Nonlinear Control Sy.. - Free

204 CHAPTER 8. PASSIVITY
Moreover, since the applied voltage is such that fo e2 (t) dt < oo, we can take limits as
T -+ o0 on both sides of thinequality,, and we have
j
00
R2 i(t)2 dt+ J v2(t) dt 0.
(v) (x, x) = 0 if and only if x = 0.
The function X x X -+ IR is called the inner product of the space X. If the space X is
complete, then the inner product space is said to be a Hilbert space. Using these proper
we can define a norm for each element of the space X as follows:
llxjfx = (x,x).
An important property of inner product space is the so-called Schwarz inequality:
I (x)y)l