Stabilization of Hamiltonian systems

Abstract

1. HAMILTONIAN SYSTEMS IN THIS paper we will be concerned with the stabilization by feedback of Hamiltonian systems. In order to facilitate our discussions (especially when applying Lyapunov’s second method) we will restrict ourselves to a particular, although natural, subclass of Hamiltonian systems given in the following way [l]. Let Q be an n-dimensional smooth manifold, denoting the configuration space, and let T*Q be the cotangent bundle, denoting the phase space or srufe space. Furthermore there is a smooth m-dimensional output manifold Y (m < n) and a smooth output map C : Q- Y. (Smooth will mean C” or Ck, with k sufficiently big, although we shall restrict ourselves in the second part of Section 2 to analytic data.) For simplicity we take C to be submersive. so rank dC(q) = m. We assume that the system on T*Q has an internal energy which is the sum of a kinetic energy K and a porentiul energy V. This means that there exists a Riemannian metric (,) on Q, in local coordinates (q,, . . . , q,J for Q given by