Time–dependent Closed Loop Control of the Radiative Transfer Equations with Applications

Patient motion during treatment is one of the future challenges in the field of external beam radiotherapy. In this paper, we address this problem by considering a time–dependent Boltzmann transport model for dose calculation and by deriving closed-loop control laws for the treatment planning problem. We formulate an optimal control problem for the de-sired dose using boundary and distributed control and derive optimality conditions. For the construction of the closed-loop control laws we use an inexact variant of model predictive control called instantaneous con-trol. We compare numerical results obtained with instantaneous control to those obtained by optimal open-loop control, and present numerical simulations in one and two spatial dimensions. Key words. Radiative transfer; optimal control; first-order optimality; radiotherapy