×

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

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.

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
PDF BibTeX XML Cite
Full Text: DOI