Dynamic programming and viscosity solutions for the optimal control of quantum spin systems

Srinivas Sridharan; Matthew R. James

doi:10.1016/j.sysconle.2011.05.010

Dynamic programming and viscosity solutions for the optimal control of quantum spin systems

Srinivas Sridharan^*, Matthew R. James

^*Corresponding author for this work

Research output: Contribution to journal › Article › peer-review

4 Citations (Scopus)

Abstract

The purpose of this paper is to describe the application of the notion of viscosity solutions to solve the HamiltonJacobiBellman (HJB) equation associated with an important class of optimal control problems for quantum spin systems. The HJB equation that arises in the control problems of interest is a first-order nonlinear partial differential equation defined on a Lie group. Hence we employ recent extensions of the theory of viscosity solutions to Riemannian manifolds in order to interpret possibly non-differentiable solutions to this equation. Results from differential topology on the triangulation of manifolds are then used develop a finite difference approximation method for numerically computing the solution to such problems. The convergence of these approximations is proven using viscosity solution methods. In order to illustrate the techniques developed, these methods are applied to an example problem.

Original language	English
Pages (from-to)	726-733
Number of pages	8
Journal	Systems and Control Letters
Volume	60
Issue number	9
DOIs	https://doi.org/10.1016/j.sysconle.2011.05.010
Publication status	Published - Sept 2011

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10.1016/j.sysconle.2011.05.010

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title = "Dynamic programming and viscosity solutions for the optimal control of quantum spin systems",

abstract = "The purpose of this paper is to describe the application of the notion of viscosity solutions to solve the HamiltonJacobiBellman (HJB) equation associated with an important class of optimal control problems for quantum spin systems. The HJB equation that arises in the control problems of interest is a first-order nonlinear partial differential equation defined on a Lie group. Hence we employ recent extensions of the theory of viscosity solutions to Riemannian manifolds in order to interpret possibly non-differentiable solutions to this equation. Results from differential topology on the triangulation of manifolds are then used develop a finite difference approximation method for numerically computing the solution to such problems. The convergence of these approximations is proven using viscosity solution methods. In order to illustrate the techniques developed, these methods are applied to an example problem.",

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T1 - Dynamic programming and viscosity solutions for the optimal control of quantum spin systems

AU - Sridharan, Srinivas

AU - James, Matthew R.

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AB - The purpose of this paper is to describe the application of the notion of viscosity solutions to solve the HamiltonJacobiBellman (HJB) equation associated with an important class of optimal control problems for quantum spin systems. The HJB equation that arises in the control problems of interest is a first-order nonlinear partial differential equation defined on a Lie group. Hence we employ recent extensions of the theory of viscosity solutions to Riemannian manifolds in order to interpret possibly non-differentiable solutions to this equation. Results from differential topology on the triangulation of manifolds are then used develop a finite difference approximation method for numerically computing the solution to such problems. The convergence of these approximations is proven using viscosity solution methods. In order to illustrate the techniques developed, these methods are applied to an example problem.

KW - Dynamic programming

KW - Numerical method

KW - Optimal control

KW - Quantum spin systems

KW - Viscosity solution

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DO - 10.1016/j.sysconle.2011.05.010

M3 - Article

SN - 0167-6911

VL - 60

SP - 726

EP - 733

JO - Systems and Control Letters

JF - Systems and Control Letters

IS - 9

ER -

Dynamic programming and viscosity solutions for the optimal control of quantum spin systems

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