---
title: Stabilizability Radius in Control Systems
url: https://www.emergentmind.com/topics/stabilizability-radius
type: topic
---

# Stabilizability Radius in Control Systems

The stabilizability radius is a quantitative measure of proximity to the loss of stabilizability. In the arXiv literature it appears in several precise but related forms. For discrete-time linear switched systems, it is the smallest exponential rate that can be enforced for every initial condition by state-dependent switching; the condition \(\tilde{\rho}(\mathcal{M})<1\) is the criterion for feedback stabilizability [2002.10369]. For affinely perturbed continuous-time LTI systems, the real structured stabilizability radius \(r_{stz}(A(\theta),B(\theta))\) is the minimum perturbation norm that renders \((A(\theta),B(\theta))\) unstabilizable [2201.01112]. In both settings, the radius quantifies how much geometric or parametric freedom remains before stabilizability is lost.

## 1. Switched-system definition and basic properties

For a finite family of matrices \(\mathcal{M}=\{A_1,\dots,A_m\}\subset\mathbb{R}^{n\times n}\), the discrete-time switched system is
\[
x(k+1)=A_{\sigma_k}x(k), \qquad \sigma_k\in\{1,\dots,m\}.
\]
When the sole control action is the switching signal and the controller has access to the current state in real time, the stabilizability radius is defined by
\[
\tilde{\rho}(\mathcal{M})=\sup_{x_0\in\mathbb{R}^n}\tilde{\rho}_{x_0}(\mathcal{M}),
\]
where \(\tilde{\rho}_{x_0}(\mathcal{M})\) is the infimum over all \(\lambda\ge 0\) such that there exists a trajectory starting from \(x_0\) and a constant \(M>0\) satisfying \(|x(k)|\le M\lambda^k|x_0|\) for all \(k\ge 0\) [2002.10369].

This definition isolates the best achievable exponential growth rate under feedback switching. The paper also gives an equivalent global formulation in which \(M\) is common to all initial conditions, while the switching law may depend on the initial state. Two structural identities are fundamental:
\[
\tilde{\rho}(\gamma\mathcal{M})=\gamma\,\tilde{\rho}(\mathcal{M}),
\qquad
\tilde{\rho}(\mathcal{M}^k)=\tilde{\rho}(\mathcal{M})^k.
\]
They allow lower-bound constructions on products \(\mathcal{M}^k\) and clarify the scaling behavior of the radius [2002.10369].

The switched-system interpretation separates stabilizability from ordinary stability. The joint spectral radius \(\hat{\rho}(\mathcal{M})<1\) characterizes stability under arbitrary switching, whereas \(\tilde{\rho}(\mathcal{M})<1\) only requires that for every initial condition there exists a state-dependent switching strategy achieving exponential decay. This distinction is central in examples where no single open-loop switching law stabilizes every state, but feedback switching still does [2002.10369].

## 2. Lower spectral radius and minimax generalizations

A closely related invariant is the lower spectral radius
\[
\check{\rho}(\mathcal{A})=\lim_{n\to\infty}\inf\left\{\|A_n\cdots A_1\|^{1/n}:A_i\in\mathcal{A}\right\}.
\]
For uncontrolled switching systems \(x(n)=A_nx(n-1)\), \(\check{\rho}(\mathcal{A})<1\) is the criterion for uniform stabilizability, while \(\rho(\mathcal{A})<1\) is the criterion for asymptotic stability under all switching sequences [1712.06805].

For controlled switching systems with disturbances,
\[
x(n)=A_nB_nx(n-1), \qquad A_n\in\mathcal{A},\ B_n\in\mathcal{B},
\]
the minimax joint spectral radii
\[
\mu_n(\mathcal{A},\mathcal{B})=
\max_{A_1,\ldots,A_n\in\mathcal{A}}
\min_{B_1,\ldots,B_n\in\mathcal{B}}
\|A_nB_n\cdots A_1B_1\|,
\]
\[
\eta_n(\mathcal{A},\mathcal{B})=
\min_{B_1,\ldots,B_n\in\mathcal{B}}
\max_{A_1,\ldots,A_n\in\mathcal{A}}
\|A_nB_n\cdots A_1B_1\|
\]
lead to asymptotic quantities
\[
\mu(\mathcal{A},\mathcal{B})=\lim_{n\to\infty}\mu_n(\mathcal{A},\mathcal{B})^{1/n},
\qquad
\eta(\mathcal{A},\mathcal{B})=\lim_{n\to\infty}\eta_n(\mathcal{A},\mathcal{B})^{1/n}.
\]
These act as generalized stabilizability radii in the presence of antagonistic disturbance and control choices [1712.06805].

