In [CITE], Kronrod proves that the connected components of isolevelsets of a continuous function can be endowed with a treestructure. Obviously, the connected components of upper level sets are aninclusion tree, and the same is true for connected components of lower levelsets. We prove that in the case of semicontinuous functions, those trees canbe merged into a single one, which, following its use in image processing, wecall “tree of shapes”. This permits us to solve a classical representationproblem in mathematical morphology: to represent an image in such a way thatmaxima and minima can be computationally dealt with simultaneously. We provethe finiteness of the tree when the image is the result of applying anyextrema killer (a classical denoising filter in image processing). The shapetree also yields an easy mathematical definition of adaptive imagequantization.

We present, analyze, and implement a new method for the design of the stiffest structure subject to a pressure load or a given field of internal forces. Our structure is represented as a subset S of a reference domain, and the complement of S is made of two other “phases”, the “void” and a fictitious “liquid” that exerts a pressure force on its interface with the solid structure. The problem we consider is to minimize the compliance of the structure S, which is the total work of the pressure and internal forces at the equilibrium displacement. In order to prevent from homogenization we add a penalization on the perimeter of S. We propose an approximation of our problem in the framework of Γ-convergence, based on an approximation of our three phases by a smooth phase-field. We detail the numerical implementation of the approximate energies and show a few experiments.

In this paper we discuss the approximate reconstruction of inhomogeneities of small volume. The data used for the reconstruction consist of boundary integrals of the (observed) electromagnetic fields. The numerical algorithms discussed are based on highly accurate asymptotic formulae for the electromagnetic fields in the presence of small volume inhomogeneities.

On an arbitrary reflexive Banach space, we build asymptotic observers for anabstract class of nonlinear control systems with possible compact outputs. Animportant part of this paper is devoted to various examples, where we discussthe existence of persistent inputs which make the system observable. Theseresults make a wide generalization to a nonlinear framework of previous workson the observation problem in infinite dimension (see [11,18,22,26,27,38,40] and other references therein).

Lower semicontinuity results are obtained for multipleintegrals of the kind $\int _{\mathbb{R}^n}\!f(x, \!\nabla_\mu u\!){\rm d} \mu$ ,where μ is a given positive measure on $\mathbb{R}^n$ , and thevector-valued function u belongs to the Sobolev space $H^{1,p}_\mu (\mathbb{R}^n, \mathbb{R}^m)$ associated with μ. The proofs areessentially based on blow-up techniques, and a significant role isplayed therein by the concepts of tangent space and of tangentmeasures to μ. More precisely, for fully general μ, anotion of quasiconvexity for f along the tangent bundle toμ, turns out to be necessary for lower semicontinuity; thesufficiency of such condition is also shown, when μ belongs toa suitable class of rectifiable measures.

We consider minimization problems of the form ${\rm min}_{u\in \varphi +W^{1,1}_0(\Omega)}\int_\Omega [f(Du(x))-u(x)]\, {\rm d}x$ where $\Omega\subseteq \mathbb{R}^N$ is a bounded convex open set, and theBorel function $f\colon \mathbb{R}^N \to [0, +\infty]$ is assumed to beneither convex nor coercive. Under suitable assumptions involvingthe geometry of Ω and the zero level set of f, we provethat the viscosity solution of a related Hamilton–Jacobi equationprovides a minimizer for the integral functional.

In this paper we study the lower semicontinuity problem for a supremalfunctional of the form $F(u,\Omega )= \underset{x\in\Omega}{\rm ess\,sup} f(x,u(x),Du(x))$ with respect to the strong convergence in L ∞(Ω), furnishing a comparison with the analogous theory developed bySerrin for integrals. A sort of Mazur's lemma for gradients of uniformlyconverging sequences is proved.

Controlling growth at crystalline surfaces requires a detailed and quantitative understandingof the thermodynamic and kinetic parameters governing mass transport. Many of theseparameters can be determined by analyzing the isothermal wandering of steps at a vicinal[“step-terrace”] type surface [for a recent review see [4]]. In the case of orthodoxcrystals one finds that these meanderings develop larger amplitudes as the equilibriumtemperature is raised (as is consistent with the statistical mechanical view of themeanderings as arising from atomic interchanges). The classical theory due to Herring,Mullins and others [5], coupled with advances in real-time experimental microscopytechniques, has proven very successful in the applied development of such crystallinematerials. However in 1997 a series of experimental observations on vicinal defects ofheavily boron-doped Silicon crystals revealed that these crystals were quite unorthodox inthe sense that a lowering of the equilibrium temperature led to increased amplitude for theisothermal wanderings of a step edge [3]. In addition, at low temperatures the stepprofile adopted a periodic saw-tooth structure rather than the straight profile predicted by the classical theories. This article examines a stored free energy model for such crystals involving a (higher order) Landau/de Gennes type {``}order parameter{"} term and provides a proof for the existence of a minimizer.

A closed loop parametrical identification procedure forcontinuous-time constant linear systems is introduced. Thisapproach which exhibits good robustness properties with respect toa large variety of additive perturbations is based on thefollowing mathematical tools:(1) module theory;(2) differential algebra;(3) operational calculus.Several concrete case-studies with computer simulationsdemonstrate the efficiency of our on-line identification scheme.

An existence and regularity theorem is proved for integral equationsof convolution type which contain hysteresis nonlinearities. Onthe basis of this result, frequency-domain stability criteria arederived for feedback systems with a linear infinite-dimensionalsystem in the forward path and a hysteresis nonlinearity in thefeedback path. These stability criteria are reminiscent of theclassical circle criterion which applies to static sector-boundednonlinearities. The class of hysteresis operators underconsideration contains many standard hysteresis nonlinearitieswhich are important in control engineering such as backlash (orplay), plastic-elastic (or stop) and Prandtl operators. Whilst themain results are developed in the context of integral equations ofconvolution type, applications to well-posed state space systemsare also considered.

