Hamzi - Eurandom

Controllability and Observability Energies for Nonlinear
System
• Theorem (Scherpen, 1993) If the origin is an asymptotically stable
equilibrium of f(x) on a neighborhood W of the origin, then for all
x ∈ W , Lo(x) is the unique smooth solution of
∂Lo 1
(x)f(x) +
∂x 2 h⊤ (x)h(x) = 0, Lo(0) = 0
under the assumption that this equation has a smooth solution on W .
Furthermore for all x ∈ W , Lc(x) is the unique smooth solution of
∂Lc
∂x
1 ∂Lc
(x)f(x) +
2 ∂x (x)g(x)g⊤ (x) ∂⊤Lc ∂x (x) = 0, Lc(0) = 0
under the assumption that this equation has a smooth solution ¯ Lc on W
and that the origin is an asymptotically stable equilibrium of
(x)) on W .
−(f(x) + g(x)g ⊤ (x) ∂ ¯ Lc
∂x
. . . . . .
Boumediene Hamzi (Imperial College) On Control and RDS in RKHS June 4th, 2012 11 / 55