The Linear-Quadratic (LQ) optimal control problem is studied for aclass of first-order hyperbolic partial differential equation modelsby using a nonlinear infinite-dimensional (distributed parameter) Hilbert state-spacedescription. First the dynamical properties of the linearized modelaround some equilibrium profile are studied. Next the LQ-feedbackoperator is computed by using the corresponding operator Riccatialgebraic equation whose solution is obtained via a relatedmatrix Riccati differential equation in the space variable. Then thelatter is applied to the nonlinear model, and the resultingclosed-loop system dynamical performances are analyzed.

We characterize quasi-static rate-independent evolutions, by means of their graph parametrization, in terms of a couple of equations: the first gives stationarity while the second provides the energy balance. An abstract existence result is given for functionals ℱ of class C1 in reflexive separable Banach spaces. We provide a couple of constructive proofs of existence which share common features with the theory of minimizing movements for gradient flows. Moreover, considering a sequence of functionals ℱn and its Γ-limit ℱ we provide, under suitable assumptions, a convergence result for the associated quasi-static evolutions. Finally, we apply this approach to a phase field model in brittle fracture.

In this paper we prove a unique continuationresult for a cascade system of parabolic equations, in which the solution of the firstequation is (partially) used as a forcing term for the second equation. As aconsequence we prove the existence of ε-insensitizing controls for someparabolic equations when the control region and the observability region do not intersect.

In this paper we propose an extension of the proximal point method to solve minimization problems with quasiconvex objective functions on Hadamard manifolds. To reach this goal, we initially extend the concepts of regular and generalized subgradient from Euclidean spaces to Hadamard manifolds and prove that, in the convex case, these concepts coincide with the classical one. For the minimization problem, assuming that the function is bounded from below, in the quasiconvex and lower semicontinuous case, we prove the convergence of the iterations given by the method. Furthermore, under the assumptions that the sequence of proximal parameters is bounded and the function is continuous, we obtain the convergence to a generalized critical point. In particular, our work extends the applications of the proximal point methods for solving constrained minimization problems with nonconvex objective functions in Euclidean spaces when the objective function is convex or quasiconvex on the manifold.

This paper is concerned with the following periodic Hamiltonianelliptic system

$ \{ -\Delta \varphi+V(x)\varphi=G_\psi(x,\varphi,\psi)\ \hbox{in }\mathbb{R}^N, \\-\Delta \psi+V(x)\psi=G_\varphi(x,\varphi,\psi)\ \hbox{in }\mathbb{R}^N, \\\varphi(x)\to 0\ \hbox{and }\psi(x)\to0\ \hbox{as }|x|\to\infty.$

Assuming the potential V is periodic and 0 lies in a gap of $\sigma(-\Delta+V)$ , $G(x,\eta)$ is periodic in x andasymptotically quadratic in $\eta=(\varphi,\psi)$ , existence andmultiplicity of solutions areobtained via variational approach.


In the setting ofa real Hilbert space ${\cal H}$, we investigate the asymptotic behavior, as time t goes to infinity, of trajectories of second-order evolutionequations

          ü(t) + γ$\dot{u}$(t) + ∇ϕ(u(t)) + A(u(t)) = 0,

where ∇ϕ is the gradient operator of a convexdifferentiable potential function ϕ : ${\cal H}\to \R$, A : ${\cal H}\to {\cal H}$ is a maximal monotone operator which is assumed to beλ-cocoercive, and γ > 0 is a damping parameter.Potential and non-potential effects are associated respectively to∇ϕ and A. Under condition λγ2 > 1, it is proved that each trajectory asymptotically weaklyconverges to a zero of ∇ϕ + A. This condition, whichonly involves the non-potential operator and the dampingparameter, is sharp and consistent with time rescaling. Passingfrom weak to strong convergence of the trajectories is obtained byintroducing an asymptotically vanishing Tikhonov-like regularizingterm. As special cases, we recover the asymptotic analysis of theheavy ball with friction dynamic attached to a convex potential, thesecond-order gradient-projection dynamic, and the second-orderdynamic governed by the Yosida approximation of a general maximalmonotone operator. The breadth and flexibility of the proposed framework is illustrated through applications in the areas of constrained optimization,dynamical approach to Nash equilibria for noncooperative games, and asymptotic stabilization in the case of a continuum of equilibria.

