The problem of invariant output tracking is considered: given a control system admitting a symmetry group G, design a feedback such that the closed-loop system tracks a desired output reference and is invariant under the action of G. Invariant output errors are defined as a set of scalar invariants of G; they are calculated with the Cartan moving frame method. It is shown that standard tracking methods based on input-output linearization can be applied to these invariant errors to yield the required “symmetry-preserving” feedback.

In this paper, we prove that the Lp approximants naturally associated to a supremal functionalΓ-converge to it. This yields a lower semicontinuity result for supremalfunctionals whose supremand satisfy weak coercivity assumptions aswell as a generalized Jensen inequality. The existence of minimizersfor variational problems involving such functionals (together with aDirichlet condition) then easily follows. In the scalarcase we show the existence of at least one absolute minimizer (i.e. localsolution) among these minimizers. We provide two different proofs ofthis fact relying on different assumptions and techniques.

We consider the pseudo-p-Laplacian, an anisotropicversion of the p-Laplacian operator for $p\not=2$ . We studyrelevant properties of its first eigenfunction for finite p andthe limit problem as p → ∞.

The paper is a continuation of a previous work of the same authorsdealing with homogenization processes for some energiesof integral type arising in the modeling of rubber-like elastomers.The previous paper took into account the general case of thehomogenization of energies in presence of pointwise oscillatingconstraints on the admissible deformations.In the present paper homogenization processes are treated in theparticular case of fixed constraints set, in which minimalcoerciveness hypotheses can be assumed, and in which the results canbe obtained in the general framework of BV spaces.The classical homogenization result is established for Dirichlet withaffine boundary data, Neumann, and mixedproblems, by proving that the limit energy is again of integral type,gradient constrained, and with an explicitly computedhomogeneous density.

Existence results for a class of one-dimensional abstract variational problems with volume constraints are established. The main assumptions on their energy are additivity, translation invariance and solvability of a transition problem. These general results yield existence results for nonconvex problems. A counterexample shows that a naive extension to higher dimensional situations in general fails.

In this paper, we prove someregularity results for the boundary of an open subset of $\xR^d$ whichminimizes the Dirichlet's energy among all open subsets withprescribed volume. In particular we show that, whenthe volume constraint is “saturated”,the reduced boundary of the optimal shape (and even the wholeboundary in dimension 2)is regular if the state function is nonnegative.

Turnpike theorems deal with the optimality of trajectories reaching asingular solution, in calculus of variations oroptimal control problems.For scalar calculus of variations problems in infinite horizon, linear withrespect to the derivative, we use the theory of viscosity solutions ofHamilton-Jacobi equations to obtain a unique characterization of the valuefunction.With this approach, we extend for the scalar case the classical result based onGreen theorem, when there is uniqueness of the singular solution.We provide a new necessary and sufficient condition for turnpikeoptimality, even in the presence of multiple singular solutions.

We compute numerically the minimizers of the Dirichlet energy $$E(u)=\frac{1}{2}\int_{B^2}|\nabla u|^2 {\rm d}x$$ among maps $u:B^2\to S^2$ from the unit disc into the unit sphere that satisfy a boundary condition and a degree condition. We use a Sobolev gradient algorithm for the minimization and we prove that its continuous version preserves the degree. For the discretization of the problem we use continuous P 1 finite elements. We propose an original mesh-refining strategy needed to preserve the degree with the discrete version of the algorithm (which is a preconditioned projected gradient). In order to improve the convergence, we generalize to manifolds the classical Newton and conjugate gradient algorithms. We give a proof of the quadratic convergence of the Newton algorithm for manifolds in a general setting.