The control-theoretic interpretation is sharp. Path-dependent stabilizability holds if and only if \(\mu(\mathcal{A},\mathcal{B})<1\), whereas path-independent periodic stabilizability holds if and only if \(\eta(\mathcal{A},\mathcal{B})<1\). The classical invariants are recovered as special cases:
\[
\rho(\mathcal{A})=\mu(\mathcal{A},I)=\eta(\mathcal{A},I),
\qquad
\check{\rho}(\mathcal{A})=\mu(I,\mathcal{A})=\eta(I,\mathcal{A}).
\]
Accordingly, the ordinary lower spectral radius is the one-player limit of the minimax construction [1712.06805].

## 3. Bounds, singular matrices, and irregularity phenomena

Several sharp lower bounds are known for the switched-system stabilizability radius. For nonsingular matrix sets \(\mathcal{M}=\{A_1,\dots,A_m\}\),
\[
\tilde{\rho}(\mathcal{M}) \ge \left(\sum_{h=1}^m |\det A_h|^{-1}\right)^{-1/n},
\]
and a stronger determinant/singular-value bound is obtained through the optimization formula \(\tilde{\rho}_-^*\) built from the smallest singular values \(\delta_h\), determinants \(\Delta_h\), and the simplex constraint on weights \(\nu\) [2002.10369].

For arbitrary finite matrix sets, including singular ones, a new universal inequality is
\[
\frac{\check{\rho}(\mathcal{M})}{m}\le \tilde{\rho}(\mathcal{M})\le \check{\rho}(\mathcal{M}).
\]
This shows that the pointwise stabilizability radius cannot be arbitrarily smaller than the joint spectral subradius. If \(\tilde{\rho}(\mathcal{M})=0\), then necessarily \(\check{\rho}(\mathcal{M})=0\) [2509.17799].

Singular matrices create additional structure. If an irreducible matrix set contains a singular matrix whose image is one-dimensional, then
\[
\tilde{\rho}(\mathcal{M})=\check{\rho}(\mathcal{M}),
\]
and in dimension two any irreducible set with at least one singular matrix satisfies the same equality. For the two-dimensional family consisting of a rank-one singular matrix and a rotation matrix, the paper derives the exact formula
\[
\tilde{\rho}
=
\inf_{l\in\mathbb{N}}
\left|\lambda_2\frac{\sin\big((l\alpha-\beta)\pi\big)}{\sin(\beta\pi)}\right|^{1/(l+1)}.
\]
In the special case
\[
M_1=\begin{pmatrix}2&0\\0&0\end{pmatrix},\qquad
M_2=\begin{pmatrix}\cos(\alpha\pi)&\sin(\alpha\pi)\\-\sin(\alpha\pi)&\cos(\alpha\pi)\end{pmatrix},
\]
this reduces to
\[
\tilde{\rho}(\alpha)=\inf_{l\in\mathbb{N}}|2\cos(l\alpha\pi)|^{1/(l+1)},
\]
with \(\tilde{\rho}(\alpha)\le 1\), and equality only for \(\alpha=1/3\) or \(\alpha=2/3\) [2509.17799].

The function \(x_0\mapsto \tilde{\rho}_{x_0}(\mathcal{M})\) can be highly irregular. Dense discontinuity occurs when different projective directions have different optimal stabilizability rates and the corresponding attainable sets overlap densely. In the Stanford–Urbano example,
\[
A_1=\frac{\sqrt{2}}{2}\begin{pmatrix}1&1\\-1&1\end{pmatrix},
\qquad
A_2=\begin{pmatrix}\frac12&0\\0&2\end{pmatrix},
\]
every product has determinant \(1\), yet feedback switching yields
\[
|x(k)|\le 2\times 0.9^{k/4}|x(0)|,
\]
so \(\tilde{\rho}(\mathcal{M})<0.9^{1/4}\approx 0.974\), while determinant- and singular-value-based lower bounds remain strictly positive [2002.10369].

## 4. Networked control interpretation

In networked control systems with lossy channels, the stabilizability radius appears as a spectral-radius threshold. For
\[
x_{t+1}=Ax_t+Bu_t+w_t,
\]
with Bernoulli packet drops of reception rate \(q\in(0,1)\), a necessary and sufficient condition for mean-square stabilizability is
\[
q>1-\frac{1}{\rho(A)^2},
\]
or equivalently
\[
\rho(A)<\frac{1}{\sqrt{1-q}}.
\]
The quantity
\[
r_{\text{stab}}(q):=\frac{1}{\sqrt{1-q}}
\]
is therefore a critical spectral-radius boundary that can be interpreted as a stabilizability radius [2103.02553].

The same framework introduces the stabilizability margin
\[
\Delta_{\text{stab}}:=\left|1-q-\frac{1}{\rho(A)^2}\right|,
\]
which measures the distance to the critical boundary
\[
q=1-\frac{1}{\rho(A)^2}.
\]
Small \(\Delta_{\text{stab}}\) means that the pair \((\rho(A),q)\) lies close to the boundary and is correspondingly sensitive to estimation error [2103.02553].

