Application of Koopman LQR to the control of nonlinear bubble dynamics

Koopman operator theory has gained interest in the past decade as a framework for analyzing nonlinear dynamics by embedding in an infinite-dimensional function space. This enables the use of linear control and estimation methods for strongly nonlinear systems. Recently, the linear quadratic regulator (LQR) problem for nonlinear dynamics was formulated in Koopman eigenfunction coordinates and its feasibility demonstrated; however, applications were limited to controlling the Hamiltonian for conservative systems. Here, we extend this framework to use multiple complex eigenfunctions and illustrate its enhanced power by driving several classical nonlinear oscillators to follow arbitrary, prescribed trajectories. We then control nonlinear bubble dynamics, as described by the well-known Rayleigh-Plesset equation, with two novel objectives: 1) stabilization of the bubble at a nonequilibrium radius, and 2) simple harmonic oscillation at amplitudes large enough to incite nonlinearities. Control is implemented through a single-frequency transducer whose amplitude is modulated by the Koopman LQR controller. This work is a step towards controlling nonspherical shape modes of encapsulated microbubbles, which has applications in biomedicine for ultrasound imaging and intravenous drug delivery.