A tracking problem is consideredin the context of a class $\mathcal{S}$ of multi-input,multi-output, nonlinear systems modelled by controlled functionaldifferential equations. The class contains, as a prototype, allfinite-dimensional, linear, m-input, m-output, minimum-phasesystems with sign-definite “high-frequency gain". The first controlobjective is tracking of reference signals r by the output y ofany system in $\mathcal{S}$ : given $\lambda \geq 0$ , construct afeedback strategy which ensures that, for every r (assumed boundedwith essentially bounded derivative) and every system of class $\mathcal{S}$ , the tracking error $e = y-r$ is such that, in the case $\lambda >0$ , $\limsup_{t\rightarrow\infty}\|e(t)\|<\lambda$ or, inthe case $\lambda=0$ , $\lim_{t\rightarrow\infty}\|e(t)\| = 0$ . Thesecond objective is guaranteed output transient performance: theerror is required to evolve within a prescribed performance funnel $\mathcal{F}_\varphi$ (determined by a function φ). Forsuitably chosen functions α, ν and θ, bothobjectives are achieved via a control structure of the form $u(t)=-\nu (k(t))\theta (e(t))$ with $k(t)=\alpha (\varphi(t)\|e(t)\|)$ , whilst maintaining boundedness of the control andgain functions u and k. In the case $\lambda=0$ , the feedbackstrategy may be discontinuous: to accommodate this feature, aunifying framework of differential inclusions is adopted in theanalysis of the general case $\lambda \geq 0$ .

We are concerned with the Lipschitz modulus of the optimal set mappingassociated with canonically perturbed convex semi-infinite optimizationproblems. Specifically, the paper provides a lower and an upper bound forthis modulus, both of them given exclusively in terms of the problem's data.Moreover, the upper bound is shown to be the exact modulus when the numberof constraints is finite. In the particular case of linear problems theupper bound (or exact modulus) adopts a notably simplified expression. Ourapproach is based on variational techniques applied to certain difference ofconvex functions related to the model. Some results of [M.J. Cánovas et al., J. Optim. Theory Appl. (2008) Online First](which go back to [M.J. Cánovas, J. Global Optim.41 (2008) 1–13] and [Ioffe, Math. Surveys55 (2000) 501–558; Control Cybern.32 (2003) 543–554])constitute the starting point of the present work.

We apply Robin penalization to Dirichlet optimal control problemsgoverned by semilinear elliptic equations. Error estimates in terms of the penalization parameter are stated. The results are compared with some previous ones in the literature and are checked by a numerical experiment. A detailed study of the regularity of the solutions of the PDEs is carried out.

In this paper we show that a linear variational inequality over an infinite dimensional real Hilbert spaceadmits solutions for every nonempty bounded closed and convex set, if and only if the linear operator involved in the variational inequality is pseudo-monotone in the sense of Brezis.

We show how to capture the gradient concentration of the solutions of Dirichlet-type problems subjected to large sources of order ${1\over \sqrt \varepsilon}$ concentrated on an ε-neighborhood of a hypersurface of the domain. To this end we define the gradient Young-concentration measures generated by sequences of finite energy and establish a very simple characterization of these measures.

We study minimal surfaces in sub-Riemannian manifolds with sub-Riemannian structures of co-rank one. These surfaces can be defined as the critical points of the so-called horizontal area functional associated with the canonical horizontal area form. We derive the intrinsic equation in the general case and then consider in greater detail 2-dimensional surfaces in contact manifolds of dimension 3. We show that in this case minimal surfaces are projections of a special class of 2-dimensional surfaces in the horizontal spherical bundle over the base manifold. The singularities of minimal surfaces turn out to be the singularities of this projection, and we give a complete local classification of them. We illustrate our results by examples in the Heisenberg group and the group of roto-translations. 


We consider a class of second-gradient elasticity models for which the internal potential energy is taken as the sum of a convex function of the second gradient of the deformation and a general function of the gradient. However, in consonance with classical nonlinear elasticity, the latter is assumed to grow unboundedly as the determinant of the gradient approaches zero. While the existence of a minimizer is routine, the existence of weak solutions is not, and we focus our efforts on that question here. In particular, we demonstrate that the determinant of the gradient of any admissible deformation with finite energy is strictly positive on the closure of the domain. With this in hand, Gâteaux differentiability of the potential energy at a minimizer is automatic, yielding the existence of a weak solution. We indicate how our results hold for a general class of boundary value problems, including “mixed” boundary conditions. For each of the two possible pure displacement formulations (in second-gradient problems), we show that the resulting deformation is an injective mapping, whenever the imposed placement on the boundary is itself the trace of an injective map.

In this work we study the structure of approximatesolutions of autonomous variational problems with a lowersemicontinuous strictly convex integrand f : Rn ×Rn $\to$ R 1 $\cup$ $\{\infty\}$ , where Rn is the n-dimensional Euclideanspace. We obtain a full description of the structure of theapproximate solutions which is independent of the length of theinterval, for all sufficiently large intervals.

We consider continuous reformulations of the Euclidean travelling salesperson problem (TSP), based on certain clustering problemformulations. These reformulations allow us to apply a generalisation with perturbations of the Weiszfeld algorithm in an attempt tofind local approximate solutions to the Euclidean TSP.

We consider a mesoscopic model for phase transitions in a periodic mediumand we construct multibump solutions.The rational perturbative case is dealt with by explicitasymptotics.

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 paper is concerned with the sampled-data based adaptivelinear quadratic (LQ) control of hybrid systems with bothunmeasurable Markov jump processes and stochastic noises.By the least matching error estimation algorithm, parameter estimatesare presented. By a double-step (DS) sampling approach and the certaintyequivalence principle, a sampled-data based adaptive LQ control isdesigned. The DS-approach is characterized by a comparatively largeestimation step for parameter estimation and a sufficient small controlstep for control updating. Under mild conditions, the closed-loop systemis shown to be stable. It is found that the key factor determining theperformance index is the estimation step rather than the control step.When the estimation step becomes too small, the system performance willbecome worse. When the estimation step is fixed, the system performancecan indeed be improved by reducing the control step, but cannot reachthe optimal value. The index difference between the sampled-data basedadaptive LQ control and the conventional LQ optimal control is asymptoticallybounded by a constant depending on the estimation step and the prioriinformation of the parameter set.