This paper is concerned with the stability of the set of trajectoriesof a patchy vector field, in the presence of impulsiveperturbations. Patchy vector fields arediscontinuous, piecewise smooth vector fieldsthat were introduced in Ancona and Bressan (1999) to study feedback stabilization problems.For patchy vector fields in the plane, with polygonalpatches in generic position, we show that the distance between a perturbed trajectory and an unperturbed one is of the same order of magnitude as the impulsive forcing term.

Let $L:\Bbb R^N\times\Bbb R^N\rightarrow\Bbb R$ be a Borelian function andconsider the following problems $$\inf\left\{F(y)=\int_a^bL(y(t),y'(t))\,{\rm d}t:\,y\in AC([a,b],\Bbb R^N), y(a)=A,\,y(b)=B\right\} \qquad\quad\! (P)$$ 
 $$ \inf\left\{F^{**}(y)=\int_a^bL^{**}(y(t),y'(t))\,{\rm d}t:\,y\in AC([a,b],\Bbb R^N), y(a)=A,\,y(b)=B\right\}\cdot \quad\;\ \! (P^{**})$$ We give a sufficient condition, weaker then superlinearity, under which $\inf F=\inf F^{**}$ if L is just continuous in x. We then extend a result of Cellina on the Lipschitz regularity of the minima of (P) when L is not superlinear.

We prove that the Paneitz energy on the standard three-sphere S 3 is bounded from below and extremal metrics must be conformally equivalent to the standard metric.

We establish an approximation theorem for a sequence oflinear elastic strains approaching a compact set in L 1 by thesequence of linear strains of mapping bounded in Sobolev space W 1,p . We apply this result to establish equalities forsemiconvex envelopes for functions defined on linear strains via a construction of quasiconvex functions with linear growth.

Let H be a real Hilbert space, $\Phi_1: H\to \xR$ a convex function of class ${\mathcal C}^1$ that we wish to minimize under the convexconstraint S.A classical approach consists in following the trajectories of the generalizedsteepest descent system (cf.   Brézis [CITE]) appliedto the non-smooth function  $\Phi_1+\delta_S$ . Following Antipin [1], it is also possible to use a continuous gradient-projection system.We propose here an alternative method as follows:given a smooth convex function  $\Phi_0: H\to \xR$ whose critical points coincidewith Sand a control parameter $\varepsilon:\xR_+\to \xR_+$ tending to zero,we consider the “Steepest Descent and Control” system \[(SDC) \qquad \dot{x}(t)+\nabla \Phi_0(x(t))+\varepsilon(t)\, \nabla \Phi_1(x(t))=0,\] where the control ε satisfies $\int_0^{+\infty} \varepsilon(t)\, {\rm d}t =+\infty$ . This last condition ensures that ε “slowly” tends to 0. When H is finite dimensional, we then prove that $d(x(t), {\rm argmin}\kern 0.12em_S \Phi_1) \to 0\quad (t\to +\infty),$ and we give sufficient conditions under which  $x(t) \to \bar{x}\in \,{\rm argmin}\kern 0.12em_S \Phi_1$ .We end the paper by numerical experiments allowing to compare the (SDC) system with the other systems already mentioned.

We describe an algorithm for computing the value function for “all source, single destination” discrete-time nonlinear optimal control problems together with approximations of associated globally optimal control strategies. The method is based on a set oriented approach for the discretization of the problem in combination with graph-theoretic techniques. The central idea is that a discretization of phase space of the given problem leads to an (all source, single destination) shortest path problem on a finite graph. The method is illustrated by two numerical examples, namely a single pendulum on a cart and a parametrically driven inverted double pendulum.

We study a finite horizon problem for a system whose evolution isgoverned by a controlled ordinary differential equation, which takesalso account of a hysteretic component: namely, the outputof a Preisach operator of hysteresis. We derive a discontinuousinfinitedimensional Hamilton–Jacobi equation and prove that, under fairlygeneral hypotheses, the value function is the unique bounded anduniformly continuous viscosity solution of the corresponding Cauchyproblem.

A homogenization problem related to the micromagnetic energy functional is studied. In particular, the existence of the integral representation for the homogenized limit of a family of energies $$ \mathcal{E}_{\varepsilon}(m)=\int_{\Omega} \phi\left(x,\frac{x}{\varepsilon},m(x)\right)\,{\rm d}x-\int_{\Omega}h_e(x)\cdot m(x)\,{\rm d}x+\frac{1}{2}\int_{\mathbb R^3}|\nabla u(x)|^2\,{\rm d}x$$ of a large ferromagnetic body is obtained.