Abstract: Abstract A MEMS model with an insulating layer is considered and its reinforced limit is derived by means of a Gamma convergence approach when the thickness of the layer tends to zero. The limiting model inherits the dielectric properties of the insulating layer. PubDate: 2020-05-14

Abstract: Abstract Fractional Navier–Stokes equations—featuring a fractional Laplacian-provide a ‘bridge’ between the Euler equations (zero diffusion) and the Navier–Stokes equations (full diffusion). The problem of whether an initially smooth flow can spontaneously develop a singularity is a fundamental problem in mathematical physics, open for the full range of models—from Euler to Navier–Stokes. The purpose of this work is to present a hybrid, geometric-analytic regularity criterion for solutions to the 3D fractional Navier–Stokes equations expressed as a balance—in the average sense—between the vorticity direction and the vorticity magnitude, key geometric and analytic descriptors of the flow, respectively. PubDate: 2020-05-12

Abstract: Abstract This paper deals with the problem of internal null-controllability of a heat equation posed on a bounded domain with Dirichlet boundary conditions and perturbed by a semilinear nonlocal term. We prove the small-time local null-controllability of the equation. The proof relies on two main arguments. First, we establish the small-time local null-controllability of a \(2 \times 2\) reaction-diffusion system, where the second equation is governed by the parabolic operator \(\tau \partial _t - \sigma \varDelta \), \(\tau , \sigma > 0\). More precisely, this controllability result is obtained uniformly with respect to the parameters \((\tau , \sigma ) \in (0,1) \times (1, + \infty )\). Secondly, we observe that the semilinear nonlocal heat equation is actually the asymptotic derivation of the reaction-diffusion system in the limit \((\tau ,\sigma ) \rightarrow (0,+\infty )\). Finally, we illustrate these results by numerical simulations. PubDate: 2020-05-12

Abstract: Abstract In the present paper, we study the existence of two non-negative solutions for a class of fractional p(x, .)-Laplacian problems with non-negative weight functions. The main tool is the Nehari manifold approach. Moreover, under some suitable assumptions, continuous and compact embeddings results are established. PubDate: 2020-05-11

Abstract: Abstract The topological derivative method is used to solve a pollution sources reconstruction problem governed by a steady-state convection-diffusion equation. To be more precise, we are dealing with a shape optimization problem which consists of reconstruction of a set of pollution sources in a fluid medium by measuring the concentration of the pollutants within some subregion of the reference domain. The shape functional measuring the misfit between the known data and solution of the state equation is minimized with respect to a set of ball-shaped geometrical subdomains representing the pollution sources. The necessary conditions for optimality are derived with the help of the topological derivative method which consists in expanding the shape functional asymptotically and then truncate it up to the second order term. The resulting expression is trivially minimized with respect to the parameters under consideration which leads to a noniterative second-order reconstruction algorithm. Two different cases are considered. Firstly, when the velocity of the leakages is given and we reconstruct the support of the unknown sources, including their locations and sizes. In the second case, we consider the size of the pollution sources to be known and find out the mean velocity of the leakages and their locations. Numerical examples are presented showing the capability of the proposed algorithm in reconstructing multiple pollution sources in both cases. PubDate: 2020-04-30

Abstract: Abstract In the present paper, we present and solve the sliding mode control (SMC) problem for a second-order generalization of the Caginalp phase-field system. This generalization, inspired by the theories developed by Green and Naghdi on one side, and Podio-Guidugli on the other, deals with the concept of thermal displacement, i.e., a primitive with respect to the time of the temperature. Two control laws are considered: the former forces the solution to reach a sliding manifold described by a linear constraint between the temperature and the phase variable; the latter forces the phase variable to reach a prescribed distribution \(\varphi ^*\). We prove existence, uniqueness as well as continuous dependence of the solutions for both problems; two regularity results are also given. We also prove that, under suitable conditions, the solutions reach the sliding manifold within finite time. PubDate: 2020-04-29

