# American Institute of Mathematical Sciences

2013, 2013(special): 437-446. doi: 10.3934/proc.2013.2013.437

## Optimal control of a linear stochastic Schrödinger equation

 1 Martin Luther University Halle-Wittenberg, Faculty of Natural Sciences II, Institute of Mathematics, D - 06099 Halle (Saale), Germany

Received  August 2012 Published  November 2013

This paper concerns a linear controlled Schrödinger equation with additive noise and corresponding initial and Neumann boundary conditions. The existence and uniqueness of the variational solution of this Schrödinger problem and some of its properties will be discussed. Furthermore, a given objective functional shall be minimized by an optimal control. Though, instead of the control only the solution of the controlled Schrödinger problem appears explicitly in the objective functional. Based on the adjoint problem of the stochastic Schrödinger problem, a gradient formula is developed.
Citation: Diana Keller. Optimal control of a linear stochastic Schrödinger equation. Conference Publications, 2013, 2013 (special) : 437-446. doi: 10.3934/proc.2013.2013.437
##### References:
 [1] G. Da Prato and J. Zabczyk, "Stochastic Equations in Infinite Dimensions",, Cambridge University Press, (1992).   Google Scholar [2] A. de Bouard and A. Debussche, The stochastic nonlinear Schrödinger equation in $H^1$,, Stochastic Analysis and Applications, 21 (2003), 97.   Google Scholar [3] M. H. Farag, A gradient-type optimization technique for the optimal control for Schrodinger equations,, International Journal on Information Theories and Applications, 10 (2003), 414.   Google Scholar [4] W. Grecksch and H. Lisei, Approximation of stochastic nonlinear equations of Schrödinger type by the splitting method,, Stochastic Analysis and Applications, 31 (2013), 314.   Google Scholar [5] W. Grecksch and H. Lisei, Stochastic nonlinear equations of Schrödinger type,, Stochastic Analysis and Applications, 29 (2011), 631.   Google Scholar [6] D. Keller, "A Problem of Optimal Control for the Linear Stochastic Schrödinger Equation",, Master's thesis, (2011).   Google Scholar [7] C. Prévôt and M. Röckner, "A Concise Course on Stochastic Partial Differential Equations",, Springer-Verlag, (2007).   Google Scholar [8] M. Subaşi, An estimate for the solution of a perturbed nonlinear quantum-mechanical problem,, Chaos, 14 (2002), 397.   Google Scholar [9] M. Subaşi, An optimal control problem governed by the potential of a linear Schrodinger equation,, Applied Mathematics and Computation, 131 (2002), 95.   Google Scholar [10] E. Zuazua, Remarks on the controllability of the Schrödinger equation,, CRM Proceedings and Lecture Notes, 33 (2003), 193.   Google Scholar

