Control of Infinite Dimensional Bilinear Systems: Applications to Quantum Control Systems

Abstract

In the dissertation, optimal control problem for bilinear systems motivated from quantum control theory are studied. Specifically, problems of quantum feedback control, control of tumor growth dynamics and time optimal control are analyzed for bilinear systems. Feedback synthesis, receding horizon synthesis and semi-smooth Newton method are developed to solve these problems. The contents of the dissertation are outlined as follows:

The first problem studied is control of quantum systems described by the linear Schr¨odinger equation. Control inputs enter through coupling operators and results in a bilinear control system. Feedback control laws with switching term are developed for the orbit tracking and the performance of the feedback control laws is demonstrated by a stable and accurate numerical integration of the closed-loop system. The asymptotic properties of the feedback laws are analyzed by the LaSalle-type invariance principle. The receding horizon control synthesis is applied to improve the performance of the feedback law. The second order accurate numerical integrations via time-splitting and the monotone convergent iterative scheme are combined to solve the optimality system.

The switching mechanism in the feedback law can be applied to a wide general class of control systems and feedback synthesis based on the Lyapunov stability. By using this principle, the problem of the tumor treatment, aiming at the reduction of the tumor cells population, is formulated in terms of optimal control theory as a state regulator problem and a feedback law with switching term is designed. Numerical evidence is shown to demonstrate the effectiveness of the feedback law to suppress the tumor growth.

A quantum system interacts with its environment. As a consequence, quantum state subject to continuous measurement can be modeled as a nonlinear stochastic differential equation by quantum filtering theory. The problem of stochastic stabilization of quantum spin systems under the noisy environment and continuous measurement via feedback control is studied. New nonlinear control law with switching term is proposed and developed to globally stabilize the quantum spin system to an arbitrary equilibrium state. Nonnegative definite preserving properties of the density matrix to measure the quantum system is very essential and a numerical method is developed to fulfill this.

Finally, time optimal and minimum effort control problems for linear and bilinear systems are studied. To overcome the difficulties of nondifferentiability in the bang-bang control, a regularized problem is formulated and the semi-smooth Newton method is applied for solving the regularized optimality system. By integrating the state and costate and variation of them in the optimality system, the nonlinear optimality system is further reduced to a nonlinear equation with some shooting parameters. The reduced Jacobian is computed for the Newton update. The initialization of the Newton method is achieved by solving a related minimum norm problem and using the standard line search strategy. The effectiveness of the proposed method is demonstrated by examples for quantum spin system and parabolic systems.