American Institute of Mathematical Sciences

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

Optimal control of a linear stochastic Schrödinger equation

Diana Keller 1,

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.
Keywords: optimal control, adjoint problem, gradient method., Linear stochastic Schrödinger equation.
Mathematics Subject Classification: Primary: 60H15, 93E20, 35Q55; Secondary: 35Q4.
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), 97-126.  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), 414-422. 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), 314-335.  Google Scholar

[5]

W. Grecksch and H. Lisei, Stochastic nonlinear equations of Schrödinger type, Stochastic Analysis and Applications, 29 (2011), 631-653.  Google Scholar

[6]

D. Keller, "A Problem of Optimal Control for the Linear Stochastic Schrödinger Equation", Master's thesis, Martin-Luther University Halle-Wittenberg, 2011. Google Scholar

[7]

C. Prévôt and M. Röckner, "A Concise Course on Stochastic Partial Differential Equations", Springer-Verlag, Berlin Heidelberg, 2007.  Google Scholar

[8]

M. Subaşi, An estimate for the solution of a perturbed nonlinear quantum-mechanical problem, Chaos, Solitons and Fractals, 14 (2002), 397-402.  Google Scholar

[9]

M. Subaşi, An optimal control problem governed by the potential of a linear Schrodinger equation, Applied Mathematics and Computation, 131 (2002), 95-106.  Google Scholar

[10]

E. Zuazua, Remarks on the controllability of the Schrödinger equation, CRM Proceedings and Lecture Notes, 33 (2003), 193-211.  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), 97-126.  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), 414-422. 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), 314-335.  Google Scholar

[5]

W. Grecksch and H. Lisei, Stochastic nonlinear equations of Schrödinger type, Stochastic Analysis and Applications, 29 (2011), 631-653.  Google Scholar

[6]

D. Keller, "A Problem of Optimal Control for the Linear Stochastic Schrödinger Equation", Master's thesis, Martin-Luther University Halle-Wittenberg, 2011. Google Scholar

[7]

C. Prévôt and M. Röckner, "A Concise Course on Stochastic Partial Differential Equations", Springer-Verlag, Berlin Heidelberg, 2007.  Google Scholar

[8]

M. Subaşi, An estimate for the solution of a perturbed nonlinear quantum-mechanical problem, Chaos, Solitons and Fractals, 14 (2002), 397-402.  Google Scholar

[9]

M. Subaşi, An optimal control problem governed by the potential of a linear Schrodinger equation, Applied Mathematics and Computation, 131 (2002), 95-106.  Google Scholar

[10]

E. Zuazua, Remarks on the controllability of the Schrödinger equation, CRM Proceedings and Lecture Notes, 33 (2003), 193-211.  Google Scholar

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