On the stability and boundedness of solutions of nonlinear vector differential equations of third order.(English)Zbl 1162.34043

Consider the vector differential equation ${d^3 x\over dt^3}+ \Psi\Biggl({dx\over dt}\Biggr) {d^2 x\over dt^2}+ B{dx\over dt}+ cx= p(t),\tag{$$*$$}$ where $$B$$ is a constant symmetric $$n\times n$$-matrix, $$c$$ is a positive constant, $$\Psi$$ is a continuous symmetric $$n\times n$$-matrix function. In case $$p\equiv 0$$, the author proves a theorem on global asymptotic stability of the equilibrium $$x= 0$$. His second theorem is concerned with boundedness of all solutions of $$(*)$$ under certain assumptions on $$p$$.
The proofs are based on the construction of suitable Lyapunov functions.

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
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References:

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