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On the Convergence of Adaptive Iterative Linearized Galerkin MethodsMay 16 2019A wide variety of different (fixed-point) iterative methods for the solution of nonlinear equations exists. In this work we will revisit a unified iteration scheme in Hilbert spaces from our previous work that covers some prominent procedures (including ... More

Necessary conditions involving Lie brackets for impulsive optimal control problemsMar 14 2019We obtain higher order necessary conditions for a minimum of a Mayer optimal control problem connected with a nonlinear, control-affine system, where the controls range on an m-dimensional Euclidean space. Since the allowed velocities are unbounded and ... More

A Higher-order Maximum Principle for Impulsive Optimal Control ProblemsMar 12 2019We consider a nonlinear control system, affine with respect to an unbounded control $u$ taking values in a closed cone of $\mathbb{R}^m$, and with drift depending on a second, ordinary control $a$, ranging on a bounded set. We provide first and higher ... More

Finite element error estimates for elliptic optimal control by BV functionsFeb 15 2019We derive a priori error estimates for two discretizations of a PDE-constrained optimal control problem that involves univariate functions of bounded variation as controls. Using, first, variational discretization of the control problem we prove $L^2$-, ... More

A priori error estimates for the optimal control of the integral fractional LaplacianOct 09 2018We design and analyze solution techniques for a linear-quadratic optimal control problem involving the integral fractional Laplacian. We derive existence and uniqueness results, first order optimality conditions, and regularity estimates for the optimal ... More

Adaptive Iterative Linearization Galerkin Methods for Nonlinear ProblemsAug 15 2018Fixed point iterations are widely used for the analysis and numerical treatment of nonlinear problems. In many cases, such schemes can be interpreted as iterative local linearization methods, which, as will be shown, can be obtained by applying a suitable ... More

Multiphase mean curvature flows with high mobility contrasts: a phase-field approach, with applications to nanowiresNov 10 2017The structure of many multiphase systems is governed by an energy that penalizes the area of interfaces between phases weighted by surface tension coefficients. However, interface evolution laws depend also on interface mobility coefficients. Having in ... More

On Some Generalized Polyhedral Convex ConstructionsMay 19 2017Generalized polyhedral convex sets, generalized polyhedral convex functions on locally convex Hausdorff topological vector spaces, and the related constructions such as sum of sets, sum of functions, directional derivative, infimal convolution, normal ... More

Numerical analysis for the pure Neumann control problem using the gradient discretisation methodMay 09 2017Nov 30 2017The article discusses the gradient discretisation method (GDM) for distributed optimal control problems governed by diffusion equation with pure Neumann boundary condition. Using the GDM framework enables to develop an analysis that directly applies to ... More

On the existence of minimisers for strain-gradient single-crystal plasticityJan 04 2017We prove the existence of minimisers for a family of models related to the single-slip-to-single-plane relaxation of single-crystal, strain-gradient elastoplasticity with $L^p$-hardening penalty. In these relaxed models, where only one slip-plane normal ... More

Gradient schemes for optimal control problems, with super-convergence for non-conforming finite elements and mixed-hybrid mimetic finite differencesAug 05 2016Oct 29 2016In this paper, optimal control problems governed by diffusion equations with Dirichlet and Neumann boundary conditions are investigated in the framework of gradient schemes. A gradient scheme is defined for the optimality system of the control problem. ... More

Gradient schemes for optimal control problems, with super-convergence for non-conforming finite elements and mixed-hybrid mimetic finite differencesAug 05 2016In this paper, optimal control problems governed by diffusion equations with Dirichlet and Neumann boundary conditions are investigated in the framework of gradient schemes. A gradient scheme is defined for the optimality system of the control problem. ... More

The gradient discretisation method for optimal control problems, with super-convergence for non-conforming finite elements and mixed-hybrid mimetic finite differencesAug 05 2016Nov 14 2017In this paper, optimal control problems governed by diffusion equations with Dirichlet and Neumann boundary conditions are investigated in the framework of the gradient discretisation method. Gradient schemes are defined for the optimality system of the ... More

Existence of Self-Cheeger sets on Riemannian manifoldsMar 01 2016Aug 08 2017Let $(\mathcal{M},g)$ be a compact Riemannian manifold of dimension $N\geq 2$. We prove the existence of a family $(\Omega_\varepsilon)_{\varepsilon\in (0,\varepsilon_0)}$ of self-Cheeger sets in $(\mathcal{M},g)$ . The domains $\Omega_\varepsilon\subset\mathcal{M}$ ... More

Numerical analysis of a family of optimal distributed control problems governed by an elliptic variational inequalityDec 31 2015The numerical analysis of a family of distributed mixed optimal control problems governed by elliptic variational inequalities (with parameter $\alpha >0$) is obtained through the finite element method when its parameter $h\rightarrow 0$. We also obtain ... More

Shape optimization for an elliptic operator with infinitely many positive and negative eigenvaluesDec 15 2015The paper deals with an eigenvalue problems possessing infinitely many positive and negative eigenvalues. Inequalities for the smallest positive and the largest negative eigenvalues, which have the same properties as the fundamental frequency, are derived. ... More

