Applied Mathematics and Optimization最新文献

This paper investigates mean field games in continuous time, state and action spaces with an infinite number of agents, where each agent aims to maximize its expected cumulative reward. Using the technique of randomized policies, we show policy evaluation and policy gradient are equivalent to the martingale conditions of a process by focusing on a representative agent. Then combined with fictitious game, we propose online and offline actor-critic algorithms for solving continuous mean field games that update the value function and policy alternatively under the given population state and action distributions. We demonstrate through two numerical experiments that our proposed algorithms can converge to the mean field equilibrium quickly and stably.

We consider a class of stochastic optimal transport, SOT for short, with given two endpoint marginals in the case where a cost function exhibits at most quadratic growth. We first study the upper and lower estimates, the short-time asymptotics, the zero-noise limits, and the explosion rate as time goes to infinity of SOT. We also show that the value function of SOT is equal to zero or infinity in the case where a cost function exhibits less than linear growth. As a by-product, we characterize the finiteness of the value function of SOT by that of the Monge–Kantorovich problem. As an application, we show the existence of a continuous semimartingale, with given initial and terminal distributions, of which the drift vector is rth integrable for (rin [1,2)). We also consider the same problem for Schrödinger’s problem where (r=2). This paper is a continuation of our previous work.

This work is concerned with the modified stochastic Swift–Hohenberg equation in a 3D thin domain. Although the diffusion motion of molecules is irregular with the interference of the film-fluid fluctuation, the invariant measure on the trajectory space reveals delicate transition of the dynamical behavior when the interior forces change. We therefore prove that the invariant measure of the system converges weakly to the unique counterpart of the stochastic Swift–Hohenberg equation in a 2D bounded domain with a concrete convergence rate, as the modified parameter and the thickness of the thin domain tend to zero. Furthermore, we address that the smooth density of the limit invariant measure fulfills a Fokker–Planck equation.

This paper concerns an optimal control problem (P) associated to a nonlinear Fokker–Planck equation via inputs with nonlocal action. The Fokker–Planck equation describes the dynamics of the probability density of a population under a control that produces a repellent vector field which displaces the population. Actually, problem (P) asks to optimally displace a population via the repellent action produced by the control. The problem is deeply related to a stochastic optimal control problem ((P_S)) for a McKean–Vlasov equation. The existence of an optimal control is obtained for the deterministic problem (P). The existence of an optimal control is established and necessary optimality conditions are derived for a penalized optimal control problem ((P_h)) related to a backward Euler approximation of the nonlinear Fokker–Planck equation (with a constant discretization step h). Using a passing-to-the-limit-like argument (as (hrightarrow 0)) one derives the necessary optimality conditions for problem (P). Some possible extensions are discussed as well.

Motivated by the multi-scheme supply chain problem, a linear-quadratic generalized Stackelberg game for time-delay is studied, in which the multi-level hierarchy structure with delay is involved. With the help of the continuity method, we first establish the unique solvability of nonlinear anticipated forward–backward stochastic delayed differential equations with a multi-level self-similar domination-monotonicity structure. Based on it, we derive the Stackelberg equilibrium in this framework. By the theoretical results, a corporate social responsibility problem is studied in the view of a multi-scheme supply chain problem, some simulations are also presented to illustrate the Stackelberg equilibrium in a special case.

We consider an abstract concept of perimeter measure space as a very general framework in which one can properly consider two of the most well-studied variational models in image processing: the Rudin–Osher–Fatemi model for image denoising (ROF) and the Mumford–Shah model for image segmentation (MS). We show the linkage between the ROF model and the two phases piecewise constant case of MS in perimeter measure spaces. We show applications of our results to nonlocal image segmentation, via discrete weighted graphs, and to multiclass classification on high dimensional spaces.

We consider the classical problem of maximizing the expected utility of terminal net wealth with a final random liability in a simple jump-diffusion model. In the spirit of Horst et al. (Stoch Process Appl 124(5):1813–1848, 2014) and Santacroce and Trivellato (SIAM J Control Optim 52(6):3517–3537, 2014), under suitable conditions the optimal strategy is expressed in implicit form in terms of a forward backward system of equations. Some explicit results are presented for the pure jump model and for exponential utilities.

In this paper, the stability of longitudinal vibrations for transmission problems of two smart-system designs are studied: (i) a serially-connected elastic–piezoelectric–elastic design with a local damping acting only on the piezoelectric layer and (ii) a serially-connected piezoelectric–elastic design with a local damping acting on the elastic part only. Unlike the existing literature, piezoelectric layers are considered magnetizable, and therefore, a fully-dynamic PDE model, retaining interactions of electromagnetic fields (due to Maxwell’s equations) with the mechanical vibrations, is considered. The design (i) is shown to have exponentially stable solutions. However, the nature of the stability of solutions of the design (ii), whether it is polynomial or exponential, is dependent entirely upon the arithmetic nature of a quotient involving all physical parameters. Furthermore, a polynomial decay rate is provided in terms of a measure of irrationality of the quotient. Note that this type of result is totally new (see Theorem 1.3 and Condition (mathrm {mathbf {(H_{Pol})}})). The main tool used throughout the paper is the multipliers technique which requires an adaptive selection of cut-off functions together with a particular attention to the sharpness of the estimates to optimize the results.

In ultrasonics, nonlocal quasilinear wave equations arise when taking into account a class of heat flux laws of Gurtin–Pipkin type within the system of governing equations of sound motion. The present study extends previous work by the authors to incorporate nonlocal acoustic wave equations with quadratic gradient nonlinearities which require a new approach in the energy analysis. More precisely, we investigate the Kuznetsov and Blackstock equations with dissipation of fractional type and identify a minimal set of assumptions on the memory kernel needed for each equation. In particular, we discuss the physically relevant examples of Abel and Mittag–Leffler kernels. We perform the well-posedness analysis uniformly with respect to a small parameter on which the kernels depend and which can be interpreted as the sound diffusivity or the thermal relaxation time. We then analyze the limiting behavior of solutions with respect to this parameter, and how it is influenced by the specific class of memory kernels at hand. Through such a limiting study, we relate the considered nonlocal quasilinear equations to their limiting counterparts and establish the convergence rates of the respective solutions in the energy norm.

In this work we define the generalized (varphi )-pullback attractors for evolution processes in complete metric spaces, which are compact and positively invariant families, that pullback attract bounded sets with a rate determined by a decreasing function (varphi ) that vanishes at infinity, called decay function. We find conditions under which a given evolution process has a generalized (varphi )-pullback attractor, both in the discrete and in the continuous cases. We present a result for the special case of generalized polynomial pullback attractors, and apply it to obtain such an object for a nonautonomous wave equation.