Second order conditions for optimality and local controllability of discrete-time systems

TL;DR

This paper develops second order necessary and sufficient conditions for optimality and local controllability in invertible discrete-time control systems, using geometric methods and Lie brackets to analyze the system's behavior.

Contribution

It introduces a geometric framework for second order optimality and controllability conditions in discrete-time systems, linking Hessians to Lie brackets.

Findings

Derived second order necessary conditions for optimality.

Established sufficient conditions for local controllability.

Expressed conditions using geometric vector fields and Lie brackets.

Abstract

We study local controllability and optimal control problems for invertible discrete-time control systems. We present second order necessary conditions for optimality and sufficient conditions for local controllability. The conditions are stated in geometric terms, using vector fields naturally associated to the system. The Hessian of the optimal problem is computed in terms of Lie brackets of vector fields of the system.