The notion of quasiconvex exposed points is introduced for compact sets of matrices, motivated from the variational approach to material microstructures. We apply the notion to give geometric descriptions of the quasiconvex extreme points for a compact set. A weak version of Straszewicz type density theorem in convex analysis is established for quasiconvex extreme points. Some examples are examined by using known explicit quasiconvex functions.

In this paper we consider an approximate controllability problem for linear parabolic equations with rapidly oscillating coefficients in a periodically perforated domain. The holes are ε-periodic and of size ε. We show that, as ε → 0, the approximate control and the corresponding solution converge respectively to the approximate control and to the solution of the homogenized problem. In the limit problem, the approximation of the final state is alterated by a constant which depends on the proportion of material in the perforated domain and is equal to 1 when there are no holes. We also prove that the solution of the approximate controllability problem in the perforated domain behaves, as ε → 0, as that of the problem posed in the perforated domain having as rigth-hand side the (fixed) control of the limit problem.

We study the local exact controllability problem for the Navier-Stokes equations that describe an incompressible fluid flow in a bounded domain Ω with control distributed in a subdomain $\omega\subset\Omega\subset \mathbb{R}^n, n\in\{2,3\}$. The result that we obtained in this paper is as follows. Suppose that $\hat v(t,x)$ is a given solution of the Navier-Stokes equations. Let $ v_0(x)$ be a given initial condition and $\Vert \hat v(0,\cdot) - v_0 \Vert < \varepsilon$ where ε is small enough. Then there exists a locally distributed control $u, \text{supp}\, u\subset (0,T)\times \omega $ such that the solution v(t,x) of the Navier-Stokes equations: $$ \partial_tv-\Delta v+(v,\nabla)v=\nabla p+u+f, \,\, \text{\rm div}\, v=0,\,\, v\vert_{\partial\Omega}=0, \,\, v \vert_{t=0} = v_0 $$ coincides with $\hat v(t,x)$ at the instant T : $v(T,x) \equiv \hat v(T,x)$.

We consider control problems governed by semilinear parabolic equations with pointwise state constraints and controls in an Lp-space (p < ∞). We construct a correct relaxed problem, prove some relaxation results, and derive necessary optimality conditions.

The purpose of this paper is to study a control by coefficients problem issued from the elastohydrodynamic lubrication. The control variable is the film thickness.The cavitation phenomenon takes place and described by the Elrod-Adams model, suggested in preference to the classical variational inequality due to its ability to describe input and output flow. The idea is to use the penalization in the state equation  by approximating the Heaviside graph whith a sequence of monotone and regular functions. We derive a necessary condition for the regularized problem,  then we establish estimates of the state and the adjoint state in the one dimensional case. Next we pass to the limit.

A Goursat structure on a manifold of dimension n is a rank two distribution Ɗ such that dim Ɗ(i) = i + 2, for 0 ≤ i ≤ n-2, where Ɗ(i) denote the elements of the derived flag of Ɗ, defined by Ɗ(0) = Ɗ and Ɗ(i+1) = Ɗ(i) + [Ɗ(i),Ɗ(i)] . Goursat structures appeared first in the work of von Weber and Cartan, who have shown that on an open and dense subset they can be converted into the so-called Goursat normal form. Later, Goursat structures have been studied by Kumpera and Ruiz. In the paper, we introduce a new local invariant for Goursat structures, called the singularity type, and prove that the growth vector and the abnormal curves of all elements of the derived flag are determined by this invariant. We provide a detailed analysis of all abnormal and rigid curves of Goursat structures. We show that neither abnormal curves, if n ≥ 6, nor abnormal curves of all elements of the derived flag, if n ≥ 9, determine the local equivalence class of a Goursat structure. The latter observation is deduced from a generalized version of Bäcklund's theorem. We also propose a new proof of a classical theorem of Kumpera and Ruiz. All results are illustrated by the n-trailer system, which, as we show, turns out to be a universal model for all local Goursat structures.

