One shows that the linearized Navier-Stokes equation in ${\mathcal{O}}{\subset} R^d,\;d \ge 2$ , around an unstable equilibriumsolution is exponentially stabilizable in probability by aninternal noise controller $V(t,\xi)=\displaystyle\sum\limits_{i=1}^{N} V_i(t)\psi_i(\xi)\dot\beta_i(t)$ , $\xi\in{\mathcal{O}}$ , where $\{\beta_i\}^N_{i=1}$ areindependent Brownian motions in a probability space and $\{\psi_i\}^N_{i=1}$ is a system of functions on ${\mathcal{O}}$ withsupport in an arbitrary open subset ${\mathcal{O}}_0\subset {\mathcal{O}}$ . Thestochastic control input $\{V_i\}^N_{i=1}$ is found in feedbackform. One constructs also a tangential boundary noise controllerwhich exponentially stabilizes in probability the equilibriumsolution.

We consider the numerical solution, in two- and three-dimensionalbounded domains, of the inverse problem for identifying the locationof small-volume, conductivity imperfections in a medium with homogeneousbackground. A dynamic approach, based on the wave equation, permitsus to treat the important case of “limited-view” data. Our numericalalgorithm is based on the coupling of a finite element solution ofthe wave equation, an exact controllability method and finally a Fourierinversion for localizing the centers of the imperfections. Numericalresults, in 2- and 3-D, show the robustness and accuracy of the approachfor retrieving randomly placed imperfections from both complete andpartial boundary measurements.

We consider a time optimal control problem arisen from the optimalmanagement of a bioreactor devoted to the treatment ofeutrophicated water. We formulate this realistic problem as astate-control constrained time optimal control problem. Afteranalyzing the state system (a complex system of coupled partialdifferential equations with non-smooth coefficients foradvection-diffusion-reaction with Michaelis-Menten kinetics,modelling the eutrophication processes) we demonstrate theexistence of, at least, an optimal solution. Then we present adetailed derivation of a first order optimality condition(involving the corresponding adjoint systems) characterizing theseoptimal solutions. Finally, a numerical example is shown.

The paper represents the first part of a series ofpapers on realization theory of switched systems. Part I presents realization theory of linear switched systems,Part II presents realization theory of bilinear switched systems.More precisely, in Part I necessary and sufficient conditionsare formulated for a family of input-output maps to berealizable by a linear switched system and a characterizationof minimal realizations is presented. The paper treats two types of switched systems.The first one is when all switching sequences are allowed. The second one is when only a subset of switching sequences isadmissible, but within this restricted set the switching times are arbitrary.The paper uses the theory of formal power series to derivethe results on realization theory.

Using the tool of two-scale convergence, we provide a rigorous mathematical setting for the homogenization result obtained by Fleck and Willis [J. Mech. Phys. Solids 52 (2004) 1855–1888] concerning the effective plastic behaviour of a strain gradient composite material. Moreover, moving from deformation theory to flow theory, we prove a convergence result for the homogenization of quasistatic evolutions in the presence of isotropic linear hardening.

We study the H –1-norm of the function 1 on tubular neighbourhoods of curves in ${\mathbb R}^{2}$ . We take the limit of small thickness ε, and we prove two different asymptotic results. The first is an asymptotic development for a fixed curve in the limit ε → 0, containing contributions from the length of the curve (at order ε 3), the ends (ε 4), and the curvature (ε 5). The second result is a Γ-convergence result, in which the central curve may vary along the sequence ε → 0. We prove that a rescaled version of the H –1-norm, which focuses on the ε 5 curvature term, Γ-converges to the L 2-norm of curvature. In addition, sequences along which the rescaled norm is bounded are compact in the W 1,2-topology. Our main tools are the maximum principle for elliptic equations and the use of appropriate trial functions in the variational characterisation of the H –1-norm. For the Γ-convergence result we use the theory of systems of curves without transverse crossings to handle potential intersections in the limit.

