Measure valued optimal control problems governed by the linear wave equation are analyzed. The space of vector measures M(Ωc, L 2 (I)) is chosen as control space and the corresponding total variation norm as the control cost functional. The support of the controls (sparsity pattern) is time-independent, which is desired in many applications, e.g., inverse problems or optimal actuator placement. New regularity results for the linear wave equation are proven and used to show the well-posedness of the control problem in all three space dimensions. Furthermore first order optimality conditions are derived and structural properties of the optimal control are investigated. Higher regularity of optimal controls in time is shown on the basis of the regularity results for the state. Finally the optimal control problem is used to solve an inverse source problem.

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In this study we propose a novel method for identifying the locations of earliest activation in the human left ventricle from activation maps measured at the epicardial surface. Electrical activation is modeled based on the viscous Eikonal equation. The sites of earliest activation are identified by solving a minimization problem. Arbitrary initial locations are assumed, which are then modified based on a shape derivative based perturbation field until a minimal mismatch between the computed and the given activation maps on the epicardial surface is achieved. The proposed method is tested in two numerical benchmarks, a generic 2D unit-square benchmark, and an anatomically accurate MRI-derived 3D human left ventricle benchmark to demonstrate potential utility in a clinical context. For unperturbed input data, our localization method is able to accurately reconstruct the earliest activation sites in both benchmarks with deviations of only a fraction of the used spatial discretization size. Further, with the quality of the input data reduced by spatial undersampling and addition of noise, we demonstrate that an accurate identification of the sites of earliest activation is still feasible.

This work is concerned with the optimal control problems governed by a 1D wave equation with variable coefficients and the control spaces M T of either measure-valued functions L 2 w * (I, M(Ω)) or vector measures M(Ω, L 2 (I)). The cost functional involves the standard quadratic tracking terms and the regularization term α u M T with α > 0. We construct and study three-level in time bilinear finite element discretizations for this class of problems. The main focus lies on the derivation of error estimates for the optimal state variable and the error measured in the cost functional. The analysis is mainly based on some previous results of the authors. The numerical results are included.

In this work we consider measure-valued optimal control problems involving the KdV-Burgers (KdVB) equation as a state equation. These optimal control problems are motivated by an inverse problem and a control problem involving the flow of water in a channel over topography. Well-posedness of the optimal control problem is established which involves the investigation of the KdVB equation for irregular source terms. Moreover optimality conditions for the control problem are derived. Efficient numerical schemes based on spectral methods are proposed for the state and adjoint equation, as well as adequate optimization methods. The theoretical results are illustrated by two numerical examples.