Existence of \(\alpha\)-mild solutions for impulsive fractional evolution equations and optimal control problems. (English) Zbl 1358.34013
Summary: In this paper, we study the existence of \(\alpha\)-mild solutions for impulsive fractional semilinear differential equations and optimal control in the \(\alpha\)-norm. We proved the existence of \(\alpha\)-mild solutions by means of fractional calculus, singular version of Gronwall inequality, and Leray-Schauder fixed point theorem. We illustrate with an example our theoretical results.
MSC:
| 34A08 | Fractional ordinary differential equations |
| 34A37 | Ordinary differential equations with impulses |
| 34G20 | Nonlinear differential equations in abstract spaces |
| 34A12 | Initial value problems, existence, uniqueness, continuous dependence and continuation of solutions to ordinary differential equations |
| 49J15 | Existence theories for optimal control problems involving ordinary differential equations |
Keywords:
fractional evolution equation; impulsive equations; \(\alpha\)-mild solutions; optimal controlsReferences:
| [1] | Ahmed, N.U.: Optimal impulse control for impulsive systems in Banach space. Int. J. Differ. Equ. Appl. 1, 37-52 (2000) · Zbl 0959.49023 |
| [2] | Ahmed, N.U., Teo, K.L., Hou, S.H.: Nonlinear impulsive systems on infinite dimensional spaces. Nonlinear Anal. 54, 907-925 (2003) · Zbl 1030.34056 · doi:10.1016/S0362-546X(03)00117-2 |
| [3] | Ahmed, N.U.: Existence of optimal controls for a general class of impulsive systems on Banach space. SIAM J. Control Optim. 42, 669-685 (2003) · Zbl 1037.49036 · doi:10.1137/S0363012901391299 |
| [4] | Agarwal, R.P., Benchohra, M., Slimani, B.A.: Existence results for differential equations with fractional order and impulses. Mem. Differ. Equ. Math. Phys. 44, 1-21 (2008) · Zbl 1178.26006 · doi:10.1134/S0012266108010011 |
| [5] | Arthi, G., Balachandran, K.: Controllability of second-order impulsive evolution systems with infinite delay. Nonlinear Anal. Hybrid Syst. 11, 139-153 (2014) · Zbl 1291.93033 · doi:10.1016/j.nahs.2013.08.001 |
| [6] | Bainov, D.D., Simeonov, P.S.: Impulsive Differential Equations: Periodic Solutions and Applications. Longman Scientific and Technical Group Limited, New York (1993) · Zbl 0815.34001 |
| [7] | Balachandran, K., Park, J.Y.: Nonlocal Cauchy problem for abstract fractional semilinear evolution equations. Nonlinear Anal. 71, 4471-4475 (2009) · Zbl 1213.34008 · doi:10.1016/j.na.2009.03.005 |
| [8] | Balachandran, K., Kiruthika, S.: Existence of solutions of abstract fractional impulsive semilinear evolution equations. Electron. J. Qual. Theory Differ. Equ. 4, 1-12 (2010) · Zbl 1201.34091 · doi:10.14232/ejqtde.2010.1.4 |
| [9] | Balachandran, K., Kiruthika, S., Trujillo, J.J.: Existence of solutions of nonlinear fractional pantograph equations. Acta Math. Sci. 33, 712-720 (2013) · Zbl 1299.34009 · doi:10.1016/S0252-9602(13)60032-6 |
| [10] | Benchohra, M., Henderson, J., Ntouyas, S.K., Ouahab, A.: Multiple solutions for impulsive semilinear functional and neutral functional differential equations in Hilbert space. J. Inequal. Appl. 2005, 189-205 (2005) · Zbl 1095.34046 · doi:10.1155/JIA.2005.189 |
| [11] | Benchohra, M., Slimani, B.A.: Existence and uniqueness of solutions to impulsive fractional differential equations. Electron. J. Differ. Equ. 10, 1-11 (2009) · Zbl 1178.34004 |
| [12] | Benchohra, M., Seba, D.: Impulsive fractional differential equations in Banach spaces. Electron. J. Qual. Theory Differ. Equ. Spec. Ed. I. 8, 1-14 (2009) · Zbl 1189.26005 · doi:10.14232/ejqtde.2009.4.8 |
| [13] | El-Borai, M.M.: Semigroups and some nonlinear fractional differential equations. Appl. Math. Comput. 149, 823-831 (2004) · Zbl 1046.34079 |