Abstract: Abstract We will consider the problem of observability inequality of the Hermite bi-cubic orthogonal spline collocation space semi-discretizations of the 2-D integro-differential equations in the square domains. We prove the uniform (with respect to mesh-size) observability inequality in a subspace of solutions generated by the low frequencies of the negative part, and the middle frequencies of the positive part. Our method uses previously known uniform observability inequalities in the 1-d case and a dyadic spectral time decomposition. Some numerical results are presented to illustrate our theoretical analysis. PubDate: 2020-04-27

Abstract: Abstract We demonstrate that difficult non-convex non-smooth optimization problems, such as Nash equilibrium problems and anisotropic as well as isotropic Potts segmentation models, can be written in terms of generalized conjugates of convex functionals. These, in turn, can be formulated as saddle-point problems involving convex non-smooth functionals and a general smooth but non-bilinear coupling term. We then show through detailed convergence analysis that a conceptually straightforward extension of the primal–dual proximal splitting method of Chambolle and Pock is applicable to the solution of such problems. Under sufficient local strong convexity assumptions on the functionals—but still with a non-bilinear coupling term—we even demonstrate local linear convergence of the method. We illustrate these theoretical results numerically on the aforementioned example problems. PubDate: 2020-04-13

Abstract: Abstract We investigate the initialization problem for the incompressible asymmetric fluids equations in two space dimensions. The problem is formulated as an optimal control problem. We prove that this problem has at least one and at most finite many solutions. Finally, we determine sufficient conditions to assure uniqueness of the solution. PubDate: 2020-04-08

Abstract: Abstract In this paper, the Kyle model of insider trading is extended by characterizing the trading volume with long memory and allowing the noise trading volatility to follow a general stochastic process. Under this newly revised model, the equilibrium conditions are determined, with which the optimal insider trading strategy, price impact and price volatility are obtained explicitly. The volatility of the price volatility appears excessive, which is a result of the fact that a more aggressive trading strategy is chosen by the insider when uninformed volume is higher. The optimal trading strategy turns out to possess the property of long memory, and the price impact is also affected by the fractional noise. PubDate: 2020-04-03

Abstract: Abstract This paper establishes the existence of a unique nonnegative continuous viscosity solution to the HJB equation associated with a linear-quadratic stochastic control problem with singular terminal state constraint and possibly unbounded cost coefficients. The existence result is based on a novel comparison principle for semi-continuous viscosity sub- and supersolutions for PDEs with singular terminal value. Continuity of the viscosity solution is enough to carry out the verification argument. PubDate: 2020-04-02

Abstract: Abstract We are concerned with a nonlinear nonautonomous model represented by an equation describing the dynamics of an age-structured population diffusing in a space habitat O, governed by local Lipschitz vital factors and by a stochastic behavior of the demographic rates possibly representing emigration, immigration and fortuitous mortality. The model is completed by a random initial condition, a flux type boundary conditions on \(\partial O\) with a random jump in the population density and a nonlocal nonlinear boundary condition given at age zero. The stochastic influence is expressed by a linear multiplicative Gaussian noise perturbation in the equation. The main result proves that the stochastic model is well-posed, the solution being in the class of path-wise continuous functions and satisfying some particular regularities with respect to the age and space. The approach is based on a rescaling transformation of the stochastic equation into a random deterministic time dependent hyperbolic-parabolic equation with local Lipschitz nonlinearities. The existence and uniqueness of a strong solution to the random deterministic equation is proved by combined semigroup, variational and approximation techniques. The information given by these results is transported back via the rescaling transformation towards the stochastic equation and enables the proof of its well-posedness. PubDate: 2020-04-01

Abstract: Abstract The shape derivability analysis of the flow of a viscous and incompressible fluid surrounding a rigid body B is considered. The novelty being in the choice of the boundary condition on the body B where we impose the so-called Navier boundary condition. Well-posedness of the time-dependent Navier–Stokes equations with mixed boundary conditions, of Navier and Dirichlet type, is established under regularity and smallness assumptions. After proving the shape differentiability of the state system, we compute the first order necessary optimality condition associated to drag shape minimization problem. PubDate: 2020-04-01

