This paper is devoted to the study of a one-dimensional optimal control problem of Lagrange type under periodic constraints. The system is governed by a Kolmogorov equation including a control term. We show that the set of initial conditions for which there exists a control such that the associated trajectory is periodic is an interval. Thanks to Pontryagin Maximum Principle, we characterize op...

Then minimization may be performed on the interval [to, tl] only, where it is understood that x and u are periodically extended to [to, co). These periodic extensions satisfy the system equation (l.l), since x, is a complete state for this functional differential system. Hence it is sufficient to consider the optimal periodic control problem (l.l), (1.2) on [to, tI] with “mixed” boundary condit...

The unidirectional acoustic boundary has a broad application prospect. Besides the acoustic metamaterial or the strongly nonlinear acoustic medium, which were applied in the realization of the unidirectional acoustic boundary, the present paper proposes another approach through the active control method, where the control source has the property of single directivity. In order to validate its e...

This paper studies the local exact controllability and the local stabilization of the semilinear Schrödinger equation posed on a product of n intervals (n ≥ 1). Both internal and boundary controls are considered, and the results are given with periodic (resp. Dirichlet or Neumann) boundary conditions. In the case of internal control, we obtain local controllability results which are sharp as fa...

The general problem under consideration in this paper is the stability analysis of hyperbolic systems. Some sufficient criteria on the boundary conditions exist for the stability of a system of conservation laws. We investigate the problem of the stability of such a system in presence of boundary errors that have a small C1-norm. Two types of perturbations are considered in this work: the error...

This article is devoted to analyze control properties for the heat equation with singular potential −μ/|x| arising at the boundary of a smooth domain Ω ⊂ R , N ≥ 1. This problem was firstly studied by Vancostenoble and Zuazua [18] and then generalized by Ervedoza [8] in the context of interior singularity. Roughly speaking, these results showed that for any value of parameters μ ≤ μ(N) := (N − ...

We present some results concerning the controllability of a quasi-linear parabolic equation (with linear principal part) in a bounded domain of RN with Dirichlet boundary conditions. We analyze the controllability problem with distributed controls (supported on a small open subset) and boundary controls (supported on a small part of the boundary). We prove that the system is null and approximat...

This paper concerns second order sufficient conditions of optimality, involving the Riccati equation, for optimal control problems with periodic boundary conditions. The problems considered involve no pathwise constraints and are ‘regular’, in the sense that the strengthened Legendre-Clebsch condition is assumed to be satisfied. A well-known sufficient condition, which we refer to as the Riccat...

We study a 2D and a 3D incompressible magnetohydrodynamic (MHD) channel flow, also known as Hartmann flow, which is electrically conducting and subject to an external transverse magnetic field. We consider periodic boundary condition along one axis (two axis, for the three-dimensional case). We consider the flow at low magnetic Reynolds number and obtain the so-called simplified magnetohydrodyn...

In this article, we study the large time behavior of solutions of first-order Hamilton-Jacobi Equations, set in a bounded domain with nonlinear Neumann boundary conditions, including the case of dynamical boundary conditions. We establish general convergence results for viscosity solutions of these Cauchy-Neumann problems by using two fairly different methods : the first one relies only on part...