| Linear time-invariant control systems of the form {dollar}E{lcub}dover dt{rcub}x=Ax+Bu{dollar} as well as their discrete-time analogue are considered, where {dollar}E{dollar} is a singular square matrix. Such systems are called generalized state-space systems or descriptor systems. It is assumed that for any admissible initial descriptor vector {dollar}x(0-)=xsb0{dollar} any control vector u(t) yields one and only one descriptor vector x(t).; This thesis deals with optimal control of descriptor systems. First we consider the optimal control problem where the final time is free, that is, we are looking for a control vector that drives the descriptor from a given initial vector to some, not necessarily fixed, final vector in some, not a priori prescribed, final time while minimizing a cost functional of the form {dollar}J=intsbsp{lcub}0{rcub}{lcub}tsb f{rcub}{dollar} L(x,u)dt. Necessary conditions are derived for the existence of minima of {dollar}J{dollar}. The problem of finding sufficient conditions for the existence of minima of {dollar}J{dollar} is not considered.; We also investigate optimal control of state-space systems involving time-derivative of the input vector. These systems arise naturally in various linear control systems, for instance, in degenerate networks and in the approximation of nonlinear control systems by linear mathematical models. The state-space system containing input derivatives is re-written in the form of a descriptor-variable system containing no input derivatives, and it is shown that by proper choice of the cost-functional weighting matrices, the original problem is reduced to an optimal control problem of a descriptor-variable system of simple structure. Various formulas are obtained for designing open-loop controls corresponding to different cost functionals and different boundary conditions.; Finally, an algorithm is developed to drive discrete-time descriptor vectors to the origin of the descriptor space. The algorithm is based on the fact that the descriptor system may have a state-space system "buried" in it. Using this state-space system, a descriptor-feedback control is designed to drive the descriptor vector to zero in a number of steps equal, at most, to the rank of {dollar}E{dollar}. |