Abstract: Abstract This paper is concerned with the study of a transmission problem of viscoelastic waves with hereditary memory, establishing the existence, uniqueness and exponential stability for the solutions of this problem. The proof of the stabilization result combines energy estimates and results due to Gérard (Commun Partial Differ Equ 16:1761–1794 (1991)) on microlocal defect measures. PubDate: 2020-04-01

Abstract: Abstract This paper deals with a class of time inconsistent stochastic linear quadratic optimal control problems in Markovian framework. Three notions, i.e., closed-loop equilibrium strategies, open-loop equilibrium controls and open-loop equilibrium strategies, are characterized in unified manners. These results indicate clearer and deeper distinctions among these notions. For example, in particular time consistent setting, the open-loop equilibrium controls are fully characterized by first-order, second-ordernecessaryoptimalityconditions, and are not optimal in general, while the closed-loop equilibrium controls naturally reduce into closed-loopoptimalcontrols. PubDate: 2020-04-01

Abstract: Abstract This paper develops near-optimal sustainable harvesting strategies for the predator in a predator-prey system. The objective function is of long-run average per unit time type in the path-wise sense. To date, ecological systems under environmental noise are usually modeled as stochastic differential equations driven by a Brownian motion. Recognizing that the formulation using a Brownian motion is only an idealization, in this paper, it is assumed that the environment is subject to disturbances characterized by a jump process with rapid jump rates. Under broad conditions, it is shown that the systems under consideration can be approximated by a controlled diffusion system. Based on the limit diffusion system, control policies of the original systems are constructed. Such an approach enables us to develop sustainable harvesting policies leading to near optimality. To treat the underlying problems, one of the main difficulties is due to the long-run average objective function. This in turn, requires the handling of a number of issues related to ergodicity. New approaches are developed to obtain the tightness of the underlying processes based on the population dynamic systems. PubDate: 2020-04-01

Abstract: Abstract An optimal control of a steady state thermistor problem is considered where the convective boundary coefficient is taken to be the control variable. A distinct feature of this paper is that the problem is considered in \({\mathbb {R}}^d\), where \(d>2\), and the electrical conductivity is allowed to vanish above a threshold temperature value. Existence of the state system is proved. An objective functional is introduced, existence of the optimal control is proved, and the optimality system is derived. PubDate: 2020-04-01

Abstract: Abstract Many formulations of quadratic allocation problems, portfolio optimization problems, the maximum weight clique problem, among others, take the form as the well-known standard quadratic optimization problem, which consists in minimizing a homogeneous quadratic function on the usual simplex in the non negative orthant. We propose to analyze the same problem when the simplex is substituted by a convex and compact base of any pointed, closed, convex cone (so, the cone of positive semidefinite matrices or the cone of copositive matrices are particular instances). Three main duals (for which a semi-infinite formulation of the primal problem is required) are associated, and we establish some characterizations of strong duality with respect to each of the three duals in terms of copositivity of the Hessian of the quadratic objective function on suitable cones. Such a problem reveals a hidden convexity and the validity of S-lemma. In case of bidimensional quadratic optimization problems, copositivity of the Hessian of the objective function is characterized, and the case when every local solution is global. PubDate: 2020-04-01

Abstract: Abstract We study mean field games and corresponding N-player games in continuous time over a finite time horizon where the position of each agent belongs to a finite state space. As opposed to previous works on finite state mean field games, we use a probabilistic representation of the system dynamics in terms of stochastic differential equations driven by Poisson random measures. Under mild assumptions, we prove existence of solutions to the mean field game in relaxed open-loop as well as relaxed feedback controls. Relying on the probabilistic representation and a coupling argument, we show that mean field game solutions provide symmetric \(\varepsilon _N\)-Nash equilibria for the N-player game, both in open-loop and in feedback strategies (not relaxed), with \(\varepsilon _N\le \frac{\text {constant}}{\sqrt{N}}.\) Under stronger assumptions, we also find solutions of the mean field game in ordinary feedback controls and prove uniqueness either in case of a small time horizon or under monotonicity. PubDate: 2020-04-01