By A. Halanay, V. Rasvan
The 12 months 1992 marks the centennial anniversary of e-book of the prestigious monograph "The common challenge of balance of movement" written by means of A. M. Liapunov. This anniversary evokes to contemplate the way in which thought and purposes have built in this century. the 1st commentary you can actually make is that the so-called "second method", these days often called the "Liapunov functionality method", has bought extra awareness than the "first method"; allow us to additionally point out the research of severe circumstances, 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 process will be the truth that, in lots of events in technology and engineering, and never in basic terms in mechanics, which was once the most resource of notion for the paintings of Liapunov, traditional Liapunov services can be proposed, in detail hooked up with the homes of the strategies. it's one of many reasons of this e-book to recommend this concept. From the mathematical point of view, the century after the 1st seem ance of Liapunov's monograph has been characterised either through common izations and via refinements of Liapunov's principles. yet we believe that the main awesome growth is the certainty of the large probabilities open for functions via balance concept as developed through Liapunov a century in the past. we now have attempted to teach the various rules during this path by means of begin ing with our own adventure within the research of a few models.
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Additional resources for Applications of Liapunov Methods in Stability
_ _ 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, ... , n - dy =0 dt ' 1, k=1 hence H Qo + n Z Y + L. xle. =1 n L. xle. = 0, k=1 i = 1, ... , n - 1 . We have already discussed the system for Xle. and we have deduced that Xle. are constant if y is constant. Since Xle. are constant they also satisfy also hence Therefore A direct computation shows 44 Some General Results in Stability Theory = Qo(H + 3Zo) 2Zo(H + Zo) # O.
The coefficients P" P2, pI, P" for head losses are given by A'l' p' = 2gd' ' A'll Pl = 2gd' ' and we see that pI Pl l' - II ' The steady-state (equilibrium) is computed from the relations W~=W~, W2O-W" 0 Zo - Z~ = - p"(W~)2 Qo = f'W~ + f''W~ , We deduce that the equilibrium water level in the surge tank is computed from the equation and if a natural inequality is satisfied a solution exists with Zo E ( - H/3, 0). With an abuse of notation, the linearized system around the corresponding equilibrium will be written as follows lk dWk 0.
3) satisfying this condition. If F(q(t), pet»~ hand . pet) = 0 then q'(t) == °hence q is constant. al al = aq,(q(t),F(q(t),p(t») = aq,(q(t),O) = On the other ° hence the invariant set consists only of equilibrium points of the form (q,O). But if p is constant and F( q, p) . wflte A(q) = 0; :~(Sq)dS) q aA J1 a2A aq (q) = 0 q* aqaq (sq)ds = 0 we have aaA(q) = O. 3 Liapunov Function 33 oA oA If oq (q) = 0 we have also oq (q)q = O. Therefore o2A Jo' q* oqoq (sq)qds = 0 o2A o2A But we assumed that oqoq ~ all hence q* oqoq(Sq)q ~ alq*q and ~~ (q) = 0 implies q = 0 (at least locally).