On the global asymptotic stability problem and the Jacobian conjecture

• Autorzy: Drużkowski L. M.
• Języki publikacji: EN

Abstrakty

EN

In this survey, we recall the formulation of the problems and give a review of some nontrivial results in the area. Let F = (F1,...,Fn] : R^n - --> R^n be a C^1 map and let F'(x) and Jac F(x) = det F'(x) denote the Jacobian matrix and the jacobian of F at a point x belongs to R^n, respectively. The Global Asymptotic Stability Problem (GASP) reads as follows: Assume that F(0] = 0 and at any point x belongs to R^n all eigenvalues of F'(x) have negative real parts. Then consider the associated system of differential equations x'j(t] = Fj(x1(t), ...,Xn(t)), j = 1,...,n. The question is whether the solution x[t] = 0 is globally asymptotically stable. If n > 2, then the answer is negative (even if F is a a polynomial automorphism), so from now on (GASP) denotes (GASP) restricted to R^2. In 1963, Olech showed that under the (GASP) assumption (i. e., Jac F[x) > 0 and Trace F'(x) = [...] < 0 for any x belong to R^2) the conclusion of (GASP) is equivalent to the injectivity of F. In 1994, Fessler, and independently Gutierrez, proved the injectivity of F and, due to the above mentioned Olech's equivalence, gave the affirmative answer to the two-dimensional (GASP). Let K denote R or C, n > 1. The Jacobian Conjecture can be formulated as follows: If F = (F1, ... ,Fn) : K^n --> K^n is a polynomial map with a constant nonzero jacobian, then F is a polynomial automorphism (i.e. there exists F^-1 and F^-1 is also a polynomial map). Although the Jacobian Conjecture is still unsolved even in the case of n = 2, it is convenient, to consider the so called Generalized Jacobian Conjecture (for short (GJC)): the Jacobian Conjecture holds for every n > 1. We give a review of some interesting conditions equivalent to the Jacobian Conjecture, including Meisters and Olech's result on the existence of a poly-flow solution of the associated Ważewski equation x'(t) = [F'(x(t))]^-1 (a). We also present, a reduction of (GJC) to the case of F of degree 3 and of special forms, then some partial results, and (JC)'s relations with other problems.

Słowa kluczowe

EN

global stability problem; Jacobian Conjecture

PL

hipoteza jakobianu

• Wydawca: Systems Research Institute, Polish Academy of Sciences
• Czasopismo: Control and Cybernetics
• Rocznik: 2005
• Tom: Vol. 34, no 3
• Strony: 747--762
• Opis fizyczny: Bibliogr 37 poz.

Twórcy

autor

Drużkowski L. M.

• Institute of Mathematics, Jagiellonian University, Reymonta 4, PL-30-059 Kraków, Poland,

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