Computing the maximum transient energy growth James F. Whidborne and Nathalie Amar Department of Aerospace Engineering Cranfield University, Bedfordshire MK43 0AL, U.K. email:j.f.whidborne@cranfield.ac.uk Pre-print: published BIT Numerical Mathematics. 51(2):447-557, 2011. (doi:10.1007/s10543-011-0326-4) Abstract 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 ar- bitrary 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. Keywords Linear dynamical systems transient growth maximum transient energy growth 1 Introduction A number of engineering and other dynamical systems are characterized by large transient energy growth. This means that following some initial perturbation to the state of the system, the magnitude of the sys- tem state trajectory grows to a very large value before decreasing and converging to the equilibrium state. For linear systems, this is a consequence of the non-normality of the system matrix and can occur even though all the eigenvalues may have very negative real parts and small imaginary parts. For many systems, this behavior is highly undesirable, particularly for certain non-linear systems where, although linear eigenvalue analysis at an equilibrium point indicates very good stability, very small initial pertur- bations in the state variables may cause them to leave the domain of attraction resulting in instability. This phenomenon is known to occur in fluid dynamic systems [1, 2, 3]. A consequence is that laminar flow can become turbulent even for Reynolds numbers for which linear stability analysis predicts stable eigenvalues. One measure of this phenomenon is the ratio between the maximum (appropriately weighted) Euclidian distance of the state trajectory from the origin to the distance of the initial perturbation from the origin [2, 4], and has been dubbed the maximum transient energy growth [5]. The measure has also been studied for vibration systems [6], climate modeling [7] and ground vehicle dynamics [8]. It has also been used as a cost objective in determining feedback controllers for fluid flow and other systems [5, 9, 10, 11], and controller design methods have been proposed to minimize this measure [12, 13, 14]. However, the efficient and accurate calculation of this measure is sometimes problematical, these difficulties are detailed in the next section. In Section 3, a number of simple lemmas are provided from which an algorithm that guarantees the calculation of this measure to an arbitrary accuracy in a finite number of steps. Some results on a test example are presented in Section 4. 1