems in Hilbert spaces. Internat. J. Control86, No 9 (2013), 1577-1585.Suche in Google Scholar

[8] M. Feockan, J. Wang, Y. Zhou, Controllability of fractional functional evolution equations of Sobolev type via characteristic solution operators. J. Optim. Theory Appl. 156, No 1 (2013), 79-95.Suche in Google Scholar

[9] G. F. Franklin, J. D. Powell, A. Emami-Naeini, Feedback Control of Dynamic Systems. Prentice Hall, 6th Ed. (2010).Suche in Google Scholar

[10] A. Harrat, A. Debbouche, Sobolev type fractional delay impulsive equations with Alpha-Sobolev resolvent families and integral conditions. Nonlinear Stud. 20, No 4 (2013), 549-558.Suche in Google Scholar

[11] R. Hilfer, Applications of Fractional Calculus in Physics. World Sci. Publishing, River Edge, NJ (2000).10.1142/3779Suche in Google Scholar

[12] E. Hille, R. S. Phillips, Functional Analysis and Semi-Groups. American Math. Soc. Colloquium Publ., Vol. 31, Amer. Math. Soc., Providence, RI (1957).Suche in Google Scholar

[13] S. Hu, N. S. Papageorgiou, Handbook of Multivalued Analysis. Vol. I. Mathematics and its Applications, 419, Kluwer Acad. Publ., Dordrecht (1997).10.1007/978-1-4615-6359-4Suche in Google Scholar

[14] M. Jelassi, H. Mejjaoli, Fractional Sobolev type spaces associated with a singular differential operator and applications. Fract. Calc. Appl. Anal. 17, No 2 (2014), 401-423; DOI: 10.2478/s13540-014-0177-1; http://link.springer.com/article/10.2478/s13540-014-0177-1.Suche in Google Scholar

[15] M. Kerboua, A. Debbouche, D. Baleanu, Approximate controllability of Sobolev type nonlocal fractional stochastic dynamic systems in Hilbert spaces. Abstr. Appl. Anal. 2013, Art. ID 262191 (2013), 10 pp.10.1155/2013/262191Suche in Google Scholar

[16] A. A. Kilbas, H. M. Srivastava, J. J. Trujillo, Theory and Applications of Fractional Differential Equations. North-Holland Mathematics Studies, 204, Elsevier, Amsterdam (2006).Suche in Google Scholar

[17] M. Kirane, A. Kadem, A. Debbouche, Blowing-up solutions to twotimes No 17-18 (2013), 1797-1804.10.1002/mana.201200047Suche in Google Scholar

[18] V. Lakshmikantham, S. Leela, J. Vasundhara Devi, Theory of Fractional Dynamic Systems. Cambridge Scientific Publishers, Cambridge (2009).Suche in Google Scholar

[19] F. Li, J. Liang, H. K. Xu, Existence of mild solutions for fractional integrodifferential equations of Sobolev type with nonlocal conditions. J. Math. Anal. Appl. 391, No 2 (2012), 510-525.Suche in Google Scholar

[20] H. Liu, J.-C. Chang, Existence for a class of partial differential equations with nonlocal conditions. Nonlinear Anal. 70, No 9 (2009), 3076-3083.Suche in Google Scholar

[21] F. Mainardi, Fractional calculus: some basic problems in continuum and statistical mechanics. In: Fractals and Fractional Calculus in Continuum Mechanics (Udine, 1996), CISM Courses and Lectures, 378, Springer, Vienna (1997), 291-348.10.1007/978-3-7091-2664-6_7Suche in Google Scholar

[22] A. B. Malinowska, D. F.M. Torres, Introduction to the Fractional Calculus of Variations. Imp. Coll. Press, London (2012).10.1142/p871Suche in Google Scholar

[23] K. S. Miller, B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations. A Wiley-Interscience Publication, Wiley, New York (1993).Suche in Google Scholar

[24] G. M. Mophou, Weighted pseudo almost automorphic mild solutions to semilinear fractional differential equations. Appl. Math. Comput. 217, No 19 (2011), 7579-7587.Suche in Google Scholar

