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Nonlocal reduction method in differential equations theory. (English) Zbl 0755.34045

Topics in mathematical analysis, Vol. Dedicated Mem. of A. L. Cauchy, Ser. Pure Math. 11, 658-694 (1989).
[For the entire collection see Zbl 0721.00014.]
The paper is devoted to a new method of stability investigation for dynamical systems. The authors give a short historical review of basic concepts of global qualitative study of differential equations and describe the sources of the new method. These are the concept of surfaces without tangency and the comparison principle. The new Leonov’s method which is called nonlocal reduction method proposes that a Lyapunov function constructed for the initial system should contain information about a certain comparison system which is called a reduction system. The latter is of lower order than the initial one but preserves all its qualitative properties. The nonlocal reduction method gives the opportunity to prove new Lyapunov-type theorems. In particular, the paper contains a theorem about Lagrange stability which is “wider” than the well-known Yoshizawa theorem. The paper contains also frequency-domain stability and oscillations theorems for indirect control systems with periodic nonlinearities, a new frequency-domain criterion of absolute stability, different from the Popov criterion. The new method may be extended to distributed parameter systems.

MSC:

34D20 Stability of solutions to ordinary differential equations
34D99 Stability theory for ordinary differential equations
34C20 Transformation and reduction of ordinary differential equations and systems, normal forms
93D05 Lyapunov and other classical stabilities (Lagrange, Poisson, \(L^p, l^p\), etc.) in control theory
93D10 Popov-type stability of feedback systems

Biographic References:

Cauchy, A. L.

Citations:

Zbl 0721.00014
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