Authors:Maisa Khader; Belkacem Said-Houari Pages: 403 - 428 Abstract: Abstract We consider the Cauchy problem for the one-dimensional Timoshenko system coupled with the heat conduction, wherein the latter is described by the Gurtin–Pipkin thermal law. We study the decay properties of the system using the energy method in the Fourier space (to build an appropriate Lyapunov functional) accompanied with some integral estimates. We show that the number \(\alpha _g\) (depending on the parameters of the system) found in (Dell’Oro and Pata J Differ Equ 257(2):523–548, 2014), which rules the evolution in bounded domains, also plays a role in an unbounded domain and controls the behavior of the solution. In fact, we prove that if \(\alpha _g=0,\) then the \(L^2\) -norm of the solution decays with the rate \((1+t)^{-1/12}\) . The same decay rate has been obtained for \(\alpha _g\ne 0,\) but under some higher regularity assumption. This high regularity requirement is known as regularity loss, which means that in order to get the estimate for the \(H^s\) -norm of the solution, we need our initial data to be in the space \(H^{s+s_0},\ s_0>1\) . PubDate: 2017-06-01 DOI: 10.1007/s00245-016-9336-6 Issue No:Vol. 75, No. 3 (2017)

Authors:Jianliang Zhai; Tusheng Zhang Pages: 471 - 498 Abstract: Abstract In this paper, we establish a large deviation principle for stochastic models of incompressible second grade fluids. The weak convergence method introduced by Budhiraja and Dupuis (Probab Math Statist 20:39–61, 2000) plays an important role. PubDate: 2017-06-01 DOI: 10.1007/s00245-016-9338-4 Issue No:Vol. 75, No. 3 (2017)

Authors:Pooja Agarwal; Utpal Manna; Debopriya Mukherjee Abstract: Abstract In this work we first present the existence, uniqueness and regularity of the strong solution of the tidal dynamics model perturbed by Lévy noise. Monotonicity arguments have been exploited in the proofs. We then formulate a martingale problem of Stroock and Varadhan associated to an initial value control problem and establish existence of optimal controls. PubDate: 2017-07-13 DOI: 10.1007/s00245-017-9440-2

Authors:Jose R. Fernández; Antonio Magaña; María Masid; Ramón Quintanilla Abstract: Abstract In this paper we analyze an homogeneous and isotropic mixture of viscoelastic solids. We propose conditions to guarantee the coercivity of the internal energy and also of the dissipation, first in dimension two and later in dimension three. We obtain an uniqueness result for the solutions when the dissipation is positive and without any hypothesis over the internal energy. When the internal energy and the dissipation are both positive, we prove the existence of solutions as well as their analyticity. Exponential stability and impossibility of localization of the solutions are immediate consequences. PubDate: 2017-07-12 DOI: 10.1007/s00245-017-9439-8

Authors:Giacomo Albi; Young-Pil Choi; Massimo Fornasier; Dante Kalise Abstract: Abstract In this paper we model the role of a government of a large population as a mean field optimal control problem. Such control problems are constrained by a PDE of continuity-type, governing the dynamics of the probability distribution of the agent population. We show the existence of mean field optimal controls both in the stochastic and deterministic setting. We derive rigorously the first order optimality conditions useful for numerical computation of mean field optimal controls. We introduce a novel approximating hierarchy of sub-optimal controls based on a Boltzmann approach, whose computation requires a very moderate numerical complexity with respect to the one of the optimal control. We provide numerical experiments for models in opinion formation comparing the behavior of the control hierarchy. PubDate: 2017-07-06 DOI: 10.1007/s00245-017-9429-x

Authors:M. H. M. Chau; Y. Lai; S. C. P. Yam Abstract: Abstract In this article, we provide the first systemic study on discrete time partially observable mean field systems in the presence of a common noise. Each player makes decision solely based on the observable process. Both the mean field games and the related tractable mean field type stochastic control problem are studied. We first solve the mean field type control problem using classical discrete time Kalman filter with notable modifications. The unique existence of the resulted forward backward stochastic difference system is then established by separation principle. The mean field game problem is also solved via an application of stochastic maximum principle, while the existence of the mean field equilibrium is shown by the Schauder’s fixed point theorem. PubDate: 2017-07-01 DOI: 10.1007/s00245-017-9437-x

