Linear ordinary differential equations of class \(L^ pS\). (English) Zbl 0718.34070

Conti, Roberto

Rend. Semin. Mat., Torino 46, No. 2, 139-144 (1988).

Let \({\mathbb{R}}^ n\) denote the real euclidean space of n-vectors x and let \(L({\mathbb{R}}^ n)\) denote the real euclidean space of \(n\times n\) matrices A. Let \(J=(\alpha,+\infty)\), -\(\infty \leq \alpha\), and let \(A: t\to A(t),\) \(t\in J\), \(A(t)\in L({\mathbb{R}}^ n)\), be a continuous function. Let I denote the identity of \(L({\mathbb{R}}^ n)\) and let \[ E_ A(t,s)=\lim_{k}[\int^{t}_{s}A(t_ 1)dt_ 1+...+\int^{t}_{s}...\int^{t_{k-1}}_{s}A(t_ 1)...A(t_ k)dt_ k...dt_ 1] \] represent the so called Cauchy (or evolution) matrix associated with A. Then the solutions of the linear ordinary differential equation \((A)\quad \dot x=A(t)x\) are given by \(x: t\to x(t)=E_ A(t,s)x(s),\) t,s\(\in J\). Equation (A) is said to be asymptotically stable if \[ (1)\quad \lim_{t\to +\infty}E_ A(t,\tau)=0,\quad \alpha <\tau, \] (uniformly) exponentially stable if for every \(\vartheta >\alpha\) there exist \(\gamma\) (\(\vartheta\))\(\geq 1\), \(\mu (\vartheta)>0\) such that \[ (2)\quad | E_ A(t,\tau)| \leq \gamma (\vartheta)e^{-\mu (\vartheta)(t-\tau)},\quad \vartheta \leq \tau \leq t. \] We denote by AS, ES the classes of matrix functions A for which (1), (2) hold, respectively. Obviously, ES\(\subset AS\) and we consider a family, depending on a parameter \(p>0\), of classes intermediate between ES and AS.

MSC:

34D05	Asymptotic properties of solutions to ordinary differential equations
34A30	Linear ordinary differential equations and systems
34B05	Linear boundary value problems for ordinary differential equations

Keywords:

linear ordinary differential equation; asymptotically stable; exponentially stable

Citations:

Zbl 0718.34071