Download e-book for iPad: Applications of Liapunov Methods in Stability by A. Halanay, V. Rasvan

By A. Halanay, V. Rasvan

ISBN-10: 9401046972

ISBN-13: 9789401046978

ISBN-10: 9401116008

ISBN-13: 9789401116008

The 12 months 1992 marks the centennial anniversary of booklet of the prestigious monograph "The basic challenge of balance of movement" written via A. M. Liapunov. This anniversary evokes to consider the best way conception and purposes have built in this century. the 1st commentary you will make is that the so-called "second method", these days referred to as the "Liapunov functionality method", has bought extra realization than the "first method"; allow us to additionally point out the learn of severe situations, which introduced extra consciousness lately in reference to the examine of bifurcations and with nonlinear stabilization. one of many purposes of recognition of the Liapunov functionality technique should be the truth that, in lots of events in technological know-how and engineering, and never merely in mechanics, which was once the most resource of suggestion for the paintings of Liapunov, usual Liapunov services could be proposed, in detail hooked up with the homes of the techniques. it's one of many reasons of this e-book to suggest this concept. From the mathematical point of view, the century after the 1st look­ ance of Liapunov's monograph has been characterised either through common­ izations and through refinements of Liapunov's principles. yet we suppose that the main stunning development is the certainty of the extensive chances open for functions by means of balance conception as built by way of Liapunov a century in the past. we now have attempted to teach many of the principles during this course by means of begin­ ing with our own adventure within the learn of a few models.

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Extra info for Applications of Liapunov Methods in Stability

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N - 1. 4 Liapunov Functions in Hydraulic Engineering 53 We see directly that we can take Bf = II(Qi - ~t) i#k Appendix E Consider now the system n-l L.

We will describe here the study of the general situation of a surge tank fed by an arbitrary number of tunnels; this study will show that a suitable Liapunov function makes it possible to obtain an estimate for the critical area from the linearized equations. This analysis exhibits some features related to the art of constructing Liapunov functions and we do not see any other procedure that could allow to obtain the result. The mathematical model associated to the hydraulic scheme of a surge tank fed by n tunnels is dWk P W2 Z - 0 9 dt + k k + - , lk k=l, ...

_ CXt-,Dt-,) F _ ~, ~t-' Qo g(H + Zo) and the Sylvester conditions read ( ~ _ cx, D, _ ~, ... _ CXt-, Dt-, ) F _ ~t-' It is thus enough to check thet Q0 CXk D) k F (~ - n-' f, T > g(H + Zo) Qo 0 1 2 g(H+Zo) > , .. ~ . 4 43 Liapunov Functions in Hydraulic Engineering Taking into account the fact that CXo n-1 0 ' " (X,le. le. _ _ L k=1 (l - ~le. fln 1 Ole. fln-1 (l 1 ~le. (Appendix F) the following stability condition is obtained Qo F > Fer = (H + 9 Z)' 0 fl,-l ~le. fln • 1 Ole. 1). xle. =0 , i = 1, ...

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Applications of Liapunov Methods in Stability by A. Halanay, V. Rasvan

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