Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Vector-Valued Integer Optimal Control with TV Regularization: Optimality Conditions and Algorithmic Treatment

Jonas Marko, Gerd Wachsmuth

Abstract

We investigate a broad class of integer optimal control problems with vector-valued controls and switching regularization using a total variation functional involving the p-norm, which influences the structure of a solution. We derive optimality conditions of first and second order for the integer optimal control problem via a switching-point reformulation. For the numerical solution, we use a trust region method utilizing Bellman's optimality principle for the subproblems. We will show convergence properties of the method and highlight the algorithms efficacy on some benchmark examples.

Vector-Valued Integer Optimal Control with TV Regularization: Optimality Conditions and Algorithmic Treatment

Abstract

We investigate a broad class of integer optimal control problems with vector-valued controls and switching regularization using a total variation functional involving the p-norm, which influences the structure of a solution. We derive optimality conditions of first and second order for the integer optimal control problem via a switching-point reformulation. For the numerical solution, we use a trust region method utilizing Bellman's optimality principle for the subproblems. We will show convergence properties of the method and highlight the algorithms efficacy on some benchmark examples.

Paper Structure

This paper contains 4 sections, 2 theorems, 20 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Problem Formulation and Properties
  3. The $\operatorname{TV}_p$ Functional
  4. Optimality Conditions

Key Result

Lemma 10

Let $u\in U_{\mathrm{ad}}$ and $(\hat{n},\hat{a},\hat{t})$ be the data of the full representation of $u$. Then, we have Thus $\operatorname{TV}_p$ is independent of the switching time vector of the full representation.

Figures (2)

  • Figure 1: The components of $u\in L^1(0,T)^2$ with $u(t)\in \{0,1,2\}^2$.
  • Figure 2: Switching from $(0,0)$ to $(2,2)$ (red vector) touches $(1,1)$, thus, stopping at $(1,1)$ does not lead to a higher variation. The blue, cyan and violet paths have the same length w.r.t. the 1- and $\infty$-norm.

Theorems & Definitions (9)

  • proof
  • proof
  • Definition 5
  • proof
  • proof
  • Definition 9
  • Lemma 10
  • proof
  • Lemma 11