Gene

Quantum mechanics

Control (management)

Nonlinear control

Pure mathematics

Applied mathematics

Biochemistry

Computer science

Dentistry

Dynamic mode decomposition

Physics

Nonlinear system

Calculus (dental)

Mathematics

Transcription factor

Mathematical analysis

Artificial intelligence

Mathematical economics

Dynamical systems theory

Repressor

Medicine

Chemistry

Operator (biology)

Control theory (sociology)

Generalization

Algebra over a field

Machine learning

SIAM Journal on Applied Dynamical Systems

We develop a new generalization of Koopman operator theory that incorporates the effects of inputs and control. Koopman spectral analysis is a theoretical tool for the analysis of nonlinear dynamical systems. Moreover, Koopman is intimately connected to dynamic mode decomposition (DMD), a method that discovers coherent, spatio-temporal modes from data, connects local-linear analysis to nonlinear operator theory, and importantly creates an equation-free architecture for the study of complex systems. For actuated systems, standard Koopman analysis and DMD are incapable of producing input-output models; moreover, the dynamics and the modes will be corrupted by external forcing. Our new theoretical developments extend Koopman operator theory to allow for systems with nonlinear input-output characteristics. We show how this generalization is rigorously connected to a recent development called dynamic mode decomposition with control. We demonstrate this new theory on nonlinear dynamical systems, including a standard susceptible-infectious-recovered model with relevance to the analysis of infectious disease data with mass vaccination (actuation).

Generalizing Koopman Theory to Allow for Inputs and Control