Koopman Semigroups
We present here a class of positive operator semigroups that arise in studying dynamical systems. The main idea is to linearize a given (nonlinear) system by considering another state space. The linear operator which acts on this new space is called the K
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Koopman Semigroups We present here a class of positive operator semigroups that arise in studying dynamical systems. The main idea is to linearize a given (nonlinear) system by considering another state space. The linear operator which acts on this new space is called the Koopman operator. It is named after B. O. Koopman, who used this in the 1930s together with G. D. Birkhoff and J. von Neumann to prove the so-called ergodic theorems. We start with a nonlinear system of ordinary differential equations, associate a semiflow to it, and then derive the corresponding Koopman semigroup. Subsequently we present the main properties of this semigroup and its generator. At the end we show some properties of the semiflow that can be deduced from the appropriate properties of the associated Koopman semigroup or its generator. In this chapter we assume some general knowledge of measure theory.
16.1 Ordinary Differential Equations and Semiflows Consider the ordinary differential equation x(t) ˙ = F (x(t)),
t ≥ 0,
x(0) = x0 ∈ Ω,
(16.1)
where Ω ⊂ Rn is an open set. We make the following standing assumptions. Assumptions 16.1. a) F : Rn → Rn is continuously differentiable. b) Equation (16.1) has global solutions for all x0 ∈ Ω. c) Ω ⊂ Rn is positively invariant for the solution of equation (16.1), i.e., x0 ∈ Ω =⇒ x(t) ∈ Ω for t ≥ 0. © Springer International Publishing AG 2017 A. Bátkai et al., Positive Operator Semigroups, Operator Theory: Advances and Applications 257, DOI 10.1007/978-3-319-42813-0_16
253
254
Chapter 16. Koopman Semigroups
We comment on these assumptions based on standard theorems from ordinary differential equations. First recall that Assumption 16.1.a) implies the existence and uniqueness of local solutions to equation (16.1). Assumption 16.1.b) is satisfied whenever F grows at most linearly, i.e., if there are constants c, d > 0 such that F (x) ≤ c x + d. ¯ is positively invariant if the subtangent condition Finally, the set Ω lim inf h↓0
1 1 d(x + hF (x), Ω) = lim inf inf x + hF (x) − z = 0 h↓0 h z∈Ω h
holds for every x ∈ ∂Ω. If Ω is convex, then this is equivalent to the angle condition (F (x)|y) ≤ 0 for x ∈ ∂Ω and y being an outer normal vector to Ω at x (see Figure 16.1).
y x F (x)
Ω
Figure 16.1: The angle condition (y, F (x)) ≥
π 2.
The assumptions imply that there exists a continuous mapping solving the differential equation in (16.1). More precisely, there exists a function ϕ : R+ ×Ω → Ω, which is continuously differentiable in its first variable, satisfying ϕ(0, x) = x ϕ(t, ϕ(s, x)) = ϕ(t + s, x)
for all x ∈ Ω and for all t, s ≥ 0, x ∈ Ω,
(16.2)
16.1. Ordinary Differential Equations and Semiflows
255
such that the solutions of equation (16.1) are given by x(t) = ϕ(t, x0 ). Such a mapping ϕ is called a continuous semiflow . It was the fundamental observation of Koopman and von Neumann that such nonlinear dynamical systems give rise to linear ones. One motivation for this construction is that in many situations we do not see the state space Ω and the dynamics on it, but we only observe so
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