This paper is the second part of a series of papers dealing withrealization theory of switched systems.The current Part II addresses realization theory of bilinear switchedsystems. In Part I [Petreczky, ESAIM: COCV, DOI: 10.1051/cocv/2010014] we presented realizationtheory of linear switched systems.More precisely, in Part II we present necessary and sufficient conditionsfor a family of input-output maps to berealizable by a bilinear switched system, together with a characterizationof minimal realizations.Similarly to Part I, the paper deals with two types of switched systems.The first one is when all switching sequences are allowed. The second one is when only a subset of switching sequences isadmissible, but within this restricted set the switching times are arbitrary.The paper uses the theory of formal power series to derivethe results on realization theory.

A zero-sum stochastic differential gameproblem on infinite horizon with continuous and impulse controls isstudied. We obtain the existence of the value of the game andcharacterize it as the unique viscosity solution of the associatedsystem of quasi-variational inequalities. We also obtain averification theorem which provides an optimal strategy of the game.

We consider higher order functionals of the form

$F[u]=\int\limits_\Omega f(D^mu)\,{\rm d}x \qquad\text{for }u:\mathbb{R}^n\supset\Omega\to\mathbb{R}^N,$

where the integrand $f:{\textstyle \bigodot^m}(\R^{n},\R^{N})\to\mathbb{R}$, m ≥ 1 is strictly quasiconvex and satisfies a non-standard growth condition.More precisely we assume that f fulfills the (p, q)-growth condition

\[\gamma|A|^p\le f(A)\le L(1+|A|^q)\qquad \mbox{for all }A \in {\textstyle \bigodot^m}(\R^{n},\R^{N}),\]

with γ, L > 0 and $1< p \le q<\min\big\{p+\frac1n,\frac{2n-1}{2n-2}p\big\}$. We study minimizers of thefunctional $F[\cdot]$ and prove a partial $C^{m,\alpha}_{\rm loc}$-regularity result.

In this paper, a $W^{-1,N'}$ estimate of the pressure is derived when its gradient is the divergence of a matrix-valued measure on $\mathbb R^N$, or on a regular bounded open set of $\mathbb R^N$. The proof is based partially on the Strauss inequality [Strauss, Partial Differential Equations: Proc. Symp. Pure Math. 23 (1973) 207–214] in dimension two, and on a recent result of Bourgain and Brezis [J. Eur. Math. Soc. 9 (2007) 277–315] in higher dimension. The estimate is used to derive a representation result for divergence free distributions which read as the divergence of a measure, and to prove an existence result for the stationary Navier-Stokes equation when the viscosity tensor is only in L1.

We consider a model for the control of a linear network flow system with unknown but bounded demandand polytopic bounds on controlled flows. We are interested in the problem of finding a suitable objective functionthat makes robust optimal the policy represented by the so-called linear saturated feedback control. We regard the problem as a suitable differential game with switching cost and study it in the framework of the viscosity solutions theory for Bellman and Isaacs equations.

This paper studies the strong unique continuation property for theLamé system of elasticity with variable Lamé coefficientsλ, µ in three dimensions, ${\rm{div}}(\mu(\nabla u+\nablau^t))+ \nabla(\lambda{\rm{div}} u)+Vu=0$where λ and μ are Lipschitz continuous and V ∈ L∞. The method is based on the Carleman estimate with polynomial weights for the Lamé operator.

We establish a local Lipschitz regularity result for local minimizers of asymptotically convex variational integrals.

In this paper, we carry out the numerical analysis of adistributed optimal control problem governed by a quasilinearelliptic equation of non-monotone type. The goal is to prove thestrong convergence of the discretization of the problem by finiteelements. The main issue is to get error estimates for thediscretization of the state equation. One of the difficulties inthis analysis is that, in spite of the partial differentialequation has a unique solution for any given control, theuniqueness of a solution for the discrete equation is an openproblem.

