Open Question:  "Lyapunov's second method for stability – For what classes of ODEs, describing dynamical systems, does the Lyapunov’s second method formulated in the classical and canonically generalized forms define the necessary and sufficient conditions for the (asymptotical) stability of motion?"  -- Wikipedia.org.

-- Reference links:

https://en.wikipedia.org/wiki/List_of_unsolved_problems_in_mathematics#Dynamical_systems;

'Differential Equations' by Viorel Barbu;

'Differential Topology',

https://folk.uib.no/nmabd/dt/080627dt.pdf;

'ON BROCKETT’S NECESSARY CONDITION FORSTABILIZABILITY AND THE TOPOLOGY OF LIAPUNOV FUNCTIONS ON R^N*',

http://www.ims.cuhk.edu.hk/~cis/2008.4/cis_8_4_01.pdf;

'Linear Openness and Feedback Stabilization of Nonlinear Control Systems',

https://www.researchgate.net/publication/315758848_Linear_Openness_and_Feedback_Stabilization_of_Nonlinear_Control_Systems.

http://grail.cs.washington.edu/projects/flight/wu2003realistic.pdf

Article Classical Converse Theorems in Lyapunov's Second Method

Article Linear Openness and Feedback Stabilization of Nonlinear Cont...

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