Double convergence of a family of discrete distributed mixed elliptic optimal control problems with a parameterDec 11 2015We consider a bounded domain $\Omega$ in $\mathbb{R}^{n}$ whose regular boundary $\partial\Omega$ consists of the union of two disjoint portions $\Gamma_{1}$ and $\Gamma_{2}$ with $meas(\Gamma_{1})>0$. The convergence of a family of continuous distributed ... More

Finite element approximation of the parabolic fractional obstacle problemJul 07 2015We study a discretization technique for the parabolic fractional obstacle problem in bounded domains. The fractional Laplacian is realized as the Dirichlet-to-Neumann map for a nonuniformly elliptic equation posed on a semi-infinite cylinder, which recasts ... More

Control of ordinary differential equations using Bagarello's operator approach : Case of forced harmonic oscillator systemsMar 12 2015Aug 22 2015This work deals with the study of an optimal control of a system of nonlinear differential equations using the Bagarello's operator approach, recently introduced in a paper (Int. Jour. of Theoretical Physics, 43, issue 12 (2004), p. 2371 - 2394). The ... More

Numerical analysis of distributed optimal control problems governed by elliptic variational inequalitiesDec 19 2014May 15 2015A continuous optimal control problem governed by an elliptic variational inequality was considered in Boukrouche-Tarzia, Comput. Optim. Appl., 53 (2012), 375-392 where the control variable is the internal energy $g$. It was proved the existence and uniqueness ... More

A commutative diagram among discrete and continuous Neumann boundary optimal control problemsDec 19 2014We consider a bounded domain D whose regular boundary consists of the union of two portions F1 and F2. The convergence of a family of continuous Neumann boundary mixed elliptic optimal control problems (Pa), governed by elliptic variational equalities, ... More

Numerical studies of the optimization of the first eigenvalue of the heat diffusion in inhomogeneous mediaAug 22 2014Apr 22 2015In this paper, we study optimization of the first eigenvalue of the heat equation with spatially nonuniform conductivity on a bounded domain under several constraints for the conductivity. We consider this problem in various boundary conditions and various ... More

Domain perturbations for elliptic problems with Robin boundary conditions of opposite signJun 12 2014May 06 2015We consider the energy of the torsion problem with Robin boundary conditions in the case where the solution is not a minimizer. Its dependence on the volume of the domain and the surface area of the boundary is discussed. In contrast to the case of positive ... More

Maximization of Laplace-Beltrami eigenvalues on closed Riemannian surfacesMay 20 2014Mar 25 2016Let $(M,g)$ be a connected, closed, orientable Riemannian surface and denote by $\lambda_k(M,g)$ the $k$-th eigenvalue of the Laplace-Beltrami operator on $(M,g)$. In this paper, we consider the mapping $(M, g)\mapsto \lambda_k(M,g)$. We propose a computational ... More

Isoperimetric inequalities for the principal eigenvalue of a membrane and the energy of problems with Robin boundary conditionsMar 13 2014Mar 20 2014An inequality for the reverse Bossel-Daners inequality is derived by means of the harmonic transplantation and the first shape derivative. This method is then applied to elliptic boundary value problems with inhomogeneous Neumann conditions. The first ... More

Second variation of domain functionals and applications to problems with Robin boundary conditionsMar 10 2014Jul 10 2015In this paper the first and second domain variation for functionals related to elliptic boundary and eigenvalue problems with Robin boundary conditions is computed. Minimality and maximality properties of the ball among nearly circular domains of given ... More

Fre'chet Generalized Trajectories and Minimizers for Variational Problems of Low CoercivityFeb 03 2014We address consecutively two problems. First we introduce a class of so called Fre'chet generalized controls for a multi-input control-affine system with non-commuting controlled vector fields. For each control of the class one is able to define a unique ... More

Duality in convex problems of Bolza over functions of bounded variationSep 09 2013This paper studies convex problems of Bolza in the conjugate duality framework of Rockafellar. We parameterize the problem by a general Borel measure which has direct economic interpretation in problems of financial economics. We derive a dual representation ... More

Conditions for zero duality gap in convex programmingNov 21 2012Apr 28 2013We introduce and study a new dual condition which characterizes zero duality gap in nonsmooth convex optimization. We prove that our condition is weaker than all existing constraint qualifications, including the closed epigraph condition. Our dual condition ... More

A Total Variation Diminishing Interpolation Operator and ApplicationsNov 05 2012We construct, on continuous $Q_1$ finite elements over Cartesian meshes, an interpolation operator that does not increase the total variation. The operator is stable in $L^1$ and exhibits second order approximation properties. With the help of it we provide ... More

Asymptotic behaviour and numerical approximation of optimal eigenvalues of the Robin LaplacianApr 03 2012We consider the problem of minimising the $n^{th}-$eigenvalue of the Robin Laplacian in $\mathbb{R}^{N}$. Although for $n=1,2$ and a positive boundary parameter $\alpha$ it is known that the minimisers do not depend on $\alpha$, we demonstrate numerically ... More

Attouch-Théra duality revisited: paramonotonicity and operator splittingOct 21 2011The problem of finding the zeros of the sum of two maximally monotone operators is of fundamental importance in optimization and variational analysis. In this paper, we systematically study Attouch-Th\'era duality for this problem. We provide new results ... More