Stability and boundedness of solutions of nonlinear vector differential equations of third order. (English) Zbl 1340.34189

Omeike, Mathew Omonigho

Arch. Math., Brno 50, No. 2, 101-106 (2014).

The author investigates stability properties of solutions of the system \[ \dot x=y,\quad \dot y=z,\quad \dot z=-\Psi(y)z-\Phi(x)y-cx+P(t),\quad t\geq 0,\tag{1} \] where \(x,y,z\in {\mathbb R}^n\), \(\Psi,\Phi:{\mathbb R}^n\to {\mathbb R}^{n\times n}\) are symmetric, positive definite matrix-valued functions, \(P:[0,\infty)\to {\mathbb R}^n\), and \(c>0\) is a real constant. The main result of the paper reads as follows.
Theorem. Suppose that there exist positive constants \(a_0,b_0\) such that \(a_0b_0>c\), \(b_0\leq \lambda_i(\Phi(x))\leq b_0+\mu,\) and \(\lambda_i(\Psi(y))\geq a_0\), \(i=1,\dots,n\), where \(\mu=4cb_0^{-2}(a_0b_0-c)>0\), and \(\lambda_i(.)\) are eigenvalues of the matrix indicated. Then the zero solution of (1) is globally asymptotically stable.

Reviewer: Ondřej Došlý (Brno)

MSC:

34D20	Stability of solutions to ordinary differential equations
34C11	Growth and boundedness of solutions to ordinary differential equations
34D05	Asymptotic properties of solutions to ordinary differential equations

Keywords:

boundedness; stability; Liapunov function; differential equations of third order