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Research Papers

Galerkin Approximations for Retarded Delay Differential Equations With State-Dependent Delays

[+] Author and Article Information
C. P. Vyasarayani

Department of Mechanical Engineering,
Indian Institute of Technology Hyderabad,
Ordnance Factory Estate,
Andhra Pradesh 502205, India
e-mail: vcprakash@iith.ac.in

Adarsh Gupta

Indian Oil Corporation Limited,
Western Region Pipelines,
Jaipur, Rajasthan 303901, India

John McPhee

Systems Design Engineering,
University of Waterloo,
Waterloo, ON N2L 3G1, Canada

1Corresponding author.

Contributed by the Dynamic Systems Division of ASME for publication in the JOURNAL OF DYNAMIC SYSTEMS, MEASUREMENT, AND CONTROL. Manuscript received January 15, 2012; final manuscript received July 12, 2013; published online August 23, 2013. Assoc. Editor: Sean Brennan.

J. Dyn. Sys., Meas., Control 135(6), 061006 (Aug 23, 2013) (6 pages) Paper No: DS-12-1015; doi: 10.1115/1.4025152 History: Received January 15, 2012; Revised July 12, 2013

In this work, approximations for state dependent delay differential equations (DDEs) are developed using Galerkin's approach. The DDE is converted into an equivalent partial differential equation (PDE) with a moving boundary, where the length of the domain dependents on the solution of the PDE. The PDE is further reduced into a finite number of ordinary differential equations (ODEs) using Galerkin's approach with time dependent basis functions. The nonlinear boundary condition is represented by a Lagrange multiplier, whose expression is derived in closed form. We also demonstrate the validity of the developed method by comparing the numerical solution of the ODEs to that of the original DDE.

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References

Driver, R., 1977, Ordinary and Delay Differential Equations, Springer-Verlag, New York.
Tsimring, L., and Pikovsky, A., 2001, “Noise-Induced Dynamics in Bistable Systems With Delay,” Phys. Rev. Lett., 87, p. 250602. [CrossRef] [PubMed]
Yeung, M., and Strogatz, S., 1999, “Time Delay in the Kuramoto Model of Coupled Oscillators,” Phys. Rev. Lett., 82(3), pp. 648–651. [CrossRef]
Guillouzic, S., L'Heureux, I., and Longtin, A., 1999, “Small Delay Approximation of Stochastic Delay Differential Equations,” Phys. Rev. E, 59(4), pp. 3970–3982. [CrossRef]
Kuang, Y., 1993, Delay Differential Equations: With Applications in Population Dynamics, Academic, Boston.
Li, C., and Chen, G., 2004, “Synchronization in General Complex Dynamical Networks With Coupling Delays,” Physica A, 343, pp. 263–278. [CrossRef]
Bocharov, G., and Rihan, F., 2000, “Numerical Modeling in Biosciences Using Delay Differential Equations,” J. Comput. Appl. Math., 125(1–2), pp. 183–199. [CrossRef]
Richard, J., 2003, “Time-Delay Systems: An Overview of Some Recent Advances and Open Problems,” Automatica, 39(10), pp. 1667–1694. [CrossRef]
Safonov, L., Tomer, E., Strygin, V., Ashkenazy, Y., and Havlin, S., 2002, “Multifractal Chaotic Attractors in a System of Delay-Differential Equations Modeling Road Traffic,” Chaos, 12(4), pp. 1006–1014. [CrossRef] [PubMed]
Hartung, F., Krisztin, T., Walther, H., and Wu, J., 2006, “Functional Differential Equations With State-Dependent Delays: Theory and Applications,” Handbook of Differential Equations: Ordinary Differential Equations, 3, pp. 435–545.
Nayfeh, A., and Mook, D., 1995, Nonlinear Oscillations, Wiley-VCH, New York.
Koto, T., 2004, “Method of Lines Approximations of Delay Differential Equations,” Comput. Math. Appl., 48(1–2), pp. 45–59. [CrossRef]
Maset, S., 2003, “Numerical Solution of Retarded Functional Differential Equations as Abstract Cauchy Problems,” J. Comput. Appl. Math., 161(2), pp. 259–282. [CrossRef]
Khalil, H., and Grizzle, J., 2002, Nonlinear Systems, Prentice Hall, New Jersey.
Sanders, J., Verhulst, F., and Murdock, J., 2007, Averaging Methods in Nonlinear Dynamical Systems, Springer-Verlag, New York.
Nayfeh, A., 1973, Perturbation Methods, Vol. 6, Wiley, New York.
Morton, K. and Mayers, D., 2005, Numerical Solution of Partial Differential Equations: An Introduction, Cambridge University Press, Cambridge, UK.
Carr, J., 1981, Applications of Centre Manifold Theory, Springer, New York.
Wahi, P., and Chatterjee, A., 2005, “Galerkin Projections for Delay Differential Equations,” ASME J. Dyn. Syst., Meas., Control, 127(1), pp. 80–87. [CrossRef]
Ghosh, D., Saha, P., and Roy Chowdhury, A., 2007, “On Synchronization of a Forced Delay Dynamical System Via the Galerkin Approximation,” Commun. Nonlinear Sci. Numer. Simul., 12(6), pp. 928–941. [CrossRef]
de Jesus Kozakevicius, A., and Kalmár-Nagy, T., 2010, “Weak Formulation for Delay Equations,” 9th Brazilian Conference on Dynamics, Control and Their Applications.
Vyasarayani, C. P., 2012, “Galerkin Approximation for Higher Order DelayDifferential Equations,” ASME J. Comput. Nonlinear Dyn., 7(3), p. 031004. [CrossRef]
Vyasarayani, C. P., 2012, “Galerkin Approximations for Neutral Delay Differential Equations,” ASME J. Comput. Nonlinear Dyn., 8, p. 021014. [CrossRef]
Wang, P., and Wei, J., 1987, “Vibrations in a Moving Flexible Robot Arm,” J. Sound Vib., 116(1), pp. 149–160. [CrossRef]
Shampine, L., 2005, “Solving ODEs and DDEs With Residual Control,” Appl. Numer. Math., 52, pp. 113–127. [CrossRef]
Kuang, Y., and Smith, H., 1992, “Slowly Oscillating Periodic Solutions of Autonomous State-Dependent Delay Equations,” Topol. Methods Nonlinear Anal., 19(9), pp. 855–872. [CrossRef]
Roussel, M., 2004, “Delay-Differential Equations,” http://people.uleth.ca/roussel/nld/delay.pdf

