# Vector Lyapunov Functions and Stability Analysis of by V. Lakshmikantham, V.M. Matrosov, S. Sivasundaram

By V. Lakshmikantham, V.M. Matrosov, S. Sivasundaram

One provider arithmetic has rendered the 'Et moi, "', si j'avait su remark en revenir, je n'y serais aspect all."' human race. It has positioned good judgment again the place it belongs, at the topmost shelf subsequent Jules Verne to the dusty canister labelled 'discarded non sense'. The sequence is divergent; accordingly we are able to do whatever with it. Eric T. Bell O. Heaviside arithmetic is a device for notion. A hugely invaluable instrument in an international the place either suggestions and non linearities abound. equally, every kind of components of arithmetic function instruments for different components and for different sciences. utilising an easy rewriting rule to the quote at the correct above one reveals such statements as: 'One provider topology has rendered mathematical physics .. .'; 'One carrier common sense has rendered com puter technology .. .'; 'One carrier type concept has rendered arithmetic .. .'. All arguably real. And all statements accessible this fashion shape a part of the raison d'etre of this sequence.

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N, D+V p (t,x);5. -C(W(t,x)), D + Vj(t,x);5. gj(t, V(t,x)); i CEK. and # p. 1} is (ho' h)-asymptotically stable. 1) is (h o, h)-equistable. Hence it is enough to prove that given to E R +, there exists a 0 = 0o(to) > 0 such that ° =o(to, A) be associated with (ho,h)-equistability. 1) with ho(to,xo) < ° We For I; = A, let 0 0• lim t_oo first note that lim in! W(t,x(t)) t_oo V p(t,x(t))-+ - 00 as t-+oo. 0• = o. For otherwise, in view of (iii), we get 43 Why several Lyapunov functions? Suppose that lim W(t,x(t» f::.

9) X(tl' to,x o) E 8S(a), x( t*, to' xo) E 8S({3), and 29 Why several Lyapunov functions? 5) such that T2(t 1, t 1, vo) =Vo. 4). 9). 13) which is a contradiction. 9). 13). 8), for t ~ to. 8(to'p) and hence the proof is complete. We shall next consider a result concerning equi-ultimate boundedness. 2. 4) is uniformly bounded only for Uo ~ a1(p*). 1) is equi-ultimately bounded. Set a = p*. 1. 1) with Now let 1 Zo 1 ~ a for any a> p*. 1 Zo 1 ~ a. We claim that there exists a t* ( I • 5• 15) T E [to, to + T] where o(to, a) = T(t0' a) >- f3C(p*) , such that 1 z(t*) 1 < p* whenever p* ~ 1 z~ 1 ~ a.

19) D+Vl(t,x) ~ -G( I x I ),G E 11:, (t,x) E R+ xS(A). Suppose further that f(t,x) is bounded on R+ xS(A). practically asymptotically stable. 4) is automatically satisfied. 1). It is now easy to show that lim x(t) = o. Hence the proof is complete. 6. Method of vector Lyapunov functions. It is well known that using a single Lyapunov function, it is possible to investigate a variety of problems in a unified way. 5, employing two Lyapunov functions is more advantageous in improving several results.