q The
small gain theorem still holds even if D and M are infinite dimensional. This is
summarized as the following corollary.
qCorollary: The following statements
are equivalent:
• (i) The
system is well-posed and internally stable for all D ÎH¥ with ||D||¥ £1/g .
• (ii) The
system is well-posed and internally stable for all D ÎRH¥ with ||D||¥ £1/g .
• (iii) The
system is well-posed and internally stable for all D ÎRqxp with ||D|| £1/g .
• (iv) ||M||¥ £ g
•
• It can be
shown that the small gain condition is sufficient to guarantee internal
stability even if D is a nonlinear and time varying “stable” operator
with an appropriately defined stability notion, see Desoer and
Vidyasagar [1975].