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59d1c80 | 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 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 | import Mathlib.Analysis.ODE.ExistUnique
import Mathlib.Analysis.Calculus.Deriv.Comp
import Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic
import Mathlib.Tactic.Linarith
import Mathlib.Tactic.NormNum
namespace OAI
noncomputable section
open Set Metric
open scoped NNReal Interval
namespace SmoothLocal.ODE
def curvatureField (k : ℝ → ℝ) (t : ℝ) (u : ℝ × ℝ) : ℝ × ℝ :=
(u.2, -k t * u.1)
theorem curvatureField_lipschitz {k : ℝ → ℝ} {t : ℝ}
(hk : |k t| ≤ (1 : ℝ) / 1000) :
LipschitzWith 1 (curvatureField k t) := by
apply LipschitzWith.of_dist_le_mul
intro u v
rw [NNReal.coe_one, one_mul, dist_eq_norm, dist_eq_norm]
change max ‖u.2 - v.2‖ ‖-k t * u.1 - -k t * v.1‖ ≤ ‖u - v‖
refine max_le (norm_snd_le (u - v)) ?_
calc
‖-k t * u.1 - -k t * v.1‖ = |k t| * ‖u.1 - v.1‖ := by
rw [← mul_sub, norm_mul, Real.norm_eq_abs, abs_neg]
_ ≤ 1 * ‖u.1 - v.1‖ :=
mul_le_mul_of_nonneg_right (by linarith) (norm_nonneg _)
_ ≤ ‖u - v‖ := by simpa using norm_fst_le (u - v)
theorem curvatureField_norm_le {k : ℝ → ℝ} {t : ℝ}
(hk : |k t| ≤ (1 : ℝ) / 1000) {u : ℝ × ℝ}
(hu : u ∈ closedBall ((1 : ℝ), (0 : ℝ)) 2) :
‖curvatureField k t u‖ ≤ 2 := by
have hu' : ‖u - ((1 : ℝ), (0 : ℝ))‖ ≤ 2 := by
simpa only [mem_closedBall, dist_eq_norm] using hu
have hfst : |u.1 - 1| ≤ 2 := by
exact (norm_fst_le (u - ((1 : ℝ), (0 : ℝ)))).trans hu'
have hsnd : ‖u.2‖ ≤ 2 := by
simpa using (norm_snd_le (u - ((1 : ℝ), (0 : ℝ)))).trans hu'
have hu1 : |u.1| ≤ 3 := by
obtain ⟨ha, hb⟩ := abs_le.mp hfst
exact abs_le.mpr ⟨by linarith, by linarith⟩
change max ‖u.2‖ ‖-k t * u.1‖ ≤ 2
refine max_le hsnd ?_
rw [norm_mul, Real.norm_eq_abs, Real.norm_eq_abs, abs_neg]
have hh := mul_le_mul hk hu1 (abs_nonneg _) (by norm_num : (0 : ℝ) ≤ 1 / 1000)
linarith
theorem exists_curvature_state_on_closed_interval
(k : ℝ → ℝ) (r : ℝ) (hr : 0 ≤ r) (hr1 : r ≤ 1)
(hkcont : ContinuousOn k (Icc (-r) r))
(hkbound : ∀ t ∈ Icc (-r) r, |k t| ≤ (1 : ℝ) / 1000) :
∃ α : ℝ → ℝ × ℝ, α 0 = (1, 0) ∧
∀ t ∈ Icc (-r) r,
HasDerivWithinAt α (curvatureField k t (α t)) (Icc (-r) r) t := by
let t₀ : Icc (-r) r := ⟨0, by constructor <;> linarith⟩
have hPL : IsPicardLindelof (curvatureField k) t₀ (1, 0) 2 0 2 1 := by
refine ⟨?_, ?_, ?_, ?_⟩
· intro t ht
apply LipschitzOnWith.of_dist_le_mul
intro u _ v _
exact (curvatureField_lipschitz (hkbound t ht)).dist_le_mul u v
· intro u _
exact continuousOn_const.prodMk (hkcont.neg.mul continuousOn_const)
· intro t ht u hu
exact curvatureField_norm_le (hkbound t ht) hu
· change (2 : ℝ) * max (r - 0) (0 - -r) ≤ (2 : ℝ) - 0
simp only [sub_zero, zero_sub, neg_neg, max_self]
linarith
simpa only [t₀] using hPL.exists_eq_forall_mem_Icc_hasDerivWithinAt₀
theorem curvature_integral_bound (k f : ℝ → ℝ) (x F : ℝ)
(hF : 0 ≤ F)
(hk : ∀ t ∈ Set.uIoc 0 x, |k t| ≤ (1 : ℝ) / 1000)
(hf : ∀ t ∈ Set.uIoc 0 x, |f t| ≤ F) :
|∫ t in 0..x, k t * f t| ≤ ((1 : ℝ) / 1000) * F * |x| := by
have h := intervalIntegral.norm_integral_le_of_norm_le_const
(a := 0) (b := x) (C := ((1 : ℝ) / 1000) * F)
(f := fun t => k t * f t) (fun t ht => by
rw [Real.norm_eq_abs, abs_mul]
exact (mul_le_mul_of_nonneg_left (hf t ht) (abs_nonneg _)).trans
(mul_le_mul_of_nonneg_right (hk t ht) hF))
simpa only [Real.norm_eq_abs, sub_zero] using h
theorem positive_of_integral_bounds (f F : ℝ)
(hF : F ≤ 1 + ((1 : ℝ) / 1000) * F)
(hf : |f - 1| ≤ ((1 : ℝ) / 1000) * F) :
(998 : ℝ) / 999 ≤ f ∧ f ≤ (1000 : ℝ) / 999 := by
obtain ⟨ha, hb⟩ := abs_le.mp hf
constructor <;> linarith
end SmoothLocal.ODE
end
end OAI
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