Theory Linear_ODE

section ‹Linear ODE›
theory Linear_ODE

  fixes f::"real  real"
  assumes "(f has_derivative f') (at x within s)"
  shows "((λx. exp (f x *R A)) has_derivative (λh. f' h *R (exp (f x *R A) * A))) (at x within s)"
proof -
  from assms have "bounded_linear f'" by auto
  with real_bounded_linear obtain m where f': "f' = (λh. h * m)" by blast
  show ?thesis
    using vector_diff_chain_within[OF _ exp_scaleR_has_vector_derivative_right, of f m x s A] assms f'
    by (auto simp: has_vector_derivative_def o_def)

fixes A::"'a::{banach,perfect_space} blinop"

definition "linode_solution t0 x0 = (λt. exp ((t - t0) *R A) x0)"

lemma linode_solution_solves_ode:
  "(linode_solution t0 x0 solves_ode (λ_. A)) UNIV UNIV" "linode_solution t0 x0 t0 = x0"
  by (auto intro!: solves_odeI derivative_eq_intros
    simp: has_vector_derivative_def blinop.bilinear_simps exp_times_scaleR_commute
      has_vderiv_on_def linode_solution_def)

lemma "(linode_solution t0 x0 usolves_ode (λ_. A) from t0) UNIV UNIV"
  using linode_solution_solves_ode(1)
proof (rule usolves_odeI)
  fix s t1
  assume s0: "s t0 = linode_solution t0 x0 t0"
  assume sol: "(s solves_ode (λx. blinop_apply A)) {t0--t1} UNIV"

  then have [derivative_intros]:
    "(s has_derivative (λh. h *R A (s t))) (at t within {t0 -- t1})" if "t  {t0 -- t1}" for t
    using that
    by (auto dest!: solves_odeD(1) simp: has_vector_derivative_def has_vderiv_on_def)
  have "((λt. exp (-(t - t0) *R A) (s t)) has_derivative (λ_. 0)) (at t within {t0 -- t1})"
    (is "(?es has_derivative _) _")
    if "t  {t0 -- t1}" for t
    by (auto intro!: derivative_eq_intros that simp: has_vector_derivative_def
  from has_derivative_zero_constant[OF convex_closed_segment this]
  obtain c where c: "t. t  {t0 -- t1}  ?es t = c" by auto
  hence "(exp ((t - t0) *R A) * (exp (-((t - t0) *R A)))) (s t) = exp ((t - t0) *R A) c"
    if "t  {t0 -- t1}" for t
    by (metis (no_types, opaque_lifting) blinop_apply_times_blinop real_vector.scale_minus_left that)
  then have s_def: "s t = exp ((t - t0) *R A) c" if "t  {t0 -- t1}" for t
    by (simp add: exp_minus_inverse that)
  from s0 s_def
  have "exp ((t0 - t0) *R A) c = x0"
    by (simp add: linode_solution_solves_ode(2))
  hence "c = x0" by simp
  then show "s t1 = linode_solution t0 x0 t1"
    using s_def[of t1] by (simp add: linode_solution_def)
qed auto