Simultaneous Local Controllability of the Spectrum and the Lyapunov Irregularity Coefficient of Regular Systems

Abstract

System (3) is homogeneous and has bounded piecewise continuous coefficients, and so the characteristic exponents and other invariants of Lyapunov transformations are defined for this system [1, p. 42]. Our main problem is to find conditions under which the entire spectrum of Lyapunov exponents [1, p. 42] and the Lyapunov irregularity coefficient [1, p. 51] of system (3) can be changed as desired (i.e., can be controlled) in a neighborhood of the entire spectrum and the irregularity coefficient of the nonperturbed system (1) by an appropriate choice of the feedback coefficient U(·) (treated as a control). We introduce some notation. Let R be the n-dimensional Euclidean space with canonical orthonormal basis e1, . . . , en and with norm ‖ · ‖ given by the relation ‖x‖ = √ x∗x (the asterisk stands for transposition), and let Mmn be the space of real m × n matrices with spectral norm, i.e., with the operator norm induced in Mmn by the Euclidean norms in R and R. The space of bounded piecewise continuous mappings U : I → Mmn defined on an arbitrary interval I ⊂ R with the uniform norm ‖U‖C(I) := sup{‖U(t)‖ : t ∈ I} is denoted by KCmn(I). We denote the principal solution matrix of the homogeneous system (1) by X(t, s) and its entire spectrum of Lyapunov exponents by λ1(A) ≤ · · · ≤ λn(A). For an arbitrary piecewise continuous bounded function p : R→ R, we set