[Paper Review] Shift operators and stability in delayed dynamic equations

Murat Adıvar, Youssef N‎. ‎Raffoul‎|arXiv (Cornell University)|Jan 18, 2011

Nonlinear Differential Equations AnalysisMathematics18 references20 citations

TL;DR

This paper introduces shift operators to construct Lyapunov functionals for delay dynamic equations on time scales, enabling unified stability and instability analysis across differential, difference, and q-difference equations. By generalizing the Lyapunov direct method via shift operators, the authors derive explicit exponential stability conditions that extend and improve upon prior results, particularly for non-additive time scales where previous methods fail.

ABSTRACT

In this paper, we use what we call the shift operator so that general delay dynamic equations of the form \[ x^Δ(t)=a(t)x(t)+b(t)x(δ_{-}(h,t))δ_{-}^Δ% (h,t),\ \ \ t\in\lbrack t_{0},\infty)_{\mathbb{T}}% \] can be analyzed with respect to stability and existence of solutions. By means of the shift operators we define a general delay function opening an avenue for the construction of Lyapunov functional on time scales. Thus, we use the Lyapunov's direct method to obtain inequalities that lead to stability and instability. Therefore, we extend and unify stability analysis of delay differential, delay difference, delay $h-$difference, and delay $q-$difference equations which are the most important particular cases of our delay dynamic equation. extbf{Keywords}: Delay dynamic equation, instability, shift operators, stability, time scales.

Motivation & Objective

• To develop a unified framework for stability analysis of delay dynamic equations on arbitrary time scales.
• To overcome limitations of prior Lyapunov functional methods that require additive time scales.
• To extend and generalize existing stability results for delay differential, difference, h-difference, and q-difference equations.
• To establish new sufficient conditions for exponential stability and instability using shift operators.
• To provide a constructive method for building Lyapunov functionals applicable to general delay functions on time scales.

Proposed method

• Introduces the shift operator δ₋(h,t) and δ₊(h,t) to define general delay functions on time scales.
• Uses shift operators to construct a Lyapunov functional V(t) involving integrals over delayed intervals.
• Applies the Lyapunov direct method on time scales by deriving inequalities on the delta derivative of V(t).
• Derives sufficient conditions for exponential stability through inequalities involving a(t), b(t), δ₋(h,t), and δ₋Δ(h,t).
• Establishes instability criteria by analyzing the sign and growth of the Lyapunov functional derivative.
• Validates the method on specific time scales, including ℝ, ℤ, and non-additive scales like q^ℤ and √ℕ.

Experimental results

Research questions

• RQ1 Can shift operators be used to define a general delay function that enables Lyapunov functional construction on arbitrary time scales?
• RQ2 How can the Lyapunov direct method be extended to delay dynamic equations on non-additive time scales?
• RQ3 What conditions ensure exponential stability of the zero solution in delay dynamic equations on time scales?
• RQ4 In what cases does the proposed method yield stronger stability results than existing approaches?
• RQ5 Can the method detect instability when standard criteria fail?

Key findings

• The proposed method ensures exponential stability of the zero solution for the delay differential equation x′(t) = x(t) - 1.5x(t - 1/3), which previous criteria like [3, Theorem 6] and [22, Corollary 1] could not establish.
• For the equation xΔ(t) = b(t)x(δ₋(h,t))δ₋Δ(h,t) with b(t) = -0.9 and h = 2/3, the method confirms exponential stability, while [3, Theorem 7] and [22, Corollary 1] fail due to violated integral conditions.
• Condition (4.6) in the paper’s framework holds for the example with a(t) = 1, b(t) = -1.5, δ₋(h,t) = t - 1/3, ensuring exponential stability.
• The method successfully handles non-additive time scales such as q^ℤ and √ℕ, where prior Lyapunov methods are inapplicable.
• Remark 1 provides explicit sufficient conditions for exponential stability involving λ, β(t), μ(t), and b(δ₊(h,t)) that generalize existing results.
• The paper demonstrates that its stability criteria are strictly stronger than those in [3, Theorem 7] and [22, Corollary 1] for certain classes of equations.