# Optimal Control and Hamiltonian Input-Output Systems

• A. J. van der Schaft
Chapter
Part of the Mathematics and Its Applications book series (MAIA, volume 29)

## Abstract

Let us consider a smooth (i.e. C or Ck) 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 ℝm or an open subset of ℝm (or an arbitrary manifold without boundary). Let now L : X × U → ℝ and K : X → ℝ be smooth functions.

## Keywords

Hamiltonian System Optimal Control Problem Poisson Bracket Integral Manifold Nonlinear Control System
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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