Figures

Grahic Jump Location
Fig. 1

Representation of a DDE with constant delay as a boundary control problem of advection equation (PDE)

Grahic Jump Location
Fig. 2

Representation of a DDE with state dependent delay as a boundary control problem of advection equation (PDE) with a moving boundary

Grahic Jump Location
Fig. 3

Time response of z(t) of Eq. (24) obtained from direct numerical integration as compared to a0(t) obtained from Galerkin approximation with (a) N = 3 and (b) N = 5

Grahic Jump Location
Fig. 4

Maximum value of absolute error max|a0(t)-z(t)| for increasing N

Grahic Jump Location
Fig. 5

(a) Time response of z(t) of Eq. (25) obtained from direct numerical integration as compared to a0(t) obtained from Galerkin approximation with N = 6. (b) Error between approximate and actual numerical solution.

Grahic Jump Location
Fig. 6

Maximum value of absolute error |a0(t)-z(t)| for frequencies 2≤ω≤10 for N = 6

Grahic Jump Location
Fig. 7

(a) Plant output z(t) of Eq. (25) for ω = 8 as compared to a0(t) obtained from the observer with N = 6. (b) Tracking error between plant output and observed output.

Grahic Jump Location
Fig. 8

(a) Time response of z(t) of Eq. (26) obtained from direct numerical integration as compared to a0(t) obtained from Galerkin approximation with N = 8. (b) Error between approximate and actual numerical solution. The system parameters are b = 0.5, c = 0.1, k = 2.5, and ω = 2.

Grahic Jump Location
Fig. 9

(a) Time response of z(t) of Eq. (27) obtained from direct numerical integration as compared to a0(t) obtained from Galerkin approximation with N = 11. (b) Error between approximate and actual numerical solution. The system parameters are k = 2, b = 0.5, c = 0.1, ω = 2, F = 2, and Ω = 3.

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