Galerkin Approximations for Stability of Delay Differential Equations With Distributed Delays

Abstract

Delay differential equations (DDEs) are infinite-dimensional systems, therefore analyzing their stability is a difficult task. The delays can be discrete or distributed in nature. DDEs with distributed delays are referred to as delay integro-differential equations (DIDEs) in the literature. In this work, we propose a method to convert the DIDEs into a system of ODEs. The stability of the DIDEs can then be easily studied from the obtained system of ODEs. By using a space-time transformation, we convert the DIDEs into a partial differential equation (PDE) with a time-dependent boundary condition. Then, by using the Galerkin method, we obtain a finite-dimensional approximation to the PDE. The boundary condition is incorporated into the Galerkin approximation using the tau method. The resulting system of ODEs will have time-periodic coefficients, provided the coefficients of the DIDEs are time periodic. Thus we use Floquet Theory to analyze the stability of the resulting ODE systems. We study several numerical examples of DIDEs with different kernel functions. We show that the results obtained using our method are in close agreement with those existing in the literature. The theory developed in this work can also be used for the integration of DIDEs. The computational complexity of our numerical integration method is O(t), whereas the direct brute-force integration of DIDE has a computational complexity of O(t 2 ).

Spurious roots of delay differential equations using Galerkin approximations

“…We refer to this formulation procedure as the “second-order Galerkin” method. As observed by Sadath and Vyasarayani (2015), some eigenvalues of the approximating ODE system are spurious : they are not characteristic roots of the original DDE.…”

“…To determine the stability of a DDE, one must determine the locations of its characteristic roots. Several methods exist for analyzing the stability of DDEs with constant delays, including the Lambert W function (Asl and Ulsoy, 2003; Jarlebring and Damm, 2007; Surya et al, 2018; Yi et al, 2007, 2010), Galerkin approximations (Sadath and Vyasarayani, 2015; Vyasarayani, 2012; Wahi and Chatterjee, 2005), Laplace transforms (Kalmár-Nagy, 2009), semi-discretization (Insperger and Stépán, 2011), and pseudo-spectral collocation (Breda et al, 2005; Butcher et al, 2004; Wu and Michiels, 2012). The Galerkin approximation and pseudo-spectral collocation methods fall under the broad category of spectral methods.…”

“…Galerkin approximations have been highly successful in studying the stability characteristics of second-order DDEs (Sadath and Vyasarayani, 2015; Vyasarayani, 2012; Vyasarayani et al, 2014; Wahi and Chatterjee, 2005). Galerkin approximations are applied to TDSs by first converting the governing DDE into a partial differential equation (PDE) and then converting the PDE into a system of ordinary differential equations (ODEs).…”

“…Galerkin approximations are applied to TDSs by first converting the governing DDE into a partial differential equation (PDE) and then converting the PDE into a system of ordinary differential equations (ODEs). Sadath and Vyasarayani (2015) performed stability studies of second-order DDEs using Galerkin approximations, converting the second-order DDE into a second-order PDE and then into a system of second-order ODEs; only in the final step were the second-order ODEs rewritten as a system of first-order ODEs. We refer to this formulation procedure as the “second-order Galerkin” method.…”

Delay compensation position tracking control of electro-hydraulic servo systems based on a delay observer

“…In the process of observer derivation, it is always difficult to solve the delay differential equation. Sadath and Vyasarayani 8 propose transforming differential equations into ordinary differential equations to study the stability of differential equations more easily. Wang and Sun 9 extend the framework of Lyapunov–Krasovskii functional to address the problem of exponential stabilization for a class of linear distributed parameter systems (DPSs) with continuous differentiable time-varying delay and spatiotemporal control input.…”