Abstract
In this paper we study the stochastic stability of numerical solutions of a stochastic controlled Schr¨odinger equation. We investigate the boundedness in second moment, the convergence and the stability of the zero solution for this equation, using two new definitions of almost sure exponential robust stability and asymptotic stability, for the Euler-Maruyama numerical scheme. Considering that the diffusion term is controlled, by using the method of Lyapunov functions and the corresponding diffusion operator associated, we apply techniques of X. Mao and A. Tsoi for achieve our task. Finally, we illustrate this method with a problem in Nuclear Magnetic Resonance (NMR).
Highlights
The stochastic controlled Schrodinger equation depending on time considerate in our study, corresponds to a two-level quantum system describing a spin 1/2particle in a constant and longitudinal static electromagnetic field in the direction of the Z axis and two randomly time varying electromagnetic fields along the X axis and Y axis, respectively
Definition 1 is a strong variant of the definition of almost sure exponential stability given in [Tsoi, 1997], which generalizes it in the case where the diffusion term depends on the control, due to the use of λk(t) function
We focus on the stability properties of zero solution for a numerical scheme type EulerMaruyama applied to two-level stochastic quantum system
Summary
The stochastic controlled Schrodinger equation depending on time considerate in our study, corresponds to a two-level quantum system describing a spin 1/2particle in a constant and longitudinal static electromagnetic field in the direction of the Z axis and two randomly time varying electromagnetic fields along the X axis and Y axis, respectively. The Euler-Maruyama method is the simplest numerical method for solving stochastic differential equations Let || · || be the Euclidean norm in Rn and let’s consider the trace as the norm in Rnm. Denoting xk = x(tk), this method computes discrete approximations xk+1 =. TN = T and satisfying the following stochastic difference equation: xk+1 = xk+S3xk∆tk+1+(S1u1(t)+S2u2(t))xk∆Wk+1 (4) Each one of these integrals approximate the corresponding integrals of equation (2). In this process, we are interested in the zero solution or equilibrium position of this equation, by using the corresponding Euler-Maruyama numerical scheme in equation (4). In [Tocino, 2005] the authors discuss the numerical stability of stochastic differential equations and numerical methods. All of the above motivates our interest in investigating the stability and robustness analysis of numerical computations in this model
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