show all references

##### References:
 [1] G. Da Prato and J. Zabczyk, "Stochastic Equations in Infinite Dimensions",, Cambridge University Press, (1992).   Google Scholar [2] A. de Bouard and A. Debussche, The stochastic nonlinear Schrödinger equation in $H^1$,, Stochastic Analysis and Applications, 21 (2003), 97.   Google Scholar [3] M. H. Farag, A gradient-type optimization technique for the optimal control for Schrodinger equations,, International Journal on Information Theories and Applications, 10 (2003), 414.   Google Scholar [4] W. Grecksch and H. Lisei, Approximation of stochastic nonlinear equations of Schrödinger type by the splitting method,, Stochastic Analysis and Applications, 31 (2013), 314.   Google Scholar [5] W. Grecksch and H. Lisei, Stochastic nonlinear equations of Schrödinger type,, Stochastic Analysis and Applications, 29 (2011), 631.   Google Scholar [6] D. Keller, "A Problem of Optimal Control for the Linear Stochastic Schrödinger Equation",, Master's thesis, (2011).   Google Scholar [7] C. Prévôt and M. Röckner, "A Concise Course on Stochastic Partial Differential Equations",, Springer-Verlag, (2007).   Google Scholar [8] M. Subaşi, An estimate for the solution of a perturbed nonlinear quantum-mechanical problem,, Chaos, 14 (2002), 397.   Google Scholar [9] M. Subaşi, An optimal control problem governed by the potential of a linear Schrodinger equation,, Applied Mathematics and Computation, 131 (2002), 95.   Google Scholar [10] E. Zuazua, Remarks on the controllability of the Schrödinger equation,, CRM Proceedings and Lecture Notes, 33 (2003), 193.   Google Scholar
 [1] Zuliang Lu, Fei Huang, Xiankui Wu, Lin Li, Shang Liu. Convergence and quasi-optimality of $L^2-$norms based an adaptive finite element method for nonlinear optimal control problems. Electronic Research Archive, 2020, 28 (4) : 1459-1486. doi: 10.3934/era.2020077 [2] Awais Younus, Zoubia Dastgeer, Nudrat Ishaq, Abdul Ghaffar, Kottakkaran Sooppy Nisar, Devendra Kumar. On the observability of conformable linear time-invariant control systems. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020444 [3] Kihoon Seong. Low regularity a priori estimates for the fourth order cubic nonlinear Schrödinger equation. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5437-5473. doi: 10.3934/cpaa.2020247 [4] José Luis López. A quantum approach to Keller-Segel dynamics via a dissipative nonlinear Schrödinger equation. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020376 [5] Claudianor O. Alves, Rodrigo C. M. Nemer, Sergio H. Monari Soares. The use of the Morse theory to estimate the number of nontrivial solutions of a nonlinear Schrödinger equation with a magnetic field. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020276 [6] Justin Holmer, Chang Liu. Blow-up for the 1D nonlinear Schrödinger equation with point nonlinearity II: Supercritical blow-up profiles. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020264 [7] Lars Grüne, Matthias A. Müller, Christopher M. Kellett, Steven R. Weller. Strict dissipativity for discrete time discounted optimal control problems. Mathematical Control & Related Fields, 2020  doi: 10.3934/mcrf.2020046 [8] Yongge Tian, Pengyang Xie. Simultaneous optimal predictions under two seemingly unrelated linear random-effects models. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020168 [9] Xiyou Cheng, Zhitao Zhang. Structure of positive solutions to a class of Schrödinger systems. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020461 [10] Youming Guo, Tingting Li. Optimal control strategies for an online game addiction model with low and high risk exposure. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020347 [11] Stefano Bianchini, Paolo Bonicatto. Forward untangling and applications to the uniqueness problem for the continuity equation. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020384 [12] Shiqi Ma. On recent progress of single-realization recoveries of random Schrödinger systems. Electronic Research Archive, , () : -. doi: 10.3934/era.2020121 [13] Scipio Cuccagna, Masaya Maeda. A survey on asymptotic stability of ground states of nonlinear Schrödinger equations II. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020450 [14] Bernard Bonnard, Jérémy Rouot. Geometric optimal techniques to control the muscular force response to functional electrical stimulation using a non-isometric force-fatigue model. Journal of Geometric Mechanics, 2020  doi: 10.3934/jgm.2020032 [15] Marion Darbas, Jérémy Heleine, Stephanie Lohrengel. Numerical resolution by the quasi-reversibility method of a data completion problem for Maxwell's equations. Inverse Problems & Imaging, 2020, 14 (6) : 1107-1133. doi: 10.3934/ipi.2020056 [16] Gang Bao, Mingming Zhang, Bin Hu, Peijun Li. An adaptive finite element DtN method for the three-dimensional acoustic scattering problem. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020351 [17] Siyang Cai, Yongmei Cai, Xuerong Mao. A stochastic differential equation SIS epidemic model with regime switching. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020317 [18] Reza Chaharpashlou, Abdon Atangana, Reza Saadati. On the fuzzy stability results for fractional stochastic Volterra integral equation. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020432 [19] Zedong Yang, Guotao Wang, Ravi P. Agarwal, Haiyong Xu. Existence and nonexistence of entire positive radial solutions for a class of Schrödinger elliptic systems involving a nonlinear operator. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020436 [20] Denis Bonheure, Silvia Cingolani, Simone Secchi. Concentration phenomena for the Schrödinger-Poisson system in $\mathbb{R}^2$. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020447

Impact Factor: