### A nonintersection property for extremals of variational problems with vector-valued functions

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Let $\mathcal{L}$(x,u,∇u) be a Lagrangian periodic of period 1 in x1,...,xn,u. We shall study the non self intersecting functions u: Rn$\to $R minimizing $\mathcal{L}$; non self intersecting means that, if u(x0 + k) + j = u(x0) for some x0∈Rn and (k , j) ∈Zn × Z, then u(x) = u(x + k) + j$\phantom{\rule{0.277778em}{0ex}}\forall $x. Moser has shown that each of these functions is at finite distance from a plane u = ρ$\xb7$x and thus has an average slope ρ; moreover, Senn has proven that it is possible to define the average action of u, which is usually called $\beta \left(\rho \right)$ since...

The paper is divided in two parts. In the first part a deep investigation is made on some system theoretical aspects of periodic systems and control, including the notions of ${H}_{2}$ and ${H}_{\infty}$ norms, the parametrization of stabilizing controllers, and the existence of periodic solutions to Riccati differential equations and/or inequalities. All these aspects are useful in the second part, where some parametrization and control problems in ${H}_{2}$ and ${H}_{\infty}$ are introduced and solved.

This paper deals with some state-feedback ${H}_{2}/{H}_{\infty}$ control problems for continuous time periodic systems. The derivation of the theoretical results underlying such problems has been presented in the first part of the paper. Here, the parametrization and optimization problems in ${H}_{2}$, ${H}_{\infty}$ and mixed ${H}_{2}/{H}_{\infty}$ are introduced and solved.

Si studia il comportamento asintotico di una classe di funzionali integrali che possono dipendere da misure concentrate su strutture periodiche multidimensionali, quando tale periodo tende a 0. Il problema viene ambientato in spazi di Sobolev rispetto a misure periodiche. Si dimostra, sotto ipotesi generali, che un appropriato limite può venire definito su uno spazio di Sobolev usuale usando tecniche di $\mathrm{\Gamma}$-convergenza. Il limite viene espresso come un funzionale integrale il cui integrando è caratterizzato...

The paper is a continuation of a previous work of the same authors dealing with homogenization processes for some energies of integral type arising in the modeling of rubber-like elastomers. The previous paper took into account the general case of the homogenization of energies in presence of pointwise oscillating constraints on the admissible deformations. In the present paper homogenization processes are treated in the particular case of fixed constraints set, in which minimal coerciveness hypotheses...

This paper studies the periodic feedback stabilization of the controlled linear time-periodic ordinary differential equation: ẏ(t) = A(t)y(t) + B(t)u(t), t ≥ 0, where [A(·), B(·)] is a T-periodic pair, i.e., A(·) ∈ L∞(ℝ+; ℝn×n) and B(·) ∈ L∞(ℝ+; ℝn×m) satisfy respectively A(t + T) = A(t) for a.e. t ≥ 0 and B(t + T) = B(t) for a.e. t ≥ 0. Two periodic stablization criteria for a T-period pair [A(·), B(·)] are established. One is an analytic criterion which is related to the transformation over time...