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A constructive approach for investigating the stability of incommensurate fractional differential systems

Kai Diethelm, Safoura Hashemishahraki, Ha Duc Thai, Hoang The Tuan

TL;DR

This work develops a constructive spectral framework for assessing stability of incommensurate fractional differential systems with Caputo derivatives. By mapping general orders to rational ones via an explicit auxiliary matrix $B$ and leveraging an equivalence between the fractional spectrum and classical eigenvalues, the authors provide a practical stability criterion applicable to both rational and irrational orders. They introduce algorithms to compute rational approximations and to transfer stability from the approximated system back to the original, and they establish Mittag-Leffler stability results for nonlinear cases, supported by numerical simulations. The approach offers a broadly applicable, numerically tractable tool for stability analysis in fractional-order dynamics with diverse order structures.

Abstract

This paper is devoted to studying the asymptotic behaviour of solutions to generalized non-commensurate fractional systems. To this end, we first consider fractional systems with rational orders and introduce a criterion that is necessary and sufficient to ensure the stability of such systems. Next, from the fractional-order pseudospectrum definition proposed by Šanca et al., we formulate the concept of a rational approximation for the fractional spectrum of a noncommensurate fractional systems with general, not necessarily rational, orders. Our first important new contribution is to show the equivalence between the fractional spectrum of a noncommensurate linear system and its rational approximation. With this result in hand, we use ideas developed in our earlier work to demonstrate the stability of an equilibrium point to nonlinear systems in arbitrary finite-dimensional spaces. A second novel aspect of our work is the fact that the approach is constructive. Finally, we give numerical simulations to illustrate the merit of the proposed theoretical results.

A constructive approach for investigating the stability of incommensurate fractional differential systems

TL;DR

This work develops a constructive spectral framework for assessing stability of incommensurate fractional differential systems with Caputo derivatives. By mapping general orders to rational ones via an explicit auxiliary matrix and leveraging an equivalence between the fractional spectrum and classical eigenvalues, the authors provide a practical stability criterion applicable to both rational and irrational orders. They introduce algorithms to compute rational approximations and to transfer stability from the approximated system back to the original, and they establish Mittag-Leffler stability results for nonlinear cases, supported by numerical simulations. The approach offers a broadly applicable, numerically tractable tool for stability analysis in fractional-order dynamics with diverse order structures.

Abstract

This paper is devoted to studying the asymptotic behaviour of solutions to generalized non-commensurate fractional systems. To this end, we first consider fractional systems with rational orders and introduce a criterion that is necessary and sufficient to ensure the stability of such systems. Next, from the fractional-order pseudospectrum definition proposed by Šanca et al., we formulate the concept of a rational approximation for the fractional spectrum of a noncommensurate fractional systems with general, not necessarily rational, orders. Our first important new contribution is to show the equivalence between the fractional spectrum of a noncommensurate linear system and its rational approximation. With this result in hand, we use ideas developed in our earlier work to demonstrate the stability of an equilibrium point to nonlinear systems in arbitrary finite-dimensional spaces. A second novel aspect of our work is the fact that the approach is constructive. Finally, we give numerical simulations to illustrate the merit of the proposed theoretical results.
Paper Structure (11 sections, 11 theorems, 111 equations, 4 figures)

This paper contains 11 sections, 11 theorems, 111 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The $\hat{\alpha}$-order spectrum: The case $\hat{\alpha} \in \left( (0,1] \cap \mathbb Q \right)^n$
  4. The $\hat{\alpha}$-order spectrum: The case $\hat{\alpha} \in (0,1]^n$
  5. Rational approximations of a fractional spectrum
  6. Equivalence between the fractional spectrum and its rational approximation
  7. Asymptotic behavior of solutions to incommen-surate frac-tional-order nonlinear systems
  8. Inhomogeneous linear systems
  9. Nonlinear Systems
  10. Examples
  11. Conclusions

Key Result

Proposition 2.2

For some given $\epsilon > 0$, $\hat{\alpha} \in (0,1]^n$ and $A \in M_n(\mathbb R)$, the $\hat{\alpha}$-order $\epsilon$-pseudospectrum of $A$ can be expressed in the following ways:

Figures (4)

  • Figure 1: Left: Location of the eigenvalues of the matrix $B$ from Example \ref{['vd1']} in the complex plane. The blue rays are oriented at an angle of $\pm \gamma \pi / 2 = \pm \pi/24$ from the positive real axis and hence indicate the boundary of the critical sector $\{ z \in \mathbb C : |\arg z | \le \gamma \pi /2\}$. Since all eigenvalues are outside of this sector, we can derive the asymptotic stability of the system. Right: Trajectories of the solution of the system \ref{['eq1']} discussed in Example \ref{['vd1']} where $\hat{\alpha}=(\frac{1}{2}, \frac{1}{4}, \frac{1}{3}, \frac{1}{6})$ and the matrix $A$ is given in eq. \ref{['eq:ex1']} when the initial condition \ref{['eq:ic']} is chosen as $x_0= (0.1, -0.1, 0.5, -0.4)^{\mathrm T}$. Note that the horizontal axis is displayed in a logarithmic scale.
  • Figure 2: Left: Trajectories of the solution of \ref{['Eq54']} with the initial condition $x^0=(0.5, -0.3, 0.7, -0.4)^{\mathrm T}$. Right: Trajectories of the solution of \ref{['Eq58']} with the initial condition $x^0=(0.2,-0.1,0.3,-0.25)^{\mathrm T}$. As in Figure \ref{['refhinh1']}, the horizontal axes in both plots are in a logarithmic scale. Both numerical solutions have been computed with Garrappa's implementation of the trapezoidal algorithm mentioned in Remark \ref{['rmk:numsol']} using the step size $h = 0.1$.
  • Figure 3: Left: Location of the eigenvalues of the matrix $B$ from Example \ref{['ex:dh1']} in the complex plane. The blue rays are oriented at an angle of $\pm \gamma \pi / 2 = \pm \pi/20$ from the positive real axis and hence indicate the boundary of the critical sector $\{ z \in \mathbb C : |\arg z | \le \gamma \pi /2\}$. Since all eigenvalues are outside of this sector, we can derive the asymptotic stability of the system. Right: Trajectories of the solution of Example \ref{['ex:dh1']} with the initial condition $x^0 = (1, -2, 2)^{\mathrm T}$, numerically computed with the same algorithm as in the other examples with a step size of $h = 0.1$.
  • Figure 4: Left: Location of the eigenvalues of the matrix $B$ from Example \ref{['ex:dh2']} in the complex plane. The blue rays are oriented at an angle of $\pm \gamma \pi / 2 = \pm \pi/20$ from the positive real axis and hence indicate the boundary of the critical sector $\{ z \in \mathbb C : |\arg z | \le \gamma \pi /2\}$. Since all eigenvalues are outside of this sector, the system is asymptotically stable. Right: Trajectories of the solution of Example \ref{['ex:dh2']} with the initial condition $x^0 = (1, -2, 2)^{\mathrm T}$, again computed with the same numerical method and a step size $h=0.1$.

Theorems & Definitions (38)

  • Definition 2.1
  • Proposition 2.2
  • proof
  • Theorem 2.3: $\hat{\alpha}$-fractional $\epsilon$-pseudo Geršgorin sets
  • proof
  • Remark 2.4
  • Theorem 2.5: Euclidean $\hat{\alpha}$-fractional $\epsilon$-pseudo Geršgorin sets
  • proof
  • Remark 2.6
  • Theorem 3.1
  • ...and 28 more