# Optimal control of the double integrator with minimum total variation

## TL;DR

This paper derives the optimal control strategy for a double integrator that minimizes both energy and total variation, providing insights into the Pareto optimal solutions and their properties.

## Contribution

It introduces a combined optimal control problem minimizing energy and total variation, deriving the unique solution and analyzing the Pareto front.

## Key findings

- The optimal total variation asymptotically is 2/3 of the minimum-energy control.
- The optimal control is piecewise constant with a single switch.
- The paper characterizes the Pareto front for the multi-objective problem.

## Abstract

We study the well-known minimum-energy control of the double integrator, along with the simultaneous minimization of the total variation in the control variable. We derive the optimality conditions and obtain the unique optimal solution to the combined problem, where the initial and terminal boundary points are specified. We study the problem from a multi-objective optimal control viewpoint, constructing the Pareto front. We show that the unique asymptotic optimal control function, for the minimization of the total variation alone, is piecewise constant with one switching at the midpoint of the time horizon. For any instance of the boundary conditions of the problem, we prove that the asymptotic optimal total variation is exactly $2/3$ of the total variation of the minimum-energy control. We illustrate the results for a particular instance of the problem and include a link to a video which animates the solutions while moving along the Pareto front.