We prove a general relaxation theorem for multidimensional control problems of Dieudonné-Rashevsky type with nonconvex integrands (, , ) in presence of a convex control restriction. The relaxed problem, wherein the integrand has been replaced by its lower semicontinuous quasiconvex envelope with respect to the gradient variable, possesses the same finite minimal value as the original problem, and admits a global minimizer. As an application, we provide existence theorems for the image registration...

For a multidimensional control problem ${\left(P\right)}_{K}$ involving controls $u\in {L}_{\infty}$, we construct a dual problem ${\left(D\right)}_{K}$ in which the variables ν to be paired with u are taken from the measure space rca (Ω,) instead of $\left({L}_{\infty}\right)*$. For this purpose, we add to ${\left(P\right)}_{K}$ a Baire class restriction for the representatives of the controls u. As main results, we prove a strong duality theorem and saddle-point conditions.

We prove a general relaxation theorem for multidimensional control problems of Dieudonné-Rashevsky type with nonconvex integrands (, , ) in presence of a convex control restriction. The relaxed problem, wherein the integrand has been replaced by its lower semicontinuous quasiconvex envelope with respect to the gradient variable, possesses the same finite minimal value as the original problem, and admits a global minimizer. As an application, we provide existence theorems for the image registration...

Motivated by the study of multidimensional control problems of Dieudonné-Rashevsky type, we raise the question how to understand to notion of quasiconvexity for a continuous function with a convex body K $\subset {\mathbb{R}}^{nm}$ instead of the whole space ${\mathbb{R}}^{nm}$ as the range of definition. In the present paper, we trace the consequences of an infinite extension of outside K, and thus study quasiconvex functions which are allowed to take the value +∞. As an appropriate envelope, we introduce and investigate the lower semicontinuous...

We consider optimal control problems for the bidomain equations of cardiac electrophysiology together with two-variable ionic models, the Rogers–McCulloch model. After ensuring the existence of global minimizers, we provide a rigorous proof for the system of first-order necessary optimality conditions. The proof is based on a stability estimate for the primal equations and an existence theorem for weak solutions of the adjoint system.

We present algorithms for the computation of extreme binary Humbert forms in real quadratic number fields. With these algorithms we are able to compute extreme Humbert forms for the number fields $\mathbb{Q}\left(\sqrt{13}\right)$ and $\mathbb{Q}\left(\sqrt{17}\right)$. Finally we compute the Hermite-Humbert constant for the number field $\mathbb{Q}\left(\sqrt{13}\right)$.

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