Authors:René Carmona; François Delarue; Daniel Lacker Abstract: Abstract The goal of the paper is to introduce a set of problems which we call mean field games of timing. We motivate the formulation by a dynamic model of bank run in a continuous-time setting. We briefly review the economic and game theoretic contributions at the root of our effort, and we develop a mathematical theory for continuous-time stochastic games where the strategic decisions of the players are merely choices of times at which they leave the game, and the interaction between the strategic players is of a mean field nature. PubDate: 2017-06-30 DOI: 10.1007/s00245-017-9435-z

Authors:Giulia Di Nunno; Hannes Haferkorn Abstract: Abstract Time change is a powerful technique for generating noises and providing flexible models. In the framework of time changed Brownian and Poisson random measures we study the existence and uniqueness of a solution to a general mean-field stochastic differential equation. We consider a mean-field stochastic control problem for mean-field controlled dynamics and we present a necessary and a sufficient maximum principle. For this we study existence and uniqueness of solutions to mean-field backward stochastic differential equations in the context of time change. An example of a centralised control in an economy with specialised sectors is provided. PubDate: 2017-06-30 DOI: 10.1007/s00245-017-9426-0

Authors:Rene Carmona; Peiqi Wang Abstract: Abstract The goal of the paper is to introduce a formulation of the mean field game with major and minor players as a fixed point on a space of controls. This approach emphasizes naturally the role played by McKean–Vlasov dynamics in some of the players’ optimization problems. We apply this approach to linear quadratic models for which we recover the existing solutions for open loop equilibria, and we show that we can also provide solutions for closed loop versions of the game. Finally, we implement numerically our theoretical results on a simple model of flocking. PubDate: 2017-06-29 DOI: 10.1007/s00245-017-9430-4

Authors:Galo Nuño Abstract: Abstract This paper analyzes problems in which a large benevolent player, controlling a set of policy variables, maximizes aggregate welfare in a continuous-time economy populated by atomistic agents subject to idiosyncratic shocks. We first provide as a benchmark the social optimum solution, in which a planner directly determines the individual controls. Then we analyze the optimal design of social policies depending on whether the large player may credibly commit to the future path of policies. On the one hand, we analyze the open-loop Stackelberg solution, in which the optimal policy path is set at time zero and the problem is time-inconsistent. On the other hand we analyze the time-consistent feedback Stackelberg solution. PubDate: 2017-06-23 DOI: 10.1007/s00245-017-9433-1

Authors:P. Cardaliaguet Abstract: Abstract The paper studies the convergence, as N tends to infinity, of a system of N coupled Hamilton–Jacobi equations, the Nash system, when the coupling between the players becomes increasingly singular. The limit equation turns out to be a mean field game system with a local coupling. PubDate: 2017-06-23 DOI: 10.1007/s00245-017-9434-0

Authors:Zhenhai Liu; Stanisław Migórski; Biao Zeng Abstract: Abstract In this paper, by introducing a new concept of the (f, g, h)-quasimonotonicity and applying the maximal monotonicity of bifunctions and KKM technique, we show the existence results of solutions for quasi mixed equilibrium problems when the constraint set is compact, bounded and unbounded, respectively, which extends and improves several well-known results in many respects. Next, we also obtain a result of optimal control to a minimization problem. Our main results can be applied to the problems of evolution equations, differential inclusions and hemivariational inequalities. PubDate: 2017-06-22 DOI: 10.1007/s00245-017-9431-3

Authors:Truong Minh Tuyen Abstract: Abstract In this paper, we study the split common null point problem. Then, using the hybrid projection method and the metric resolvent of monotone operators, we prove a strong convergence theorem for an iterative method for finding a solution of this problem in Banach spaces. PubDate: 2017-06-22 DOI: 10.1007/s00245-017-9427-z

