Tamtam Proceedings - lamsin

Control of the chaotic advection 65We choose a pseudo inverse Γ :ΓV (A, θ) =∑k∈Z L ,ω(A).k≠0V k (A)iω(A).k eiθ.k . (9)The operator R is the projector on the resonant part of the perturbation :RV =∑V k (A)e iθ.k . (10)k∈Z L ,ω(A).k=0From these operators defined for integrable part H 0 , we construct a global control termfor the perturbed Hamiltonian H 0 + V , i.e., we construct f such that the controlled HamiltonianH c = H 0 + V + f up to canonical transformations has the same dynamics thanH 0 + RV .We give the following proposition :Proposition 1 For V ∈ A, where A is an algebra of operators in Hilbert space, and Γconstructed from H 0 , we have the following equation : e {ΓV } (H 0 + V + f) = H 0 + RV ,wheref(V ) =∞∑n=1(−1) n(n + 1)! {ΓV }n (nR + 1)V.So we stabilize the perturbed Hamiltonian by a smal control term, well adapted to theproblem.We note that if V is of order ɛ, the control term is of order ɛ 2 . In general, the control termdepends on all the variables A and θ, and acts globally on all phase space.2. Computation of the control termThe autonomous Hamiltonian of the model is H(x, y, E, τ) = E + ψ(x, y, τ), whereψ is given by Eq. (1).H(x, y, E, τ) = E + ɛ sin(x + B sin τ) sin y = H 0 (x, E) + V (x, y, τ), (11)where H 0 (x, E) = E is the integrable part and V the perturbation.The phase space is extended to (x, y, E, τ), where A = (x, E) are the action variablesand θ = (y, τ) the angle. A and θ are canonical conjugate variables.The equations of motion are :ẋ = − ∂H∂y = −∂V ∂y= −ɛ sin(x + B sin τ) cos y,TAMTAM –Tunis– 2005