We consider the exact controllability of a hybridsystem consisting of an elastic beam, clamped at one end and attached at the other end to arigid antenna. Such a system is governed by one partialdifferential equation and two ordinary differential equations. Using theHUM method, we prove that the hybrid system is exactlycontrollable in an arbitrarily short time in the usual energy space.

We consider an energy-functional describing rotating superfluids at a rotating velocity ω, and prove similar results as for the Ginzburg-Landau functional of superconductivity: mainly the existence of branches of solutions with vortices, the existence of a critical ω above which energy-minimizers have vortices, evaluations of the minimal energy as a function of ω, and the derivation of a limiting free-boundary problem.

We investigate the minimum time transfer of asatellite around the Earth. Using an optimal control model, we studythe controllability of the system and propose a geometrical analysisof the optimal command structure. Furthermore, in order to solve theproblem numerically, a new parametric technique is introduced forwhich convergence properties are established.

We study the simultaneously reachable subspace for two strings controlled from a common endpoint. We give necessary and sufficient conditions for simultaneous spectral and approximate controllability. Moreover we prove the lack of simultaneous exact controllability and we study the space of simultaneously reachable states as a function of the position of the joint. For each type of controllability result we give the sharp controllability time.

This paper studies the exact controllability of the Maxwell system in a bounded domain, controlled by a current flowing tangentially in the boundary of the region, as well as the exact controllability the same problem but perturbed by a dissipative term multiplied by a small parameter in the boundary condition. This boundary perturbation improves the regularity of the problem and is therefore a singular perturbation of the original problem. The purpose of the paper is to examine the connection, for small values of the perturbation parameter, between observability estimates for the two systems, and between the optimality systems corresponding to the problem of norm minimum exact control of the solutions of the two systems from the rest state to a specified terminal state.

We consider a finite-dimensional control system $(\Sigma)\ \ \dot x(t)=f(x(t),u(t))$, such that there exists a feedback stabilizer k that renders $\dot x=f(x,k(x))$ globally asymptotically stable. Moreover, for (H,p,q) with H an output map and $1\leq p\leq q\leq \infty$, we assume that there exists a ${\cal {K}}_{\infty}$-function α such that $\|H(x_u)\|_q\leq \alpha(\|u\|_p)$, where xu is the maximal solution of $(\Sigma)_k \ \ \dot x(t)=f(x(t),k(x(t))+u(t))$, corresponding to u and to the initial condition x(0)=0. Then, the gain function $G_{(H,p,q)}$ of (H,p,q) given by 14.5cm $$ G_{(H,p,q)}(X)\stackrel{\rm def}{=}\sup_{\|u\|_p=X}\|H(x_u)\|_q, $$ is well-defined. We call profile of k for (H,p,q) any ${\cal {K}}_{\infty}$-function which is of the same order of magnitude as $G_{(H,p,q)}$. For the double integrator subject to input saturation and stabilized by $k_L(x)=-(1\ 1)^Tx$, we determine the profiles corresponding to the main output maps. In particular, if $\sigma_0$ is used to denote the standard saturation function, we show that the L2-gain from the output of the saturation nonlinearity to u of the system $\ddot x=\sigma_0(-x-\dot x+u)$ with $x(0)= \dot x(0)=0$, is finite. We also provide a class of feedback stabilizers kF that have a linear profile for (x,p,p), $1\leq p\leq \infty$. For instance, we show that the L2-gains from x and $\dot x$ to u of the system $\ddot x=\sigma_0(-x-\dot x-(\dot x)^3+u)$ with $x(0)= \dot x(0)=0$, are finite.

Consider a Timoshenko beam that is clamped to an axis perpendicular to the axis of the beam. We study the problem to move the beam from a given initial state to a position of rest, where the movement is controlled by the angular acceleration of the axis to which the beam is clamped. We show that this problem of controllability is solvable if the time of rotation is long enough and a certain parameter that describes the material of the beam is a rational number that has an even numerator and an odd denominator or vice versa.

In this paper we consider second order evolution equations with unbounded feedbacks. Under a regularity assumption we show that observability properties for the undamped problem imply decay estimates for the damped problem. We consider both uniform and non uniform decay properties.

