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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.

Stabilization of the nonlinear Korteweg-de Vries (KdV) equation on a bounded interval by model predictive control (MPC) is investigated. This MPC strategy does not need any terminal cost or terminal constraint to guarantee the stability. The semi-global stabilizability is the key condition. Based on this condition, the suboptimality and exponential stability of the model predictive control are investigated. Finally, numerical experiment is presented which validates the theoretical results.

Using a projection-based decoupling of the Fokker−Planck equation, control strategies that allow to speed up the convergence to the stationary distribution are investigated. By means of an operator theoretic framework for a bilinear control system, two different feedback control laws are proposed. Projected Riccati and Lyapunov equations are derived and properties of the associated solutions are given. The well-posedness of the closed loop systems is shown and local and global stabilization results, respectively, are obtained. An essential tool in the construction of the controls is the choice of appropriate control shape functions. Results for a two dimensional double well potential illustrate the theoretical findings in a numerical setup.

The problem of finite-dimensional compensator design for the monodomain equations is investigated. Exponential stabilizability and detectability of the linearized infinite-dimensional control system is studied. It is shown that the system is not exactly null-controllable but still can be exponentially stabilized by finite-rank input and output operators provided the desired stability margin is small enough. Based on existing results on model order reduction of infinite-dimensional systems, a finite-dimensional compensator is obtained by LQG-balanced truncation. The compensator is shown to be locally stabilizing for the infinite-dimensional nonlinear control system. Examples motivated by cardiophysiology are used to illustrate these results in a numerical setup.

Time optimal control problems for an internally controlled heat equation with pointwisecontrol constraints are studied. By Pontryagin’s maximum principle and properties of nontrivialsolutions of the heat equation, we derive a bang-bang property for time optimal control. Using thebang-bang property and establishing certain connections between time and norm optimal control problemsfor the heat equation, necessary and sufficient conditions for the optimal time and the optimalcontrol are obtained.

An approximation procedure for time optimal control problems for the linear wave equationis analyzed. Its asymptotic behavior is investigated and an optimality system including the maximumprinciple and the transversality conditions for the regularized and unregularized problems are derived.

Optimal control problems involving hybrid binary-continuous control costs are challenging due to their lack of convexity and weak lower semicontinuity. Replacing such costs with their convex relaxation leads to a primal-dual optimality system that allows an explicit pointwise characterization and whose MoreauYosida regularization is amenable to a semismooth Newton method in function space. This approach is especially suited for computing switching controls for partial differential equations. In this case, the optimality gap between the original functional and its relaxation can be estimated and shown to be zero for controls with switching structure. Numerical examples illustrate the effectiveness of this approach.

Optimal control problems in measure spaces governed by parabolic equations with are considered. The controls appear as spatial measure in the initial condition and as space-time measures as forcing functions. First order optimality conditions are derived and certain structural properties, in particular sparsity, are discussed. An framework for approximation if these highly irregular problems is also proposed.

This work is concerned with optimal control of partial differential equations where the control enters the state equation as a coefficient and should take on values only from a given discrete set of values corresponding to available materials. A "multibang" framework based on convex analysis is proposed where the desired piecewise constant structure is incorporated using a convex penalty term. Together with a suitable tracking term, this allows formulating the problemof optimizing the topology of the distribution of material parameters as minimizing a convex functional subject to a (nonlinear) equality constraint. The applicability of this approach is validated for two model problems where the control enters as a potential and a diffusion coefficient, respectively. This is illustrated in both cases by numerical results based on a semismooth Newton method.