Comparison of optimal and geometric control methods for regulation of distributed parameter systems

This thesis is concerned with several aspects of set-point control for distributed parameter systems. We first present the geometric design methodology for set-point control. We then demonstrate this methodology for a multi-input multi-output set-point control problem for a two dimensional nonlinear convection-diffusion heat equation. Next we consider the main point of this work, to investigate the relationship between the geometric design method and the classical method of PDE constrained minimization from optimal control. We show that for a linear stationary system the geometric method produces an optimal control gain. Then we consider the special case when the number of control inputs is less than the number of measured outputs, for which we obtain an ``overdetermined system". With fewer control inputs than outputs we do not expect to achieve exact asymptotic tracking so we seek a solution with smallest least-squares error. In this case we employ the Moore-Penrose pseudo-inverse in the geometric design strategy to obtain the smallest least-squares error with smallest possible gain. Two examples are given to demonstrate this method. The first example is a stationary control problem for a Laplace equation and the second example involves temperature control of a two dimensional incompressible Navier-Stokes flow.