Asymptotic decay rates from the growth properties of Lyapunov functions near singular attractors
Asymptotic stability of the origin and estimates on the rate of decay to the zero solution of the autonomous differential equation (1) \(\dot x=G(x)\), \(x\in X\), where X is a real Hilbert space, can be obtained from a Lyapunov function \(F\in C^ 1(X,R)\) such that: \(a\| x\|^ k\leq F(x)\leq b\| x\|^ k\), \(<\nabla F(x),G(x)>\leq -c\| x\|^{k+m}.\) Similar estimates are derived from a somewhat different set of conditions, namely (2) \(F(0)=0\), \(F(x)\geq a\| x\|^ k\), (3) \(\| \nabla F(x)\| \geq bF^{\ell}(x)\), \(<\nabla F(x),G(x)>\leq - c\| \nabla F(x)\|^ m.\) Lyapunov functions of this type arise naturally in systems such as \(\dot x=-H(x)\nabla F(x)\), where H(x) is a bounded self-adjoint positive definite linear operator. Such an equation generalizes the continuous steepest descent equation for F. The author explores circumstances under which (2) implies (3). In particular a detailed study is made in case (i) \(F(0)=0\), \(F(x)>a\| x\|^ 4\); (ii) the quadratic part of F is positive definite on its range i.e. \[ \exists \lambda_ 2>0,\quad \forall y\in (Ker \nabla_ 2F(0))^{\perp},\quad <y,\nabla_ 2F(0)y>\geq \lambda_ 2\| y\|^ 2; \] (iii) \(<\nabla F(x),G(x)>\leq -c\| \nabla F(x)\|^ m\), \(m>4/3\). For such systems, the origin is asymptotically stable and \(\| x(t)\| =O(t^{-1/(3m-4)})\) as \(t\to \infty\). Examples for \(X={\mathbb{R}}\) or \({\mathbb{R}}^ 2\) are discussed. They motivate the theory.
scientific article; zbMATH DE number 4027940
scientific article