Spectral Bounds on the Solution of Linear Time‐Periodic Systems

Abstract

Linear time-periodic systems arise whenever a nonlinear system is linearized about a periodic trajectory. Stability of the solution may be proven by rigorous bounds on the solution. The key idea of this paper is to derive Chebyshev projection bounds on the original system by solving an approximated system. Depending on the smoothness of the original function, we formulate two upper bounds. The theoretical results are illustrated and compared to trigonometric spline bounds by means of two examples which include an anisotropic rotor-bearing system. 1 Linear time-periodic system and rigorous bounds on the solution A linear time-periodic system is a set of linear ordinary differential equations (ODEs) with time-periodic coefficients with some periodicity T and a given initial conditioṅwhere x : R → R n and A : R → R n×n . Under well known assumptions a unique solution to (1) is guaranteed, see e.g.[1]. The idea is to replace each function a ij (t) of the matrix function 1−t 2 dt for 1 < k ≤ m 1 and i, j = 1, . . . , n, see e.g. [2]. Determining the solution x(t) in the interval [0, T ] is sufficient due to its semigroup property [3], hence the Chebyshev polynomials are shifted to the interval [0, T ] and we obtain the following system of ODEṡwhere (S T m1 A) denotes the componentwise Chebyshev projection of the matrix function A. We use the tau method, introduced by S.A. Orszag and his co-authors [4], to obtain a Chebyshev polynomial approximation y(t) ≈ (S [2]. In order to prove rigorous upper bounds on x(t) we need to assume, that y(t) satisfies |y(t)| ≤ M y in the Bernstein ellipse E ρy . The ellipse has foci at 0 and T , and ρ y > 1 is defined as the sum of the semimajor and semiminor axis. Further, let us define γ(A) := max Depending on the smoothness of the matrix function A, we can prove:

Trigonometric spline and spectral bounds for the solution of linear time-periodic systems

“…We use the spectral method [11,34] in the setting of polynomial approximation of linear ordinary differential equations [3,10]. The solution of the approximated system is entire and hence, the truncation error of the approximated solution can be given.…”