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On Infinite-horizon Minimum Energy Control

Mohamed-Ali Belabbas, Xudong Chen

TL;DR

The paper resolves the infinite-horizon minimum energy control problem for finite-dimensional LTI systems by deriving a necessary and sufficient condition: the problem has a solution for all $x_0$ if and only if $(A,B)$ is stabilizable and $A$ has no imaginary eigenvalues. The authors analyze the controllability Gramian $W_{A,B}(T)$ as $T\to\infty$, decompose the dynamics via Jordan form, and show that the limit $W_{A,B}(T)^{-1}$ converges to a Riccati-based matrix $K$ on the unstable subspace, while the admissible initial-state subspace is exactly the center/anti-stable part $\mathbf{V}_{\mathsf{a}}$. When imaginary eigenvalues are present, solvability holds only for initial states in $\mathbf{V}_{\mathsf{a}}$, with the optimal cost $x_0^\dagger K x_0$ and feedback $u^*(t)=-B^\dagger K x(t)$. The paper also develops sophisticated asymptotic techniques (including buffer terms and Schur complements) to handle cases with non-positive or purely imaginary spectra, yielding a complete, constructive characterization and connecting infinite-horizon optimal control to a Riccati equation and pole-placement interpretation. This work clarifies when energy-optimal control is possible in the long run and provides explicit optimal laws and costs in the stabilizable regime.

Abstract

We address the infinite-horizon minimum energy control problem for linear time-invariant finite-dimensional systems $(A, B)$. We show that the problem admits a solution if and only if $(A, B)$ is stabilizable and $A$ does not have imaginary eigenvalues.

On Infinite-horizon Minimum Energy Control

TL;DR

The paper resolves the infinite-horizon minimum energy control problem for finite-dimensional LTI systems by deriving a necessary and sufficient condition: the problem has a solution for all if and only if is stabilizable and has no imaginary eigenvalues. The authors analyze the controllability Gramian as , decompose the dynamics via Jordan form, and show that the limit converges to a Riccati-based matrix on the unstable subspace, while the admissible initial-state subspace is exactly the center/anti-stable part . When imaginary eigenvalues are present, solvability holds only for initial states in , with the optimal cost and feedback . The paper also develops sophisticated asymptotic techniques (including buffer terms and Schur complements) to handle cases with non-positive or purely imaginary spectra, yielding a complete, constructive characterization and connecting infinite-horizon optimal control to a Riccati equation and pole-placement interpretation. This work clarifies when energy-optimal control is possible in the long run and provides explicit optimal laws and costs in the stabilizable regime.

Abstract

We address the infinite-horizon minimum energy control problem for linear time-invariant finite-dimensional systems . We show that the problem admits a solution if and only if is stabilizable and does not have imaginary eigenvalues.

Paper Structure

This paper contains 14 sections, 22 theorems, 131 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Main Results
  3. Proof of Theorem \ref{['thm:main2p']}
  4. Proof of sufficiency
  5. Proof of necessity
  6. The case of imaginary eigenvalues
  7. On repeated eigenvalues
  8. Proof of Theorem \ref{['thm:pimlimit']}
  9. The case of non-positive eigenvalues
  10. Asymptotic convergence of matrix products
  11. Proof of Theorem \ref{['th:theorem6prop2']}
  12. Proof of Theorem \ref{['thm:main']}
  13. Proof of item 2 of Theorem \ref{['thm:limit0']}
  14. Proof of item 3 of Theorem \ref{['thm:limit0']}

Key Result

Theorem 1.1

Let $\mathbb{F}$ be either $\mathbb{R}$ or $\mathbb{C}$. Let $(A,B)\in \mathbb{F}^{n\times n}\times \mathbb{F}^{n\times m}$ be a stabilizable pair. The infinite-horizon minimum energy control problem admits a solution for all $x_0 \in \mathbb{F}^n$ if and only if $A$ has no imaginary eigenvalue.

Figures (2)

  • Figure 1: For the controllable matrix pair $(A,B)$ with $A$ a $3$-by-$3$ Jordan block with zero eigenvalues, and $B=[0;0;1]$, we plot in (a) the absolute value of the optimal control $u_T(t)$ in \ref{['eq:optimfinTu']} which drives the system from $x_0=[1;1;1]$ to $x(T)=0$ for increasing $T$, and in (b) the $\mathrm{L}^2$ norm of $u_T$. We see that as $T$ grows, $\|u_T\|$ become smaller and, in the limit, convergence to $0$. However, since the system is not stabilizable, a zero control does not drive $x_0$ to the origin. This illustrates the issue with imaginary eigenvalues.
  • Figure 2: Illustration of the convergence of the inverse of the controllability Gramian for different matrix pairs $(A_\epsilon, B)$.

Theorems & Definitions (40)

  • Definition 1: Admissible control
  • Theorem 1.1
  • Theorem 2.1
  • Proposition 2.2
  • proof
  • Theorem 2.3
  • Corollary 2.4: The real case
  • proof
  • Proposition 3.1
  • proof
  • ...and 30 more