Geodesic
On solving the problems of stability by Lyapunov's direct method
Izvestiâ vysših učebnyh zavedenij. Matematika, no. 6 (2017), pp. 33-43

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We consider problems of stability and instability of the trivial solution of non-autonomous systems of differential equations. We suggest new theorems of Lyapunov's direct method with the use of semi-definite auxiliary functions. The idea is based on the use of the additional function that evaluates the rate of convergence of the solutions to the set, where Lyapunov's function vanishes. We formulate theorems on the non-asymptotic stability and instability. The results are adduced by the examples, where we give a comparison with known results.
Keywords: system of differential equations, equilibrium, stability, Lyapunov's function. 
@article{IVM_2017_6_a3,
     author = {B. S. Kalitine},
     title = {On solving the problems of stability by {Lyapunov's} direct method},
     journal = {Izvesti\^a vys\v{s}ih u\v{c}ebnyh zavedenij. Matematika},
     pages = {33--43},
     publisher = {mathdoc},
     number = {6},
     year = {2017},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/IVM_2017_6_a3/}
}
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B. S. Kalitine. On solving the problems of stability by Lyapunov's direct method. Izvestiâ vysših učebnyh zavedenij. Matematika, no. 6 (2017), pp. 33-43. http://geodesic.mathdoc.fr/item/IVM_2017_6_a3/