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We study the asymptotic behavior of solutions to the constrained MFG system as the time horizon T goes to infinity. For this purpose, we analyze first Hamilton–Jacobi equations with state constraints from the viewpoint of weak KAM theory, constructing a Mather measure for the associated variational problem. Using these results, we show that a solution to the constrained ergodic mean field games system exists and the ergodic constant is unique. Finally, we prove that any solution of the first-order constrained MFG problem on [0, T] converges to the solution of the ergodic system as T goes to infinity. [ABSTRACT FROM AUTHOR]

The aim of this paper is to study the long-time behavior of solutions to deterministic mean field games systems on Euclidean space. This problem was addressed on the torus T n in Cardaliaguet (Dyn Games Appl 3:473–488, 2013), where solutions are shown to converge to the solution of a certain ergodic mean field games system on T n . By adapting the approach in Fathi and Maderna (Nonlinear Differ Equ Appl NoDEA 14:1–27, 2007), we identify structural conditions on the Lagrangian, under which the corresponding ergodic system can be solved in R n . Then, we show that time-dependent solutions converge to the solution of such a stationary system on all compact subsets of the whole space. [ABSTRACT FROM AUTHOR]

In this paper, we focus on action-minimizing methods for contact Hamiltonian systems. Based on implicit variational principles introduced in Wang et al. (Nonlinearity 30:492-515, 2017), we generalize some fundamental results of Aubry-Mather theory and weak KAM theory from Hamiltonian systems to contact Hamiltonian systems with moderate increasing on the contact variable. [ABSTRACT FROM AUTHOR]

Consider the Cauchy problem of a time-periodic Hamilton-Jacobi equation on a closed manifold, where the Hamiltonian satisfies the condition: The Aubry set of the corresponding Hamiltonian system consists of one hyperbolic 1-periodic orbit. It is proved that the unique viscosity solution of Cauchy problem converges exponentially fast to a 1-periodic viscosity solution of the Hamilton-Jacobi equation as the time tends to infinity. [ABSTRACT FROM AUTHOR]

This paper introduces a modified random sample consensus (M-RANSAC) and short-time fractional Fourier transform (STFRFT)-based algorithm for feature extraction of synthetic aperture radar (SAR) scattering centers. In this algorithm, the range migration curve (RMC) of a scattering center is formulated as a parametric model. By estimating these parameters, the backscattering envelope of scattering center, corresponding to the backscattering variation in synthetic aperture time, is extracted directly from a time-domain range-compressed signal. The estimated parameters can also reconstruct the geographical location and along-track velocity of scattering centers. Thus, even without knowing explicit knowledge of platform velocity and forming a SAR image, this algorithm is capable of realizing feature extraction. To estimate parameters scatter by scatter, M-RANSAC approach is proposed as an implementary method with iterative procedure. In the iterations, fitting precision indicator (FPI) works cooperatively with construction fitness coefficient (CFC) to determine the optimal parameters of different scattering centers. Adapting this method to more general cases, STFRFT is introduced to separate the overlapped trajectories of RMCs of scattering centers. The root mean squared errors (RMSEs) of parameter estimation are close to their Cramér-Rao lower bounds (CRLB). The effectiveness of feature extraction based on the devised algorithm is validated by both simulated and real SAR data. [ABSTRACT FROM AUTHOR]

In this paper we introduce a new kind of Lax-Oleinik type operator with parameters associated with positive definite Lagrangian systems for both the time-periodic case and the time-independent case. On one hand, the family of new Lax-Oleinik type operators with an arbitrary $${u \in C(M, \mathbb{R}^1)}$$ as initial condition converges to a backward weak KAM solution in the time-periodic case, while it was shown by Fathi and Mather that there is no such convergence of the Lax-Oleinik semigroup. On the other hand, the family of new Lax-Oleinik type operators with an arbitrary $${u \in C(M, \mathbb{R}^1)}$$ as initial condition converges to a backward weak KAM solution faster than the Lax-Oleinik semigroup in the time-independent case. [ABSTRACT FROM AUTHOR]

For the standard Lagrangian in classical mechanics, which is defined as the kinetic energy of the system minus its potential energy, we study the rate of convergence of the corresponding Lax-Oleinik semigroup. Under the assumption that the unique global minimum point of the Lagrangian is a degenerate fixed point, we provide an upper bound estimate of the rate of convergence of the semigroup. [ABSTRACT FROM AUTHOR]