Hamzi - Eurandom

Controllability and Observability Energies of Nonlinear
Systems in RKHSes
• Towards deriving an equivalent but computable expression for ˆ Lc defined
in terms of kernels, we recall the sampling operator Sx introduced by
Smale.
• Let x = {xi} m i=1 denote a generic sample of m data points. To x we can
associate the operators
Sx : H → R m , h ∈ H ↦→ ( h(x1), . . . , h(xm) )
S ∗ x : R m → H, c ∈ R m ↦→ ∑m ciKxi i=1 .
If x is the collection of m = Np controllability samples, one can check
that Wc = 1
m S∗ xSx and Kc = SxS ∗ x.
. . . . . .
Boumediene Hamzi (Imperial College) On Control and RDS in RKHS June 4th, 2012 30 / 55