A new approach to irreversible quasistatic fracture growth is given, by means of Young measures.The study concerns a cohesive zone model with prescribed crack path, when the material gives different responses to loading and unloading phases.In the particular situation of constant unloading response,the result contained in [G. Dal Maso and C. Zanini,Proc. Roy. Soc. Edinburgh Sect. A137 (2007) 253–279] is recovered.In this case, the convergence of the discrete time approximationsis improved.

We consider the flow of gas through pipelines controlled by a compressorstation. Under a subsonic flow assumption we prove the existenceof classical solutions for a given finite time interval.The existence result is used to construct Riemannian feedback laws and to prove a stabilization result for a coupled system of gas pipes with a compressorstation. We introduce a Lyapunov function and prove exponential decay with respect to the L2-norm.

We investigate a global-in-time variational approach to abstract evolution by means of the weighted energy-dissipation functionals proposed by Mielke and Ortiz [ESAIM: COCV14 (2008) 494–516]. In particular, we focus on gradient flows in Hilbert spaces. The main result is the convergence of minimizers and approximate minimizers of these functionals to the unique solution of the gradient flow. Sharp convergence rates are provided and the convergence analysis is combined with time-discretization. Applications of the theory to various classes of parabolic PDE problems are presented. In particular, we focus on two examples of microstructure evolution from [S. Conti and M. Ortiz, J. Mech. Phys. Solids56 (2008) 1885–1904.].

We provide a detailed analysis of the minimizers of the functional $u \mapsto \int_{\Bbb R^n} |\nabla u|^2 + D \int_{\Bbb R^n} |u|^{\gamma}$ , $\gamma \in (0, 2)$ , subject to the constraint $\|u\|_{L^2} = 1$ . This problem, e.g., describes the long-time behavior of the parabolic Anderson in probability theory or ground state solutions of a nonlinear Schrödinger equation. While existence can be proved with standard methods, we show that the usual uniqueness results obtained with PDE-methods can be considerably simplified by additional variational arguments. In addition, we investigate qualitative properties of the minimizers and also study their behavior near the critical exponent 2.

We study the stabilization of global solutions of theKawahara (K) equation in a bounded interval, under the effect ofa localized damping mechanism. The Kawahara equation is a modelfor small amplitude long waves. Using multiplier techniques andcompactness arguments we prove the exponential decay of the solutions of the (K) model. The proofrequires of a unique continuation theorem and the smoothing effectof the (K) equation on the real line, which are proved in this work.

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 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.

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.

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

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 consider the following classical autonomous variational problem

\[ \textrm{minimize\,} \left\{F(v)=\int_a^b f(v(x),v'(x))\ \dx\,:\,v\in AC([a,b]), \;v(a)=\alpha,\; v(b)=\beta\right\},\]

where the Lagrangian f is possibly neither continuous, nor convex, nor coercive.We prove a monotonicity property of the minimizers stating that they satisfy the maximum principle or the minimum one. By virtue of such a property, applying recent results concerning constrained variational problems, we derive a relaxation theorem, the DuBois-Reymond necessary condition and some existenceor non-existence criteria.

Convex duality is a powerful framework for solving non-smooth optimal control problems. However, for problems set in non-reflexive Banach spaces such as L1(Ω) or BV(Ω), the dual problem is formulated in a space which has difficult measure theoretic structure. The predual problem, on the other hand, can be formulated in a Hilbert space and entails the minimization of a smooth functional with box constraints, for which efficient numerical methods exist. In this work, elliptic control problems with measures and functions of bounded variation as controls are considered. Existence and uniqueness of the corresponding predual problems are discussed, as is the solution of the optimality systems by a semismooth Newton method. Numerical examples illustrate the structural differences in the optimal controls in these Banach spaces, compared to those obtained in corresponding Hilbert space settings.

The motivation of this article is double. First of all we provide a geometrical framework to the application of the smooth continuation method in optimal control, where the concept of conjugate points is related to the convergence of the method. In particular, it can be applied to the analysis of the global optimality properties of the geodesic flows of a family of Riemannian metrics. Secondly, this study is used to complete the analysis of two-level dissipative quantum systems, where the system is depending upon three physical parameters, which can be used as homotopy parameters, and the time-minimizing trajectory for a prescribed couple of extremities can be analyzed by making a deformation of the Grushin metric on a two-sphere of revolution.