[25] D. Mozyrska, D. F.M. Torres, Minimal modified energy control for fractional linear control systems with the Caputo derivative. Carpathian J. Math. 26, No 2 (2010), 210-221.Suche in Google Scholar

[26] D. Mozyrska, D. F.M. Torres, Modified optimal energy and initial memory of fractional continuous-time linear systems. Signal Process. 91, No 3 (2011), 379-385.Suche in Google Scholar

[27] G. M. N'Gu'er'ekata, A Cauchy problem for some fractional abstract differential equation with non local conditions. Nonlinear Anal. 70, No 5 (2009), 1873-1876.Suche in Google Scholar

[28] M. D. Ortigueira, On the initial conditions in continuous-time fractional linear systems. Signal Process. 83, No 11 (2003), 2301-2309.Suche in Google Scholar

[29] N. Ozdemir, D. Karadeniz, B. B. M-I skender, Fractional optimal control problem of a distributed system in cylindrical coordinates. Phys. Lett. A373, No 2 (2009), 221-226.Suche in Google Scholar

[30] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations. Applied Mathematical Sciences, 44, Springer, New York (1983).10.1007/978-1-4612-5561-1Suche in Google Scholar

[31] I. Podlubny, Fractional Differential Equations. Mathematics in Science and Engineering, 198, Academic Press, San Diego, CA (1999).Suche in Google Scholar

[32] J. Sabatier, O. P. Agrawal, J. A. Tenreiro Machado, Advances in Fractional Calculus. Springer, Dordrecht (2007).10.1007/978-1-4020-6042-7Suche in Google Scholar

[33] R. Sakthivel, N. I. Mahmudov, J. J. Nieto, Controllability for a class of fractional-order neutral evolution control systems. Appl. Math. Comput. 218, No 20 (2012), 10334-10340.Suche in Google Scholar

[34] S. G. Samko, A. A. Kilbas, O. I. Marichev, Fractional Integrals and Derivatives. Gordon and Breach, Yverdon (1993); Transl. from the 1987 Russian original, Nauka i Tekhnika, Moscow (1987).Suche in Google Scholar

[35] M. R. Sidi Ammi, E. H. El Kinani, D. F.M. Torres, Existence and uniqueness of solutions to functional integro-differential fractional equations. Electron. J. Differential Equations2012, No 103 (2012), 9 pp.Suche in Google Scholar

[36] J. Wang, Y. Zhou, A class of fractional evolution equations and optimal controls. Nonlinear Anal. Real World Appl. 12, No 1 (2011), 262-272.Suche in Google Scholar

[37] J. R. Wang, Y. Zhou, W. Wei, H. Xu, Nonlocal problems for fractional integrodifferential equations via fractional operators and optimal controls. Comput. Math. Appl. 62, No 3 (2011), 1427-1441.Suche in Google Scholar

[38] S. D. Zaidman, Abstract Differential Equations. Research Notes in Mathematics, 36, Pitman, Boston, MA (1979).Suche in Google Scholar

[39] E. Zeidler, Nonlinear Functional Analysis and Its Applications. II/A. Transl. from German by the author and Leo F. Boron, Springer, New York (1990).10.1007/978-1-4612-0981-2Suche in Google Scholar

[40] L. Zhang, B. Ahmad, G. Wang, R. P. Agarwal, Nonlinear fractional integro-differential equations on unbounded domains in a Banach space. J. Comput. Appl. Math. 249 (2013), 51-56.Suche in Google Scholar

[41] X. Zhao, N. Duan, B. Liu, Optimal control problem of a generalized GinzburgLandau model equation in population problems. Math. Meth. Appl. Sci. 37, No 3 (2014), 435-446.Suche in Google Scholar

[42] Y. Zhou, F. Jiao, Nonlocal Cauchy problem for fractional evolution equations. Nonlinear Anal. Real World Appl. 11, No 5 (2010), 4465-4475.Suche in Google Scholar

[43] Y. Zhou, F. Jiao, Existence of mild solutions for fractional neutral evolution equations. Comput. Math. Appl. 59, No 3 (2010), 1063-1077.Suche in Google Scholar