International Journal of Dynamics and Control

, Volume 2, Issue 2, pp 234-246

Spectrum design using distributed delay

  • Feng WeiAffiliated withDepartment of Mathematics, University of Michigan
  • , Dániel BachrathyAffiliated withMTA-BME Research Group on Dynamics of Machines and Vehicles, Hungarian Academy of Sciences
  • , Gábor OroszAffiliated withDepartment of Mechanical Engineering, University of Michigan Email author 
  • , A. Galip UlsoyAffiliated withDepartment of Mechanical Engineering, University of Michigan


This paper focuses on the problem of designing the rightmost eigenvalues of linear scalar distributed delay systems. We consider two different but complementary methods: generalized stability charts and matrix Lambert W functions. The generalized stability charts are based on the intersection of Hopf and fold surfaces that provide important insight into the problem, but the geometry of the surfaces may become complicated for certain delay distributions. The Lambert W function approach can be applied to general delay distributions, but requires numerical solutions which can suffer from convergence problems. We present some examples using both approaches.


Spectrum design Distributed delay Generalized stability chart Lambert W function