By A. Halanay, V. Rasvan

ISBN-10: 9401046972

ISBN-13: 9789401046978

ISBN-10: 9401116008

ISBN-13: 9789401116008

The yr 1992 marks the centennial anniversary of e-book of the distinguished monograph "The normal challenge of balance of movement" written by way of A. M. Liapunov. This anniversary conjures up to consider the best way conception and functions have built in this century. the 1st statement it is easy to make is that the so-called "second method", these days often called the "Liapunov functionality method", has obtained extra awareness than the "first method"; allow us to additionally point out the research of serious circumstances, which introduced extra recognition lately in reference to the research of bifurcations and with nonlinear stabilization. one of many purposes of recognition of the Liapunov functionality procedure could be the truth that, in lots of events in technology and engineering, and never purely in mechanics, which was once the most resource of proposal for the paintings of Liapunov, average Liapunov services could be proposed, in detail hooked up with the houses of the methods. it really is one of many reasons of this ebook to suggest this concept. From the mathematical point of view, the century after the 1st seem ance of Liapunov's monograph has been characterised either by means of common izations and by way of refinements of Liapunov's rules. yet we consider that the main incredible growth is the knowledge of the extensive chances open for functions via balance concept as built by way of Liapunov a century in the past. we've got attempted to teach a number of the principles during this path via commence ing with our own adventure within the examine of a few models.

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We assume even that ~: (x, u) has also all eigenvalues with strictly negative real parts, that is x is exponentially stable. Can one be sure that by passing directly from the value 11 of the control to the value U, the evolution will be "directed to" x? 4) for u = u ; even if the condition for the first approximation is satisfied one knows only that the evolution will lead to x if the initial condition is close enough to X, hence the change of regime will be ensured only if x is close enough to x.

4) which is associated to the system in a natural way. We have ov aq-(q,p)F(q,p) + [atoq(q, F(q,p)) - oAoq(q)- -F*(q,p)R(q,F(q,p)) ] av ap al = [aF Paq(q,p)- aq(q,F(q,p))- 1 al q, F( q, p))aq aF (q, p) + aq-(q) aA - aq'( F( q, p) + [al aq (q, F( q, p))- 1[F(q,p) + a:p(q,p)p*aF* - aA aq(q) - F*(q,p)R(q, F(q,p)) aF* ( aq,(q,F(q,p)) al ) *] -a:p(q,p) = = Some General Results in Stability Theory 32 1 aF al aF aA = [Paq(q,p)- aq(q,F(q,p))-Paq(q,p)+ aq(q) F(q,p)+ 1[F(q,p)+ + [ al aq(q, F(q,p)) - aA aq(q) - P(q,p)R(q, F(q,p)) 1 ap ap +ap(q,p)p* - ap(q,p)p* = = -F*(q,p)R(q,F(q,p»F(q,p):S; -(XIF(q,p)1 2 In the above computation, the use was made of the identity al a q' (q, F( q, P» =p which defines F( q , P ).

It thus appears useful to weaken these requirements while preserving the conclusion concerning asymptotic stability. Such results were first obtained by BarbaSin and Krasovskii (1952) and have been further developed by La Salle (1968). 1) if it is C1 , ~~ (x)f(x) :::; for all XED, and if for every xED there exists a neighbourhood U of x and Il E R such that Vex) 2: Il for all x E U. Let G = { XED, ~~ ( x)f( x) = o} . 1), G defined as above. 1)) contained in ° G. 1) if it follows from x(to) E M that x(t) E M for all t for which the solution is defined.

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

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