One proves that the steady-state solutions to Navier–Stokes equations with internal controllers are locally exponentially stabilizable by linear feedback controllers provided by a LQ control problem associated with the linearized equation.

For systems with slowly varying parameters the controllability behavior isstudied and the relation to the control sets for the systems with frozenparameters is clarified.

We minimize, with respect to shape, the moment of inertia of a turbine having the givenlowest eigenfrequency of the torsional oscillations. The necessary conditions of optimality in conjunction with certainphysical parameters admit a unique optimal design.

We study the large-time behaviour of thenonlinear oscillator \[\hskip-20mm m\,x'' + f(x') + k\,x=0\,,\] where m, k>0 and f is a monotone real function representingnonlinear friction. We are interested in understanding thelong-time effect of a nonlinear damping term, with specialattention to the model case   $f(x')= A\,|x'|^{\alpha-1}x'$  withα real, A>0. We characterize the existence and behaviourof fast orbits, i.e., orbits that stop in finite time.

Let A 0 be a possibly unbounded positive operator on the Hilbert space H, which is boundedly invertible. Let C 0 be a bounded operator from ${\cal D}\Big(A_0^{\frac{1}{2}}\Big)$ to another Hilbertspace U. We prove that the system of equations $$\ddot z(t)+A_0 z(t) + {\frac{1}{2}}C_0^*C_0\dot z(t) =C_0^*u(t) $$ $$y(t) =-C_0 \dot z(t)+u(t),$$ determines a well-posed linear system with input u and output y.The state of this system is $$x(t) = \left[\begin{matrix}\, z(t) \\ \dot z(t)\end{matrix}\right] \in{\cal D}\left(A_0^{\frac{1}{2}}\right)\times H = X ,$$ where X is the state space. Moreover, we have the energy identity $$\|x(t)\|^2_X-\|x(0)\|_X^2 =\int_0^T\| u(t)\|^2_U {\rm d}t - \int_0^T \|y(t)\|_U^2 {\rm d}t.$$ We show that the system described above is isomorphic to its dual, sothat a similar energy identity holds also for the dual system andhence, the system is conservative. We derive various other propertiesof such systems and we give a relevant example: a wave equation on abounded n-dimensional domain with boundary control and boundaryobservation on part of the boundary.

In this paper we consider a free boundary problem for a nonlinearparabolic partial differential equation. In particular, we areconcerned with the inverse problem, which means we know thebehavior of the free boundary a priori and would like a solution,e.g. a convergent series, in order to determine what thetrajectories of the system should be for steady-state tosteady-state boundary control. In this paper we combine twoissues: the free boundary (Stefan) problem with a quadraticnonlinearity. We prove convergence of a series solution and give adetailed parametric study on the series radius of convergence.Moreover, we prove that the parametrization can indeed can be usedfor motion planning purposes; computation of the open loop motionplanning is straightforward. Simulation results are given and weprove some important properties about the solution. Namely, a weakmaximum principle is derived for the dynamics, stating that themaximum is on the boundary. Also, we prove asymptotic positivenessof the solution, a physical requirement over the entire domain, asthe transient time from one steady-state to another gets large.

Let X be a Banach space and X'its continuous dual. C(X) (resp. C(X')) denotes the set of nonempty convex closed subsets of X (resp. ω*-closed subsets of X') endowed with the topologyof uniform convergence of distance functions on bounded sets. This topologyreduces to the Hausdorff metric topology on the closed and bounded convexsets [16] and in general has a Hausdorff-like presentation [11]. Moreover,this topology is well suited for estimations and constructive approximations [6-9]. We prove here, that under natural qualification conditions, the stability ofthe convergence associated to the topology defined on C(X) (resp. C(X')) is preserved by aclass of linear transformations. Building on these results, and byidentifing each convex function with its epigraph, the stability at thefunctional level is acquired towards some operations of convex analysiswhich play a basic role in convex optimization and duality theory. The keyhypothesis in the qualification conditions ensuring the functional stabilityis the notion of inf-local compactness of a convex function introduced in [28] and expressed in the space X' by the quasi-continuity of itsconjugate. Then we generalize the stability results of McLinden andBergstrom [31] and the ones of Beer and Lucchetti [17] in infinite dimensioncase. Finally we give some applications in convex optimization andmathematical programming in general Banach spaces.

We study the sequence u n , which is solutionof $-{\rm div}(a(x,{\nabla}u_n))+ \Phi''(|u_n|)\,u_n= f_n+ g_n$ in Ω anopen boundedset of R N and un = 0 on ∂Ω, when f n tends to ameasure concentrated on a set of null Orlicz-capacity. We consider the relationbetween this capacity and the N-function Φ, and prove a non-existenceresult.

We construct explicitly an homogeneous feedback for a class ofsingle input, two dimensional and homogeneous systems.

This paper deals with theobservability analysis and the observer synthesis of a class ofnonlinear systems. In the single output case, it is known [4-6] that systems whichare observable independently of the inputs, admit an observablecanonical form. These systems are called uniformly observablesystems. Moreover, a high gain observer for these systems can bedesigned on the basis of this canonical form. In this paper, weextend the above results to multi-output uniformly observablesystems. Corresponding canonical forms are presented andsufficient conditions which permit the design of constant and highgain observers for these systems aregiven.