| 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 |
|
|
