Casimir elements and optimal control

Abstract

Introduction. Optimal problems on Lie groups are an important class of problems linking modern control theory with its classical predecessors mechanics and differential geometry. My previous papers [5], [6] and [7] showed that even the most classical problems of mechanics, such as the motion of the rigid body, or its geometric companion, the equations describing the equilibrium configurations of an elastic rod can be effectively analyzed on Lie groups through the Maximum Principle and its associated Hamiltonian formalism. This paper will further illustrate the importance of modern geometric techniques by concentrating on time optimal control problems of mechanical systems recently studied in [3], [10] and [12]. Each of these studies are inspired by an important paper of Dubins [4] who considered and solved, the following geometric problem: Among all C curves γ(t) in the plane, which are parametrized by arc length, and which further satisfy the condition that d 2γ dt2 (t) is a measurable function with || 2γ dt2 (t)|| ≤ 1 almost everywhere, find the curve of minimal length which connects two arbitrary points in the plane and has prescribed tangent vectors at these points. Following H. J. Sussmann [12] we shall refer to this problem as Dubins’ problem. Each admissible curve γ(t) in the problem of Dubins can be parametrized by the angle θ, which the tangent vector makes with the horizontal direction to yield: