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$Continuous-time random walks, fractional calculus and ... - Eurandom$

Example (Fokker-Planck Integrable Systems) • Consider the SDE dxt = (αxt + βx 3 t ) dt + √ σ dWt • Provided α and β are such that lim |x|→∞ Φ(x) = ∞ and e −γΦ ∈ L 1 (R) for all γ > 0, the invariant density is given by 2 − ρ∞(x) = Ze σ Φ(x) = Ze 2 α ( σ 2 x2 + β 4 x4 ) , with Z being the appropriate normalization constant. . . . . . . Boumediene <strong>Hamzi</strong> (Imperial College) On Control and RDS in RKHS June 4th, 2012 52 / 55
Example (A Fokker-Planck Integrable System) • The stationary Fokker-Planck equation for this example reads f(x) = σρ ′ ∞(x)/(2ρ∞(x)), so that we find αx + βx3 = σ ρ 2 ′ ∞(x) ρ∞(x) , ∀ x ∈ supp(ρ∞) . • To obtain the parameters as a least squares fit to the observations, let {xi} m i=1 ⊂ R, with xi ∈ supp(ˆρ∞), denote a (finite) sequence of samples. Since the preceding equation holds for all x ∈ supp(ρ∞) ⊃ supp(ˆρ∞) we have that αxi + βx 3 i = ˆσ ˆρ 2 ′ ∞(xi) ˆρ∞(xi) , for 1 ≤ i ≤ m. Consequently, the estimators ˆα and ˆ β obtained by a least squares fit solve the system of linear equations (∑m i=1 x2 ∑ i m i=1 x4i ∑m i=1 x4 ∑ i m i=1 x6i ) ( ) ˆα ˆβ = ˆσ 2 m∑ i=1 ˆρ ′ ∞(xi) ˆρ∞(xi) ( xi which can be solved explicitly provided the system matrix is invertible. x 3 i ) . . . . . . Boumediene <strong>Hamzi</strong> (Imperial College) On Control and RDS in RKHS June 4th, 2012 53 / 55
Page 1 and 2: Preliminary Results on Control and
Page 3 and 4: Outline • Review of Some Concepts
Page 5 and 6: Review of Some Concepts for Linear
Page 11 and 12: Controllability and Observability E
Page 13 and 14: Balancing of Linear Control Systems
Page 15 and 16: Balancing of Nonlinear Systems •
Page 17 and 18: Empirical Estimates of the Gramians
Page 19 and 20: Reproducing Kernel Hilbert Spaces
Page 21 and 22: Reproducing Kernel Hilbert Spaces T
Page 33 and 34: Balanced Reduction of Nonlinear Con
Page 35 and 36: Balancing in RKHS • “Balancing
Page 37 and 38: Experiment: Inputs ◮ Excite with
Page 39 and 40: Experiment diag(TKcT T ) 10 −1 10
Page 45 and 46: Nonlinear SDEs in RKHSes • Assump
Page 47 and 48: Nonlinear SDEs in RKHSes • The st
Page 49 and 50: Parameter Estimation for SDEs • I
Page 51: Parameter Estimation for SDEs • I
Page 55: Open Questions • Derivation of da

Hamzi - Eurandom

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