### A free boundary problem with a volume penalization

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We present below a new series of conjectures and open problems in the fields of (global) Optimization and Matrix analysis, in the same spirit as our recently published paper [J.-B. Hiriart-Urruty, Potpourri of conjectures and open questions in Nonlinear analysis and Optimization. SIAM Review 49 (2007) 255–273]. With each problem come a succinct presentation, a list of specific references, and a view on the state of the art of the subject.

We present below a new series of conjectures and open problems in the fields of (global) Optimization and Matrix analysis, in the same spirit as our recently published paper [J.-B. Hiriart-Urruty, Potpourri of conjectures and open questions in Nonlinear analysis and Optimization. SIAM Review49 (2007) 255–273]. With each problem come a succinct presentation, a list of specific references, and a view on the state of the art of the subject.

We consider the integral functional $J\left(u\right)={\int}_{\Omega}\left[f\right(\left|Du\right|\left)-u\right]dx$, $u\in {W}_{0}^{1,1}\left(\Omega \right)$, where $\Omega \subset {\mathbb{R}}^{n}$, $n\ge 2$, is a nonempty bounded connected open subset of ${\mathbb{R}}^{n}$ with smooth boundary, and $\mathbb{R}\ni s\mapsto f\left(\right|s\left|\right)$ is a convex, differentiable function. We prove that if $J$ admits a minimizer in ${W}_{0}^{1,1}\left(\Omega \right)$ depending only on the distance from the boundary of $\Omega $, then $\Omega $ must be a ball.

We propose a variational model to describe the optimal distributions of residents and services in an urban area. The functional to be minimized involves an overall transportation cost taking into account congestion effects and two aditional terms which penalize concentration of residents and dispersion of services. We study regularity properties of the minimizers and treat in details some examples.

Integral functionals based on convex normal integrands are minimized subject to finitely many moment constraints. The integrands are finite on the positive and infinite on the negative numbers, strictly convex but not necessarily differentiable. The minimization is viewed as a primal problem and studied together with a dual one in the framework of convex duality. The effective domain of the value function is described by a conic core, a modification of the earlier concept of convex core. Minimizers...

This paper concerns an obstacle control problem for an elastic (homogeneous) and isotropic) pseudoplate. The state problem is modelled by a coercive variational inequality, where control variable enters the coefficients of the linear operator. Here, the role of control variable is played by the thickness of the pseudoplate which need not belong to the set of continuous functions. Since in general problems of control in coefficients have no optimal solution, a class of the extended optimal control...

In this paper we study the compact and convex sets K ⊆ Ω ⊆ ℝ2that minimize$${\int}_{\Omega}(,K)\phantom{\rule{0.166667em}{0ex}}\mathrm{d}+{\lambda}_{1}\mathrm{Vol}\left(K\right)+{\lambda}_{2}\mathrm{Per}\left(K\right)$$∫ Ω dist ( x ,K ) d x + λ 1 Vol ( K ) + λ 2 Per ( K ) for some constantsλ1 and λ2, that could possibly be zero. We compute in particular the second order derivative of the functional and use it to exclude smooth points of positive curvature for the problem with volume constraint. The problem with perimeter constraint behaves differently since polygons are never minimizers. Finally using a purely geometrical argument from...

We consider a general class of mathematical models P for cancer chemotherapy described as optimal control problems over a fixed horizon with dynamics given by a bilinear system and an objective which is linear in the control. Several two- and three-compartment models considered earlier fall into this class. While a killing agent which is active during cell division constitutes the only control considered in the two-compartment model, Model A, also two three-compartment models, Models B and C, are...

The motivation for this work is the real-time solution of a standard optimal control problem arising in robotics and aerospace applications. For example, the trajectory planning problem for air vehicles is naturally cast as an optimal control problem on the tangent bundle of the Lie Group $SE\left(3\right),$ which is also a parallelizable riemannian manifold. For an optimal control problem on the tangent bundle of such a manifold, we use frame co-ordinates and obtain first-order necessary conditions employing calculus...