We investigate the diffusion limit for general conservative Boltzmann equations with oscillating coefficients. Oscillations have a frequency of the same order as the inverse of the mean free path, and the coefficients may depend on both slow and fast variables. Passing to the limit, we are led to an effective drift-diffusion equation. We also describe the diffusive behaviour when the equilibrium function has a non-vanishing flux.

We prove Lipschitz continuity for local minimizers of integral functionals of the Calculus of Variations in the vectorial case, where the energy density depends explicitly on the space variables and has general growth with respect to the gradient. One of the models is $$ F\left(u \right)=\int_{\Omega}a(x)[h\left(|Du|\right)]^{p(x)}{\rm d}x $$ with h a convex function with general growth (also exponential behaviour is allowed).

Motion planning and boundary control for a class of linear PDEs with constant coefficients is presented. With the proposed method transitions from rest to rest can be achieved in a prescribed finite time. When parameterizing the system by a flat output, the system trajectories can be calculated from the flat output trajectory by evaluating definite convolution integrals. The compact kernels of the integrals can be calculated using infinite series. Explicit formulae are derived employing Mikusiński's operational calculus. The method is illustrated through an application to a model of a Timoshenko beam, which is clamped on a rotating disk and carries a load at its free end.

An optimal control problem when controls act on the boundary can also be understood as a variational principle under differential constraints and no restrictions on boundary and/or initial values. From this perspective, some existence theorems can be proved when cost functionals depend on the gradient of the state. We treat the case of elliptic and non-elliptic second order state laws only in the two-dimensional situation. Our results are based on deep facts about gradient Young measures.

We investigate the asymptotic behaviour, as ε → 0, of a class of monotone nonlinear Neumann problems, with growth p-1 (p ∈]1, +∞[), on a bounded multidomain $\Omega_\varepsilon\subset \mathbb{R}^N$ (N ≥ 2). The multidomain ΩE is composed of two domains. The first one is a plate which becomes asymptotically flat, with thickness hE in the xN direction, as ε → 0. The second one is a “forest" of cylinders distributed with ε-periodicity in the first N - 1 directions on the upper side of the plate. Each cylinder has a small cross section of size ε and fixed height (for the case N=3, see the figure). We identify the limit problem, under the assumption: ${\lim_{\varepsilon\rightarrow 0} {\varepsilon^p\over h_\varepsilon}=0}$. After rescaling the equation, with respect to hE, on the plate, we prove that, in the limit domain corresponding to the “forest" of cylinders, the limit problem identifies with a diffusion operator with respect to xN, coupled with an algebraic system. Moreover, the limit solution is independent of xN in the rescaled plate and meets a Dirichlet transmission condition between the limit domain of the “forest" of cylinders and the upper boundary of the plate.


The aim of this paper is to study a class of domains whose geometry strongly depends on time namely. More precisely, we consider parabolic equations in perforated domains with rapidly pulsing (in time) periodic perforations, with a homogeneous Neumann condition on the boundary of the holes. We study the asymptotic behavior of the solutions as the period ε of the holes goes to zero. Since standard conservation laws do not hold in this model, a first difficulty is to get a priori estimates of the solutions. We obtain them in a weighted space where the weight is the principal eigenfunction of an “adjoint” periodic time-dependent eigenvalue problem. This problem is not a classical one, and its investigation is an important part of this work. Then, by using the multiple scale method, we construct the leading terms of a formal expansion (with respect to ε) of the solution and give the limit “homogenized” problem. An interesting peculiarity of the model is that, depending on the geometry of the holes, a large convection term may appear in the limit equation.

