On the solvability and optimal controls of fractional integrodifferential evolution systems with infinite delay. (English) Zbl 1357.49018

Wang, JinRong; Zhou, Yong; Medveď, Milan

J. Optim. Theory Appl. 152, No. 1, 31-50 (2012).

Summary: In this paper, we study the solvability and optimal controls of a class of fractional integrodifferential evolution systems with infinite delay in Banach spaces. Firstly, a more appropriate concept for mild solutions is introduced. Secondly, existence and continuous dependence of mild solutions are investigated by utilizing the techniques of a priori estimation and extension of step by steps. Finally, existence of optimal controls for system governed by fractional integrodifferential evolution systems with infinite delay is proved.

MSC:

49J21	Existence theories for optimal control problems involving relations other than differential equations
34A08	Fractional ordinary differential equations
34K37	Functional-differential equations with fractional derivatives

Keywords:

fractional integrodifferential evolution systems; infinite delay; solvability; continuous dependence; optimal controls

References:

[1]	Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and applications of fractional differential equations. In: North-Holland Mathematics Studies, vol. 204. Elsevier Science, Amsterdam (2006) · Zbl 1092.45003
[2]	Miller, K.S., Ross, B.: An Introduction to the Fractional Calculus and Differential Equations. Wiley, New York (1993) · Zbl 0789.26002
[3]	Podlubny, I.: Fractional Differential Equations. Academic Press, San Diego (1999) · Zbl 0924.34008
[4]	Lakshmikantham, V., Leela, S., Devi, J.V.: Theory of Fractional Dynamic Systems. Scientific Publishers Cambridge, Cambridge (2009) · Zbl 1188.37002
[5]	Agarwal, R.P., Benchohra, M., Hamani, S.: A survey on existence results for boundary value problems of nonlinear fractional differential equations and inclusions. Acta Appl. Math. 109, 973–1033 (2010) · Zbl 1198.26004 · doi:10.1007/s10440-008-9356-6
[6]	Agarwal, R.P., Belmekki, M., Benchohra, M.: A survey on semilinear differential equations and inclusions involving Riemann–Liouville fractional derivative. Adv. Differ. Equ. 2009, e1–e47 (2009) · Zbl 1182.34103
[7]	Agarwal, R.P., Zhou, Y., He, Y.: Existence of fractional neutral functional differential equations. Comput. Math. Appl. 59, 1095–1100 (2010) · Zbl 1189.34152 · doi:10.1016/j.camwa.2009.05.010
[8]	Belmekki, M., Benchohra, M.: Existence results for fractional order semilinear functional differential with nondense domain. Nonlinear Anal. TMA 72, 925–932 (2010) · Zbl 1179.26018 · doi:10.1016/j.na.2009.07.034
[9]	Benchohra, M., Henderson, J., Ntouyas, S.K., Ouahab, A.: Existence results for fractional order functional differential equations with infinite delay. J. Math. Anal. Appl. 338, 1340–1350 (2008) · Zbl 1209.34096 · doi:10.1016/j.jmaa.2007.06.021
[10]	Benchohra, M., Henderson, J., Ntouyas, S.K., Ouahab, A.: Existence results for fractional functional differential inclusions with infinite delay and application to control theory. Fract. Calc. Appl. Anal. 11, 35–56 (2008) · Zbl 1149.26010
[11]	Henderson, J., Ouahab, A.: Fractional functional differential inclusions with finite delay. Nonlinear Anal. TMA 70, 2091–2105 (2009) · Zbl 1159.34010 · doi:10.1016/j.na.2008.02.111
[12]	Hu, L., Ren, Y., Sakthivel, R.: Existence and uniqueness of mild solutions for semilinear integro-differential equations of fractional order with nonlocal initial conditions and delays. Semigroup Forum 79, 507–514 (2009) · Zbl 1184.45006 · doi:10.1007/s00233-009-9164-y
[13]	Mophou, G.M., N’Guérékata, G.M.: Existence of mild solutions of some semilinear neutral fractional functional evolution equations with infinite delay. Appl. Math. Comput. 216, 61–69 (2010) · Zbl 1191.34098 · doi:10.1016/j.amc.2009.12.062
[14]	Ren, Y., Qin, Y., Sakthivel, R.: Existence results for fractional order semilinear integro-differential evolution equations with infinite delay. Integral Equ. Oper. Theory 67, 33–49 (2010) · Zbl 1198.45009 · doi:10.1007/s00020-010-1767-x
