File size: 3,111 Bytes
37bcf52 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 | import Mathlib.Analysis.Calculus.ContDiff.Comp
import Mathlib.Analysis.Calculus.FDeriv.Symmetric
import Mathlib.Analysis.Calculus.Deriv.Mul
import Mathlib.Analysis.Calculus.Deriv.Prod
namespace OAI
/-! Commutation of a time derivative and an initial-point derivative.
These lemmas turn joint smoothness of an actual flow into its variational
equations, without an additional smooth-dependence input. -/
noncomputable section
namespace ForcedComputation.Flow
open scoped ContDiff
variable {E F : Type} [NormedAddCommGroup E] [NormedSpace ℝ E]
[NormedAddCommGroup F] [NormedSpace ℝ F]
theorem spaceDerivative_eq {G : ℝ × E → F} (hG : ContDiff ℝ ∞ G)
(t : ℝ) (x v : E) :
fderiv ℝ (fun y => G (t, y)) x v = fderiv ℝ G (t, x) (0, v) := by
have hd := ((hG.differentiable (by simp) (t, x)).hasFDerivAt.comp x
(hasFDerivAt_prodMk_right t x)).fderiv
exact congrArg (fun A : E →L[ℝ] F => A v) hd
theorem timeDerivative_eq {G : ℝ × E → F} (hG : ContDiff ℝ ∞ G)
(t : ℝ) (x : E) :
deriv (fun s => G (s, x)) t = fderiv ℝ G (t, x) (1, 0) := by
exact ((hG.differentiable (by simp) (t, x)).hasFDerivAt.comp_hasDerivAt t
((hasDerivAt_id t).prodMk (hasDerivAt_const t x))).deriv
theorem spaceDerivative_smooth {G : ℝ × E → F} (hG : ContDiff ℝ ∞ G) (v : E) :
ContDiff ℝ ∞ (fun p : ℝ × E => fderiv ℝ (fun y => G (p.1, y)) p.2 v) := by
have he : (fun p : ℝ × E => fderiv ℝ (fun y => G (p.1, y)) p.2 v) =
fun p => fderiv ℝ G p (0, v) := by
funext p
exact spaceDerivative_eq hG p.1 p.2 v
rw [he]
exact (hG.fderiv_right (by simp)).clm_apply contDiff_const
theorem time_space_commute {G : ℝ × E → F} (hG : ContDiff ℝ ∞ G)
(t : ℝ) (x v : E) :
HasDerivAt (fun s => fderiv ℝ (fun y => G (s, y)) x v)
(fderiv ℝ (fun y => deriv (fun s => G (s, y)) t) x v) t := by
have hD : ContDiff ℝ ∞ (fderiv ℝ G) := hG.fderiv_right (by simp)
have ht := (hD.differentiable (by simp) (t, x)).hasFDerivAt.comp_hasDerivAt t
((hasDerivAt_id t).prodMk (hasDerivAt_const t x))
have htv := ht.clm_apply (hasDerivAt_const t (0, v))
simp only [map_zero, add_zero, Function.comp_def, id_eq] at htv
have he : (fun s => fderiv ℝ (fun y => G (s, y)) x v) =
fun s => fderiv ℝ G (s, x) (0, v) := by
funext s
exact spaceDerivative_eq hG s x v
rw [he]
have htime : (fun y => deriv (fun s => G (s, y)) t) =
fun y => fderiv ℝ G (t, y) (1, 0) := by
funext y
exact timeDerivative_eq hG t y
rw [htime]
have hx := (hD.differentiable (by simp) (t, x)).hasFDerivAt.comp x
(hasFDerivAt_prodMk_right t x)
have hxv := congrArg (fun A : E →L[ℝ] F => A v)
(hx.clm_apply (hasFDerivAt_const (1, 0) x)).fderiv
simp only [add_apply, ContinuousLinearMap.comp_apply,
zero_apply, map_zero, zero_add, ContinuousLinearMap.flip_apply,
ContinuousLinearMap.inr_apply, Function.comp_def] at hxv
rw [hxv]
have hs := (hG.contDiffAt (x := (t, x))).isSymmSndFDerivAt (by simp) (1, 0) (0, v)
rw [← hs]
exact htv
end ForcedComputation.Flow
end
end OAI
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