koopman operator theory

**Koopman Operator Theory** is a **mathematical framework that represents nonlinear dynamical systems as linear operators in an infinite-dimensional function space** — enabling the use of powerful linear analysis tools (eigenvalues, modes, spectral decomposition) on inherently nonlinear systems. **What Is the Koopman Operator?** - **Concept**: Instead of tracking the state $x(t)$ directly, track "observables" $g(x(t))$ — functions of the state. - **Linearity**: The Koopman operator $mathcal{K}$ advances observables linearly: $mathcal{K}g(x) = g(f(x))$, even if $f$ is nonlinear. - **Approximation**: In practice, approximate the infinite-dimensional operator using data-driven methods (Dynamic Mode Decomposition, deep learning). **Why It Matters** - **Linear Control**: Once the nonlinear system is "linearized" via Koopman, standard linear control methods apply. - **Process Modeling**: Used in semiconductor manufacturing for modeling plasma etch dynamics and other nonlinear processes. - **Interpretability**: Koopman modes provide physical insight into the dominant dynamics of complex systems. **Koopman Operator Theory** is **seeing nonlinear systems through a linear lens** — a mathematical transformation that makes the intractable tractable.