We characterize the geometry of a path in a sub-Riemannian manifold using two metric invariants, the entropy and the complexity. The entropy of a subset A of a metric space is the minimum number of balls of a given radius ε needed to cover A. It allows one to compute the Hausdorff dimension in some cases and to bound it from above in general. We define the complexity of a path in a sub-Riemannian manifold as the infimum of the lengths of all trajectories contained in an ε-neighborhood of the path, having the same extremities as the path. The concept of complexity for paths was first developed to model the algorithmic complexity of the nonholonomic motion planning problem in robotics. In this paper, our aim is to estimate the entropy, Hausdorff dimension and complexity for a path in a general sub-Riemannian manifold. We construct first a norm $\| \cdot \|_{\varepsilon}$ on the tangent space that depends on a parameter ε > 0. Our main result states then that the entropy of a path is equivalent to the integral of this ε-norm along the path. As a corollary we obtain upper and lower bounds for the Hausdorff dimension of a path. Our second main result is that complexity and entropy are equivalent for generic paths. We give also a computable sufficient condition on the path for this equivalence to happen.

In this paper, we consider general nonlinear systems with observations, containing a (single) unknown function φ. We study the possibility to learn about this unknown function via the observations: if it is possible to determine the [values of the] unknown function from any experiment [on the set of states visited during the experiment], and for any arbitrary input function, on any time interval, we say that the system is “identifiable”. For systems without controls, we give a more or less complete picture of what happens for this identifiability property. This picture is very similar to the picture of the “observation theory” in [7]: Contrarily to the case of the observability property, in order to identify in practice, there is in general no hope to do something better than using “approximate differentiators”, as show very elementary examples. However, a practical methodology is proposed in some cases. It shows very reasonable performances.
As an illustration of what may happen in controlled cases, we consider the equations of a biological reactor, [2,4], in which a population is fed by some substrate. The model heavily depends on a “growth function”, expressing the way the population grows in presence of the substrate. The problem is to identify this “growth function”. We give several identifiability results, and identification methods, adapted to this problem.

In this paper it is shown that the generalized smoothing spline obtained by solving an optimal control problem for a linear control system converges to a deterministic curve even when the data points are perturbed by random noise. We furthermore show that such a spline acts as a filter for white noise. Examples are constructed that support the practical usefulness of the method as well as gives some hints as to the speed of convergence.

We consider the stabilization of Maxwell's equations with space-time variable coefficients in a bounded region with a smooth boundary by means of linear or nonlinear Silver–Müller boundary condition. This is based on some stability estimates that are obtained using the “standard" identity with multiplier and appropriate properties of the feedback. We deduce an explicit decay rate of the energy, for instance exponential, polynomial or logarithmic decays are available for appropriate feedbacks.

We intend to conduct a fairly complete study on Timoshenko beams with pointwise feedback controls and seek to obtain information about the eigenvalues, eigenfunctions, Riesz-Basis-Property, spectrum-determined-growth-condition, energy decay rate and various stabilities for the beams. One major difficulty of the present problem is the non-simplicity of the eigenvalues. In fact, we shall indicate in this paper situations where the multiplicity of the eigenvalues is at least two. We build all the above-mentioned results from an effective asymptotic analysis on both the eigenvalues and the eigenfunctions, and conclude with the Riesz-Basis-Property and the spectrum-determined-growth-condition. Finally, these results are used to examine the stability effects on the system by the location of the pointwise control relative to the length of the whole beam.

In this paper we establish a variant and generalized weak linking theorem, which contains more delicate result and insures the existence of bounded Palais–Smale sequences of a strongly indefinite functional. The abstract result will be used to study the semilinear Schrödinger equation $-\Delta u+V(x)u=K(x)|u|^{2^\ast-2}u+g(x, u), u\in W^{1,2}({\bf R}^N)$, where N ≥ 4; V,K,g are periodic in xj for 1 ≤ j ≤ N and 0 is in a gap of the spectrum of -Δ + V; K>0. If $0<g(x, u)u\leq c|u|^{2^\ast}$ for an appropriate constant c, we show that this equation has a nontrivial solution.

We consider the Laplace equation in a smooth bounded domain. We prove logarithmic estimates, in the sense of John [5] of solutions on a part of the boundary or of the domain without known boundary conditions. These results are established by employing Carleman estimates and techniques that we borrow from the works of Robbiano [8,11]. Also, we establish an estimate on the cost of an approximate control for an elliptic model equation.