We consider the shape optimization problem \hbox{$\min\big\{\E(\Gamma)\ :\ \Gamma\in\A,\ \H^1(\Gamma)=l\\big\},$}min{ℰ(Γ):Γ ∈ 𝒜,ℋ1(Γ) = l}, where ℋ1 is the one-dimensional Hausdorffmeasure and𝒜is an admissible class of one-dimensional setsconnecting some prescribed set of points \hbox{$\D=\{D_1,\dots,D_k\}\subset\R^d$}𝒟 =  { D1,...,Dk }  ⊂ Rd. The cost functional ℰ(Γ) is theDirichlet energy of Γ defined through the Sobolev functions onΓ vanishing on the pointsDi. We analyze the existence of a solutionin both the families of connected sets and of metric graphs. At the end, several explicitexamples are discussed.

This paper is concerned with the internal distributed control problem for the 1D Schrödinger equation, i ut(x,t) = −uxx+α(x) u+m(u) u, that arises in quantum semiconductor models. Here m(u) is a non local Hartree–type nonlinearity stemming from the coupling with the 1D Poisson equation, and α(x) is a regular function with linear growth at infinity, including constant electric fields. By means of both the Hilbert Uniqueness Method and the contraction mapping theorem it is shown that for initial and target states belonging to a suitable small neighborhood of the origin, and for distributed controls supported outside of a fixed compact interval, the model equation is controllable. Moreover, it is shown that, for distributed controls with compact support, the exact controllability problem is not possible.

A 3D-2D dimension reduction for −Δ1 is obtained. A power law approximation from −Δp as p → 1 in terms of Γ-convergence, duality and asymptotics for least gradient functions has also been provided.

This paper addresses a new differential game problem with forward-backward doubly stochastic differential equations. There are two distinguishing features. One is that our game systems are initial coupled, rather than terminal coupled. The other is that the admissible control is required to be adapted to a subset of the information generated by the underlying Brownian motions. We establish a necessary condition and a sufficient condition for an equilibrium point of nonzero-sum games and a saddle point of zero-sum games. To illustrate some possible applications, an example of linear-quadratic nonzero-sum differential games is worked out. Applying stochastic filtering techniques, we obtain an explicit expression of the equilibrium point.

This paper provides a convergent numerical approximation of the Pareto optimal set for finite-horizon multiobjective optimal control problems in which the objective space is not necessarily convex. Our approach is based on Viability Theory. We first introduce a set-valued return function V and show that the epigraph of V equals the viability kernel of a certain related augmented dynamical system. We then introduce an approximate set-valued return function with finite set-values as the solution of a multiobjective dynamic programming equation. The epigraph of this approximate set-valued return function equals to the finite discrete viability kernel resulting from the convergent numerical approximation of the viability kernel proposed in [P. Cardaliaguet, M. Quincampoix and P. Saint-Pierre. Birkhauser, Boston (1999) 177–247. P. Cardaliaguet, M. Quincampoix and P. Saint-Pierre, Set-Valued Analysis 8 (2000) 111–126]. As a result, the epigraph of the approximate set-valued return function converges to the epigraph of V. The approximate set-valued return function finally provides the proposed numerical approximation of the Pareto optimal set for every initial time and state. Several numerical examples illustrate our approach.

The aim of this paper is to answer the question: Do the controls of a vanishing viscosity approximation of the one dimensional linear wave equation converge to a control of the conservative limit equation? The characteristic of our viscous term is that it contains the fractional power α of the Dirichlet Laplace operator. Through the parameter α we may increase or decrease the strength of the high frequencies damping which allows us to cover a large class of dissipative mechanisms. The viscous term, being multiplied by a small parameter ε devoted to tend to zero, vanishes in the limit. Our analysis, based on moment problems and biorthogonal sequences, enables us to evaluate the magnitude of the controls needed for each eigenmode and to show their uniform boundedness with respect to ε, under the assumption that α∈[0,1)\{½}. It follows that, under this assumption, our starting question has a positive answer.

We prove uniform continuity ofradially symmetric vector minimizersuA(x) = UA(|x|)to multiple integrals ∫BRL**(u(x), |Du(x)|) dx on aball BR ⊂ ℝd,among the Sobolev functions u(·) in A+W01,1 (BR, ℝm), using ajointly convex lsc L∗∗ : ℝm×ℝ → [0,∞]withL∗∗(S,·) evenand superlinear. Besides such basic hypotheses,L∗∗(·,·) is assumed to satisfy alsoa geometrical constraint, which we callquasi − scalar; the simplest example being thebiradial caseL∗∗(|u(x)|,|Du(x)|).Complete liberty is given forL∗∗(S,λ)to take the ∞ value, so that our minimization problem implicitly also representse.g. distributed-parameteroptimal control problems, onconstrained domains, under PDEs or inclusions inexplicit or implicit form. While generic radial functionsu(x) = U(|x|) inthis Sobolev space oscillate wildly as |x| → 0, our minimizingprofile-curve UA(·) is, incontrast, absolutely continuous andtame, in the sense that its“static level” L∗∗(UA(r),0)always increases with r, a original feature of our result.

