DDE Note Week 2

第二週：DDE的性質──維度、跟ODE和PDE的比較與關係、FDE form

和ODE的比較

一個普通的線性ODE:

{\begin{cases} \dot{x} (t) = A x (t) & for t \in R \\ x (0) = x_{0} \in R^{n} \end{cases}

一個普通的DDE:

{\begin{cases} \dot{T} (t) = - k T (t - τ) & for t \geq 0 \\ φ (t) \in C^{0} [- τ, 0] \end{cases}

Backward continuation

ODE

Unique backward continuation.

DDE

Haha, no way!

(no backward continuation) Consider $\dot{T} (t) = a T (t) + b T (t - τ)$ for $t \geq 0$ with $T (t) = c$ on $[- τ, 0]$ (prehistory).
Assume that it can be continued to $[- τ - ϵ, - τ]$ . Then for $t \in [- τ, 0]$ : $\dot{T} (t) = 0 = a c + b T (t - τ) ⟹ T (t - τ) = - \frac{a c}{b} \neq c if - \frac{a}{b} \neq 1.$ That is, $T (t) = - a c / b$ on $[- τ - ϵ, - τ]$ , and $T$ has a discontinuity at $t = - τ$ , which contradicts with stepwise continuity.
(many backward continuation)

FDE form of the DDE

Define $X = C^{0} ([- τ, 0], R^{N})$ . The the DDE can be written as

{\begin{cases} \dot{u} (t) = F (u_{t}) & for t \geq 0, \\ u_{0} = φ \in X, \end{cases}

where $F : X \to R^{N}$ is a linear functional, and $u_{t} (θ) = u (t + θ)$ is a function lives on $[- τ, 0]$ .

(semi-)flow 自然律/Dynamical system 動力系統

ODE的自然律(flow)

$Φ : R \times R^{n} \to R^{n}$ , defined by

Φ (t, x_{0}) = Φ^{t} (x_{0}) = e^{t A} x_{0} .

DDE的自然律(semiflow)

$Ψ : [0, \infty) \times X \to X$ , defined by

Ψ (t, φ) = Ψ^{t} (φ) = u_{t},

where

u_{t} (θ) = u (t + θ) = {\begin{cases} φ (t + θ) & if t + θ \in [- τ, 0] \\ φ (0) + \int_{0}^{t + θ} F (u_{s}) d s & if t + θ > 0. \end{cases}

solution segment/window

就是 $u_{t}$ 們

無窮小生成元(infinitesimal generator)

Infinitesimal generator 無窮小生成元

Let $X$ be a Banach space. Let $Φ$ be a flow on $X$ . The infinitesimal generator associated with $Φ$ is defined by
$G v := \partial_{t} Φ^{t} (v) |_{t = 0} = lim_{t ↘ 0} \frac{Φ^{t} (v) - v}{t}$
with the domain
$D (G) = {v \in X : lim_{t ↘ 0} \frac{Φ^{t} (v) - v}{t} exists in X} .$
(c.f. NTHU MATH 526500 DDE Note week 2)

ODE的無窮小生成元

The matrix $A$ .

DDE的無窮小生成元

$\partial_{θ}$ .

Compatibility conditions (相容條件)

Compatibility conditions 相容條件

為了確認FDE form of the DDE的無窮小生成元的domain，我們注意到在定義中，當 $θ = 0$ 的時候：
$G φ (0) = φ^{'} (0) = \dot{u} (0) = F u_{0} = F φ .$
所以 $G$ 的domain有一個限制條件就是 $φ^{'} (0) = F φ$ .

(c.f. NTHU MATH 526500 DDE Note week 2)

Transport PDEs on $X$

boundary condition

DDE是無窮維的

這句話是什麼意思？就是像空間(phase space)是有限維還是無窮維的意思

FDE form 和 ODE form 的優勢

(2024-09-12), (2024-09-17)

當天的筆記

[[IMG-20240913045502596.pdf]]