Applied Mathematics and Optimization最新文献

In this paper, we are concerned with the following Schrödinger-Poisson systems

with prescribed (L^{2})-norm mass

where (2< q<ple 6), (alpha in mathbb {R}) is a parameter, c is a prescribed value, (lambda in mathbb {R}) is a Lagrange multiplier, (Omega subset mathbb {R}^3) is a smooth bounded domain and (p=6) is the Sobolev critical exponent. We first prove that the problem has a positive normalized solution, which is a local minimizer. Next, under the assumption that (Omega ) is star-shaped, we show the existence of a second normalized solution for (alpha <0) and (4le p< 6) by using Jeanjean’s theory, Pohozaev identity and Mountain pass theorem. Additionally, we give asymptotic behavior of the local minimizer as (crightarrow 0).

This paper studies uniform measure attractors for non-autonomous stochastic lattice systems with delay driven by higher-order nonlinear noise. While previous studies have investigated attractors for stochastic lattice systems with delay, the existence of uniform measure attractors for systems with higher-order nonlinear drift and diffusion terms remains unresolved due to the inherent difficulty in obtaining uniform closed absorbing sets under higher-order nonlinearities. To address this challenge, we establish an equivalent theoretical framework for uniform measure attractors via (omega )-limit compactness and uniform asymptotic tightness, which removes the reliance on uniform closed absorbing sets. Within this novel framework, we prove the existence and uniqueness of uniform measure attractors for non-autonomous stochastic delay lattice systems with almost periodic forcing and higher-order nonlinear terms.

In this paper we study the problem of optimal multiple mode switching in finite horizon when switching a system from a regime i to another one j incurs a payment (-g_{ij}) which is not necessarily negative, i.e., switching the controlled system could incur a subsidy. Under the monotonicity and the triangle inequality properties of the switching payments (g_{ij}), we show that the problem is well-posed and we exhibit the optimal strategy. We use probabilistic tools relying on the notion of Snell envelope of processes and systems of reflected backward stochastic differential equations with inter-connected obstacles. At the end of the paper, in the Markovian framework, we show that the vector of value functions of the problem is a unique viscosity solution to its associated HJB system of variational inequalities (or PDEs) with inter-connected obstacles.

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.