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On the stability of second order parametric ordinary differential equations and applications

Z. Mazgouri, A. El Ayoubi

Abstract

This work deals with Lipschitz stability for a parametric version of the general second order Ordinary Differential Equation (ODE) initial-value Cauchy problem. We first establish a Lipschitz stability result for this problem under a partial variation of the data. Then, we apply our abstract result to second order differential equations governed by cocoercive operators. Furthermore, we discuss more concrete applications of the stability for two specific applied mathematical models inherent in electricity and control theory. Finally, we provide numerical tests based on the software source Scilab, which are done with respect to parametric linear control systems, illustrating henceforth the validity of our abstract theoretical result.

On the stability of second order parametric ordinary differential equations and applications

Abstract

This work deals with Lipschitz stability for a parametric version of the general second order Ordinary Differential Equation (ODE) initial-value Cauchy problem. We first establish a Lipschitz stability result for this problem under a partial variation of the data. Then, we apply our abstract result to second order differential equations governed by cocoercive operators. Furthermore, we discuss more concrete applications of the stability for two specific applied mathematical models inherent in electricity and control theory. Finally, we provide numerical tests based on the software source Scilab, which are done with respect to parametric linear control systems, illustrating henceforth the validity of our abstract theoretical result.
Paper Structure (8 sections, 9 theorems, 48 equations, 3 figures)

This paper contains 8 sections, 9 theorems, 48 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Main result
  4. Applications
  5. Second order dynamical systems governed by cocoercive operators
  6. Differential equation of RLC circuit's current
  7. Linear control systems
  8. Numerical experiment

Key Result

Theorem 2.1

Let $T>0,$$x_0\in\mathbb{R}^{n}$ and let $g : [0,T]\times B(x_0,r)\longrightarrow \mathbb{R}^{n}$ be a $L^{1}$-Carathéodory function. Then, for any real number $d$ such that $0<d\leq T$ and $\int_{0}^{d}m(s)ds\leq r,$ the Cauchy problem $S(g,x_{0})$ admits a unique solution on $[0,d].$

Figures (3)

  • Figure 1: RLC parallel circuit
  • Figure 2: A series RLC circuit
  • Figure 3: The uniform convergence of $z_{\lambda}$ to $z$.

Theorems & Definitions (16)

  • Theorem 2.1: F
  • Theorem 2.2: F
  • Corollary 2.1
  • Theorem 2.3: M.P.F
  • Lemma 2.1
  • proof
  • Theorem 3.1
  • proof
  • Theorem 4.1
  • proof
  • ...and 6 more