Convergence of the Time - Discretized Monotonic Schemes

Many numerical simulations in (bilinear) quantum control use the monotonically convergent Krotov algorithms (introduced by Tannor et al. [Time Dependent Quantum Molecular Dynamics (1992) 347–360]), Zhu and Rabitz [J. Chem. Phys. (1998) 385–391] or their unified form described in Maday and Turinici [J. Chem. Phys. (2003) 8191–8196]. In Maday et al. [Num. Math. (2006) 323–338], a time discretization which preserves the property of monotonicity has been presented. This paper introduces a proof of the convergence of these schemes and some results regarding their rate of convergence. Mathematics Subject Classification. 49J20, 68W40. Received: January 2, 2006. Revised: November 3, 2006. Introduction Quantum control has recently been subjected to significant developments through encouraging experimental results [7, 15]. At computational level [4], the introduction of monotonic Krotov algorithms (introduced by Tannor et al. in [18]), Zhu and Rabitz [20] or the unified form proposed by Maday and Turinici in [10] has enabled us to design efficient methods being implemented to obtain laser fields that control the molecular dynamics [13]. From a mathematical point of view, it can be proved that these procedures monotonically increase the values of a given criterium. Yet, few results are available about the convergence of the control sequences obtained by these schemes. In [14], a first necessary condition for convergence has been obtained in case of large penalization of the L-norm of the control. A functional analysis of the schemes has been displayed in [6] in the abstract framework of semi-group theory. To complete these works, a first analysis of a time-discretized version of the monotonic algorithms is presented here. Let us briefly introduce the model and the corresponding optimal control framework used in this paper. Consider a quantum system described by its wavefunction ψ = ψ(x, t). The relevant spatial coordinates, denoted by x, belong to a general space Ω ⊂ R, where p is the number of particles considered (the symbol x will often be omitted in the following developments in order to make it simple). Absorbing boundary conditions [17] can be used to treat numerically the case of unbounded domains. We assume homogeneous Dirichlet