This paper investigates the output controllability problem of temporal Boolean networkswith inputs (control nodes) and outputs (controlled nodes). A temporal Boolean network isa logical dynamic system describing cellular networks with time delays. Using semi-tensorproduct of matrices, the temporal Boolean networks can be converted into discrete timelinear dynamic systems. Some necessary and sufficient conditions on the outputcontrollability via two kinds of inputs are obtained by providingcorresponding reachable sets. Two examples are given to illustrate the obtainedresults.

We consider a damped abstract second order evolution equation with an additionalvanishing damping of Kelvin–Voigt type. Unlike the earlier work by Zuazua and Ervedoza, wedo not assume the operator defining the main damping to be bounded. First, using aconstructive frequency domain method coupled with a decomposition of frequencies and theintroduction of a new variable, we show that if the limit system is exponentially stable,then this evolutionary system is uniformly − with respect to the calibration parameter −exponentially stable. Afterwards, we prove uniform polynomial and logarithmic decayestimates of the underlying semigroup provided such decay estimates hold for the limitsystem. Finally, we discuss some applications of our results; in particular, the case ofboundary damping mechanisms is accounted for, which was not possible in the earlier workmentioned above.

In this paper we consider a new kind of Mumford–Shah functionalE(u, Ω) for mapsu : ℝm → ℝnwith m ≥ n. The most important novelty is that theenergy features a singular set Su ofcodimension greater than one, defined through the theory of distributional jacobians.After recalling the basic definitions and some well established results, we prove anapproximation property for the energy E(u, Ω) viaΓ −convergence, in the same spirit of the work by Ambrosio andTortorelli [L. Ambrosio and V.M. Tortorelli, Commun. Pure Appl. Math.43 (1990) 999–1036].

The liner parabolic equation \hbox{$\frac{\pp y}{\pp t}-\frac12\,\D y+F\cdot\na y={\vec{1}}_{\calo_0}u$}$\frac{\mathit{\partial y}}{\mathit{\partial t}}\mathrm{-}\frac{\mathrm{1}}{\mathrm{2}}\hspace{0.17em}\mathit{\Delta y}\mathrm{+}\mathit{F}\mathrm{·}\mathrm{\nabla }\mathit{y}\mathrm{=}{{1}}_{{\mathrm{𝒪}}_{\mathrm{0}}}\mathit{u}$ with Neumann boundary condition on a convex open domain 𝒪 ⊂ ℝdwith smooth boundary is exactly null controllable on each finite interval if 𝒪0is an open subset of 𝒪which contains a suitable neighbourhood of the recession cone of \hbox{$\ov\calo$}$\overline{\mathrm{𝒪}}$. Here, F : ℝd → ℝd is a bounded, C1-continuous function, and F = ∇g, where g is convex and coercive.

In this article, we consider a swimmer (i.e. a self-deformable body) immersed in a fluid, the flow of which is governed by the stationary Stokes equations. This model is relevant for studying the locomotion of microorganisms or micro robots for which the inertia effects can be neglected. Our first main contribution is to prove that any such microswimmer has the ability to track, by performing a sequence of shape changes, any given trajectory in the fluid. We show that, in addition, this can be done by means of arbitrarily small body deformations that can be superimposed to any preassigned sequence of macro shape changes. Our second contribution is to prove that, when no macro deformations are prescribed, tracking is generically possible by means of shape changes obtained as a suitable combination of only four elementary deformations. Eventually, still considering finite dimensional deformations, we state results about the existence of optimal swimming strategies on short time intervals, for a wide class of cost functionals.

This paper studies the periodic feedback stabilization of the controlled linear time-periodic ordinary differential equation: ẏ(t) = A(t)y(t) + B(t)u(t), t ≥ 0, where [A(·), B(·)] is a T-periodic pair, i.e., A(·) ∈ L∞(ℝ+; ℝn×n) and B(·) ∈ L∞(ℝ+; ℝn×m) satisfy respectively A(t + T) = A(t) for a.e. t ≥ 0 and B(t + T) = B(t) for a.e. t ≥ 0. Two periodic stablization criteria for a T-period pair [A(·), B(·)] are established. One is an analytic criterion which is related to the transformation over time T associated with A(·); while another is a geometric criterion which is connected with the null-controllable subspace of [A(·), B(·)]. Two kinds of periodic feedback laws for a T-periodically stabilizable pair [ A(·), B(·) ] are constructed. They are accordingly connected with two Cauchy problems of linear ordinary differential equations. Besides, with the aid of the geometric criterion, we find a way to determine, for a given T-periodic A(·), the minimal column number m, as well as a time-invariant n×m matrix B, such that the pair [A(·), B] is T-periodically stabilizable.