Formal Model Reduction

ion An abstraction (V , ψ,F) is given by: • V : a finite set of observables, • ψ: a mapping from V → R into V ] → R, • F: a C∞ mapping from V ] → R into V ] → R; such that: • ψ is linear with positive coefficients, and for any sequence (xn) ∈ (V → R+)N such that (||xn||) diverges towards +∞, then (||ψ(xn)||) diverges as well (for arbitrary norms || · || and || · ||), • the following diagram commutes: (V → R) F −→ (V → R) ψ y yψ `∗ `∗ (V ] → R) F] −→ (V ] → R) i.e. ψ ◦ F = F ◦ψ. Jérôme Feret 25 Friday, September the 16th Abstraction example • V ∆ = {[(u,u,u)], [(u,p,u)], [(p,p,u)], [(u,p,p)], [(p,p,p)]} • F(ρ) ∆ =  [(u,u,u)] 7→ −k·ρ([(u,u,u)]) [(u,p,u)] 7→ −k·ρ([(u,p,u)]) + k·ρ([(u,u,u)]) − k·ρ([(u,p,u)]) [(u,p,p)] 7→ −k·ρ([(u,p,p)]) + k·ρ([(u,p,u)]) · · · • V ] ∆ = {[(u,u,u)], [(?,p,u)], [(?,p,p)], [(u,p,?)], [(p,p,?)]} • ψ(ρ) ∆ =  [(u,u,u)] 7→ ρ([(u,u,u)]) [(?,p,u)] 7→ ρ([(u,p,u)]) + ρ([(p,p,u)]) [(?,p,p)] 7→ ρ([(u,p,p)]) + ρ([(p,p,p)]) . . . • F(ρ) ∆ =  [(u,u,u)] 7→ −k·ρ([(u,u,u)]) [(?,p,u)] 7→ −k·ρ([(?,p,u)]) + k·ρ([(u,u,u)]) [(?,p,p)] 7→ k·ρ([(?,p,u)]) . . . (Completeness can be checked analytically.) Jérôme Feret 26 Friday, September the 16thion example • V ∆ = {[(u,u,u)], [(u,p,u)], [(p,p,u)], [(u,p,p)], [(p,p,p)]} • F(ρ) ∆ =  [(u,u,u)] 7→ −k·ρ([(u,u,u)]) [(u,p,u)] 7→ −k·ρ([(u,p,u)]) + k·ρ([(u,u,u)]) − k·ρ([(u,p,u)]) [(u,p,p)] 7→ −k·ρ([(u,p,p)]) + k·ρ([(u,p,u)]) · · · • V ] ∆ = {[(u,u,u)], [(?,p,u)], [(?,p,p)], [(u,p,?)], [(p,p,?)]} • ψ(ρ) ∆ =  [(u,u,u)] 7→ ρ([(u,u,u)]) [(?,p,u)] 7→ ρ([(u,p,u)]) + ρ([(p,p,u)]) [(?,p,p)] 7→ ρ([(u,p,p)]) + ρ([(p,p,p)]) . . . • F(ρ) ∆ =  [(u,u,u)] 7→ −k·ρ([(u,u,u)]) [(?,p,u)] 7→ −k·ρ([(?,p,u)]) + k·ρ([(u,u,u)]) [(?,p,p)] 7→ k·ρ([(?,p,u)]) . . . (Completeness can be checked analytically.) Jérôme Feret 26 Friday, September the 16th Abstract continuous trajectories Let (V,F) be a concrete system. Let (V , ψ,F) be an abstraction of the concrete system (V,F). Let X0 ∈ V → R be an initial (concrete) state. We know that the following system: Yψ(X0)(T) = ψ(X0) + ∫ T t=0 F ( Yψ(X0)(t) ) ·dt has a unique maximal solution Yψ(X0) such that Yψ(X0) = ψ(X0). Theorem 1 Moreover, this solution is the projection of the maximal solution XX0 of the system XX0(T) = X0 + ∫ T t=0 F ( XX0(t) ) ·dt. (i.e. Yψ(X0) = ψ(XX0)) Jérôme Feret 27 Friday, September the 16thcontinuous trajectories Let (V,F) be a concrete system. Let (V , ψ,F) be an abstraction of the concrete system (V,F). Let X0 ∈ V → R be an initial (concrete) state. We know that the following system: Yψ(X0)(T) = ψ(X0) + ∫ T t=0 F ( Yψ(X0)(t) ) ·dt has a unique maximal solution Yψ(X0) such that Yψ(X0) = ψ(X0). Theorem 1 Moreover, this solution is the projection of the maximal solution XX0 of the system XX0(T) = X0 + ∫ T t=0 F ( XX0(t) ) ·dt. (i.e. Yψ(X0) = ψ(XX0)) Jérôme Feret 27 Friday, September the 16th Abstract continuous trajectories Proof sketch Given an abstraction (V , ψ,F), we have: XX0(T) = X0 + ∫T t=0F ( XX0(t) ) ·dt ψ ( XX0(T) ) = ψ ( X0 + ∫T t=0F ( XX0(t) ) ·dt ) ψ ( XX0(T) ) = ψ(X0) + ∫T t=0[ψ ◦ F] ( XX0(t) ) ·dt (ψ is linear) ψ ( XX0(T) ) = ψ(X0) + ∫T t=0F ] ( ψ ( XX0(t) )) ·dt (F is ψ-complete) We set Y0 ∆ = ψ(X0) and YY0 ∆ = ψ ◦ XX0. Then we have: YY0(T) = Y0 + ∫T t=0F ] ( YY0(t) ) ·dt The assumption about || · ||, || · ||, and ψ ensures that ψ ◦ XX0 is a maximal solution. Jérôme Feret 28 Friday, September the 16thcontinuous trajectories Proof sketch Given an abstraction (V , ψ,F), we have: XX0(T) = X0 + ∫T t=0F ( XX0(t) ) ·dt ψ ( XX0(T) ) = ψ ( X0 + ∫T t=0F ( XX0(t) ) ·dt ) ψ ( XX0(T) ) = ψ(X0) + ∫T t=0[ψ ◦ F] ( XX0(t) ) ·dt (ψ is linear) ψ ( XX0(T) ) = ψ(X0) + ∫T t=0F ] ( ψ ( XX0(t) )) ·dt (F is ψ-complete) We set Y0 ∆ = ψ(X0) and YY0 ∆ = ψ ◦ XX0. Then we have: YY0(T) = Y0 + ∫T t=0F ] ( YY0(t) ) ·dt The assumption about || · ||, || · ||, and ψ ensures that ψ ◦ XX0 is a maximal solution. Jérôme Feret 28 Friday, September the 16th