[Paper Review] Nonoscillation and Stability of the Second Order Ordinary Differential Equations with a Damping Term

Leonid Berezansky, Elena Braverman+1 more|arXiv (Cornell University)|Jul 14, 2008

Nonlinear Differential Equations Analysis13 references3 citations

TL;DR

This paper establishes new nonoscillation and exponential stability criteria for second-order linear ordinary differential equations with a damping term, using measurable coefficients without requiring continuity. By analyzing nonoscillation intervals via Green's functions and Floquet theory, it derives conditions for exponential stability based on integral properties of the damping coefficient, extending classical results to broader classes of equations with minimal regularity assumptions.

ABSTRACT

In this paper we consider the linear ordinary equation of the second order $$ L x(t)\equiv \ddot{x}(t) +a(t)\dot{x}(t)+b(t)x(t)=f(t), \eqno{(1)} $$ and the corresponding homogeneous equation $$ \ddot{x}(t) +a(t)\dot{x}(t)+b(t)x(t)=0. \eqno{(2)} $$ Note that $[α,β]$ is called a nonoscillation interval if every nontrivial solution has at most one zero on this interval. Many investigations which seem to have no connection such as differential inequalities, the Polia-Mammana decomposition (i.e. representation of the operator $L$ in the form of products of the first order differential operators), unique solvability of the interpolation problems, kernels oscillation, separation of zeros, zones of Lyapunov's stability and some others have a certain common basis - nonoscillation. Presumably Sturm was the first to consider the two problems which naturally appear here: to develop corollaries of nonoscillation and to find methods to check nonoscillation. In this paper we obtain several tests for nonoscillation on the semiaxis and apply them to propose new results on asymptotic properties and the exponential stability of the second order equation (2). Using the Floquet representations and upper and lower estimates of nonoscillation intervals of oscillatory solutions we deduce results on the exponential and Lyapunov's stability and instability of equation (2).

Motivation & Objective

• To develop nonoscillation criteria for second-order linear ODEs with measurable coefficients, extending beyond classical continuity assumptions.
• To establish new conditions for exponential stability of the homogeneous equation using integral properties of the damping coefficient.
• To connect nonoscillation theory with stability analysis via Floquet theory and Green's function positivity.
• To generalize the Bohl-Perron theorem for stability by using nonoscillatory solutions as model equations.
• To provide explicit bounds on nonoscillation intervals through estimates of the distance between zeros of solutions.

Proposed method

• Utilizes the Cauchy function and Green's function to represent solutions and analyze boundary value problems.
• Applies differential inequality theorems to deduce positivity of Green's functions, implying nonoscillation.
• Employs Floquet theory to analyze the monodromy matrix and spectral radius for periodic coefficients.
• Derives upper and lower bounds on the distance between consecutive zeros of solutions to exclude periodic behavior.
• Uses the transformation $ z''(t) + p(t)z(t) = 0 $ with $ p(t) = b(t) - a^2(t)/4 - a'(t)/2 $ to reduce the damped equation to a canonical form.
• Applies spectral radius estimates of integral operators defined by Green's functions to infer stability properties.

Experimental results

Research questions

• RQ1 Under what conditions is the solution of a second-order ODE with measurable coefficients nonoscillatory on the semiaxis?
• RQ2 How can the exponential stability of the homogeneous equation be determined without assuming continuity of coefficients?
• RQ3 What role does the integral of the damping coefficient $ \int_0^\omega a(t)dt $ play in determining stability?
• RQ4 How do estimates of the distance between zeros of solutions relate to the stability or instability of the system?
• RQ5 Can nonoscillation intervals be characterized using Green's function positivity and spectral radius analysis?

Key findings

• Equation (7.1) is exponentially stable if $ \int_0^\omega a(t)dt > 0 $, as shown by the spectral radius of the monodromy operator being less than one.
• If $ \int_0^\omega a(t)dt < 0 $, the fundamental solutions grow exponentially, indicating instability with $ |\lambda_1| > 1 $.
• When $ \int_0^\omega a(t)dt = 0 $, the fundamental solutions remain bounded, corresponding to $ |\lambda_1| = 1 $.
• Theorem 12 provides explicit sufficient conditions for oscillation: if $ P = \mathrm{ess\,inf}\,p(t) > 0 $, $ Q = \mathrm{ess\,sup}\,p(t) $, and $ \omega \in \left(0, \frac{\pi}{2\sqrt{Q}}\right] \cup \cdots \cup \left(\frac{k-1}{2}\frac{\pi}{\sqrt{P}}, \frac{k\pi}{2\sqrt{Q}}\right) $ with $ \frac{k-1}{k} < \sqrt{P/Q} $, then the equation is oscillatory and the distance between zeros differs from $ 2\omega $.
• The example in Section 7.1 shows that even if $ \int_0^\pi a(t)dt > 0 $, exponential stability may fail if the solution structure allows unbounded components, such as $ x_1 = e^{-t}\cos t $, highlighting the necessity of excluding periodic zero patterns.