Skip to main content

Computing the maximum transient energy growth


The calculation of maximum transient energy growth is a problem of interest in several areas of science and engineering. An algorithm that guarantees the calculation of this measure to an arbitrary accuracy in a finite number of steps is proposed for finite-dimensional linear-time-invariant dynamical systems. The algorithm is illustrated with a numerical example.

This is a preview of subscription content, access via your institution.


  1. Amar, N.: Peaking in control systems: some numerical issues. Master’s thesis, Cranfield University, Bedfordshire, UK (2008)

  2. Bewley, T., Liu, S.: Optimal and robust control and estimation of linear paths to transition. J. Fluid Mech. 365, 305–349 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  3. Boyd, S., El Ghaoui, L., Feron, E., Balakrishnan, V.: Linear Matrix Inequalities in System and Control Theory. SIAM, Philadelphia (1994)

    MATH  Google Scholar 

  4. Farrell, B.: Optimal excitation of perturbations in viscous shear flow. Phys. Fluids 31(8), 2093–2102 (1988)

    Article  Google Scholar 

  5. Hémon, P., Noger, C.: Transient growth of energy and aeroelastic stability of ground vehicles. C. R., Méc. 332, 175–180 (2004)

    Article  Google Scholar 

  6. Hinrichsen, D., Pritchard, A.: Mathematical Systems Theory I: Modelling, State Space Analysis, Stability and Robustness. Texts in Applied Mathematics, vol. 48. Springer, Berlin (2005)

    MATH  Google Scholar 

  7. Kohaupt, L.: Differential calculus for some p-norms of the fundamental matrix with applications. J. Comput. Appl. Math. 135(1), 1–21 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  8. Kohaupt, L.: New upper bounds for free linear and nonlinear vibration systems with applications of the differential calculus of norms. Appl. Math. Model. 28, 367–388 (2004). doi:10.1016/j.apm.2003.08.004

    Article  MATH  Google Scholar 

  9. McKernan, J., Whidborne, J., Papadakis, G.: Linear quadratic control of plane Poiseuille flow—the transient behaviour. Int. J. Control 80(12), 1912–1930 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  10. Moler, C., Van Loan, C.: Nineteen dubious ways to compute the exponential of a matrix, twenty-five years later. SIAM Rev. 45(1), 3–49 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  11. Plischke, E.: Transient effects of linear dynamical systems. Ph.D. thesis, Universität Bremen (2005)

  12. Plischke, E., Wirth, F.: Stabilization of linear systems with prescribed transient bounds. In: Proc. 16th Int. Symp. Math. Theory Networks & Syst. (MTNS2004), Leuven, Belgium (2004).

    Google Scholar 

  13. Reddy, S., Henningson, D.: Energy growth in viscous channel flows. J. Fluid Mech. 252, 209–238 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  14. Schmid, P., Henningson, D.: Stability and Transition in Shear Flows. Applied Mathematical Sciences, vol. 142. Springer, New York (2001)

    Book  MATH  Google Scholar 

  15. Sempf, M., Merkel, P., Strumberger, E., Tichmann, C., Günter, S.: Robust control of resistive wall modes using pseudospectra. New J. Phys. 11(5), 053015 (2009).

    Article  Google Scholar 

  16. Thompson, C., Battisti, D.: A linear stochastic dynamical model of ENSO. Part I: Model development. J. Climate 13(15), 2818–2832 (2000)

    Article  Google Scholar 

  17. Trefethen, L., Embree, M.: Spectra and Pseudospectra: The Behavior of Nonnormal Matrices and Operators. Princeton University Press, Princeton (2005)

    MATH  Google Scholar 

  18. Trefethen, L., Trefethen, A., Reddy, S., Driscoll, T.: Hydrodynamic stability without eigenvalues. Science 261, 578–584 (1993)

    Article  MathSciNet  Google Scholar 

  19. van Loan, C.: The sensitivity of the matrix exponential. SIAM J. Numer. Anal. 14(6), 971–981 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  20. Veselić, K.: Bounds for exponentially stable semigroups. Linear Algebra Appl. 358, 309–333 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  21. Whidborne, J., McKernan, J.: On minimizing maximum transient energy growth. IEEE Trans. Autom. Control 52(9), 1762–1767 (2007)

    Article  MathSciNet  Google Scholar 

  22. Whidborne, J., McKernan, J., Papadakis, G.: Minimising transient energy growth in plane Poiseuille flow. Proc. Inst. Mech. Eng., Part I, J. Syst. Control Eng. 222(5), 323–331 (2008). doi:10.1243/09596518JSCE493

    Article  Google Scholar 

  23. Zhao, H., Bau, H.: Limitations of linear control of thermal convection in a porous medium. Phys. Fluids 18(7), 074109 (2006)

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations


Corresponding author

Correspondence to James F. Whidborne.

Additional information

Communicated by Anna-Karin Tornberg.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Whidborne, J.F., Amar, N. Computing the maximum transient energy growth. Bit Numer Math 51, 447–457 (2011).

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI:


Mathematics Subject Classification (2000)