[15]	Chang, Y.K., Kavitha, V., Arjunan, M.M.: Existence and uniqueness of mild solutions to a semilinear integrodifferential equation of fractional order. Nonlinear Anal. TMA 71, 5551–5559 (2009) · Zbl 1179.45010 · doi:10.1016/j.na.2009.04.058
[16]	Jaradat, O.K., Al-Omari, A., Momani, S.: Existence of the mild solution for fractional semilinear initial value problems. Nonlinear Anal. TMA 69, 3153–3159 (2008) · Zbl 1160.34300 · doi:10.1016/j.na.2007.09.008
[17]	El-Borai, M.M.: Some probability densities and fundamental solutions of fractional evolution equations. Chaos Solitons Fractals 14, 433–440 (2002) · Zbl 1005.34051 · doi:10.1016/S0960-0779(01)00208-9
[18]	El-Borai, M.M.: The fundamental solutions for fractional evolution equations of parabolic type. J. Appl. Math. Stoch. Anal. 3, 197–211 (2004) · Zbl 1081.34053 · doi:10.1155/S1048953304311020
[19]	Hernández, E., O’Regan, D., Balachandran, K.: On recent developments in the theory of abstract differential equations with fractional derivatives. Nonlinear Anal. TMA 73, 3462–3471 (2010) · Zbl 1229.34004 · doi:10.1016/j.na.2010.07.035
[20]	Wang, J., Zhou, Y.: Study of an approximation process of time optimal control for fractional evolution systems in Banach spaces. Adv. Differ. Equ. 2011, e1–e16 (2011) · Zbl 1222.49006
[21]	Wang, J., Yang, Y., Wei, W.: Nonlocal impulsive problems for fractional differential equations with time-varying generating operators in Banach spaces. Opusc. Math. 30, 361–381 (2010) · Zbl 1232.34013 · doi:10.7494/OpMath.2010.30.3.361
[22]	Wang, J., Wei, W., Yang, Y.: On some impulsive fractional differential equations in Banach spaces. Opusc. Math. 30, 507–525 (2010) · Zbl 1242.34011 · doi:10.7494/OpMath.2010.30.4.507
[23]	Wang, J., Wei, W., Yang, Y.: Fractional nonlocal integrodifferential equations and its optimal control in Banach spaces. J. SIAM 14, 79–91 (2010) · Zbl 1243.34007
[24]	Wang, J., Zhou, Y., Wei, W.: A class of fractional delay nonlinear integrodifferential controlled systems in Banach spaces. Commun. Nonlinear Sci. Numer. Simul. 16, 4049–4059 (2011) · Zbl 1223.45007 · doi:10.1016/j.cnsns.2011.02.003
[25]	Wang, J., Zhou, Y.: A class of fractional evolution equations and optimal controls. Nonlinear Anal., Real World Appl. 12, 262–272 (2011) · Zbl 1214.34010 · doi:10.1016/j.nonrwa.2010.06.013
[26]	Zhou, Y.: Existence and uniqueness of fractional functional differential equations with unbounded delay. Int. J. Dyn. Differ. Equ. 1, 239–244 (2008) · Zbl 1175.34081
[27]	Zhou, Y., Jiao, F., Li, J.: Existence and uniqueness for fractional neutral differential equations with infinite delay. Nonlinear Anal. TMA 71, 3249–3256 (2009) · Zbl 1177.34084 · doi:10.1016/j.na.2009.01.202
[28]	Zhou, Y., Jiao, F.: Existence of extremal solutions for discontinuous fractional functional differential equations. Int. J. Dyn. Differ. Equ. 2, 237–252 (2009) · Zbl 1188.34108
[29]	Zhou, Y., Jiao, F.: Existence of mild solutions for fractional neutral evolution equations. Comput. Math. Appl. 59, 1063–1077 (2010) · Zbl 1189.34154 · doi:10.1016/j.camwa.2009.06.026
[30]	Zhou, Y., Jiao, F.: Nonlocal Cauchy problem for fractional evolution equations. Nonlinear Anal., Real World Appl. 11, 4465–4475 (2010) · Zbl 1260.34017 · doi:10.1016/j.nonrwa.2010.05.029
[31]	Hale, J., Kato, J.: Phase spaces for retarded equations with infinite delay. Funkc. Ekvacioj 21, 11–41 (1978) · Zbl 0383.34055
[32]	Henry, D.: Geometric Theory of Semilinear Parabolic Equations. LNM, vol. 840. Springer, Berlin (1981) · Zbl 0456.35001
[33]	Ye, H., Gao, J., Ding, Y.: A generalized Gronwall inequality and its application to a fractional differential equation. J. Math. Anal. Appl. 328, 1075–1081 (2007) · Zbl 1120.26003 · doi:10.1016/j.jmaa.2006.05.061
[34]	Zeidler, E.: Nonlinear Functional Analysis and Its Application II/A. Springer, New York (1990) · Zbl 0684.47029
[35]	Balder, E.: Necessary and sufficient conditions for L 1-strong-weak lower semicontinuity of integral functional. Nonlinear Anal., Real World Appl. 11, 1399–1404 (1987) · Zbl 0638.49004