### An Extremal Problem Involving Functions and Their Inverses.

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This paper presents a new observability estimate for parabolic equations in $\Omega \times (0,T)$, where $\Omega $ is a convex domain. The observation region is restricted over a product set of an open nonempty subset of $\Omega $ and a subset of positive measure in $(0,T)$. This estimate is derived with the aid of a quantitative unique continuation at one point in time. Applications to the bang-bang property for norm and time optimal control problems are provided.

In this paper, we introduce an iterative algorithm for finding a common element of the set of solutions of a mixed equilibrium problem, the set of fixed points of a nonexpansive mapping, and the the set of solutions of a variational inclusion in a real Hilbert space. Furthermore, we prove that the proposed iterative algorithm converges strongly to a common element of the above three sets, which is a solution of a certain optimization problem related to a strongly positive bounded linear operator....

For optimal control problems with ordinary differential equations where the ${L}^{\infty}$-norm of the control is minimized, often bang-bang principles hold. For systems that are governed by a hyperbolic partial differential equation, the situation is different: even if a weak form of the bang-bang principle still holds for the wave equation, it implies no restriction on the form of the optimal control. To illustrate that for the Dirichlet boundary control of the wave equation in general not even feasible...

For a class of 2-D elastic energies we show that a radial equilibrium solution is the unique global minimizer in a subclass of all admissible maps. The boundary constraint is a double cover of ${S}^{1}$; the minimizer $u$ is ${C}^{1}$ and is such that $det\nabla u$ vanishes at one point.

For a class of 2-D elastic energies we show that a radial equilibrium solution is the unique global minimizer in a subclass of all admissible maps. The boundary constraint is a double cover of S1; the minimizer u is C1 and is such that $det\nabla u$ vanishes at one point.

We apply four different methods to study an intrinsically bang-bang optimal control problem. We study first a relaxed problem that we solve with a naive nonlinear programming approach. Since these preliminary results reveal singular arcs, we then use Pontryagin’s Minimum Principle and apply multiple indirect shooting methods combined with homotopy approach to obtain an accurate solution of the relaxed problem. Finally, in order to recover a purely bang-bang solution for the original problem, we...

We deal with the problems of four boundary points conditions for both differential inclusions and differential equations with and without moving constraints. Using a very recent result we prove existence of generalized solutions for some differential inclusions and some differential equations with moving constraints. The results obtained improve the recent results obtained by Papageorgiou and Ibrahim-Gomaa. Also by means of a rather different approach based on an existence theorem due to O. N. Ricceri...

This paper is a proceedings version of the ongoing work [20], and has been the object of the talk of the second author at Journées EDP in 2012.In this work we investigate optimal observability properties for wave and Schrödinger equations considered in a bounded open set $\Omega \subset {\mathbb{R}}^{n}$, with Dirichlet boundary conditions. The observation is done on a subset $\omega $ of Lebesgue measure $\left|\omega \right|=L\left|\Omega \right|$, where $L\in (0,1)$ is fixed. We denote by ${\mathcal{U}}_{L}$ the class of all possible such subsets. Let $T\>0$. We consider first the benchmark problem of maximizing...