| [14] | El-Sayed, M.A.: Fractional order diffusion wave equation. Int. J. Theoret. Phys. 35, 311-322 (1996) · Zbl 0846.35001 · doi:10.1007/BF02083817 |
| [15] | Hernández, E., O’Regan, D., Balachandran, K.: On recent developments in the theory of abstract differential equations with fractional derivatives. Nonlinear Anal. 73, 3462-3471 (2010) · Zbl 1229.34004 · doi:10.1016/j.na.2010.07.035 |
| [16] | Jaradat, O.K., Al-Omari, A., Momani, S.: Existence of the mild solution for fractional semilinear initial value problems. Nonlinear Anal. 69, 3153-3159 (2008) · Zbl 1160.34300 · doi:10.1016/j.na.2007.09.008 |
| [17] | Kilbas, A.A., Sirvastava, H.M., Trujillo, J.J.: Theory and Applications of Fractional Differential Equations. North-Holand Mathematics Studies, vol. 204. Elsevier Science B.V., Amsterdam (2006) · Zbl 1121.34063 |
| [18] | Lakshmikantham, V., Bainov, D.D., Simeonov, P.S.: Theory of Impulsive Differential Equations. World Scientific, Singapore (1989) · Zbl 0719.34002 · doi:10.1142/0906 |
| [19] | Lakshmikantham, V., Leela, S., Vasundhara Devi, J.: Theory of Fractional Dynamic Systems. Cambridge Scientific Publishers, Cambridge (2009) · Zbl 1188.37002 |
| [20] | Liu, H., Chang, J.C.: Existence for a class of partial differential equations with nonlocal conditions. Nonlinear Anal. 70, 3076-083 (2009) · Zbl 1170.34346 · doi:10.1016/j.na.2008.04.009 |
| [21] | Miller, K.S., Ross, B.: An Introduction to the Fractional Calculus and Differential Equations. Wiley, New York (1993) · Zbl 0789.26002 |
| [22] | Mophou, G.M., N’Guerekata, G.M.: Mild solution for semilinear differential equations. Electron. J. Differ. Equ. 21, 1-9 (2009) · Zbl 1180.34006 · doi:10.1007/s10884-008-9127-0 |
| [23] | Mophou, G.M.: Existence and uniqueness of mild solution to impulsive fractional differential equations. Nonlinear Anal. 72, 1604-1615 (2010) · Zbl 1187.34108 · doi:10.1016/j.na.2009.08.046 |
| [24] | Ozdemir, N., Karadeniz, D., Iskender, B.B.: Fractional optimal control problem of a distributed system in cylindrical coordinates. Phys. Lett. A 373, 221-226 (2009) · Zbl 1227.49007 · doi:10.1016/j.physleta.2008.11.019 |
| [25] | Pazy, A.: Semigroups of Linear Operators and Applications to Partial Differential Equations. Springer, New York (1983) · Zbl 0516.47023 · doi:10.1007/978-1-4612-5561-1 |
| [26] | Podlubny, I.: Fractional Differential Equations. Academic, New York (1999) · Zbl 0924.34008 |
| [27] | Wang, J.R., Yang, Y.L., Wei, W.: Nonlocal impulsive problems for fractional differential equations with time-varying generating operators in Banach spaces. Opusucula Math. 30, 361-381 (2010) · Zbl 1232.34013 · doi:10.7494/OpMath.2010.30.3.361 |
| [28] | Wei, W., Xiang, X., Peng, Y.: Nonlinear impulsive integro-differential equation of mixed type and optimal controls. Optimization 55, 141-156 (2006) · Zbl 1101.45002 · doi:10.1080/02331930500530401 |
| [29] | Xiang, X., Wei, W.: Mild solution for a class of nonlinear impulsive evolution inclusion on Banach space. Southeast Asian. Bull. Math. 30, 367-376 (2006) · Zbl 1121.34063 |
| [30] | Xiang, X., Wei, W., Jiang, Y.: Strongly nonlinear impulsive system and necessary conditions of optimality. Dyn. Cont. Discrete Impuls. Syst. A Math. Anal. 12, 811-824 (2005) · Zbl 1081.49027 |
| [31] | Yu, X. L., Xiang, X., Wei, W.: Solution bundle for class of impulsive differential inclusions on Banach spaces. J. Math. Anal. Appl. 327, 220-232 (2007) · Zbl 1113.34046 · doi:10.1016/j.jmaa.2006.03.075 |
| [32] | Zhou, Y., Jiao, F.: Existence of mild solution for fractional neutral evolution equations. Comput. Math. Appl. 59, 1063-1077 (2010) · Zbl 1189.34154 · doi:10.1016/j.camwa.2009.06.026 |
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