Sensitivity analysis (with respect to the regularization parameter)of the solution of a class of regularized state constrainedoptimal control problems is performed. The theoretical results arethen used to establish an extrapolation-based numerical scheme forsolving the regularized problem for vanishing regularizationparameter. In this context, the extrapolation technique providesexcellent initializations along the sequence of reducingregularization parameters. Finally, the favorable numericalbehavior of the new method is demonstrated and a comparison toclassical continuation methods is provided.

This paper deals with topology optimization of domains subject to a pointwise constraint on the gradient of the state. To realize this constraint, a class of penalty functionals is introduced and the expression of the corresponding topological derivative is obtained for the Laplace equation in two space dimensions. An algorithm based on these concepts is proposed. It is illustrated by some numerical applications.

This paper is devoted to an analysis of vortex-nucleationfor a Ginzburg-Landau functional withdiscontinuous constraint. This functional has been proposedas a model for vortex-pinning, and usuallyaccounts for the energyresulting from the interface of two superconductors. Thecritical applied magnetic field for vortex nucleation is estimated inthe London singular limit,and as a by-product, results concerning vortex-pinning andboundary conditions on the interface are obtained.

Optimal control problems for semilinear elliptic equationswith control constraints and pointwise state constraints arestudied. Several theoretical results are derived, which arenecessary to carry out a numerical analysis for this class ofcontrol problems. In particular, sufficient second-order optimalityconditions, some new regularity results on optimal controls and asufficient condition for the uniqueness of the Lagrange multiplierassociated with the state constraints are presented.

In terms of the normal cone and the coderivative,we provide some necessary and/or sufficient conditions of metric subregularity for(not necessarily closed) convex multifunctions in normed spaces. As applications, we present someerror bound results for (not necessarily lower semicontinuous) convex functions on normedspaces. These results improve and extend some existing error bound results.

The level set method has become widely used in shape optimization where it allows a popular implementation of the steepest descent method. Once coupled with a ersatz material approximation [Allaire et al., J. Comput. Phys.194 (2004) 363–393], a single mesh is only used leading to very efficient and cheap numerical schemes in optimization of structures. However, it has some limitations and cannot be applied in every situation. This work aims at exploring such a limitation. We estimate the systematic error committed by using the ersatz material approximation and, on a model case, explain that they amplifies instabilities by a second order analysis of the objective function.

The paper studies optimal portfolio selection for discrete timemarket models in mean-variance and goal achieving setting. Theoptimal strategies are obtained for models with an observed processthat causes serial correlations of price changes. The optimalstrategies are found to be myopic for the goal-achieving problem andquasi-myopic for the mean variance portfolio.

We prove some new upper and lower bounds for the first Dirichlet eigenvalue of triangles and quadrilaterals. In particular, we improvePólya and Szegö's [Annals of Mathematical Studies 27 (1951)] lower bound for quadrilaterals and extendHersch's [Z. Angew. Math. Phys. 17 (1966) 457–460] upper bound for parallelograms to general quadrilaterals.

In this paper, we study thecontrol system associated with the incompressible 3D Euler system.We show that the velocity field and pressure of the fluid areexactly controllable in projections by the same finite-dimensionalcontrol. Moreover, the velocity is approximately controllable. We also prove that 3D Eulersystem is not exactly controllable by a finite-dimensionalexternal force.

Given a continuous viscosity solution of a Dirichlet-type Hamilton-Jacobi equation, we show that the distance function to the conjugate locus which is associated to this problem is locally semiconcave on its domain. It allows us to provide a simple proof of the fact that the distance function to the cut locus associated to this problem is locally Lipschitz on its domain. This result, which was already an improvement of a previous one by Itoh and Tanaka [Trans. Amer. Math. Soc. 353 (2001) 21–40], is due to Li and Nirenberg [Comm. Pure Appl. Math. 58 (2005) 85–146]. Finally, we give applications of our results in Riemannian geometry. Namely, we show that the distance function to the conjugate locus on a Riemannian manifold is locally semiconcave. Then, we show that if a Riemannian manifold is a C 4-deformation of the round sphere, then all its tangent nonfocal domains are strictly uniformly convex.

In the framework of the linear fracture theory, a commonly-used toolto describe the smooth evolution of a crack embedded in a bounded domain Ω is the so-calledenergy release rate defined as the variation of the mechanicalenergy with respect to the crack dimension. Precisely, thewell-known Griffith's criterion postulates the evolution of thecrack if this rate reaches a critical value. In this work, in the anti-plane scalar case, weconsider the shape design problem which consists in optimizing thedistribution of two materials with different conductivities in Ω in order to reducethis rate. Since this kind of problem is usually ill-posed, wefirst derive a relaxation by using the classical non-convexvariational method. The computation of the quasi-convex envelope ofthe cost is performed by using div-curl Young measures, leads to anexplicit relaxed formulation of the original problem, and exhibits fine microstructure in the form offirst order laminates. Finally, numerical simulations suggest thatthe optimal distribution permits to reduce significantly the value of the energy release rate.

This article is devoted to the optimal control of state equations with memory of the form:

$\dot{x}(t)=F(x(t),u(t), \int_0^{+\infty} A(s) x(t-s) {\rm d}s), \; t>0,$

with initial conditions $x(0)=x, \; x(-s)=z(s), s>0$ .

Denoting by $y_{x, z, u}$ the solution of the previous Cauchy problem and:

$v(x,z):=\inf_{u\in V} \lbrace \int_0^{+\infty} {\rm e}^{-\lambda s } L(y_{x,z,u}(s), u(s)){\rm d}s\rbrace$

where V is a class of admissible controls, we prove that v is the only viscosity solution of an Hamilton-Jacobi-Bellman equation of the form:

$\lambda v(x,z)+H(x,z,\nabla_x v(x,z))+\langle D_z v(x,z), \dot{z} \rangle=0$

in the sense of the theory of viscosity solutions in infinite-dimensions of Crandall and Lions.

Being a unique phenomenon in hybrid systems, mode switchis of fundamental importance in dynamic and control analysis. Inthis paper, we focus on global long-time switching and stabilityproperties of conewise linear systems (CLSs), which are a class oflinear hybrid systems subject to state-triggered switchingsrecently introduced for modeling piecewise linear systems. Byexploiting the conic subdivision structure, the “simple switchingbehavior” of the CLSs is proved. The infinite-time mode switchingbehavior of the CLSs is shown to be critically dependent on twoattracting cones associated with each mode; fundamental propertiesof such cones are investigated. Verifiable necessary andsufficient conditions are derived for the CLSs with infinite modeswitches. Switch-free CLSs are also characterized by exploringthe polyhedral structure and the global dynamical properties. Theequivalence of asymptotic and exponential stability of the CLSs isestablished via the uniform asymptotic stability of the CLSs thatin turn is proved by the continuous solution dependence on initialconditions. Finally, necessary and sufficient stability conditionsare obtained for switch-free CLSs.

The paper deals with the genericity of domain-dependent spectral properties of the Laplacian-Dirichlet operator. In particular we prove that, generically, the squares of the eigenfunctions form a free family. We also show that the spectrum is generically non-resonant. The results are obtained by applying global perturbations of the domains and exploiting analytic perturbation properties.The work is motivated by two applications: an existence result for the problem of maximizing the rate of exponential decay of a damped membrane and an approximate controllability result for the bilinear Schrödinger equation.