Extension of the Perron-frobenius Theorem: 1 to Homogeneous from Linear
نویسندگان
چکیده
This paper deals with homogeneous cooperative sys-terns, a class of positive systems. It is shown that they admit a fairly simple asymptotic behavior, thereby generalizing the well-known Perron-Frobenius theorem from linear to homogeneous systems. As a corollary a simple criterion for global asymptotic stability is established. Then these systems are subject to constant inputs and we prove that asymptotic stability of the uncontrolled system is inherited by the new equilibrium point of the controlled system. Recent results on monotone control systems indicate the importance of this property in proving small gain theorems.
منابع مشابه
The Sign-Real Spectral Radius for Real Tensors
In this paper a new quantity for real tensors, the sign-real spectral radius, is defined and investigated. Various characterizations, bounds and some properties are derived. In certain aspects our quantity shows similar behavior to the spectral radius of a nonnegative tensor. In fact, we generalize the Perron Frobenius theorem for nonnegative tensors to the class of real tensors.
متن کاملSome results on the block numerical range
The main results of this paper are generalizations of classical results from the numerical range to the block numerical range. A different and simpler proof for the Perron-Frobenius theory on the block numerical range of an irreducible nonnegative matrix is given. In addition, the Wielandt's lemma and the Ky Fan's theorem on the block numerical range are extended.
متن کاملPERRON-FROBENIUS THEORY ON THE NUMERICAL RANGE FOR SOME CLASSES OF REAL MATRICES
We give further results for Perron-Frobenius theory on the numericalrange of real matrices and some other results generalized from nonnegative matricesto real matrices. We indicate two techniques for establishing the main theorem ofPerron and Frobenius on the numerical range. In the rst method, we use acorresponding version of Wielandt's lemma. The second technique involves graphtheory.
متن کاملThe Perron-frobenius Theorem for Homogeneous, Monotone Functions
If A is a nonnegative matrix whose associated directed graph is strongly connected, the Perron-Frobenius theorem asserts that A has an eigenvector in the positive cone, (R). We associate a directed graph to any homogeneous, monotone function, f : (R) → (R), and show that if the graph is strongly connected then f has a (nonlinear) eigenvector in (R). Several results in the literature emerge as c...
متن کاملA primer of Perron–Frobenius theory for matrix polynomials
We present an extension of Perron–Frobenius theory to the spectra and numerical ranges of Perron polynomials, namely, matrix polynomials of the form L(λ) = Iλ − Am−1λm−1 − · · · − A1λ− A0, where the coefficient matrices are entrywise nonnegative. Our approach relies on the companion matrix linearization. First, we recount the generalization of the Perron–Frobenius Theorem to Perron polynomials ...
متن کامل