In this paper, using direct and inverse images for fractional stochastic tangent sets, weestablish the deterministic necessary and sufficient conditions which control that thesolution of a given stochastic differential equation driven by the fractional Brownianmotion evolves in some particular sets K. As a consequence, a comparisontheorem is obtained.

In this paper, we study the multiplicity of solutions for a class of noncooperativep-Laplacian operator elliptic system. Under suitable assumptions, weobtain a sequence of solutions by using the limit index theory.

Under an appropriate oscillating behaviour either at zero or at infinity of the nonlinearterm, the existence of a sequence of weak solutions for an eigenvalue Dirichlet problem onthe Sierpiński gasket is proved. Our approach is based on variational methods and on someanalytic and geometrical properties of the Sierpiński fractal. The abstract results areillustrated by explicit examples.

This paper concerns continuous dependence estimates for Hamilton-Jacobi-Bellman-Isaacsoperators. We establish such an estimate for the parabolic Cauchy problem in the wholespace  [0, +∞) × ℝn and, under some periodicity and eitherellipticity or controllability assumptions, we deduce a similar estimate for the ergodicconstant associated to the operator. An interesting byproduct of the latter result will bethe local uniform convergence for some classes of singular perturbation problems.

Traffic flow is modeled by a conservation law describing the density of cars. It isassumed that each driver chooses his own departure time in order to minimize the sum of adeparture and an arrival cost. There are N groups of drivers, Thei-th group consists of κidrivers, sharing the same departure and arrival costsϕi(t),ψi(t).For any given population sizesκ1,...,κn,we prove the existence of a Nash equilibrium solution, where no driver can lower his owntotal cost by choosing a different departure time. The possible non-uniqueness, and acharacterization of this Nash equilibrium solution, are also discussed.

For robust discretizations of the Navier-Stokes equations with small viscosity, standardGalerkin schemes have to be augmented by stabilization terms due to the indefiniteconvective terms and due to a possible lost of a discrete inf-sup condition. For optimalcontrol problems for fluids such stabilization have in general an undesired effect in thesense that optimization and discretization do not commute. This is the case for thecombination of streamline upwind Petrov-Galerkin (SUPG) and pressure stabilizedPetrov-Galerkin (PSPG). In this work we study the effect of different stabilized finiteelement methods to distributed control problems governed by singular perturbed Oseenequations. In particular, we address the question whether a possible commutation error inoptimal control problems lead to a decline of convergence order. Therefore, we givea priori estimates for SUPG/PSPG. In a numerical study for a flow withboundary layers, we illustrate to which extend the commutation error affects theaccuracy.

In this paper, we study one kind of stochastic recursive optimal control problem for thesystems described by stochastic differential equations with delay (SDDE). In ourframework, not only the dynamics of the systems but also the recursive utility depend onthe past path segment of the state process in a general form. We give the dynamicprogramming principle for this kind of optimal control problems and show that the valuefunction is the viscosity solution of the corresponding infinite dimensionalHamilton-Jacobi-Bellman partial differential equation.

This paper analyzes the continuum model/complete electrode model in the electricalimpedance tomography inverse problem of determining the conductivity parameter fromboundary measurements. The continuity and differentiability of the forward operator withrespect to the conductivity parameter inLp-norms are proved. These analytical resultsare applied to several popular regularization formulations, which incorporate apriori information of smoothness/sparsity on the inhomogeneity through Tikhonovregularization, for both linearized and nonlinear models. Some important properties,e.g., existence, stability, consistency andconvergence rates, are established. This provides some theoretical justifications of theirpractical usage.

In this paper we study the compact and convex setsK ⊆ Ω ⊆ ℝ2 that minimize

\begin{equation*} \int_{\Omega} \dist(\x,K) \,{\rm d}\x +\lambda_1 {\rm Vol}(K)+\lambda_2 {\rm Per}(K) \end{equation*}${\mathrm{\int }}_{\mathit{\Omega }}\mathrm{dist}\mathrm{\left(}{x}\mathit{,K}\mathrm{\right)}\mathrm{d}{x}\mathrm{+}{\mathit{\lambda }}_{\mathrm{1}}\mathrm{Vol}\mathrm{\left(}\mathit{K}\mathrm{\right)}\mathrm{+}{\mathit{\lambda }}_{\mathrm{2}}\mathrm{Per}\mathrm{\left(}\mathit{K}\mathrm{\right)}$

This paper deals with the optimal control problem in which the controlled system isdescribed by a fully coupled anticipated forward-backward stochastic differential delayedequation. The maximum principle for this problem is obtained under the assumption that thediffusion coefficient does not contain the control variables and the control domain is notnecessarily convex. Both the necessary and sufficient conditions of optimality are proved.As illustrating examples, two kinds of linear quadratic control problems are discussed andboth optimal controls are derived explicitly.

We study the initial value problem for the drift-diffusion model arising in semiconductordevice simulation and plasma physics. We show that the corresponding stationary problem inthe whole space ℝn admits a unique stationary solution in ageneral situation. Moreover, it is proved that when n ≥ 3, a uniquesolution to the initial value problem exists globally in time and converges to thecorresponding stationary solution as time tends to infinity, provided that the amplitudeof the stationary solution and the initial perturbation are suitably small. Also, we showthe sharp decay estimate for the perturbation. The stability proof is based on the timeweighted Lp energy method.

We examine shape optimization problems in the context of inexact sequential quadraticprogramming. Inexactness is a consequence of using adaptive finite element methods (AFEM)to approximate the state and adjoint equations (via the dual weightedresidual method), update the boundary, and compute the geometric functional. We present anovel algorithm that equidistributes the errors due to shape optimization anddiscretization, thereby leading to coarse resolution in the early stages and fineresolution upon convergence, and thus optimizing the computational effort. We discuss theability of the algorithm to detect whether or not geometric singularities such as cornersare genuine to the problem or simply due to lack of resolution – a new paradigm inadaptivity.

In this paper we investigate analytic affine control systems \hbox{$\dot{q}$}q̇ = X + uY, u ∈  [a,b] , whereX,Y is an orthonormal frame for a generalized Martinet sub-Lorentzianstructure of order k of Hamiltonian type. We construct normal forms forsuch systems and, among other things, we study the connection between the presence of thesingular trajectory starting at q0 on the boundary of thereachable set from q0 with the minimal number of analyticfunctions needed for describing the reachable set from q0.

We discuss a variational problem defined on couples of functions that are constrained totake values into the 2-dimensional unit sphere. The energy functional contains, besidesstandard Dirichlet energies, a non-local interaction term that depends on the distancebetween the gradients of the two functions. Different gradients are preferred or penalizedin this model, in dependence of the sign of the interaction term. In this paper we studythe lower semicontinuity and the coercivity of the energy and we find an explicitrepresentation formula for the relaxed energy. Moreover, we discuss the behavior of theenergy in the case when we consider multifunctions with two leaves rather than couples offunctions.

We discuss several new results on nonnegative approximate controllability for theone-dimensional Heat equation governed by either multiplicative or nonnegative additivecontrol, acting within a proper subset of the space domain at every moment of time. Ourmethods allow us to link these two types of controls to some extend. The main resultsinclude approximate controllability properties both for the static and mobile controlsupports.