## Fractional Schrödinger equations with potential and optimal controls.(English)Zbl 1253.35205

Summary: In this paper, we study fractional Schrödinger equations with potential and optimal controls. The first novelty is a suitable concept on a mild solution for our problems. Existence, uniqueness, local stability and attractivity, and data continuous dependence of mild solutions are also presented respectively. The second novelty is an initial study on the optimal control problems for the controlled fractional Schrödinger equations with potential. Existence and uniqueness of optimal pairs for the standard Lagrange problem are obtained.

### MSC:

 35R11 Fractional partial differential equations 49J20 Existence theories for optimal control problems involving partial differential equations 35Q93 PDEs in connection with control and optimization 35Q40 PDEs in connection with quantum mechanics
Full Text:

### References:

 [1] Sulem, P.-L.; Sulem, C., The nonlinear Schrödinger equation: self-focusing and wave collapse, (1999), Hardback · Zbl 0928.35157 [2] Cazenave, T., (), New York University, Courant Institute of Mathematical Sciences, New York [3] Cazenave, T.; Lions, P.-L., Orbital stability of standing waves for some nonlinear Schrödinger equations, Comm. math. phys., 85, 549-561, (1982) · Zbl 0513.35007 [4] Cazenave, T., Stable solutions of the logarithmic Schrödinger equation, Nonlinear anal.: TMA, 7, 1127-1140, (1983) · Zbl 0529.35068 [5] Floer, A.; Weinstein, A., Nonspreading wave packets for the cubic Schrödinger equation with a bounded potential, J. funct. anal., 69, 397-408, (1986) · Zbl 0613.35076 [6] Guo, B.L.; Wu, Y.P., Orbital stability of solitary waves for the nonlinear derivative Schrödinger equation, J. differential equations, 123, 35-55, (1995) · Zbl 0844.35116 [7] Cuccagna, S., Stabilization of solutions to nonlinear Schrödinger equations, Comm. pure appl. math., 54, 1110-1145, (2001) · Zbl 1031.35129 [8] Tsai, T.-P., Asymptotic dynamics of nonlinear Schrödinger equations with many bound states, J. differential equations, 192, 225-282, (1995) · Zbl 1038.35128 [9] Buslaev, V.S.; Sulem, C., On asymptotic stability of solitary waves for nonlinear Schrödinger equations, Ann. inst. Henri poincare (C) non linear anal., 20, 419-475, (2003) · Zbl 1028.35139 [10] Wang, Y., Global existence and blow up of solutions for the inhomogeneous nonlinear Schrödinger equation in $$\mathbb{R}^2$$, J. math. anal. appl., 338, 1008-1019, (2008) · Zbl 1135.35080 [11] Fibich, G., Singular solutions of the subcritical nonlinear Schrödinger equation, Physica D: nonlinear phenom., 240, 1119-1122, (2011) · Zbl 1225.35216 [12] Eid, R.; Muslih, S.I.; Baleanu, D.; Rabei, E., On fractional Schrödinger equation in $$\alpha$$-dimensional fractional space, Nonlinear anal.: RWA, 10, 1299-1304, (2009) · Zbl 1162.35344 [13] Guerrero, P.; López, J.L.; Nieto, J.J., Global $$H^1$$ solvability of the 3D logarithmic Schrödinger equation, Nonlinear anal.: RWA, 11, 79-87, (2010) · Zbl 1180.81071 [14] Yildiz, B.; Subaşi, M., On the optimal control problem for linear Schrödinger equation, Appl. math. comput., 121, 373-381, (2001) · Zbl 1020.49002 [15] Subaşi, M., An optimal control problem governed by the potential of a linear Schrödinger equation, Appl. math. comput., 131, 95-106, (2002) · Zbl 1019.49004 [16] Baudouin, L.; Kavian, O.; Puel, J.-P., Regularity for a Schrödinger equation with singular potentials and application to bilinear optimal control, J. differential equations, 216, 188-222, (2005) · Zbl 1109.35094 [17] Baudouin, L.; Salomon, J., Constructive solution of a bilinear optimal control problem for a Schrödinger equation, Systems control lett., 57, 453-464, (2008) · Zbl 1153.49023 [18] Yetişkin, H.; Subaşi, M., On the optimal control problem for Schrödinger equation with complex potential, Appl. math. comput., 216, 1896-1902, (2010) · Zbl 1193.49005 [19] Diethelm, K., The analysis of fractional differential equations, (2010), Springer-Verlag Berlin [20] Kilbas, A.A.; Srivastava, H.M.; Trujillo, J.J., () [21] Lakshmikantham, V.; Leela, S.; Vasundhara Devi, J., Theory of fractional dynamic systems, (2009), Cambridge Scientific Publishers · Zbl 1188.37002 [22] Miller, K.S.; Ross, B., An introduction to the fractional calculus and differential equations, (1993), John Wiley New York · Zbl 0789.26002 [23] Podlubny, I., Fractional differential equations, (1999), Academic Press San Diego · Zbl 0918.34010 [24] Tarasov, V.E., Fractional dynamics: application of fractional calculus to dynamics of particles, fields and media, (2010), Springer HEP · Zbl 1214.81004 [25] 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 [26] Agarwal, R.P.; Belmekki, M.; Benchohra, M., A survey on semilinear differential equations and inclusions involving riemann – liouville fractional derivative, Adv. differential equations, 2009, 47, (2009), Article ID 981728 · Zbl 1182.34103 [27] Chen, F.; Nieto, J.J.; Zhou, Y., Global attractivity for nonlinear fractional differential equations, Nonlinear anal.: RWA, 13, 287-298, (2012) · Zbl 1238.34011 [28] Ahmad, B.; Nieto, J.J.; Alsaedi, A.; El-Shahed, M., A study of nonlinear Langevin equation involving two fractional orders in different intervals, Nonlinear anal.: RWA, 13, 599-606, (2012) · Zbl 1238.34008 [29] Wang, J.; Zhou, Y., Analysis of nonlinear fractional control systems in Banach spaces, Nonlinear anal.: TMA, 74, 5929-5942, (2011) · Zbl 1223.93059 [30] Wang, J.; Zhou, Y., A class of fractional evolution equations and optimal controls, Nonlinear anal.: RWA, 12, 262-272, (2011) · Zbl 1214.34010 [31] Wang, J.; Zhou, Y., Existence and controllability results for fractional semilinear differential inclusions, Nonlinear anal.: RWA, 12, 3642-3653, (2011) · Zbl 1231.34108 [32] Wang, J.; Zhou, Y.; Medved˘, M., On the solvability and optimal controls of fractional integrodifferential evolution systems with infinite delay, J. optim. theory appl., 152, 31-50, (2012) · Zbl 1357.49018 [33] Wang, J.; Zhou, Y.; Wei, W., Optimal feedback control for semilinear fractional evolution equations in Banach spaces, Systems control lett., 61, 472-476, (2012) · Zbl 1250.49035 [34] Zhou, Y.; Jiao, F., Nonlocal Cauchy problem for fractional evolution equations, Nonlinear anal.: RWA, 11, 4465-4475, (2010) · Zbl 1260.34017 [35] Zhou, Y.; Jiao, F., Existence of mild solutions for fractional neutral evolution equations, Comput. math. appl., 59, 1063-1077, (2010) · Zbl 1189.34154 [36] Pazy, A., Semigroups of linear operators and applications to partial differential equations, (1983), Springer-Verlag New York, Berlin, Heidelberg, Tokyo · Zbl 0516.47023 [37] Henry, D., () [38] 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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.