Time optimal control for a reaction diffusion system arising in cardiac electrophysiology – a monolithic approach

Motivated by the termination of undesirable arrhythmia, a time optimal control formulation for the monodomain equations is proposed. It is shown that, under certain conditions, the optimal solutions of this problem steer the system into an appropriate stable neighborhood of the resting state. Towards this goal, some new regularity results and asymptotic properties for the monodomain equations with the Rogers−McCulloch ionic model are obtained. For the numerical realization, a monolithic approach, which simultaneously optimizes for the optimal times and optimal controls, is presented and analyzed. Its practical realization is based on a semismooth Newton method. Numerical examples and comparisons are included

Nonconvex penalization of switching control of partial differential equations

This paper is concerned with optimal control problems for parabolic partial differential equations with pointwise in time switching constraints on the control. A standard approach to treat constraints in nonlinear optimization is penalization, in particular using L1-type norms. Applying this approach to the switching constraint leads to a nonsmooth and nonconvex infinite-dimensional minimization problem which is challenging both analytically and numerically. Adding H1 regularization or restricting to a finite-dimensional control space allows showing existence of optimal controls. First-order necessary optimality conditions are then derived using tools of nonsmooth analysis. Their solution can be computed using a combination of MoreauYosida regularization and a semismooth Newton method. Numerical examples illustrate the properties of this approach.ERC advanced Grant 668998 (OCLOC)(VLID)251239

Magnetic Resonance RF pulse design by optimal control with physical constraints

Optimal control approaches have proved useful in designing RF pulses for large tip-angle applications. A typical challenge for optimal control design is the inclusion of constraints resulting from physiological or technical limitations, that assure the realizability of the optimized pulses. In this work we show how to treat such inequality constraints, in particular, amplitude constraints on the B1 field, the slice-selective gradient and its slew rate, as well as constraints on the slice profile accuracy. For the latter a pointwise profile error and additional phase constraints are prescribed. Here, a penalization method is introduced that corresponds to a higher-order tracking instead of the common quadratic tracking. The order is driven to infinity in the course of the optimization. We jointly optimize for the RF and slice-selective gradient waveform. The amplitude constraints on these control variables are treated efficiently by semismooth Newton or quasi-Newton methods. The method is flexible, adapting to many optimization goals. As an application we reduce the power of refocusing pulses, which is important for spin echo based applications with a short echo spacing. Here, the optimization method is tested in numerical experiments for reducing the pulse power of simultaneous multislice refocusing pulses. The results are validated by phantom and in-vivo experiments. Keywords: RF pulse design, slice-selective, optimal control, physical constraints, inequality constraints.F3209-N1