Hurwitz Stability of Interval Tensors

• Hurwitz stability of interval tensors is defined by requiring that every tensor instance, with entries varying within prescribed bounds, has eigenvalues with negative real parts.
• The methodology reduces complexity through auxiliary tensors and eigenvalue inclusion sets, combining multilinear algebra, spectral theory, and polynomial analysis.
• The criteria extend classical matrix stability concepts using algebraic positivity, LMI representations, and spectrahedral certificates, offering practical tools for robust control and optimization.

Hurwitz stability of interval tensors concerns the robust spectral properties of high-order structured arrays (tensors) whose entries are not fixed but allowed to vary independently within prescribed intervals. This concept generalizes the classical Hurwitz stability of interval matrices, providing criteria and computational methods to ensure that all admissible instances of a multilinear system preserve stability or positive definiteness despite parameter uncertainty. The theoretical landscape combines algebraic tensor analysis, eigenvalue inclusion theory, stability criteria via polynomial forms, and recent advances in determinantal representations and spectral certificates, culminating in explicit sufficient and necessary conditions for both analysis and synthesis in robust control, systems theory, and applied mathematics.

1. Preliminaries: Definitions and Spectral Criteria

An interval tensor is defined by a lower-bound and an upper-bound tensor, $\underline{\mathcal{A}}$ and $\overline{\mathcal{A}}$, yielding the set

$$\mathcal{A}^I = [\underline{\mathcal{A}},\overline{\mathcal{A}}] := \{ \mathcal{A}\ |\ \underline{\mathcal{A}} \leq \mathcal{A} \leq \overline{\mathcal{A}} \},$$

with the (possibly symmetric) center and deviation given by $$\mathcal{A}^c = (\underline{\mathcal{A}}+\overline{\mathcal{A}})/2,$$ $$\Delta = (\overline{\mathcal{A}}-\underline{\mathcal{A}})/2$$ (Ye et al., 15 Sep 2025). Hurwitz stability for a real tensor $\mathcal{A}\in T_{m,n}$ is the property that every E-eigenvalue $\lambda$ (i.e., the spectral roots of $\mathcal{A}x^{m-1}=\lambda x^{[m-1]}$) satisfies $\operatorname{Re}(\lambda)<0$. $\mathcal{A}^I$ is said to be Hurwitz stable if $\mathcal{A}$ is Hurwitz stable for all $\mathcal{A}\in \mathcal{A}^I$. This interval generalization addresses uncertainties in system parameters of multidimensional, nonlinear, or polynomial dynamical models.

2. Reduction to Auxiliary Tensors and Equivalence Conditions

The combinatorial complexity of explicitly verifying definiteness or Hurwitz stability for all $2^{n^m}$ extreme points of an $n$-dimensional, $m$-order tensor is intractable. Recent results introduce auxiliary tensors to achieve a feasible reduction (Ye et al., 15 Sep 2025). Let $Y=\{z\in\mathbb{R}^n : |z_j|=1,\,1\leq j\leq n\}$, with $|Y|=2^n$ ($2^{n-1}$ for even $m$, due to sign symmetry). The auxiliary tensors are constructed as

$$\mathcal{A}^z = \mathcal{A}^c - \Delta \times_1 T_z \times_2 T_z \cdots \times_m T_z,$$

where $T_z$ is a diagonal matrix with diagonal $z$, and (componentwise) $$a^z_{i_1\ldots i_m} = a^c_{i_1\ldots i_m} - \delta_{i_1\ldots i_m}$$ if $z_{i_1}\cdots z_{i_m}=1$, or $a^c_{i_1\ldots i_m}+\delta_{i_1\ldots i_m}$ otherwise.

For symmetric interval tensors, positive (semi-)definiteness of $\mathcal{A}^I$ is equivalent to that of all auxiliary tensors $\mathcal{A}^z$, i.e.,

$$\mathcal{A}^I\ \text{pos. definite} \iff \mathcal{A}^s \text{ pos. definite} \iff \mathcal{A}^z\ \forall z\in Y\ \text{pos. definite}.$$

For Hurwitz stability, analogous equivalence is established:

• $\mathcal{A}^I$ is Hurwitz stable if the associated symmetric interval tensor $\mathcal{A}_s^I$ is Hurwitz stable,
• and the latter holds iff each auxiliary tensor $$\widetilde{\mathcal{A}}^z = \mathcal{A}^c + \Delta \times_1 T_z \cdots \times_m T_z$$ is Hurwitz stable (Ye et al., 15 Sep 2025).

Thus, analysis is reduced to at most $2^{n-1}$ principal auxiliary tensors for computational tractability.

3. Spectral Bounds via Eigenvalue Inclusion Sets

The location of all real $H$-eigenvalues of even-order symmetric tensors can be contained in computable regions: double $\overline{B}$-intervals $\Lambda(\mathcal{A})$ and quasi-double $\overline{B}$-intervals $\Psi(\mathcal{A})$ (Jin et al., 2015). These intervals are constructed using generalized diagonal dominance and off-diagonal spread parameters:

• For each $i$, define

$$\Lambda_i = [ a_{i\ldots i} - \beta_i(\mathcal{A}),\ a_{i\ldots i} - \gamma_i(\mathcal{A}) ],\quad \tilde{\Lambda}_i = ( a_{i\ldots i} - \beta_i(\mathcal{A}) - \Delta_i,\ a_{i\ldots i} - \gamma_i(\mathcal{A}) - \Theta_i )$$

where $\beta_i,\gamma_i,\Delta_i,\Theta_i$ summarize worst-case off-diagonal coupling.

