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Constrained polynomial roots and a modulated approach to Schur stability

Ziyad AlSharawi, Jose S. Cánovas, Sadok Kallel

TL;DR

This work reframes Schur stability as a global-stability problem for delayed linear difference equations and develops a systematic, incremental algorithm that tests coefficient conditions starting from the classical $\ell_1$-norm criterion $\|p\|_1<2$. By embedding the root-coefficient relationship into a semialgebraic framework and leveraging mixed-monotone maps, it derives a sequence of increasingly strong sufficient conditions that certify all roots lie in $\mathbb{D}$. The approach yields a semialgebraic filtration of the coefficient locus $\mathcal{C}_n(\mathbb{D})$, provides explicit results for degree-2 polynomials and practical applications (e.g., Cournot models, population dynamics), and offers a computationally attractive alternative to Jury’s algorithm in parameter-rich or high-degree settings. The method elegantly combines real algebraic geometry with dynamical-systems techniques to produce fast, conservative stability certificates with clear geometric interpretation and potential for higher-dimensional extension.

Abstract

It is common in stability analysis to linearize a system and investigate the spectrum of the Jacobian matrix. This approach faces the challenge of determining the matrix spectrum when the coefficients depend on parameters or when the characteristic polynomial is more than quartic. In this paper, we reverse the classical process and use the authors' work on global stability to find sufficient conditions on the coefficients that ensure the zeros of the characteristic polynomial are in the open unit disk. This leads to an algorithm that begins by testing the $\ell_1$-norm of the polynomial, and if it is not less than two, perform an iteration process that can be implemented with moderate effort. We give examples that show the effectiveness of our method when compared with the Jury's algorithm. Last, we formalize our constructions in terms of semialgebraic sets.

Constrained polynomial roots and a modulated approach to Schur stability

TL;DR

This work reframes Schur stability as a global-stability problem for delayed linear difference equations and develops a systematic, incremental algorithm that tests coefficient conditions starting from the classical -norm criterion . By embedding the root-coefficient relationship into a semialgebraic framework and leveraging mixed-monotone maps, it derives a sequence of increasingly strong sufficient conditions that certify all roots lie in . The approach yields a semialgebraic filtration of the coefficient locus , provides explicit results for degree-2 polynomials and practical applications (e.g., Cournot models, population dynamics), and offers a computationally attractive alternative to Jury’s algorithm in parameter-rich or high-degree settings. The method elegantly combines real algebraic geometry with dynamical-systems techniques to produce fast, conservative stability certificates with clear geometric interpretation and potential for higher-dimensional extension.

Abstract

It is common in stability analysis to linearize a system and investigate the spectrum of the Jacobian matrix. This approach faces the challenge of determining the matrix spectrum when the coefficients depend on parameters or when the characteristic polynomial is more than quartic. In this paper, we reverse the classical process and use the authors' work on global stability to find sufficient conditions on the coefficients that ensure the zeros of the characteristic polynomial are in the open unit disk. This leads to an algorithm that begins by testing the -norm of the polynomial, and if it is not less than two, perform an iteration process that can be implemented with moderate effort. We give examples that show the effectiveness of our method when compared with the Jury's algorithm. Last, we formalize our constructions in terms of semialgebraic sets.

Paper Structure

This paper contains 10 sections, 10 theorems, 76 equations, 5 figures, 3 tables.

Table of Contents

  1. Introduction
  2. The algebraic theory
  3. The embedding theory
  4. Degree two polynomials
  5. The linear approximation
  6. Increasing delay
  7. The Algorithm
  8. Running the algorithm: applications
  9. Stratifying the coefficient locus
  10. Conclusion

Key Result

Theorem 2.1

Let $B$ be a semialgebraic set of ${\mathbb C}$ and consider the set $\mathcal{C}_n(B)\subset{\mathbb R}^n$ of all coefficients $(a_0,\ldots, a_{n-1})\in{\mathbb R}^n$ of real monic polynomials $p(z) = z^n + a_{n-1}z^{n-1}+\cdots +a_1z+a_0$ of degree $n\geq 1$ whose roots belong to $B$. Then $\mathc

Figures (5)

  • Figure 1: The region with solid shading indicates where the eigenvalues are real and within $\mathbb{D}$. In contrast, the shaded region with dots indicates the region where the eigenvalues are non-real and within $\mathbb{D}$.
  • Figure 3: This figure illustrates the conditions obtained on $\alpha$ and $\beta$ in contrast with Fig. \ref{['Fig-TD']}. The cases (i) to (iv) are based on Table \ref{['Tab-TD1']}. The colors reflect different types of monotonicity in Definition \ref{['ordering']}
  • Figure 5: The total shaded region in this figure illustrates the feasible region of Inequality \ref{['In-ThirdExpansion']}. The colored regions illustrate the type of monotonicity we obtain in the second expansion. Again here, the coloring matches the coloring in Fig. \ref{['Fig-TD2']} and Fig. \ref{['Fig-TD3']} to reflect the type of monotonicity in the expanded map.
  • Figure 6: This figure shows the stability regions in the $(\alpha,\beta)$-plane obtained after computationally applying Proposition \ref{['Th-Order2']} for $(s_0,t_0)$ to $(s_3,t_3)$. Here, the colors differ from those in Fig. \ref{['Fig-TD2']} to Fig. \ref{['Fig-TD4']} because this figure's goal is to illustrate how the stability region extends overall, but not necessarily from one step to the next.
  • Figure 7: The red boundary in both figures represent the stability region obtained by the necessary and sufficient conditions. The shaded regions represents the regions obtained by the first two steps of our algorithm as obtained in Proposition \ref{['Pr-MathBiologyExample']}

Theorems & Definitions (28)

  • Theorem 2.1
  • Example 2.2
  • Example 2.3
  • Lemma 2.4
  • proof
  • proof
  • Example 2.5
  • Definition 3.1
  • Example 3.2
  • Theorem 3.3
  • ...and 18 more