Control theory for distributed parameter systems

In this thesis problems in control theory for distributed parameter systems are studied, using a semigroup approach. Firstly the control problem is formulated for systems z = Az + Bu on a Banach space Z, when the control operator B is unbounded on Z. The semigroup Tt generated by the system operator A is required to be smoothing so that the resultant operator TtB is bounded by an LÞfunction. The finite and infinite time regulator, and the tracking problems for such systems are then solved. By constructing an iterative sequence of sub-optimal controls it is shown that for the regulator problems the optimal control is feedback, and for the tracking problem is feedback plus open-loop. It is further shown that the feedback operator, time independent in the infinite time case, is the unique solution to an Integral Riccatl equation which is differentiable. The differential equation has unique solution also, where we make additional assumptions on the system operators in the infinite time case. The open-loop control of the tracking problem is also shown to be associated with the unique solutions to integral and differential equations. Arising out of the solution to the infinite time regulator problem, the stabilizability result, exact null controllability implies stabillzability, is also proved for these systems. The results obtained are then compared with those of other authors and applications given. Observer theory for distributed parameter systems described by semigroups is then considered. Conditions are found, in terms of the system operators, for an observer to be an asymptotic state estimator under feedback and general control action. The increase in cost due to using an observer as feedback in the regulator is studied and found to be dependent on the Initial state of the system, in general unknown