The parabolic equations driven by linearly multiplicative Gaussian noise are stabilizable in probability by linear feedback controllers with support in a suitably chosen open subset of the domain. This procedure extends to Navier − Stokes equations with multiplicative noise. The exact controllability is also discussed.

We consider the problem of minimising the nth-eigenvalue of the RobinLaplacian in RN. Although for n = 1,2 and apositive boundary parameter α it is known that the minimisers do notdepend on α, we demonstrate numerically that this will not always be thecase and illustrate how the optimiser will depend on α. We derive aWolf–Keller type result for this problem and show that optimal eigenvalues grow at mostwith n1/N, which is in sharp contrast withthe Weyl asymptotics for a fixed domain. We further show that the gap between consecutiveeigenvalues does go to zero as n goes to infinity. Numerical results thensupport the conjecture that for each n there exists a positive value ofαn such that the ntheigenvalue is minimised by n disks for all0 < α < αnand, combined with analytic estimates, that this value is expected to grow withn1/N.

In this paper, we are concerned with the existence of multi-bump solutions for anonlinear Schrödinger equations with electromagnetic fields. We prove under some suitableconditions that for any positive integer m, there existsε(m) > 0 such that, for0 < ε < ε(m),the problem has an m-bump complex-valued solution. As a result, whenε → 0, the equation has more and more multi-bumpcomplex-valued solutions.

This article is the starting point of a series of works whose aim is the study of deterministic control problems where the dynamic and the running cost can be completely different in two (or more) complementary domains of the space ℝN. As a consequence, the dynamic and running cost present discontinuities at the boundary of these domains and this is the main difficulty of this type of problems. We address these questions by using a Bellman approach: our aim is to investigate how to define properly the value function(s), to deduce what is (are) the right Bellman Equation(s) associated to this problem (in particular what are the conditions on the set where the dynamic and running cost are discontinuous) and to study the uniqueness properties for this Bellman equation. In this work, we provide rather complete answers to these questions in the case of a simple geometry, namely when we only consider two different domains which are half spaces: we properly define the control problem, identify the different conditions on the hyperplane where the dynamic and the running cost are discontinuous and discuss the uniqueness properties of the Bellman problem by either providing explicitly the minimal and maximal solution or by giving explicit conditions to have uniqueness.

We first prove existence and uniqueness of optimal transportation maps for the Monge’s problem associated to a cost function with a strictly convex constraint in the Euclidean plane ℝ2. The cost function coincides with the Euclidean distance if the displacement y − x belongs to a given strictly convex set, and it is infinite otherwise. Secondly, we give a sufficient condition for existence and uniqueness of optimal transportation maps for the original Monge’s problem in ℝ2. Finally, we get existence of optimal transportation maps for a cost function with a convex constraint, i.e. y − x belongs to a given convex set with at most countable flat parts.

We study the partial differential equation

        max{Lu − f, H(Du)} = 0

where u is the unknownfunction, L is a second-order elliptic operator, f is agiven smooth function and H is a convex function. This is a modelequation for Hamilton-Jacobi-Bellman equations arising in stochastic singular control. Weestablish the existence of a unique viscosity solution of the Dirichlet problem that has aHölder continuous gradient. We also show that if H is uniformlyconvex, the gradient of this solution is Lipschitz continuous.

Time optimal control problems for an internally controlled heat equation with pointwisecontrol constraints are studied. By Pontryagin’s maximum principle and properties ofnontrivial solutions of the heat equation, we derive a bang-bang property for time optimalcontrol. Using the bang-bang property and establishing certain connections between timeand norm optimal control problems for the heat equation, necessary and sufficientconditions for the optimal time and the optimal control are obtained.

