Acoustic control of nonspherical shape oscillations in microbubbles using a Koopman linear quadratic regulator

Large-amplitude oscillations of spherical bubbles are known to be destabilizing and to induce nonspherical shape dynamics. Acoustic driving of microbubbles can often reach this threshold for excitation, whether the bubbles are in extended free fluid media or near boundaries. In many medical applications, such as diagnostic ultrasound and therapeutic drug delivery, nonspherical shape oscillations produce important secondary effects, such as microstreaming, jetting, or acoustic backscatter. The active control of these shape oscillations is therefore an important open problem. This work continues an ongoing application of Koopman operator theory to the acoustic control of bubble dynamics using a Koopman linear quadratic regulator (KLQR). We drive an uncoated, nonspherical bubble to follow two types of targets: (1) simple harmonic oscillations at prescribed frequencies in individual shape modes, (2) destabilizing nonspherical shape oscillations to promote bubble rupture at a prescribed time. Compared to previous work by the authors, the KLQR controller is modified to use Koopman eigenfunctions defined in terms of the observables of the bubble, rather than the state of the bubble. This vastly generalizes the control framework and allows for more flexible target specification.