Optimal Control and Hamiltonian Input-Output Systems

Abstract

Let us consider a smooth (i.e. C^∞ or C^k) nonlinear control system

$$ \dot{x} = f(x,u)\quad x \in X, u \in U $$

((1))

where f is a smooth mapping. For simplicity of exposition we will take X to be ℝⁿ or an open subset of ℝⁿ. (X could be an arbitrary manifold.) Furthermore we will make the (restrictive) assumption that U equals ℝ^m or an open subset of ℝ^m (or an arbitrary manifold without boundary). Let now L : X × U → ℝ and K : X → ℝ be smooth functions.