We establish some new results about the Γ-limit, with respect to the L1-topology, of two different (but related) phase-field approximations \hbox{$\{\mathcal E_\eps\}_\eps,\,\{\widetilde{\mathcalE}_\eps\}_\eps$}$\mathrm{\{}{\mathrm{ℰ}}_{\mathit{\epsilon }}{\mathrm{\}}}_{\mathit{\epsilon }}\mathit{,}\hspace{0.17em}\mathrm{\{}\begin{array}{}{}_{\mathrm{􏽥}}\\ {\mathrm{ℰ}}_{\mathit{\epsilon }}\end{array}{\mathrm{\}}}_{\mathit{\epsilon }}$ of the so-called Euler’s Elastica Bending Energy for curves in the plane. In particular we characterize the Γ-limit as ε → 0 of ℰε, and show that in general the Γ-limits of ℰε and \hbox{$\widetilde{\mathcal E}_\eps$}$\begin{array}{}{\mathrm{􏽥}}_{}\\ {\mathrm{ℰ}}_{\mathit{\epsilon }}\end{array}$ do not coincide on indicator functions of sets with non-smooth boundary. More precisely we show that the domain of the Γ-limit of \hbox{$\widetilde{\mathcal E}_\eps$}$\begin{array}{}{\mathrm{􏽥}}_{}\\ {\mathrm{ℰ}}_{\mathit{\epsilon }}\end{array}$ strictly contains the domain of the Γ-limit of ℰε.

We compute the Γ-limit of a sequence of non-local integral functionalsdepending on a regularization of the gradient term by means of a convolution kernel. Inparticular, as Γ-limit, we obtain free discontinuity functionals withlinear growth and with anisotropic surface energy density.

This paper is concerned with the study of a model case of first order Hamilton-Jacobi equations posed on a “junction”, that is to say the union of a finite number of half-lines with a unique common point. The main result is a comparison principle. We also prove existence and stability of solutions. The two challenging difficulties are the singular geometry of the domain and the discontinuity of the Hamiltonian. As far as discontinuous Hamiltonians are concerned, these results seem to be new. They are applied to the study of some models arising in traffic flows. The techniques developed in the present article provide new powerful tools for the analysis of such problems.

This paper deals with the numerical computation of boundary null controls for the 1D waveequation with a potential. The goal is to compute approximations of controls that drivethe solution from a prescribed initial state to zero at a large enough controllabilitytime. We do not apply in this work the usual duality arguments but explore instead adirect approach in the framework of global Carleman estimates. More precisely, we considerthe control that minimizes over the class of admissible null controls a functionalinvolving weighted integrals of the state and the control. The optimality conditions showthat both the optimal control and the associated state are expressed in terms of a newvariable, the solution of a fourth-order elliptic problem defined in the space-timedomain. We first prove that, for some specific weights determined by the global Carlemaninequalities for the wave equation, this problem is well-posed. Then, in the framework ofthe finite element method, we introduce a family of finite-dimensional approximate controlproblems and we prove a strong convergence result. Numerical experiments confirm theanalysis. We complete our study with several comments.

The adjoint method, recently introduced by Evans, is used to study obstacle problems,weakly coupled systems, cell problems for weakly coupled systems of Hamilton − Jacobiequations, and weakly coupled systems of obstacle type. In particular, new results aboutthe speed of convergence of some approximation procedures are derived.

We address in this article the computation of the convex solutions of the Dirichletproblem for the real elliptic Monge − Ampère equation for general convex domains in twodimensions. The method we discuss combines a least-squares formulation with a relaxationmethod. This approach leads to a sequence of Poisson − Dirichlet problems and anothersequence of low dimensional algebraic eigenvalue problems of a new type. Mixed finiteelement approximations with a smoothing procedure are used for the computer implementationof our least-squares/relaxation methodology. Domains with curved boundaries are easilyaccommodated. Numerical experiments show the convergence of the computed solutions totheir continuous counterparts when such solutions exist. On the other hand, when classicalsolutions do not exist, our methodology produces solutions in a least-squares sense.

