Parametric Koopman
Parametric Dynamical Systems¶
Let \((X, \mathcal{S}, m)\), with \(X \subseteq \mathbb{R}^{N_x}\) be a finite measure space where \(\mathcal{S}\) is the Borel \(\sigma\)-algebra and \(m\) is a measure on \((X, \mathcal{S})\). Consider a set \(U \subseteq \mathbb{R}^{N_u}\) of parameters, and the parametric discrete-time dynamics
\[ \mathbf{x}_{n+1} = \mathbf{f}(\mathbf{x}_n, \mathbf{u}), \quad n = 0,1,\cdots \]
where \(\mathbf{u} \in U\) remains static, or control systems
\[ \mathbf{x}_{n+1} = \mathbf{f}(\mathbf{x}_n, \mathbf{u}_n), \quad n = 0,1,\cdots, \]
where \(\mathbf{u}_n \in U\) changes in discrete steps dynamically.
Parametric Koopman Operator¶
Definition. Consider the Hilber space
\[ L^2(X, m) = \left\{ \phi: X \rightarrow \mathbb{R}: \|\phi\|_{L^2(X, m)} < \infty \right\}. \]
An element \(\phi \in L^2(X, m)\) is called an observable.
Definition. The parametric Koopman operator maps an observable \(\phi \in L^2(X, m)\) to another observable \(\mathcal{K}\phi\) defined by:
\[ \mathcal{K}(\mathbf{u}) \phi(\mathbf{x}) := \phi \circ \mathbf{f}(\mathbf{x}, \mathbf{u}). \]