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Abstract
The miniaturization of electronic circuits and devices, and recent advances in laser technology, have brought to the forefront the need for, and possibility of, controlling systems which exhibit quantum mechanical features. This thesis aims to develop some concepts of the classical control theory so that they become applicable to quantum systems. In particular, it considers the bounded real properties for a class of linear quantum systems which can be defined by complex quantum stochastic differential equations in terms of annihilation operators only. It develops complex quantum versions of the Bounded Real Lemma, the Strict Bounded Real Lemma and the Lossless Bounded Real Lemma. For the class of quantum systems under consideration, it is shown that the question of physical realizability is related to the bounded real and lossless bounded real properties. Furthermore, it considers a coherent H∞ control problem for this same class of linear quantum systems and shows that a solution to the H∞ control problem can be obtained in terms of a pair of complex algebraic Riccati equations. In addition, the question of physical realizability of the resulting quantum controllers is related to a bounded real property. Also, the thesis considers a coherent LQG control problem for this class of linear quantum systems and develops a method for designing an LQG controller which satisfies a physical realizability condition. The main idea involves constructing a suitable cost function for the LQG problem such that a physically realizable controller is obtained. The resulting controller provides robust stability as well as optimal performance. Finally, a finite horizon H∞ control problem is solved for a class of linear quantum systems using a dynamic game approach. Three cases are considered: the continuous time case, the sampled-data measurements case and the delayed measurement case. The methodology adopted in these three cases, involves an equivalence between the quantum control problem and an auxiliary classical stochastic H∞ problem. Then, by solving the finite horizon H∞ control problem for the equivalent stochastic systems using some results from a corresponding deterministic problem, following a dynamic game approach, the finite horizon H∞ control problem for linear quantum systems is solved separately for each case.