By V.I. Vorotnikov
Unlike the traditional examine for the final idea of balance, this mono graph bargains with difficulties on balance and stabilization of dynamic platforms with appreciate to not all yet simply to a given a part of the variables characterizing those platforms. Such difficulties are usually known as the issues of partial balance (stabilization). They certainly come up in purposes both from the requirement of right functionality of a procedure or in assessing method capa bility. additionally, loads of genuine (or wanted) phenomena may be formulated by way of those difficulties and be analyzed with those difficulties taken because the foundation. the subsequent multiaspect phenomena and difficulties could be indicated: • "Lotka-Volterra ecological precept of extinction;" • focusing and acceleration of debris in electromagnetic fields; • "drift" of the gyroscope axis; • stabilization of a spacecraft by way of in particular prepared relative movement of rotors attached to it. additionally very potent is the method of the matter of balance (stabilization) with admire to the entire variables according to initial research of partial sta bility (stabilization). A. M. Lyapunov, the founding father of the fashionable concept of balance, used to be the 1st to formulate the matter of partial balance. Later, works by way of V. V. Rumyan tsev drew the eye of many mathematicians and mechanicians around the globe to this challenge, which led to its being intensively labored out. the tactic of Lyapunov capabilities grew to become the major investigative procedure which grew to become out to be very powerful in examining either theoretic and utilized problems.
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Extra resources for Partial Stability and Control
It was obtained employing Bellman's dynamic programming technique . Rumyantsev [1970J, [1971b] extended this theorem to the problem of optimal stabilization with respect to part of the variables. ) to denote the dot product of the corresponding vector functions. 12. 2), the inequalities b(llwll), Ilwll = (xi + ... + x~/h, m ~ k ~ n a(IIyll) ~ VO(t,x) ~ and a vector function u = UO(t,x) E U such that (1) W(t,x) == -w(t,x,UO(t,x)) ~ -c(llwll); O()] oV o (oVO _ . 5. THE PROBLEM OF STABILITY OF LINEAR SYSTEMS 41 (3) B[VO, t, x, u] 2": 0 for any u E U; and (4) for any u* (t, x) E U that ensures asymptotic y-stability of the unperturbed motion x = 0 the following equality is true: lim VO(t,x*[t]) = 0, t -+ 00.
Zubov's method of constructing auxiliary systems . 1) which then can be used as a basis for constructing a certain auxiliary system of differential equations. Then the Lyapunov function for the auxiliary system, having been constructed, solves the PSt-problem for the original system. 3. The method of constructing different-type limiting systems and limiting V-functions. Of course, this method is not really a method of constructing V-functions, though the possibilities for finding "appropriate" V-function are extended within its scope.
The intensive development of this method resulted in a rather general principle of comparison with the Lyapunov vector function (Lakshmikantham et al. ) being worked out. This principle has been successfully used in both theoretical and applied research on the dynamics of nonlinear systems, especially high-dimensional complex systems. Matrosov [1965J applied the MLVF to the PSt-problem. The same problem was considered by Peiffer , Peiffer and Rouche ' Shoichi et al. [1982J, and Vorotnikov [1991aJ.
Partial Stability and Control by V.I. Vorotnikov