Authors:Nguyen Quang Huy; Do Sang Kim; Nguyen Van Tuyen Abstract: Abstract In the present paper, we focus on the vector optimization problems with constraints, where objective functions and constrained functions are Fréchet differentiable, and whose gradient mapping is locally Lipschitz. By using the second-order symmetric subdifferential and the second-order tangent set, we introduce some new types of second-order regularity conditions in the sense of Abadie. Then we establish some second-order necessary optimality conditions Karush–Kuhn–Tucker-type for local efficient (weak efficient, Geoffrion properly efficient) solutions of the considered problem. In addition, we provide some sufficient conditions for a local efficient solution to such problem. PubDate: 2017-06-20 DOI: 10.1007/s00245-017-9432-2

Authors:Rong Liu; Guirong Liu Abstract: Abstract This paper investigates the optimal contraception control for a nonlinear size-structured population model with three kinds of mortality rates: intrinsic, intra-competition and female sterilant. First, we transform the model to a system of two subsystems, and establish the existence of a unique non-negative solution by means of frozen coefficients and fixed point theory, and show the continuous dependence of the population density on control variable. Then, the existence of an optimal control strategy is proved via compactness and extremal sequence. Next, necessary optimality conditions of first order are established in the form of an Euler–Lagrange system by the use of tangent-normal cone technique and adjoint system. Moreover, a numerical result for the optimal control strategy is presented. Our conclusions would be useful for managing the vermin. PubDate: 2017-06-19 DOI: 10.1007/s00245-017-9428-y

Authors:Nacira Agram; Bernt Øksendal Abstract: Abstract By a memory mean-field process we mean the solution \(X(\cdot )\) of a stochastic mean-field equation involving not just the current state X(t) and its law \(\mathcal {L}(X(t))\) at time t, but also the state values X(s) and its law \(\mathcal {L}(X(s))\) at some previous times \(s<t.\) Our purpose is to study stochastic control problems of memory mean-field processes. We consider the space \(\mathcal {M}\) of measures on \(\mathbb {R}\) with the norm \( \cdot _{\mathcal {M}}\) introduced by Agram and Øksendal (Model uncertainty stochastic mean-field control. arXiv:1611.01385v5, [2]), and prove the existence and uniqueness of solutions of memory mean-field stochastic functional differential equations. We prove two stochastic maximum principles, one sufficient (a verification theorem) and one necessary, both under partial information. The corresponding equations for the adjoint variables are a pair of (time-advanced backward stochastic differential equations (absdes), one of them with values in the space of bounded linear functionals on path segment spaces. As an application of our methods, we solve a memory mean–variance problem as well as a linear–quadratic problem of a memory process. PubDate: 2017-06-17 DOI: 10.1007/s00245-017-9425-1

Authors:Stefan Ankirchner; Maike Klein; Thomas Kruse Abstract: Abstract We consider the problem of optimally stopping a continuous-time process with a stopping time satisfying a given expectation cost constraint. We show, by introducing a new state variable, that one can transform the problem into an unconstrained control problem and hence obtain a dynamic programming principle. We characterize the value function in terms of the dynamic programming equation, which turns out to be an elliptic, fully non-linear partial differential equation of second order. We prove a classical verification theorem and illustrate its applicability with several examples. PubDate: 2017-06-10 DOI: 10.1007/s00245-017-9424-2

Authors:Qingying Hu; Hongwei Zhang; Gongwei Liu Abstract: Abstract We consider the initial boundary value problem for a class of logarithmic wave equations with linear damping. By constructing a potential well and using the logarithmic Sobolev inequality, we prove that, if the solution lies in the unstable set or the initial energy is negative, the solution will grow as an exponential function in the \(H^1_0(\Omega )\) norm as time goes to infinity. If the solution lies in a smaller set compared with the stable set, we can estimate the decay rate of the energy. These results are extensions of earlier results. PubDate: 2017-06-07 DOI: 10.1007/s00245-017-9423-3