Robust optimal control of quantum molecular systems in the presence of disturbances and uncertainties

Abstract

Recently, systematic optimal control theory has been successfully applied to design optical fields which act on a molecule to guide or transform it into a specifically prescribed state. Under realistic laboratory conditions and for real molecules, it is inevitable that there will exist disturbances and uncertainties in the control optical field, in the molecular Hamiltonian, and in the field-coupling coefBcient (e.g. , the dipole moment), etc. Therefore, it is important to design an optimal field which is robust to disturbances. In this paper, we provide a method through which the worst possible disturbance to the control process, under certain possible constraints, can be calculated and a corresponding optimal control field designed to be robust to the disturbances. Under disturbances of small amplitude, a perturbation solution to this problem can be applied. Illustrations are given for control of the wave-packet evolution of a diatomic molecule in the presence of disturbances.

Robustness Issues in Quantum Control

“…In the minimax optimal control approach to robust open-loop control of quantum systems, the uncertainties in the Hamiltonian are represented in terms of a vector quantity w which is subject to constraints. Then, the robust control problem is the minimax optimal control problem min u max w J.u; w/ where J.u; w/ is a suitable cost function, and the problem is subject to the constraints defined by the system dynamics (1) and the constraints on the uncertainty w; see Zhang and Rabitz (1994). Some standard numerical procedures have been proposed to solve this minimax optimal control problem with applications in chemical physics; see Zhang and Rabitz (1994).…”

“…These include some recent results on robust open-loop control of quantum systems; see Zhang and Rabitz (1994). Also considered are some recent results on robust stability analysis results for uncertain quantum systems, which amount to quantum versions of the classical small gain theorem; see Petersen et al (2012).…”

Closed‐Loop and Robust Control of Quantum Systems

“…A general formalism of quantum robust optimal control problem was given in [16], which pointed out that to design a control field that achieves the best objective functional under possible worst uncertainties is in essence a minimax problem. Reference [16] also provided a method to calculate the worst possible disturbance to the control process and to design a corresponding robust optimal control field.…”

“…In realistic environment, the quantum system is unavoidable to be subject to disturbances, uncertainties, and incomplete knowledge. These factors can all be viewed as uncertainties in the control field, in the Hamiltonian system, in the field-coupling coefficient (e.g., the dipole moment), and so forth and might affect the control results [16]. In order to achieve robustness in control method and to develop new insights into complicated quantum plants (such as quantum networks), it is desirable to apply classical robust control theory into quantum domain.…”

Optimal control of quantum systems

“…Energy-optimal designs have been considered in the Quantum Control literature for a variety of problems in both isolated and open quantum systems [2,3,4,28,29,30,31,32,33,34,35]. Except for the few cases where an analytical solution can be found (for two-and three-dimensional systems), one must resort to numerical solution techniques.…”