Date of Award
Doctor of Philosophy (PhD)
A certain parametrization of substantial planar curves yields a centroaffine arclength s and a centroaffine curvature ks that remain invariant under GL(2, R ) motions. In Chapter 4 we search for those substantial curves with predetermined position and velocity at the initial and terminal points, which minimize the total square curvature 0T k2s 2ds as k varies over all square summable functions on each interval [0, T]. These curves are called centroaffine elastic curves. Thinking of the curvature k as a control, we pose our problem as an optimal control problem over the Lie group GL(2, R ) with fixed initial and terminal values but with free terminal time T. To find information about the elastic curves we apply a geometric version of the Pontryagin maximum principle. We find that the optimal k for these curves must satisfy the third order nonlinear differential equation d3ds3k =&parl0;32k2-4e &parr0;ddsk. To study the nonconstant solutions of this equation we consider it as a second order conservative differential equation depending upon parameters. Using this necessary condition, numerical experiments are carried out to graph representative extremals for the case e = 1. We also pose the minimal centroaffine arclength problem by using the same framework. We apply the maximum principle and the generalized Legendre-Clebsch condition for optimality of singular extremals to show that the minimal centroaffine arclength problem has no solution. This improves a result by Mayer and Myller. Motivated by the discussion of the curves with minimal centroaffine arclength, we look in Chapter 5 at an extended optimal control problem in which the centroaffine arclength is regarded as an additional control function. This problem serves as a model for an extension of the minimal arclength problem for which unbounded nonnegative controls are allowed. In this dissertation we show that, in the absence of chattering controls, extremal trajectories for this problem are concatenations of trajectories determined by impulsive controls and null controls. We also describe the trajectories and costs associated with the null control and the impulsive controls for our dynamics.
Cruz delgado, Angel L., "On Some Optimal Control Problems for the Centroaffine Geometry on the Plane." (2000). LSU Historical Dissertations and Theses. 7320.