On the asymptotic behavior of solutions of nonautonomous differential inclusions with a set of several Lyapunov functions

For non-autonomous differential inclusions, the issues of attraction and asymptotic behavior of solutions are considered. The basis of the research is the development of the method of limit differential equations in combination with the direct Lyapunov method with several Lyapunov functions. This makes it possible to more accurately localize and determine the structure of $\omega$-limit sets of solutions. The main problems of the research are the absence of properties of the invariance type of $\omega$-limit sets of non-autonomous systems and the construction of limit differential relations. They are solved using limit differential inclusions constructed using shifts (translations) of the main differential inclusions. The results have the form of generalizations of the LaSalle invariance principle and provide preliminary information on the limit behavior of solutions. A set of additional Lyapunov functions allows one to refine this behavior and to single out those points from the set of zeros of the derivative of the main Lyapunov function that obviously do not belong to the $\omega$-limit sets. The results are illustrated by the example of a linear oscillator with dry friction.