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Local negative feedback) asStability ). results of the sociated with (YJ,which is not an integral part of (9, type presented herein for composite systems with unstable subsystems and without local stabilizing feedback have apparently not been established at this time. This problem is of great practical importance and needs to be pursued further. (Results for systems with unstable subsystems are given in Thompson . , /), the easier it is to satisfy the negative definiteness requirement of matrix S.
G ( x , t ) ,t+h] - u ( x , t ) } . 12) h+0+ If u is continuously differentiable with respect to all of its arguments, then the total derivative of u with respect t o t along solutions of Eq. 13) where V v ( x ,t ) denotes the gradient vector of the scalar function v and &/at represents the partial derivative of u with respect to t. Whether u is continuous or continuously differentiable will either be clear from context or it will be specified. In the former case Du(,, is specified by Eq. 12) while in the latter case Do(,) is given by Eq.
B) If N = L, the equilibrium of ( Y )is completely unstable. Proof. Given ui of hypothesis (i) and CI of hypothesis (iii) we choose u(x,t) = C aiui(zi,t). 2, we obtain 1 Du(,)(x,t) 5 C $i3(IziI) i= 1 for all zi E Bi(ri), i E L, and t E J . Since test matrix S is negative definite, we have A M ( S )< 0 and Du(,,(x, t ) is negative definite. $ N , and t E J } . , in every neighborhood of the origin x = 0, there is at least one point x’ # 0 for which u(x’, t ) < 0 for all t E J . Furthermore, on the set D, u ( x , t ) is bounded from below.