We apply four different methods to study an intrinsically bang-bang optimal control problem. We study first a relaxed problem that we solve with a naive nonlinear programming approach. Since these preliminary results reveal singular arcs, we then use Pontryagin’s Minimum Principle and apply multiple indirect shooting methods combined with homotopy approach to obtain an accurate solution of the relaxed problem. Finally, in order to recover a purely bang-bang solution for the original problem, we use once again a nonlinear programming approach.

We provide a deterministic-control-based interpretation for a broad class of fullynonlinear parabolic and elliptic PDEs with continuous Neumann boundary conditions in asmooth domain. We construct families of two-person games depending on a small parameterε which extend those proposed by Kohn and Serfaty [21]. These new games treat a Neumann boundary conditionby introducing some specific rules near the boundary. We show that the value functionconverges, in the viscosity sense, to the solution of the PDE as ε tendsto zero. Moreover, our construction allows us to treat both the oblique and the mixed typeDirichlet–Neumann boundary conditions.

In the context of a variational model for the epitaxial growth of strained elastic films,we study the effects of the presence of anisotropic surface energies in the determinationof equilibrium configurations. We show that the threshold effect that describes thestability of flat morphologies in the isotropic case remains valid for weak anisotropies,but is no longer present in the case of highly anisotropic surface energies, where we showthat the flat configuration is always a local minimizer of the total energy. Following theapproach of [N. Fusco and M. Morini, Equilibrium configurations of epitaxially strainedelastic films: second order minimality conditions and qualitative properties of solutions.Preprint], we obtain these results by means of a minimality criterion based on thepositivity of the second variation.

In a recent article [B. Bonnard, J.-B. Caillau, R. Sinclair and M. Tanaka, Ann. Inst. Henri Poincaré, Anal. Non Linéaire 26 (2009) 1081–1098], we relate the computation of the conjugate and cut loci of a family of metrics on two-spheres of revolution whose polar form is g = dϕ2 + m(ϕ)dθ2 to the period mapping of the ϕ-variable. One purpose of this article is to use this relation to evaluate the cut and conjugate loci for a family of metrics arising as a deformation of the round sphere and to determine the convexity properties of the injectivity domains of such metrics. These properties have applications in optimal control of space and quantum mechanics, and in optimal transport.

The paper studies discrete/finite-difference approximations of optimal control problemsgoverned by continuous-time dynamical systems with endpoint constraints. Finite-differencesystems, considered as parametric control problems with the decreasing step ofdiscretization, occupy an intermediate position between continuous-time and discrete-time(with fixed steps) control processes and play a significant role in both qualitative andnumerical aspects of optimal control. In this paper we derive an enhanced version of theApproximate Maximum Principle for finite-difference control systems, which is new even forproblems with smooth endpoint constraints on trajectories and occurs to be the firstresult in the literature that holds for nonsmooth objectives and endpoint constraints. Theresults obtained establish necessary optimality conditions for constrained nonconvexfinite-difference control systems and justify stability of the Pontryagin MaximumPrinciple for continuous-time systems under discrete approximations.

The ill-posed problem of solving linear equations in the space of vector-valued finiteRadon measures with Hilbert space data is considered. Approximate solutions are obtainedby minimizing the Tikhonov functional with a total variation penalty. The well-posednessof this regularization method and further regularization properties are mentioned.Furthermore, a flexible numerical minimization algorithm is proposed which convergessubsequentially in the weak* sense and with rate 𝒪(n-1) in terms of the functional values. Finally, numerical results for sparsedeconvolution demonstrate the applicability for a finite-dimensional discrete data spaceand infinite-dimensional solution space.

We study the Fokker–Planck equation as the many-particle limit of a stochastic particlesystem on one hand and as a Wasserstein gradient flow on the other. We write thepath-space rate functional, which characterises the large deviations from the expectedtrajectories, in such a way that the free energy appears explicitly. Next we use thisformulation via the contraction principle to prove that the discrete time rate functionalis asymptotically equivalent in the Gamma-convergence sense to the functional derived fromthe Wasserstein gradient discretization scheme.