The problem of partial exact boundary controllability and exponential stability for the higher-dimensional linear system of thermoelasticity isconsidered. By introducing a velocity feedback on part of the boundary of the thermoelastic body, which is clamped along the rest of its boundary, toincrease the loss of energy, we prove that the energy in the system of thermoelasticity decays to zero exponentially. We also give a positive answer toa related open question raised by Alabau and Komornik for the Lamé system. Via Russell's “Controllability via Stabilizability" principle, we thenprove that the thermoelastic system is partially controllable with boundary controls without smallness restrictions on the coupling parameters. 


This paper is addressed to proving a new Carleman estimate for stochastic parabolic equations. Compared to the existing Carleman estimate in this respect (see [S. Tang and X. Zhang, SIAM J. Control Optim. 48 (2009) 2191–2216.], Thm. 5.2), one extra gradient term involving in that estimate is eliminated. Also, our improved Carleman estimate is established by virtue of the known Carleman estimate for deterministic parabolic equations. As its application, we prove the existence of insensitizing controls for backward stochastic parabolic equations. As usual, this insensitizing control problem can be reduced to a partial controllability problem for a suitable cascade system governed by a backward and a forward stochastic parabolic equation. In order to solve the latter controllability problem, we need to use our improved Carleman estimate to establish a suitable observability inequality for some linear cascade stochastic parabolic system, while the known Carleman estimate for forward stochastic parabolic equations seems not enough to derive the desired inequality.

Using some results proved in De Pascale and Pratelli [Calc. Var. Partial Differ. Equ.14 (2002) 249-274] (and De Pascale et al. [Bull. London Math. Soc.36 (2004) 383-395]) and a suitable interpolation technique, we show that the transport density relative to an Lp source is also an Lp function for any $1\leq p\leq +\infty$ .

Our concern is the computation of optimal shapes in problems involving(−Δ)1/2. We focus on the energyJ(Ω) associated to the solution uΩ of thebasic Dirichlet problem( − Δ)1/2uΩ = 1in Ω, u = 0 in Ωc. We show that regularminimizers Ω of this energy under a volume constraint are disks. Our proof goes throughthe explicit computation of the shape derivative (that seems to be completely new in thefractional context), and a refined adaptation of the moving plane method.

This paper considers the inversion problem related to themanipulation of quantumsystems using laser-matter interactions. The focusis on the identification of the field free Hamiltonian and/orthe dipole moment of aquantum system. The evolution of the system is given by the Schrödingerequation. The available data are observations as a function of timecorresponding to dynamics generated by electric fields. Thewell-posedness of the problem is proved, mainly focusing on the uniqueness ofthe solution. A numerical approach is also introduced with anillustration of its efficiency on a test problem.

In this paper we prove uniqueness results for the renormalized solution, if it exists, of a class ofnon coercive nonlinear problems whose prototype is
 $$\left\{- \hbox{div}( a(x)(1+|\nabla u|^{2})^{\frac{p-2}{2}}\nabla u) +b(x)(1+|\nabla u|^{2})^{\frac{\lambda}{2}} =f\hbox{in}\quad\Omega, u=0\hbox{on} \quad \partial\Omega,\right.$$ 
where Ω is a bounded open subset of ${\mathbb{R}}^N$ , N > 2, 2-1/N < p < N , a belongs to L ∞(Ω), $a(x)\ge\alpha_0>0$ ,f is a function inL 1(Ω), b is a function in $L^r(\Omega)$ and 0 ≤ λ < λ *(N,p,r), for some r and λ *(N,p,r).


