Stabilizability Radius in Control Systems

• Stabilizability radius is a quantitative measure that defines the minimal perturbation or feedback switching needed to render a control system unstabilizable.
• It connects key spectral invariants—such as the lower spectral radius and joint spectral radius—with state-dependent strategies in both discrete-time switched and continuous-time systems.
• This concept is vital for networked control and robust design, impacting performance analysis, sensitivity studies, and NP-hard computational methodologies.

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 (Dettmann et al., 2020). 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 (Zhang et al., 2022). 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 $r_{stz}(A(\theta),B(\theta))$0 for all $r_{stz}(A(\theta),B(\theta))$1 (Dettmann et al., 2020).

This definition isolates the best achievable exponential growth rate under feedback switching. The paper also gives an equivalent global formulation in which $r_{stz}(A(\theta),B(\theta))$2 is common to all initial conditions, while the switching law may depend on the initial state. Two structural identities are fundamental: $r_{stz}(A(\theta),B(\theta))$3 They allow lower-bound constructions on products $r_{stz}(A(\theta),B(\theta))$4 and clarify the scaling behavior of the radius (Dettmann et al., 2020).

The switched-system interpretation separates stabilizability from ordinary stability. The joint spectral radius $r_{stz}(A(\theta),B(\theta))$5 characterizes stability under arbitrary switching, whereas $r_{stz}(A(\theta),B(\theta))$6 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 (Dettmann et al., 2020).

2. Lower spectral radius and minimax generalizations

A closely related invariant is the lower spectral radius

$r_{stz}(A(\theta),B(\theta))$7

For uncontrolled switching systems $r_{stz}(A(\theta),B(\theta))$8, $r_{stz}(A(\theta),B(\theta))$9 is the criterion for uniform stabilizability, while $(A(\theta),B(\theta))$0 is the criterion for asymptotic stability under all switching sequences (Kozyakin, 2017).

For controlled switching systems with disturbances,

$(A(\theta),B(\theta))$1

the minimax joint spectral radii

$(A(\theta),B(\theta))$2

$(A(\theta),B(\theta))$3

lead to asymptotic quantities

$(A(\theta),B(\theta))$4

These act as generalized stabilizability radii in the presence of antagonistic disturbance and control choices (Kozyakin, 2017).

The control-theoretic interpretation is sharp. Path-dependent stabilizability holds if and only if $(A(\theta),B(\theta))$5, whereas path-independent periodic stabilizability holds if and only if $(A(\theta),B(\theta))$6. The classical invariants are recovered as special cases: $(A(\theta),B(\theta))$7 Accordingly, the ordinary lower spectral radius is the one-player limit of the minimax construction (Kozyakin, 2017).

3. Bounds, singular matrices, and irregularity phenomena

Several sharp lower bounds are known for the switched-system stabilizability radius. For nonsingular matrix sets $(A(\theta),B(\theta))$8,

$(A(\theta),B(\theta))$9

and a stronger determinant/singular-value bound is obtained through the optimization formula $$\mathcal{M}=\{A_1,\dots,A_m\}\subset\mathbb{R}^{n\times n}$$0 built from the smallest singular values $$\mathcal{M}=\{A_1,\dots,A_m\}\subset\mathbb{R}^{n\times n}$$1, determinants $$\mathcal{M}=\{A_1,\dots,A_m\}\subset\mathbb{R}^{n\times n}$$2, and the simplex constraint on weights $$\mathcal{M}=\{A_1,\dots,A_m\}\subset\mathbb{R}^{n\times n}$$3 (Dettmann et al., 2020).

For arbitrary finite matrix sets, including singular ones, a new universal inequality is

$$\mathcal{M}=\{A_1,\dots,A_m\}\subset\mathbb{R}^{n\times n}$$4

This shows that the pointwise stabilizability radius cannot be arbitrarily smaller than the joint spectral subradius. If $$\mathcal{M}=\{A_1,\dots,A_m\}\subset\mathbb{R}^{n\times n}$$5, then necessarily $$\mathcal{M}=\{A_1,\dots,A_m\}\subset\mathbb{R}^{n\times n}$$6 (Dettmann et al., 22 Sep 2025).

Singular matrices create additional structure. If an irreducible matrix set contains a singular matrix whose image is one-dimensional, then

$$\mathcal{M}=\{A_1,\dots,A_m\}\subset\mathbb{R}^{n\times n}$$7

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

$$\mathcal{M}=\{A_1,\dots,A_m\}\subset\mathbb{R}^{n\times n}$$8

In the special case

$$\mathcal{M}=\{A_1,\dots,A_m\}\subset\mathbb{R}^{n\times n}$$9

this reduces to

$$x(k+1)=A_{\sigma_k}x(k), \qquad \sigma_k\in\{1,\dots,m\}.$$0

with $$x(k+1)=A_{\sigma_k}x(k), \qquad \sigma_k\in\{1,\dots,m\}.$$1, and equality only for $$x(k+1)=A_{\sigma_k}x(k), \qquad \sigma_k\in\{1,\dots,m\}.$$2 or $$x(k+1)=A_{\sigma_k}x(k), \qquad \sigma_k\in\{1,\dots,m\}.$$3 (Dettmann et al., 22 Sep 2025).