We describe precisely, under generic conditions, the contact of the accessibility set at time T with an abnormal direction, first for a single-input affine control system with constraint on the control, and then as an application for a sub-Riemannian system of rank 2. As a consequence we obtain in sub-Riemannian geometry a new splitting-up of the sphere near an abnormal minimizer γ into two sectors, bordered by the first Pontryagin's cone along γ, called the L∞-sector and the L2-sector. Moreover we find again necessary and sufficient conditions of optimality of an abnormal trajectory for such systems, for any optimization problem.

We study the Bellman equation for undiscounted exit time optimal control problems with fully nonlinear Lagrangians and fully nonlinear dynamics using the dynamic programming approach. We allow problems whose non-Lipschitz dynamics admit more than one solution trajectory for some choices of open loop controls and initial positions. We prove a uniqueness theorem which characterizes the value functions of these problems as the unique viscosity solutions of the corresponding Bellman equations that satisfy appropriate boundary conditions. We deduce that the value function for Sussmann's Reflected Brachystochrone Problem for an arbitrary singleton target is the unique viscosity solution of the corresponding Bellman equation in the class of functions which are continuous in the plane, null at the target, and bounded below. Our results also apply to degenerate eikonal equations, and to problems whose targets can be unbounded and whose Lagrangians vanish for some points in the state space which are outside the target, including Fuller's Example.

This work is devoted to the numerical comparison of four active control techniques in order to increase the pressure recovery generated by thedeceleration of a slightly compressible viscous flow over a dihedral plane. It is performed by theuse of vortex generator jets and intrusive sensors. The governing equations,the two-dimensional direct numerical simulation code and the flow configuration are first briefly recalled. Then, the objective of thecontrol is carefully displayed, and the uncontrolled flow described. The main part of this work deals with the explanation, the implementation andthe comparison of four active control strategies: closed loop control,adaptative control, physical ramp control and sub-optimalcontrol. Each of these techniques is of different nature,and results are very formative to understand what is important– or less – to make the control efficient.

We consider optimal control problems for convection-diffusion equations with a pointwise control or a control localized on a smooth manifold. We prove optimality conditions for the control variable and for the position of the control. We do not suppose that the coefficient of the convection term is regular or bounded, we only suppose that it has the regularity of strong solutions of the Navier–Stokes equations. We consider functionals with an observation on the gradient of the state. To obtain optimality conditions we have to prove that the trace of the adjoint state on the control manifold belongs to the dual of the control space. To study the state equation, which is an equation with measures as data, and the adjoint equation, which involves the divergence of Lp-vector fields, we first study equations without convection term, and we next use a fixed point method to deal with the complete equations.

Γ-convergence techniques and relaxation results of constrained energy functionals are used to identify the limiting energy as the thickness ε approaches zero of a ferromagnetic thin structure $\Omega_\varepsilon=\omega\times(-\varepsilon,\varepsilon)$, $\omega\subset\mathbb R^2$, whose energy is given by $$ {\cal E}_{\varepsilon}({\overline{m}})=\frac{1}{\varepsilon} \int_{\Omega_{\varepsilon}}\left(W({\overline{m}},\nabla{\overline{m}}) +{\frac{1}{2}}\nabla {\overline{u}}\cdot {\overline{m}}\right)\,{\rm d}x $$ subject to $$ \hbox{div}(-\nabla {\overline{u}}+{\overline{m}}\chi_{\Omega_\varepsilon})=0 \quad\hbox{ on }\mathbb R^3, $$ and to the constraint $$ |\overline{m}|=1 \hbox{ on }\Omega_\varepsilon, $$ where W is any continuous function satisfying p-growth assumptions with p> 1. Partial results are also obtained in the case p=1, under an additional assumption on W.

In the present paper, we study the problem of small-time local attainability (STLA) of a closed set. For doing this, we introduce a new concept of variations of the reachable set well adapted to a given closed set and prove a new attainability result for a general dynamical system. This provide our main result for nonlinear control systems. Some applications to linear and polynomial systems are discussed and STLA necessary and sufficient conditions are obtained when the considered set is a hyperplane.