Lyapunov equations: a (fixed) point of view

TL;DR

This work establishes a fixed-point viewpoint on the discrete-time Lyapunov equation, showing that asymptotic stability, observability, and Lyapunov solvability are intimately linked: any two imply the third. The authors recast $A^{T}QA - Q + C^{T}C=0$ as a fixed-point equation on the set of positive semidefinite matrices with unit trace and apply Brouwer's fixed-point theorem to obtain a family of fixed points $X_{\alpha}$, from which a Lyapunov solution is recovered as $Q=X_{\alpha}/\alpha$ when $\lambda_{\alpha}=1$, with $\lambda_{\alpha}=\mathrm{tr}(A^{T}X_{\alpha}A+\alpha C^{T}C)$. The argument proceeds by showing observability implies $X_{\alpha}\succ 0$, proving uniqueness of the fixed point, and using a continuity/ IVT argument to establish the existence of $\alpha$ with $\lambda_{\alpha}=1$, thereby producing a unique $Q\succ 0$ solving the Lyapunov equation. The approach highlights a symmetric, elementary route to Lyapunov theory and suggests natural extensions to cone-structured and nonlinear settings, including internally positive systems via analogous constructions.

Abstract

The Lyapunov equation is the gateway drug of nonlinear control theory. In these notes we revisit an elegant statement connecting the concepts of asymptotic stability and observability, to the solvability of Lyapunov equations, and discuss how this statement can be proved using the Brouwer fixed-point theorem.

Figures (4)

• Figure 1: Euler diagram showing the interdependence of the statements in \ref{['thm:main']}. The theorem shows that it is not possible for any pair of the statements to be true without the third also being true.
• Figure 2: Points on the red surface and above in the top axis satisfy the inequalities $xz\geq{}y^2$, $x\geq{}0$, and $z\geq{}0$. Points on the surface in the middle axis satisfy $x+z=1$. The circle on the third axis shows the intersection of these two shapes. The circle corresponds to the compact convex set $\mathcal{C}$ for the two by two matrices as defined by \ref{['eq:unittr']}. Points on this circle correspond to values of $x$, $y$, and $z$ such that the matrix in \ref{['eq:2x2']} is positive semi-definite with unit trace, and the function $f\left(\cdot\right)$ in \ref{['eq:fixptorig']} maps points on this set back onto itself. Since the set is compact and convex, the Brouwer fixed-point theorem guarantees the existence of at least one fixed-point for this function (that is, a point on the circle that is mapped back onto itself by $f\left(\cdot\right)$).
• Figure 3: Sketch of $\mathcal{C}$, two distinct points $X_\alpha,Y_\alpha\in\mathcal{C}$, and the line $\theta X_{\alpha} + \left(1-\theta\right)Y_{\alpha}$. Points in the interior of $\mathcal{C}$ are positive definite, and points on the boundary are only semi-definite. Since $\mathcal{C}$ is compact, the line must pass through its boundary. This means that there is a value of $\theta$ such that $g\left(\theta\right)$ is positive semi-definite, but not positive definite. It also follows that when $\theta$ becomes sufficiently large, $g\left(\theta\right)\notin\mathcal{C}$. These two observations are used in \ref{['sec:unique']} to deduce that for each $\alpha>0$, the function $f\left(\cdot\right)$ has a unique fixed point.
• Figure 4: The blue curve shows the function $g^{-1}\left(f\left(g\left(\theta\right)\right)\right)$, where without loss of generality, it has been assumed that $\lambda_\alpha>\gamma_\alpha$ (more specifically, $\lambda_\alpha=1.2$ and $\gamma_\alpha=0.8$). The red and cyan lines show the cobweb plots of function iterates starting at points slightly to the left and right of $\theta=0$ (the points $\theta_0^{\mathrm{a}}$ and $\theta_0^{\mathrm{b}}$ respectively). The function has a repelling fixed point at $\theta=0$ and an attracting fixed point at $\theta=1$. Since the function has an asymptote at $\theta=\tfrac{-\gamma_a}{\lambda_a-\gamma_a}$ (the dotted line), if we iteratively apply the function starting slightly to the left of the repelling fixed point, the iterates can grow arbitrarily large. This implies that for sufficiently large $n$, there exists a $\theta_0$ such that $f\left(\theta_n\right)\notin\mathcal{C}$ (as $\theta_k$ varies, $f\left(\theta_k\right)$ will move along the line in \ref{['fig:3']} and eventually leave $\mathcal{C}$, since $\mathcal{C}$ is compact).

Theorems & Definitions (6)

• Theorem 1
• Remark 1
• Theorem 2: The Brouwer fixed-point theorem
• Remark 2
• Remark 3
• Remark 4