• For $i\neq j$, define refined intervals via inequalities involving pairwise cross-couplings.

The total inclusion set

$$\Lambda(\mathcal{A}) = \bigcup_i [\Lambda_i \cup \tilde{\Lambda}_i] \cup \bigcup_{i\neq j}\Lambda_{ij}$$

contains all $H$-eigenvalues. Tighter inclusion is provided by $\Psi(\mathcal{A})$. For positive definiteness (all eigenvalues positive), it suffices that $\Lambda(\mathcal{A})\subset(0,\infty)$. These intervals generalize Brauer-type and Gershgorin-type inclusions and are practical for certifying robustness in face of interval uncertainty.

4. Algebraic and Positivity-Based Hurwitz Criteria

For uncertain systems where entries depend rationally on interval parameters, Hurwitz stability is translated to the positivity of certain forms (polynomials) on the admissible parameter domain, frequently $[0,1]^m$ (Hou et al., 2010). Concretely:

• The uncertain coefficients in the characteristic polynomial and principal minors are rational functions of the parameters.
• By algebraic manipulation (homogenization), these are mapped to polynomial forms whose positivity is equivalent to Hurwitz stability.
• The parameter domain is subdivided into simplices, e.g., as

$$S_\theta = \{(x_1,\ldots,x_m)\ |\ 1\geq x_{k_1}\geq \cdots \geq x_{k_m}\geq 0 \}$$

for all permutation $\theta$.

• Positivity is then certified either by Pólya's theorem (with its associated degree bounds) or more efficiently via the Weighted Difference Substitution (WDS) method. If every WDS-transformed form has strictly positive coefficients, positivity holds on the simplex.

This approach yields a finite-step, computationally efficient necessary and sufficient test for robust Hurwitz stability in families of uncertain matrices or tensors parameterized by intervals of rational functions.

5. Determinantal Representations and Spectrahedral Certificates

A multivariate polynomial strictly Hurwitz stable on a tube domain admits (up to a strictly stable factor) a determinantal representation of the form

$$p(z)q(z) = \det\left(A_0 + z_1 A_1 + \cdots + z_d A_d\right)$$

where $A_0$ satisfies $\frac{A_0-A_0^*}{2i}>0$, $A_j\geq 0$, and $\sum_j A_j>0$ (Vinnikov et al., 26 Nov 2024). The Cayley transform is instrumental in moving between Schur and Hurwitz settings, while the matrix-valued Hermitian Positivstellensatz guarantees existence of a spectrahedral (LMI) certificate. In the context of interval tensors, if every realization of the $A_j$ (with tensor entries varying in their intervals) satisfies the required positivity, the associated polynomial (e.g., characteristic or spectral polynomial) is Hurwitz stable for the entire family. This LMI approach lends itself to modern semidefinite programming.

6. Hurwitz Stability Criteria for Interval Matrix and Tensor Polynomials

Generalizations of classical Hurwitz stability to interval matrix polynomials employ the positivity of block-Hankel matrices built from matricial Markov parameters and the existence of continued fraction/Stieltjes representations (Zhan et al., 2019, Choque-Rivero, 15 Jul 2025). For interval data, sufficient conditions include:

• All contiguous block Hankel minors or “quasiminors” (formed from interval extensions of Markov parameters) are strictly positive definite across the entire interval.
• For Hurwitz-type matrix polynomials, coprimeness conditions and explicit Bezoutians (subject to coefficient commutativity) allow construction of algebraic certificates—these may plausibly extend to the tensor setting through appropriately defined tensor polynomials, Bezoutians, and continued fractions.
• Matricial Stieltjes continued fractions with interval positive definite coefficients $C_k$ provide robust Hurwitz certificates for interval families.

For symmetric parametric interval matrices, Hurwitz stability is equivalent to strong positive definiteness of $-A^I$ and can be checked at all interval vertices, linking deterministic and interval-valued spectral analysis (Skalna, 2017).

7. Practical Implications and Computational Complexity

The survey of methods highlights tractable procedures for robust stability certification:

• For interval tensors of moderate dimension, explicit enumeration and testing of $2^{n-1}$ auxiliary tensors is computationally feasible (Ye et al., 15 Sep 2025).
• Interval inclusion sets (double/quasi-double $\overline{B}$-intervals, Brauer sets) give analytic tools for quickly establishing stability/definiteness or ruling out instability for a given family (Jin et al., 2015).
• Algebraic positivity and LMI-based determinations scale to higher-dimension problems and facilitate integration with optimization frameworks.
• Application scenarios include robust control, signal processing, multiagent systems, and any context where parametric uncertainty propagates through high-order interactions.
• Attested sufficiency and necessity of conditions, when available, provide strong assurance absent in conservative vertex-only or traditional “worst-case” methods.

A plausible implication is that methods based on spectrahedral certificates (determinantal representations, Hermitian Positivstellensätze) represent a unifying framework capable of addressing both classical and modern robust stability problems for high-dimensional uncertain systems.

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In sum, Hurwitz stability of interval tensors is a richly structured topic integrating multilinear algebra, spectral theory, algebraic geometry, and optimization. The field has matured to the point where both sharp theoretical criteria and practical computational tools are available for certifying stability under multidimensional uncertainty, with open directions in the generalization of algebraic and spectral methods from matrices to tensors, the development of tractable LMI frameworks, and the tight bounding of spectra in high-complexity settings.