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]

Cambridge University Press, 1992.  Google Scholar

[2]

Stochastic Analysis and Applications, 21 (2003), 97-126.  Google Scholar

[3]

International Journal on Information Theories and Applications, 10 (2003), 414-422. Google Scholar

[4]

Stochastic Analysis and Applications, 31 (2013), 314-335.  Google Scholar

[5]

Stochastic Analysis and Applications, 29 (2011), 631-653.  Google Scholar

[6]

Master's thesis, Martin-Luther University Halle-Wittenberg, 2011. Google Scholar

[7]

Springer-Verlag, Berlin Heidelberg, 2007.  Google Scholar

[8]

Chaos, Solitons and Fractals, 14 (2002), 397-402.  Google Scholar

[9]

Applied Mathematics and Computation, 131 (2002), 95-106.  Google Scholar

[10]

CRM Proceedings and Lecture Notes, 33 (2003), 193-211.  Google Scholar

show all references

References:
[1]

Cambridge University Press, 1992.  Google Scholar

[2]

Stochastic Analysis and Applications, 21 (2003), 97-126.  Google Scholar

[3]

International Journal on Information Theories and Applications, 10 (2003), 414-422. Google Scholar

[4]

Stochastic Analysis and Applications, 31 (2013), 314-335.  Google Scholar

[5]

Stochastic Analysis and Applications, 29 (2011), 631-653.  Google Scholar

[6]

Master's thesis, Martin-Luther University Halle-Wittenberg, 2011. Google Scholar

[7]

Springer-Verlag, Berlin Heidelberg, 2007.  Google Scholar

[8]

Chaos, Solitons and Fractals, 14 (2002), 397-402.  Google Scholar

[9]

Applied Mathematics and Computation, 131 (2002), 95-106.  Google Scholar

[10]

CRM Proceedings and Lecture Notes, 33 (2003), 193-211.  Google Scholar

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