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.

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.

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.

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.

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.

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.

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.

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.

The numerical solution of ill-posed problems requires suitableregularization techniques. One possible option is to consider timeintegration methods to solve the Showalter differential equationnumerically. The stopping time of the numerical integrator correspondsto the regularization parameter. A number of well-knownregularization methods such as the Landweber iteration or theLevenberg-Marquardt method can be interpreted as variants of theEuler method for solving the Showalter differential equation. Motivated by an analysis of the regularization properties of theexact solution of this equation presented by [U. Tautenhahn, Inverse Problems10 (1994) 1405–1418], we consider a variant of the exponential Euler methodfor solving the Showalter ordinary differential equation. We discuss asuitable discrepancy principle for selecting the step sizes withinthe numerical method and we review the convergence properties of [U. Tautenhahn, Inverse Problems10 (1994) 1405–1418], and of our discrete version [M. Hochbruck et al., Technical Report (2008)].Finally, we present numerical experiments which show that thismethod can be efficiently implemented by using Krylov subspacemethods to approximate the product of a matrix function with avector.

The plane wave stability properties of the conservative schemes of Besse [SIAM J. Numer. Anal.42 (2004) 934–952] and Fei et al. [Appl. Math. Comput.71 (1995) 165–177] for the cubic Schrödinger equation are analysed. Although the two methods possess many of the same conservation properties, we show that their stability behaviour is very different.An energy preserving generalisation of the Fei method with improved stability is presented.

In this paper, we consider linear ordinary differential equations originating in electronic engineering, which exhibit exceedingly rapid oscillation. Moreover, the oscillation model is completely different from the familiar framework of asymptotic analysis of highly oscillatory integrals. Using a Bessel-function identity, we expand the oscillator into asymptotic series, and this allows us to extend Filon-type approach to this setting. The outcome is a time-stepping method that guarantees high accuracy regardless of the rate of oscillation.

We prove the existence of a positive solution to the BVP $$(\Phi(t)u'(t))'=f(t,u(t)),\,\,\,\,\,\,\,\,\,\,\,u'(0)=u(1)=0, $$ imposing some conditions on Φ and f. In particular, weassume $\Phi(t)f(t,u)$ to be decreasing in t. Our methodcombines variational and topological arguments and can be appliedto some elliptic problems in annular domains. An $L_\infty$ boundfor the solution is provided by the $L_\infty$ norm of any testfunction with negative energy.

We present a Monte Carlo technique for sampling from thecanonical distribution in molecular dynamics. The method is built uponthe Nosé-Hoover constant temperature formulation and the generalizedhybrid Monte Carlo method. In contrast to standard hybrid Monte Carlo methodsonly the thermostat degree of freedom is stochastically resampled during a Monte Carlo step.

In this paper, we present an abstract framework which describes algebraically the derivation of order conditions independently of the nature of differential equations considered or the type of integrators used to solve them. Our structure includes a Hopf algebra of functions, whose properties are used to answer several questions of prime interest in numerical analysis. In particular, we show that, under some mild assumptions, there exist integrators of arbitrarily high orders for arbitrary (modified) vector fields.

In long-time numerical integration of Hamiltonian systems,and especially in molecular dynamics simulation,it is important that the energy is well conserved. For symplecticintegrators applied with sufficiently small step size, thisis guaranteed by the existence of a modifiedHamiltonian that is exactly conserved up to exponentially smallterms. This article is concerned with the simplifiedTakahashi-Imada method, which is a modificationof the Störmer-Verlet method that is as easy to implement buthas improved accuracy. This integrator is symmetric andvolume-preserving, but no longer symplectic. We study itslong-time energy conservation and give theoreticalarguments, supported by numerical experiments, whichshow the possibility of a drift in the energy (linear or like a random walk).With respect to energy conservation, this article provides empiricaland theoretical data concerning the importance of using a symplecticintegrator.

In [C.W. Gear, T.J. Kaper, I.G. Kevrekidis and A. Zagaris,SIAM J. Appl. Dyn. Syst. 4 (2005) 711–732],we developeda class of iterative algorithmswithin the contextof equation-free methodsto approximatelow-dimensional,attracting,slow manifoldsin systemsof differential equationswith multiple time scales.For user-specified valuesof a finite numberof the observables,the mth memberof the classof algorithms( $m = 0, 1, \ldots$ )finds iterativelyan approximationof the appropriate zeroof the (m+1)st time derivativeof the remaining variablesanduses this rootto approximate the locationof the pointon the slow manifoldcorresponding to these valuesof the observables.This articleis the firstof two articlesin whichthe accuracy and convergenceof the iterative algorithmsare analyzed.Here,we work directlywith fast-slow systems,in which there isan explicit small parameter, ε,measuring the separationof time scales.We show that,for each $m = 0, 1, \ldots$ ,the fixed pointof the iterative algorithmapproximates the slow manifoldup to and includingterms of ${\mathcal O}(\varepsilon^m)$ .Moreover,for each m,we identify explicitlythe conditionsunder whichthe mth iterative algorithmconverges to this fixed point.Finally,we show thatwhenthe iterationis unstable(orconverges slowly)it may be stabilized(orits convergencemay be accelerated)by applicationof the Recursive Projection Method.Alternatively,the Newton-KrylovGeneralized Minimal Residual Methodmay be used.In the subsequent article,we will considerthe accuracy and convergenceof the iterative algorithmsfor a broader classof systems – in whichthere need not bean explicitsmall parameter – to whichthe algorithms also apply.

This paper deals with modeling the passivebehavior of skeletal muscle tissue includingcertain microvibrations at the cell level. Our approach combines a continuum mechanics model with large deformation and incompressibility at the macroscale with chains of coupled nonlinear oscillators.The model verifies that an externally appliedvibration at the appropriate frequency is able to synchronize microvibrations in skeletal muscle cells.From the numerical analysis point of view, one faces here a partial differential-algebraic equation (PDAE) that after semi-discretization in space by finite elements possessesan index up to three, depending on certain physicalparameters. In this context, the consequences forthe time integration as well as possible remedies are discussed.