A data-driven version is also developed. The spectral radius \(\rho(A)\) is estimated from finitely many samples by least squares, and the packet reception rate is estimated from the packet-drop sequence by
\[
\hat q=\frac{1}{N_q}\sum_{k=0}^{N_q-1}\gamma_k,
\qquad
|q-\hat q|\le f_5(N_q,\delta_q)
\]
with high probability. The resulting conservative effective threshold is
\[
r_{\text{stab}}^{\text{eff}}:=\frac{1}{\sqrt{1-\hat q+f_5}},
\]
and the test compares \(\rho(\hat A)+\epsilon\) against that threshold. The sample complexity of the decision problem is inversely proportional to the square of the stabilizability margin [2103.02553].

## 5. Affine-structured perturbation radii and computational complexity

For continuous-time LTI systems with affine perturbations
\[
A(\theta)=A+\sum_{i=1}^p\theta_iA_i,\qquad
B(\theta)=B+\sum_{i=1}^p\theta_iB_i,
\]
the real structured stabilizability radius is
\[
r_{stz}(A(\theta),B(\theta))=
\min\left\{
\big\|{\bf\Gamma}(\theta)\big\|:
\theta\in\mathbb{R}^p,\ (A(\theta),B(\theta))\ \text{is unstabilizable}
\right\}.
\]
The same affine framework also defines the real structured controllability radius \(r_{con}\) and the real structured stability radius \(r_{stb}\), producing the trio RSCR, RSSZR, and RSSR [2201.01112].

The perturbation model is general: \({\bf\Gamma}(\theta)\) may be a full matrix, a diagonal matrix, or a vector, and the hardness results are independent of whether the Frobenius norm or the spectral norm is used. Determining the feasibility of RSCR is NP-complete, checking the feasibility of RSSZR is NP-hard, and computing both radii is NP-hard. A related RSSR decision problem is NP-hard as well [2201.01112].

The computational formulation uses real block matrices. For RSSZR one introduces
\[
Z=
\begin{bmatrix}
A(\theta)-\mu I & B(\theta) & -\lambda I & 0\\
\lambda I & 0 & A(\theta)-\mu I & B(\theta)
\end{bmatrix},
\qquad \mu\ge 0,
\]
so that rank deficiency of \(Z\) encodes the existence of an uncontrollable unstable mode. The resulting low-rank optimization problem is relaxed by replacing the rank constraint with a regularized truncated nuclear norm term
\[
\|Z\|_*-\|Z\|_{F_{2n-1}}.
\]
A unified majorization–minimization framework then handles RSCR, RSSZR, and RSSR for both Frobenius- and \(2\)-norm objectives [2201.01112].

The same framework yields two-stage algorithms with performance specifications. After an initialization step that drives the relevant block matrix near singularity, the regularization parameter is chosen as \(\gamma=g(\theta^{(0)})/\epsilon\), and the limit point satisfies a residual singular-value bound such as
\[
\sigma_{\min}\big([A(\theta^*)-(\mu^*+{\bf j}\lambda^*)I,\ B(\theta^*)]\big)\le \epsilon,
\qquad \mu^*\ge 0.
\]
This produces local optima that are explicitly close to the unstabilizable set [2201.01112].

## 6. Relation to adjacent radius notions

The stabilizability radius sits inside a broader family of distance-to-loss-of-property measures. Closely related notions include the real structured stability radius of sparse LTI systems under Frobenius-norm-bounded perturbations [1810.10578], large-scale structured real stability radius computed by one-sided interpolatory model reduction with quadratic convergence [2105.01001], and approximate stability radius analysis and redesign based on eigenvalue sensitivity and successive linear approximation [2403.12006]. In positive-systems settings, the nearest stable or unstable nonnegative matrix is characterized through Frobenius-norm distance and singular-value formulas [1802.03054].

Energy-based and infinite-dimensional variants extend the same robustness logic. Dissipative-Hamiltonian systems admit structured and unstructured stability radii, with formulations through \(\mathcal{H}_\infty\) norms, Rayleigh quotients, or eigenvalue optimization [1808.03574, 2511.14935]. For exponentially stable infinite-dimensional interconnected well-posed systems, the stability radius with respect to static perturbations is bounded, and for regular systems with zero feedthrough and finite-dimensional output spaces it is given exactly by a transfer-function formula [1912.00644].

These neighboring theories do not redefine the stabilizability radius, but they provide the computational vocabulary in which it is often studied: joint spectral characteristics, transfer-function norms, spectral value sets, rank-deficiency certificates, Rayleigh-quotient minimization, and projection-based reduction. The literature therefore treats stabilizability radius not as a single invariant with one universal formula, but as a family of technically precise radius concepts adapted to switching, uncertainty structure, network constraints, and admissible control mechanisms [1712.06805].

Source: https://www.emergentmind.com/topics/stabilizability-radius