We consider, in an open subset Ω of ${\mathbb R}^N$ , energies depending on the perimeter of a subset $E\subset\Omega$ (or some equivalent surface integral) and on a function u which isdefined only on $\Omega\setminus E$ . We compute the lower semicontinuous envelopeof such energies. This relaxation has to take intoaccount the fact that in the limit, the “holes” E maycollapse into a discontinuity of u, whose surface will be countedtwice in the relaxed energy. We discuss some situations where suchenergies appear, and give, as an application, a new proofof convergence for an extensionof Ambrosio-Tortorelli's approximation to the Mumford-Shah functional.


We study existence and approximation of non-negative solutions of partial differential equations of the type 
 $$\partial_t u - \div (A(\nabla (f(u))+u\nabla V )) = 0 \qquad \mbox{in } (0,+\infty )\times \mathbb{R}^n,\qquad\qquad (0.1)$$ where A is a symmetric matrix-valued function of the spatial variable satisfying a uniform ellipticity condition, $f:[0,+\infty) \rightarrow[0,+\infty)$ is a suitable non decreasing function, $V:\mathbb{R}^n \rightarrow\mathbb{R}$ is a convex function.Introducing the energy functional $\phi(u)=\int_{\mathbb{R}^n} F(u(x))\,{\rm d}x+\int_{\mathbb{R}^n}V(x)u(x)\,{\rm d}x$ ,where F is a convex function linked to f by $f(u) = uF'(u)-F(u)$ ,we show that u is the “gradient flow” of ϕ with respect to the2-Wasserstein distance between probability measures onthe space $\mathbb{R}^n$ , endowed with the Riemannian distance induced by $A^{-1}.$ In the case of uniform convexity of V, long time asymptotic behaviour and decay rate to the stationary statefor solutions of equation (0.1) are studied.A contraction property in Wasserstein distance for solutions of equation (0.1)is also studied in a particular case.

We study the local exponential stabilization of the 2D and 3DNavier-Stokes equations in a bounded domain, around a givensteady-state flow, by means of a boundary control. We look for acontrol so that the solution to the Navier-Stokes equations be astrong solution. In the 3D case, such solutions may exist if theDirichlet control satisfies a compatibility condition with theinitial condition. In order to determine a feedback law satisfyingsuch a compatibility condition, we consider an extended systemcoupling the Navier-Stokes equations with an equation satisfied bythe control on the boundary of the domain. We determine a linearfeedback law by solving a linear quadratic control problem for thelinearized extended system. We show that this feedback law alsostabilizes the nonlinear extended system.

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.

The problem of boundary stabilization for the isotropic linear elastodynamic system and the wave equation with Ventcel's conditions are considered (see [12]). The boundary observability and the exact controllability were etablished in [11]. We prove here the enegy decay to zero for the elastodynamic system with stationary Ventcel's conditions by introducing a nonlinear boundary feedback. We also give a boundary feedback leading to arbitrarily large energy decay rates for the elastodynamic system with evolutive Ventcel's conditions. A spectral study proves, finally, that the natural feedback is not sufficient to assure the exponential decay in the case of the wave equation with Ventcel's conditions.

We consider the spatial behavior of the velocity field u(x, t) of a fluid filling the whole space $\xR^n$ ($n\ge2$) for arbitrarily small values of the time variable. We improve previous results on the spatial spreading by deducing the necessary conditions $\int u_h(x,t)u_k(x,t)\,{\rm d}x=c(t)\delta_{h,k}$ under more general assumptions on the localization of u. We also give some new examples of solutions which have a stronger spatial localization than in the generic case.

For a one-dimensional nonlocal nonconvex singular perturbation problemwith a noncoercive periodic well potential,we prove a Γ-convergence theorem and show compactnessup to translationin all Lp and the optimal Orlicz space for sequences of boundedenergy. This generalizes work of Alberti, Bouchitté and Seppecher(1994) for the coercive two-well case.The theorem has applications to a certain thin-film limit ofthe micromagnetic energy.

We study a quasiconvex conjugation that transforms the level sum of functions into the pointwise sum of their conjugates and derivenew duality results for the minimization of the max of two quasiconvex functions. Following Barron and al., we show that the levelsum provides quasiconvex viscosity solutions for Hamilton-Jacobi equations in which the initial condition is a general continuousquasiconvex function which is not necessarily Lipschitz or bounded.