Applied Mathematics and Optimization最新文献

We study the existence and uniqueness of the barycenter of a signed distribution of probability measures on a Hilbert space. The barycenter is found, as usual, as a minimum of a functional. In the case where the positive part of the signed measure is a singleton, we can show also uniqueness. In the one-dimensional case, we characterize the quantile function of the unique minimum as the orthogonal projection of the (L^2)-barycenter of the quantiles on the cone of nonincreasing functions in (L^2(0,1)). Further, we provide a stability estimate in dimension one and a counterexample to uniqueness in (mathbb {R}^2). Finally, we address the consistency of the barycenters and we prove that barycenters of a sequence of approximating measures converge (up to subsequences) to a barycenter of the limit measure.

We investigate the singular stochastic optimal control problem with model uncertainty, where the necessary conditions determined by the corresponding maximum principle are trivial. We derive robust integral form and pointwise second-order necessary optimality conditions involving integrals with respect to certain reference probabilities, under specific compactness and monotonicity. Both the drift and diffusion terms depend on the control, and the control regions are assumed to be convex because saddle point analyses are crucial for deriving the variational inequality with a common reference probability for any admissible control. Other main technical components for the integral type conditions include weak convergence arguments and the minimax theorem, while the pointwise conditions involve the Clark-Ocone formula and Lebesgue differentiation type theorem. Additionally, an example is provided to demonstrate the motivation and effectiveness of the results.

In this paper, we prove that the model of stochastic lattice dynamical system driven by a fractional Brownian motion with Hurst parameter (Hin (1/2,1)) proposed in [3, Bessaih et al. 2017] under the same proper conditions possesses a random attractor, which turns out to have component singleton sets. The result improves the one in [14, Gu 2013].

We are interested in the study of stochastic games for which each player faces an optimal stopping problem. In our setting, the players may interact through the criterion to optimise as well as through their dynamics. After briefly discussing the N-player game, we formulate the corresponding mean-field problem. In particular, we introduce a weak formulation of the game for which we are able to prove existence of Nash equilibria for a large class of criteria. We also prove that equilibria for the mean-field problem provide approximated Nash equilibria for the N-player game, and we formally derive the master equation associated with our mean-field game.

The paper establishes the exponential turnpike property for a class of mean-field stochastic linear-quadratic (LQ) optimal control problems with periodic coefficients. It first introduces the concepts of stability, stabilizability, and detectability for stochastic linear systems. Then, the long-term behavior of the associated Riccati equations is analyzed under stabilizability and detectability conditions. Subsequently, a periodic mean-field stochastic LQ problem is formulated and solved. Finally, its optimal pair is shown to be the turnpike limit of the initial optimal control problem.

In this paper, we consider a quasilinear two-species chemotaxis system with two chemicals in a bounded domain with smooth boundary. Under specific parameter conditions, we show that for any sufficiently regular initial data, the associated initial-boundary value problem admits a globally bounded classical solution. Our results provide a more in-depth understanding of the chemotaxis system and significantly improve previously known ones.

The aim of this present paper is to study an optimal control problem for a quasistatic viscoelastic contact problem with long memory and multi-zones which can be formulated by a parabolic quasi-variational hemivariational inequality with history-dependent term. We first tackle the solvability of the abstract parabolic quasi-variational hemivariational inequality problem, which is regarded as difficult and typically abandoned by most researchers. Then based on this abstract result we establish the existence of weak solution to the contact problem. Finally, under the existence of weak solution we further show the solvability of optimal control problem by employing the stability of the abstract parabolic quasi-variational hemivariational inequality.

This paper deals with the global existence and asymptotic behavior of positive solutions for the following chemotaxis competition system with loop and singular sensitivity

under homogeneous Neumann boundary conditions, where (Omega subset mathbb {R}^{N}(Nge 2)) is a bounded domain with smooth boundary, (f_{1}(u,w)=u(a_{1}-b_{1}u-c_{1}w)) and (f_{2}(u,w)=w(a_{2}-b_{2}w-c_{2}u), chi _{i},,xi _{i}, a_{i}, b_{i}, c_{i}>0(i=1,2)). It is shown that if the parameters satisfy certain conditions, then the problem possesses a unique global-in-time classical bounded solution. Furthermore, by the method of Lyapunov functionals, the global stability of steady states is established.

This article concerns the existence and multiplicity of multi-bump type nodal solutions for a class of fractional p-Laplacian Schrödinger equations involving logarithmic nonlinearity and deepening potential well. We apply suitable variational arguments to show that the equation has at least (2^{k}-1) multi-bump type nodal solutions as the parameter becomes large enough.

This paper investigates a chemotaxis-generalized Navier–Stokes system with rotational flux

Here, (alpha in (0,1)) and the matrix-valued sensitivity function S(x, n, c) represents the rotational effect and satisfies (|S(x,n,c)|le C_{S}), where (C_{S}) is a positive constant. Suppose that the initial data ((n_{0},c_{0},textbf{u}_{0})in C^{0}(mathbb {T}^2)times W^{1,q}(mathbb {T}^2)times C^{0}(mathbb {T}^2)) satisfy a smallness condition on (Vert c_{0}Vert _{L^{infty }(mathbb {T}^2)}), we establish the existence and uniqueness of global classical solution to (0.1). Furthermore, we show that the solution ((n,c,textbf{u})) converge to ((bar{n},0,0)) as (trightarrow infty ), where (bar{n}=frac{1}{|mathbb {T}^2|}int _{mathbb {T}^2}n_{0}(x)). Our results cover and optimize the conclusions regarding classical Laplacian diffusion problem.