Definition of eigenvalues for a nonlinear system

Nonlinear Eigenvalue Approach to Differential Riccati Equations for Contraction Analysis

“…Next, we show the definition of the nonlinear eigenvalues and eigenvectors [13], [14], [17]. ⊳ Nonlinear eigenvalues have similar properties to those in linear algebra.…”

“…Our main concern in this paper is extending the eigenvalue method to the DRE in contraction analysis in terms of recently introduced nonlinear eigenvalues and eigenvectors [13], [14]. First, we demonstrate that solutions to the DRE can be expressed as functions of nonlinear right eigenvectors of the corresponding Hamiltonian matrix as in the linear case.…”

Eigenvalues for a nonlinear time-delay system

“…The idea of eigenvalues has been extended also to the case of nonlinear systems, both continuous-and discrete-time, and it is known to the control community under various names and approaches, see e.g. [1], [2], [3], [4]. In comparison to the linear systems these objects are, in general, functions of the system variables.…”

“…The notion of an eigenvalue introduced within the algebraic approaches to the problem formulation, i.e. [2], [3], is equivalent to the notion of an eigenvalue introduced in the so-called pseudo-linear algebra, see e.g. [5], [6], [7].…”

Stability criteria with nonlinear eigenvalues for diagonalizable nonlinear systems

“…The proof for the diagonalizable system can be separated into two steps: diagonalizing the system and deriving the asymptotic stability conditions for the one-dimensional systems. In this paper, we generalize this result to diagonalizable nonlinear systems [5] by using the recently defined nonlinear eigenvalues, which are functions of the state variables [5,6,7]. The diagonalizable nonlinear system is the system that can be transformed into n one-dimensional subsystems, called the diagonal form, which is a generalization of the diagonal linear system, and even if the original system is real, its diagonal form can be complex.…”