Galerkin Approximation for Stability of Delay Differential Equations

Anwar Sadath, K T (2016) Galerkin Approximation for Stability of Delay Differential Equations. PhD thesis, Indian Institute of Technology Hyderabad.

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Abstract

Delay differential equations (DDEs) are used as mathematical models to describe time delay effects in engineering, biology, and control theory. DDEs are infinitedimensional systems and analyzing their stability is a difficult task. The problem is even more challenging if time-periodic coefficients or time-periodic delays are present in the DDEs. If by some means these DDEs can be approximated using a system of ODEs, the existing stability theory of ODEs can be applied for analyzing the stability of these DDEs. By using a shift of time transformation, the DDEs are converted into partial differential equations (PDEs) along with time-dependent boundary conditions. Sometimes, these PDEs can have time periodic coefficients depending on the nature of the DDE. Using Galerkin approximations with Legendre polynomials as basis functions, reduced order ODE approximations are developed for these PDEs. The boundary conditions are incorporated into the ODEs using the tau or Lagrange multiplier method. The obtained finite-dimensional ODEs approximate the infinite-dimensional DDEs. These ODEs can be used to analyze the stability of the given DDEs. In this thesis, ODE approximations were developed for DDEs with time-periodic coefficients, time-periodic delays, and distributed delays. The developed theory for distributed delays can also be used for efficient time integration of these equations. Due to the time periodic nature of the approximating ODEs, Floquet theory has been used to analyze the stability of these systems. It is well known that obtaining a Floquet transition matrix for a large system of ODEs is computationally expensive. To mitigate this problem, the Arnoldi algorithm is applied for the stability analysis. This resulted in an efficient procedure for determining the stability of these DDEs. Our algorithm takes only 10-20% of the computational time as compared to classical Floquet theory for determining the stability. The methods developed in this thesis were applied to study the stability of several problems relevant to mechanical engineering.

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