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Optimal Control on Positive Cones

Richard Pates, Anders Rantzer

TL;DR

This work develops a general cone-based framework for infinite-horizon optimal control on finite-dimensional positive cones, showing that under a cone-compatibility condition the Bellman equation admits a linear solution $J^*(x)=\lambda^\top x$ that can be computed by convex optimization. The main result establishes equivalences between finite value, a fixed-point equation $\lambda_* = s + A^T\lambda_* + \phi(r + B^T\lambda_*)$, and a dual-cone convex program, with an explicit control law obtained by minimizing $(r+B^T\lambda_*)^T u$ over feasible $(x,u)\in\mathcal{P}$; the Bellman iterations produce $J_k(x)=\lambda_k^T x$. The framework is illustrated across three examples—LQR on the PSD cone, a polyhedral-cone regulator for positive systems, and a structured LQR preserving circulant structure—recovering algebraic Riccati equations, LP/SDP formulations, and structure-preserving controllers, respectively. This approach unifies classical control theory with positive-system and network-oriented methods, offering a scalable toolbox for explicit solutions in broader cone-based settings.

Abstract

An optimal control problem on finite-dimensional positive cones is stated. Under a critical assumption on the cone, the corresponding Bellman equation is satisfied by a linear function, which can be computed by convex optimization. A separate theorem relates the assumption on the cone to the existence of minimal elements in certain subsets of the dual cone. Three special cases are derived as examples. The first one, where the positive cone is the set of positive semi-definite matrices, reduces to standard linear quadratic control. The second one, where the positive cone is a polyhedron, reduces to a recent result on optimal control of positive systems. The third special case corresponds to linear quadratic control with additional structure, such as spatial invariance.

Optimal Control on Positive Cones

TL;DR

This work develops a general cone-based framework for infinite-horizon optimal control on finite-dimensional positive cones, showing that under a cone-compatibility condition the Bellman equation admits a linear solution that can be computed by convex optimization. The main result establishes equivalences between finite value, a fixed-point equation , and a dual-cone convex program, with an explicit control law obtained by minimizing over feasible ; the Bellman iterations produce . The framework is illustrated across three examples—LQR on the PSD cone, a polyhedral-cone regulator for positive systems, and a structured LQR preserving circulant structure—recovering algebraic Riccati equations, LP/SDP formulations, and structure-preserving controllers, respectively. This approach unifies classical control theory with positive-system and network-oriented methods, offering a scalable toolbox for explicit solutions in broader cone-based settings.

Abstract

An optimal control problem on finite-dimensional positive cones is stated. Under a critical assumption on the cone, the corresponding Bellman equation is satisfied by a linear function, which can be computed by convex optimization. A separate theorem relates the assumption on the cone to the existence of minimal elements in certain subsets of the dual cone. Three special cases are derived as examples. The first one, where the positive cone is the set of positive semi-definite matrices, reduces to standard linear quadratic control. The second one, where the positive cone is a polyhedron, reduces to a recent result on optimal control of positive systems. The third special case corresponds to linear quadratic control with additional structure, such as spatial invariance.
Paper Structure (9 sections, 3 theorems, 74 equations)

This paper contains 9 sections, 3 theorems, 74 equations.

Table of Contents

  1. Introduction
  2. Main Result
  3. Examples
  4. The Linear Quadratic Regulator
  5. A Linear Regulator on a Polyhedral Cone
  6. A Structured Linear Quadratic Regulation Problem
  7. Conclusions
  8. Acknowledgment
  9. Proof of \ref{['lem:1']}

Key Result

Theorem 1

Let $\mathcal{P}$ be a proper cone in $\mathbb{R}^{n+m}$ with dual cone $\mathcal{P}^*$. Given $A\in\mathbb{R}^{n\times n}$, $B\in\mathbb{R}^{n\times m}$, suppose that for every $(x,u)\in\mathcal{P}$ there exists $v$ such that $(Ax+Bu,v)\in\mathcal{P}$. Let $(s,r)$ be an interior point of $\mathcal{ for all $(x,v)\in\mathcal{P}$, $(\eta,\mu)\in\mathcal{P}^*$. Then the following three statements ar

Theorems & Definitions (8)

  • Theorem 1
  • proof
  • Theorem 2
  • proof
  • Definition 1
  • Lemma 1
  • proof
  • proof