# Towards Velocity Turnpikes in Optimal Control of Mechanical Systems

## TL;DR

This paper introduces the concepts of velocity steady states and hyperbolic velocity turnpike properties to analyze and control mechanical systems, linking optimal solutions to trim primitives and time-varying turnpikes.

## Contribution

It formalizes velocity turnpikes and introduces velocity steady states, combining trim primitives with time-varying turnpike analysis for the first time.

## Key findings

- Optimal solutions correspond to trim solutions for all finite horizons.
- Velocity steady states serve as partial steady states in control.
- First formalization of velocity turnpikes in mechanical systems.

## Abstract

The paper proposes first steps towards the formalization and characterization of time-varying turnpikes in optimal control of mechanical systems. We propose the concepts of velocity steady states, which can be considered as partial steady states, and hyperbolic velocity turnpike properties for analysis and control. We show for a specific example that, for all finite horizons, both the (essential part of the) optimal solution and the orbit of the time-varying turnpike correspond to (optimal) trim solutions. Hereby, the present paper appears to be the first to combine the concepts of trim primitives and time-varying turnpike properties.