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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:
35R11Fractional partial differential equations
49J20Optimal control problems with PDE (existence)
35Q93PDEs in connection with control and optimization
35Q40PDEs in connection with quantum mechanics
References:
[1]Sulem, P. -L.; Sulem, C.: The nonlinear Schrödinger equation: self-focusing and wave collapse, (1999)
[2]Cazenave, T.: Semilinear Schrödinger equations, Courant lecture notes in mathematics 10 (2003) · Zbl 1055.35003
[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 · doi:10.1007/BF01403504
[4]Cazenave, T.: Stable solutions of the logarithmic Schrödinger equation, Nonlinear anal.: TMA 7, 1127-1140 (1983) · Zbl 0529.35068 · doi:10.1016/0362-546X(83)90022-6
[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 · doi:10.1016/0022-1236(86)90096-0
[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 · doi:10.1006/jdeq.1995.1156
[7]Cuccagna, S.: Stabilization of solutions to nonlinear Schrödinger equations, Comm. pure appl. Math. 54, 1110-1145 (2001) · Zbl 1031.35129 · doi:10.1002/cpa.1018
[8]Tsai, T. -P.: Asymptotic dynamics of nonlinear Schrödinger equations with many bound states, J. differential equations 192, 225-282 (1995) · Zbl 1038.35128 · doi:10.1016/S0022-0396(03)00041-X
[9]Buslaev, V. S.; Sulem, C.: On asymptotic stability of solitary waves for nonlinear Schrödinger equations, Ann. inst. Henri Poincarè (C) non linear anal. 20, 419-475 (2003) · Zbl 1028.35139 · doi:10.1016/S0294-1449(02)00018-5 · doi:numdam:AIHPC_2003__20_3_419_0
[10]Wang, Y.: Global existence and blow up of solutions for the inhomogeneous nonlinear Schrödinger equation in R2, J. math. Anal. appl. 338, 1008-1019 (2008) · Zbl 1135.35080 · doi:10.1016/j.jmaa.2007.05.057
[11]Fibich, G.: Singular solutions of the subcritical nonlinear Schrödinger equation, Physica D: Nonlinear phenom. 240, 1119-1122 (2011) · Zbl 1225.35216 · doi:10.1016/j.physd.2011.04.004
[12]Eid, R.; Muslih, S. I.; Baleanu, D.; Rabei, E.: On fractional Schrödinger equation in α-dimensional fractional space, Nonlinear anal.: RWA 10, 1299-1304 (2009) · Zbl 1162.35344 · doi:10.1016/j.nonrwa.2008.01.007
[13]Guerrero, P.; López, J. L.; Nieto, J. J.: Global H1 solvability of the 3D logarithmic Schrödinger equation, Nonlinear anal.: RWA 11, 79-87 (2010) · Zbl 1180.81071 · doi:10.1016/j.nonrwa.2008.10.017
[14]Yildiz, B.; Subaşi, M.: On the optimal control problem for linear Schrödinger equation, Appl. math. Comput. 121, 373-381 (2001) · Zbl 1020.49002 · doi:10.1016/S0096-3003(00)00013-8
[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 · doi:10.1016/S0096-3003(01)00161-8
[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 · doi:10.1016/j.jde.2005.04.006
[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 · doi:10.1016/j.sysconle.2007.11.002
[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 · doi:10.1016/j.amc.2010.03.039
[19]Diethelm, K.: The analysis of fractional differential equations, (2010)
[20]Kilbas, A. A.; Srivastava, H. M.; Trujillo, J. J.: Theory and applications of fractional differential equations, North-holland mathematics studies 204 (2006)
[21]Lakshmikantham, V.; Leela, S.; Devi, J. Vasundhara: Theory of fractional dynamic systems, (2009)
[22]Miller, K. S.; Ross, B.: An introduction to the fractional calculus and differential equations, (1993)
[23]Podlubny, I.: Fractional differential equations, (1999)
[24]Tarasov, V. E.: Fractional dynamics: application of fractional calculus to dynamics of particles, fields and media, (2010)
[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 · doi:10.1007/s10440-008-9356-6
[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) · Zbl 1182.34103 · doi:10.1155/2009/981728
[27]Chen, F.; Nieto, J. J.; Zhou, Y.: Global attractivity for nonlinear fractional differential equations, Nonlinear anal.: RWA 13, 287-298 (2012)
[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)
[29]Wang, J.; Zhou, Y.: Analysis of nonlinear fractional control systems in Banach spaces, Nonlinear anal.: TMA 74, 5929-5942 (2011) · Zbl 1223.93059 · doi:10.1016/j.na.2011.05.059
[30]Wang, J.; Zhou, Y.: A class of fractional evolution equations and optimal controls, Nonlinear anal.: RWA 12, 262-272 (2011) · Zbl 1214.34010 · doi:10.1016/j.nonrwa.2010.06.013
[31]Wang, J.; Zhou, Y.: Existence and controllability results for fractional semilinear differential inclusions, Nonlinear anal.: RWA 12, 3642-3653 (2011) · Zbl 1231.34108 · doi:10.1016/j.nonrwa.2011.06.021
[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)
[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)
[34]Zhou, Y.; Jiao, F.: Nonlocal Cauchy problem for fractional evolution equations, Nonlinear anal.: RWA 11, 4465-4475 (2010)
[35]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
[36]Pazy, A.: Semigroups of linear operators and applications to partial differential equations, (1983)
[37]Henry, D.: Geometric theory of semilinear parabolic equations, Lnm 840 (1981) · Zbl 0456.35001
[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 · doi:10.1016/j.jmaa.2006.05.061