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Definition. A function u is said to be positive definite if there exists rp E K such that u(x, t ) 2 (~(1x1)for all t E J and for all x E B ( r ) for some r > 0. 17. Definition. A function v is said to be negative definite if - u is positive definite. 18. Definition. A function u, defined on R ” x J , is said to be radially unbounded if there exists cp E K R such that u(x, t ) 2 cp(lx1) for all x E R” and t E J . (Recall the assumption u(0, t ) = 0 for all t E J . 19. Definition. A function u is said to be decrescent if there exists cp E K such that Iu(x, t)l I cp(lxl) for all t E J and for all x E B ( r ) for some r > 0.
I$i2(l~il) + 1 1 aiui(zi,t). , r , are sufficiently large. , I , there exist constants K,, K2, K3, and Mi such that I K1 2 Let M,’ = . C J=r+l IaijIC$j3(IzjI>I”2, [$,3((zll), . . , $ r 3 ( z) , \ ) ] , l e t t h e r x r m a t r i x R = [ri] bespecifiedby cci(ai+aii), and let Mi 2 I ~ ~ i ~ ~ ~ ~ ( z i ~ ~ ) I s” = ( R + RT)/2. , r are sufficiently large, the sign of D u ( , , ( x , r ) is determined by L,(s”)C:=,$i3(1zil), and Du(,)(x,t) < 0 for all t E J . , 1. For any other combination of indices, the proof is similar.
1) is stable. 22. Theorem. If there exists a positive definite, decrescent function u with a negative semidefinite derivative Du(,,, then the equilibrium x = 0 of Eq. (I) is uniformly stable. 23. Theorem. If there exists a positive definite, decrescent function u with a negative definite derivative Du(,,, then the equilibrium x = 0 of Eq. (I) is uniformly asymptotically stable. In this case there exist cpl, cpz, (p3 E K such that cpt(lxl> for all x E B ( r ) and t u(x,t> I cpz(lxl), E Du(l,(x,t)5 -cp3(IxI) J.
A Hacker's Revenge