The function $$x(k+1)=A_{\sigma_k}x(k), \qquad \sigma_k\in\{1,\dots,m\}.$$4 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,

$$x(k+1)=A_{\sigma_k}x(k), \qquad \sigma_k\in\{1,\dots,m\}.$$5

every product has determinant $$x(k+1)=A_{\sigma_k}x(k), \qquad \sigma_k\in\{1,\dots,m\}.$$6, yet feedback switching yields

$$x(k+1)=A_{\sigma_k}x(k), \qquad \sigma_k\in\{1,\dots,m\}.$$7

so $$x(k+1)=A_{\sigma_k}x(k), \qquad \sigma_k\in\{1,\dots,m\}.$$8, while determinant- and singular-value-based lower bounds remain strictly positive (Dettmann et al., 2020).

4. Networked control interpretation

In networked control systems with lossy channels, the stabilizability radius appears as a spectral-radius threshold. For

$$x(k+1)=A_{\sigma_k}x(k), \qquad \sigma_k\in\{1,\dots,m\}.$$9

with Bernoulli packet drops of reception rate $$\tilde{\rho}(\mathcal{M})=\sup_{x_0\in\mathbb{R}^n}\tilde{\rho}_{x_0}(\mathcal{M}),$$0, a necessary and sufficient condition for mean-square stabilizability is

$$\tilde{\rho}(\mathcal{M})=\sup_{x_0\in\mathbb{R}^n}\tilde{\rho}_{x_0}(\mathcal{M}),$$1

or equivalently

$$\tilde{\rho}(\mathcal{M})=\sup_{x_0\in\mathbb{R}^n}\tilde{\rho}_{x_0}(\mathcal{M}),$$2

The quantity

$$\tilde{\rho}(\mathcal{M})=\sup_{x_0\in\mathbb{R}^n}\tilde{\rho}_{x_0}(\mathcal{M}),$$3

is therefore a critical spectral-radius boundary that can be interpreted as a stabilizability radius (Xu et al., 2021).

The same framework introduces the stabilizability margin

$$\tilde{\rho}(\mathcal{M})=\sup_{x_0\in\mathbb{R}^n}\tilde{\rho}_{x_0}(\mathcal{M}),$$4

which measures the distance to the critical boundary

$$\tilde{\rho}(\mathcal{M})=\sup_{x_0\in\mathbb{R}^n}\tilde{\rho}_{x_0}(\mathcal{M}),$$5

Small $$\tilde{\rho}(\mathcal{M})=\sup_{x_0\in\mathbb{R}^n}\tilde{\rho}_{x_0}(\mathcal{M}),$$6 means that the pair $$\tilde{\rho}(\mathcal{M})=\sup_{x_0\in\mathbb{R}^n}\tilde{\rho}_{x_0}(\mathcal{M}),$$7 lies close to the boundary and is correspondingly sensitive to estimation error (Xu et al., 2021).

A data-driven version is also developed. The spectral radius $$\tilde{\rho}(\mathcal{M})=\sup_{x_0\in\mathbb{R}^n}\tilde{\rho}_{x_0}(\mathcal{M}),$$8 is estimated from finitely many samples by least squares, and the packet reception rate is estimated from the packet-drop sequence by

$$\tilde{\rho}(\mathcal{M})=\sup_{x_0\in\mathbb{R}^n}\tilde{\rho}_{x_0}(\mathcal{M}),$$9

with high probability. The resulting conservative effective threshold is

$\tilde{\rho}_{x_0}(\mathcal{M})$0

and the test compares $\tilde{\rho}_{x_0}(\mathcal{M})$1 against that threshold. The sample complexity of the decision problem is inversely proportional to the square of the stabilizability margin (Xu et al., 2021).

5. Affine-structured perturbation radii and computational complexity

For continuous-time LTI systems with affine perturbations

$\tilde{\rho}_{x_0}(\mathcal{M})$2

the real structured stabilizability radius is

$\tilde{\rho}_{x_0}(\mathcal{M})$3

The same affine framework also defines the real structured controllability radius $\tilde{\rho}_{x_0}(\mathcal{M})$4 and the real structured stability radius $\tilde{\rho}_{x_0}(\mathcal{M})$5, producing the trio RSCR, RSSZR, and RSSR (Zhang et al., 2022).

The perturbation model is general: $\tilde{\rho}_{x_0}(\mathcal{M})$6 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 (Zhang et al., 2022).

The computational formulation uses real block matrices. For RSSZR one introduces

$\tilde{\rho}_{x_0}(\mathcal{M})$7

so that rank deficiency of $\tilde{\rho}_{x_0}(\mathcal{M})$8 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

$\tilde{\rho}_{x_0}(\mathcal{M})$9

A unified majorization–minimization framework then handles RSCR, RSSZR, and RSSR for both Frobenius- and $\lambda\ge 0$0-norm objectives (Zhang et al., 2022).

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 $\lambda\ge 0$1, and the limit point satisfies a residual singular-value bound such as

$\lambda\ge 0$2

This produces local optima that are explicitly close to the unstabilizable set (Zhang et al., 2022).

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 (Katewa et al., 2018), large-scale structured real stability radius computed by one-sided interpolatory model reduction with quadratic convergence (Aliyev, 2021), and approximate stability radius analysis and redesign based on eigenvalue sensitivity and successive linear approximation (Rai et al., 2024). In positive-systems settings, the nearest stable or unstable nonnegative matrix is characterized through Frobenius-norm distance and singular-value formulas (Guglielmi et al., 2018).

Energy-based and infinite-dimensional variants extend the same robustness logic. Dissipative-Hamiltonian systems admit structured and unstructured stability radii, with formulations through $\lambda\ge 0$3 norms, Rayleigh quotients, or eigenvalue optimization (Aliyev et al., 2018, Benner et al., 18 Nov 2025). 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 (Jacob et al., 2019).

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 (Kozyakin, 2017).