Numerical Solution of the Dynamic Programming Equation for the Optimal Control of Quantum Spin Systems

Details Numerical Solution of the Dynamic Programming Equation for the Optimal Control of Quantum Spin Systems Numerical Solution of the Dynamic Programming Equation for the Optimal Control of Quantum Spin Systems

2010-02-16

arxiv.org/abs/1002.3067v2

Physics

Quantum Physics

11 pages, 5 figures

Scientific paper

10.1016/j.sysconle.2011.05.010

The purpose of this paper is to describe the numerical solution of the Hamilton-Jacobi-Bellman (HJB) for an optimal control problem for quantum spin systems. This HJB equation is a first order nonlinear partial differential equation defined on a Lie group. We employ recent extensions of the theory of viscosity solutions from Euclidean space to Riemannian manifolds to interpret possibly non-differentiable solutions to this equation. Results from differential topology on the triangulation of manifolds are then used to develop a finite difference approximation method, which is shown to converge using viscosity solution techniques. An example is provided to illustrate the method.

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