The Herglotz principle and vakonomic dynamics

TL;DR

This paper develops a variational approach to vakonomic dynamics on contact systems with nonlinear constraints, linking it to optimal control problems with implicit cost functions and providing a geometric framework for their analysis.

Contribution

It introduces a novel variational principle for vakonomic contact systems using the Herglotz action, extending Lagrangian methods to nonlinear constrained dynamics with applications in control theory.

Findings

Derived vakonomic dynamics via Herglotz principle on contact systems.

Connected vakonomic contact dynamics to optimal control with implicit costs.

Provided a geometric framework for analyzing symmetries in these systems.

Abstract

In this paper we study vakonomic dynamics on contact systems with nonlinear constraints. In order to obtain the dynamics, we consider a space of admisible paths, which are the ones tangent to a given submanifold. Then, we find the critical points of the Herglotz action on this space of paths. This dynamics can be also obtained through an extended Lagrangian, including Lagrange multiplier terms. This theory has important applications in optimal control theory for Herglotz control problems, in which the cost function is given implicitly, through an ODE, instead of by a definite integral. Indeed, these control problems can be considered as particular cases of vakonomic contact systems, and we can use the Lagrangian theory of contact systems in order to understand their symmetries and dynamics.