This work studies the null-controllability of a class of abstract parabolic equations. The main contribution in the general caseconsists in giving a short proof of an abstract version of a sufficient condition for null-controllability which has been proposed by Lebeau and Robbiano. We do not assume that the control operator is admissible. Moreover, we give estimates of the control cost.In the special case of the heat equation in rectangular domains, we provide an alternative way to check the Lebeau-Robbiano spectral condition. We then show that the sophisticated Carleman and interpolation inequalities used in previous literature may be replaced by a simple result of Turán. In this case, we provide explicit values for the constants involved in the above mentioned spectral condition. As far as we are aware, this is the first proof of the null-controllability of the heat equation with arbitrary control domain in a n-dimensional open set which avoids Carleman estimates.

We prove that the critical points of the 3d nonlinear elasticity functionalon shells of small thickness h and around the mid-surface S of arbitrary geometry, converge as h → 0to the critical points of the vonKármán functional on S, recently proposed in [Lewicka et al., Ann. Scuola Norm. Sup. Pisa Cl. Sci. (to appear)].This result extends the statement in [Müller and Pakzad, Comm. Part. Differ. Equ. 33 (2008) 1018–1032], derived for the case of plates when $S\subset\mathbb{R}^2$.The convergence holds provided the elastic energies of the 3d deformations scale like h4 and the external body forces scale like h3.

Exact controllabilityresults for a multilayer plate system are obtained from the method of Carleman estimates.The multilayer plate system is a natural multilayer generalization of a classical three-layer “sandwichplate” system due to Rao and Nakra. The multilayer version involves a number ofLamé systems for plane elasticity coupled with a scalar Kirchhoff plate equation. The plate is assumed to be either clamped or hinged and controlsare assumed to be locally distributed in a neighborhood of a portion of the boundary. The Carleman estimates developed for thecoupled system are based on some new Carleman estimates for the Kirchhoff plate as well as some known Carleman estimates due to Imanuvilov and Yamamoto for the Lamé system.

We prove a general relaxation theorem for multidimensional control problems of Dieudonné-Rashevsky type with nonconvex integrands f(t, ξ, v) in presence of a convex control restriction. The relaxed problem, wherein the integrand f has been replaced by its lower semicontinuous quasiconvex envelope with respect to the gradient variable, possesses the same finite minimal value as the original problem, and admits a global minimizer. As an application, we provide existence theorems for the image registration problem with convex and polyconvex regularization terms.

We are interested in the feedback stabilization of a fluid flow over a flat plate, around a stationary solution, in the presence of perturbations. More precisely, we want to stabilize the laminar-to-turbulent transition location of a fluid flow over a flat plate. For that we study the Algebraic Riccati Equation (A.R.E.) of a control problem in which the state equation is a doubly degenerate linear parabolic equation. Because of the degenerate character of the state equation, the classical existence results in the literature of solutions to algebraic Riccati equations do not apply to this class of problems. Here taking advantage of the fact that the semigroup of the state equation is exponentially stable and that the observation operator is a Hilbert-Schmidt operator, we are able to prove the existence and uniqueness of solution to the A.R.E. satisfied by the kernel of the operator which associates the 'optimal adjoint state' with the 'optimal state'. In part 2 [Buchot and Raymond, Appl. Math. Res. eXpress (2010) doi:10.1093/amrx/abp007], we study problems in which the feedback law is determined by the solution to the A.R.E. and another nonhomogeneous term satisfying an evolution equation involving nonhomogeneous perturbations of the state equation, and a nonhomogeneous term in the cost functional.

This paper is devoted to the study of a coupled system which consists ofa wave equation and a heat equation coupled through a transmission conditionalong a steady interface. This system is a linearized model forfluid-structure interaction introduced by Rauch, Zhang and Zuazuafor a simple transmission condition and by Zhang and Zuazua for anatural transmission condition. Using an abstract theorem of Burq and a new Carleman estimate proved near the interface, wecomplete the results obtained by Zhang and Zuazua and by Duyckaerts.We prove, without a Geometric Control Condition, a logarithmic decayof the energy.