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### A Bellman approach for two-domains optimal control problems in ℝN

ESAIM: Control, Optimisation and Calculus of Variations

This article is the starting point of a series of works whose aim is the study of deterministic control problems where the dynamic and the running cost can be completely different in two (or more) complementary domains of the space ℝN. As a consequence, the dynamic and running cost present discontinuities at the boundary of these domains and this is the main difficulty of this type of problems. We address these questions by using a Bellman approach: our aim is to investigate how to define properly...

### A boundary value problem for Beltrami differential equation

Banach Center Publications

Solutions to Beltrami differential equation with prescribed boundary correspondence in some plane domains are given.

### A boundary value problem of the Dirichlet type for Hamilton-Jacobi equations

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

### A Cauchy Problem fo Analytic Functionals.

Mathematica Scandinavica

### A continuous finite element method with face penalty to approximate Friedrichs' systems

ESAIM: Mathematical Modelling and Numerical Analysis

A continuous finite element method to approximate Friedrichs' systems is proposed and analyzed. Stability is achieved by penalizing the jumps across mesh interfaces of the normal derivative of some components of the discrete solution. The convergence analysis leads to optimal convergence rates in the graph norm and suboptimal of order ½ convergence rates in the L2-norm. A variant of the method specialized to Friedrichs' systems associated with elliptic PDE's in mixed form and reducing the number...

### A differential inclusion : the case of an isotropic set

ESAIM: Control, Optimisation and Calculus of Variations

In this article we are interested in the following problem: to find a map $u:\Omega \to {ℝ}^{2}$ that satisfies$\left\{\begin{array}{cc}Du\in E\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\hfill & \mathit{\text{a.e.}}\phantom{\rule{4.0pt}{0ex}}\text{in}\phantom{\rule{4.0pt}{0ex}}\Omega \hfill \\ u\left(x\right)=\varphi \left(x\right)\hfill & x\in \partial \Omega \phantom{\rule{85.35826pt}{0ex}}\hfill \end{array}\right\$where $\Omega$ is an open set of ${ℝ}^{2}$ and $E$ is a compact isotropic set of ${ℝ}^{2×2}$. We will show an existence theorem under suitable hypotheses on $\varphi$.

### A differential inclusion: the case of an isotropic set

ESAIM: Control, Optimisation and Calculus of Variations

In this article we are interested in the following problem: to find a map $u:\Omega \to {ℝ}^{2}$ that satisfies $\left\{\begin{array}{cc}Du\in E\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\hfill & \mathit{\text{a.e.}}\phantom{\rule{4.0pt}{0ex}}\text{in}\phantom{\rule{4.0pt}{0ex}}\Omega \hfill \\ u\left(x\right)=\varphi \left(x\right)\hfill & x\in \partial \Omega \hfill \end{array}\right\$ where Ω is an open set of ${ℝ}^{2}$ and E is a compact isotropic set of ${ℝ}^{2×2}$. We will show an existence theorem under suitable hypotheses on φ.

### A direct approach to the Weiss conjecture for bounded analytic semigroups

Czechoslovak Mathematical Journal

We give a new proof of the Weiss conjecture for analytic semigroups. Our approach does not make any recourse to the bounded ${H}^{\infty }$-calculus and is based on elementary analysis.

### A direct proof of the equivalence between the entropy solutions of conservation laws and viscosity solutions of Hamilton-Jacobi equations in one-space variable.

JIPAM. Journal of Inequalities in Pure &amp; Applied Mathematics [electronic only]

### A first order partial differential equation with an integral boundary condition

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

In questo lavoro si considera un’equazione alle derivate parziali del primo ordine con una condizione sulla frontiera di tipo integrale. Si studia resistenza, l'unicità e il comportamento asintotico delle soluzioni.

### A general existence theorem for differential inclusions in the vector valued case.

Portugaliae Mathematica. Nova Série

### A general Hamilton-Jacobi framework for non-linear state-constrained control problems

ESAIM: Control, Optimisation and Calculus of Variations

The paper deals with deterministic optimal control problems with state constraints and non-linear dynamics. It is known for such problems that the value function is in general discontinuous and its characterization by means of a Hamilton-Jacobi equation requires some controllability assumptions involving the dynamics and the set of state constraints. Here, we first adopt the viability point of view and look at the value function as its epigraph. Then, we prove that this epigraph can always be described...

### A Geometric Intrepertation for the Hans Lewy Operator

Δελτίο της Ελληνικής Μαθηματικής Εταιρίας

### A Hamilton-Jacobi approach to junction problems and application to traffic flows

ESAIM: Control, Optimisation and Calculus of Variations

This paper is concerned with the study of a model case of first order Hamilton-Jacobi equations posed on a “junction”, that is to say the union of a finite number of half-lines with a unique common point. The main result is a comparison principle. We also prove existence and stability of solutions. The two challenging difficulties are the singular geometry of the domain and the discontinuity of the Hamiltonian. As far as discontinuous Hamiltonians are concerned, these results seem to be new. They...

### A local version of the generalized retract theorem of Ważewski with applications in the theory of partial differential equations of first order

Annales Polonici Mathematici

### A mathematical model describing cellular division with a proliferating phase duration depending on the maturity of cells.

Electronic Journal of Differential Equations (EJDE) [electronic only]

### A necessary and sufficient condition for the diagonalization of multi-dimensional quasilinear systems.

Electronic Journal of Differential Equations (EJDE) [electronic only]

### A new approach to the existence of almost everywhere solutions of nonlinear PDEs

Archivum Mathematicum

We discuss the existence of almost everywhere solutions of nonlinear PDE’s of first (in the scalar and vectorial cases) and second order.

### A new LMI-based robust finite-time sliding mode control strategy for a class of uncertain nonlinear systems

Kybernetika

This paper presents a novel sliding mode controller for a class of uncertain nonlinear systems. Based on Lyapunov stability theorem and linear matrix inequality technique, a sufficient condition is derived to guarantee the global asymptotical stability of the error dynamics and a linear sliding surface is existed depending on state errors. A new reaching control law is designed to satisfy the presence of the sliding mode around the linear surface in the finite time, and its parameters are obtained...

### A new technique in constructing closed-form solutions for nonlinear PDEs appearing in fluid mechanics and gas dynamics.

Mathematical Problems in Engineering

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