Applied Mathematics and Optimization最新文献

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.

This paper introduces a novel stochastic control framework to enhance the capabilities of automated investment managers, or robo-advisors, by accurately inferring clients’ investment preferences from past activities. Our approach leverages a continuous-time model that incorporates utility functions and a generic discounting scheme with a time-varying rate, which can be tailored to each client’s risk tolerance, valuation of daily consumption, and significant life goals; this general discounting scheme is also referred to as the time preference. Through state augmentation, the new stochastic control framework allows for the joint inference of utilities and time preferences. We establish the corresponding dynamic programming principle and the verification theorem. Additionally, we provide sufficient conditions for the identifiability of client investment preferences. To complement our theoretical developments, we propose a learning algorithm based on maximum likelihood estimation within a discrete-time Markov Decision Process framework, augmented with entropy regularization. We prove that the Hessian matrix of the log-likelihood function is negative semi-definite at the client’s investment parameters, facilitating fast convergence of our proposed algorithm. Practical effectiveness and efficiency are showcased through two numerical examples, including Merton’s problem and an investment problem with unhedgeable risks. Our proposed framework not only advances financial technology by improving personalized investment advice but also contributes broadly to other fields such as healthcare, economics, and artificial intelligence, where understanding individual preferences is crucial.

In this paper, we conduct a numerical analysis of the strong stabilization and polynomial decay of solutions for the initial boundary value problem associated with a system that models the dynamics of a mixture of two rigid solids with porosity. This mathematical model accounts for the complex interactions between the rigid components and their porous structure, providing valuable information on the mechanical behavior of such systems. Our primary objective is to establish conditions under which stabilization is ensured and to rigorously quantify the rate of decay of the solutions. Using numerical simulations, we assess the effectiveness of different stabilization mechanisms and analyze the influence of key system parameters on the overall dynamics.

The exponential characterization of a partially dissipative Timoshenko system with memory effects acting on the shear component is addressed in this paper. A wider class of admissible kernels that may exhibit flat zones and jump discontinuities is explored. Then, by the introduction of a novel condition, referred to as the shape condition, it is proven that the previous classical criteria for the system’s exponential characterization are no longer sufficient in our extended setting. Here, the geometric structure of the kernel, particularly its flatness and the distribution of discontinuities, is seen to play a crucial role in determining the characterization of exponential stability. Hence, a refined characterization is established in terms of three simultaneous conditions: dissipation, equal wave speeds, and the shape condition. The results not only generalize previous findings based on smooth kernels but also offer a deeper understanding of the interplay between the geometric aspects of the memory kernel and the system’s asymptotic behavior.

In this paper, we investigate the well-posedness, complete regularity and longtime dynamics for the Kirchhoff-type wave equations with degenerate energy fractional damping on a bounded domain (Omega subset {mathbb {R}}^N: u_{tt}-phi (Vert nabla uVert ^2)Delta u+[E_U(t)]^q(-Delta )^theta u_t+f(u)=0), together with the Dirichlet boundary condition, where (theta in (0,1)) is a dissipative index determining the strength of the dissipation, (q>0) and (E_U(t)) is the energy of the system which degenerates at the point (U=(u,u_t)=(0,0)). This type of dissipations is connected with energy damping models proposed by Balakrishnan et al. (Stabilization of flexible structures, 1988; Proceedings Damping 89, Flight Dynamics Lab and Air Force Wright Aeronautical Labs, WPAFB, 1989). The main results are as follows: (i) the existence and the uniqueness of the global weak solution in natural energy space ({mathcal {H}}); (ii) the complete regularity of the weak solutions as (t>0); (iii) the existence of the global attractor of the related solution semigroup (S^theta (t)) for each (theta ). The most distinct and challenging aspect of this paper is the “parabolic like" nature of the dynamics under the degenerate energy level nonlinear damping. The method used here allows overcoming the difficulties arising from the degenerated energy damping and improving greatly the results on this issue for this type of models in literature before.

This article investigates the instability and stability of hydrostatic equilibria of the three-dimensional nonhomogeneous incompressible viscous flows subject to the Navier-slip boundary conditions in a bounded domain. First, we establish the linear instability of a steady-state solution with a density profile that increases in a specific region. Then, we prove the nonlinear instability of the hydrostatic equilibrium in the Hadamard sense by constructing a nonlinearly perturbed solution, achieved through a sequence of refined energy estimates. Finally, we also show the global linear and non-linear stability of the steady-state solution when the density profile decreases in the direction opposite to the gravitational force. Our findings provide important extensions to previous results on both (a) the two-dimensional case involving flat boundaries and constant slip coefficients for incompressible viscous flows, and (b) the more complex three-dimensional scenario with no-slip boundary conditions.

Reinforcement learning for multi-agent games has attracted lots of attention recently. However, given the challenge of solving Nash equilibria, existing works with guaranteed polynomial complexities either focus on variants of zero-sum and potential games, or aim at solving (coarse) correlated equilibria, or require access to simulators, or rely on certain assumptions that are hard to verify. This work proposes MF-OML (Mean-Field Occupation-Measure Learning), an online mean-field reinforcement learning algorithm for computing approximate Nash equilibria of large population sequential symmetric games. MF-OML is the first fully polynomial multi-agent reinforcement learning algorithm for provably solving Nash equilibria (up to mean-field approximation gaps that vanish as the number of players N goes to infinity) beyond variants of zero-sum and potential games. When evaluated by the cumulative deviation from Nash equilibria, the algorithm is shown to achieve a high probability regret bound of (tilde{O}(M^{3/4}+N^{-1/2}M)) for games with the strong Lasry-Lions monotonicity condition, and a regret bound of (tilde{O}(M^{11/12}+N^{-1/6}M)) for games with only the Lasry-Lions monotonicity condition, where M is the total number of episodes and N is the number of agents of the game. As a by-product, we also obtain the first tractable globally convergent computational algorithm for computing approximate Nash equilibria of monotone mean-field games.

We study a fluid–structure interaction problem between a viscous incompressible fluid and an elastic beam with fixed endpoints in a static setting. The 3D fluid domain is bounded, nonsmooth and non simply connected, the fluid is modeled by the stationary Navier–Stokes equations subject to inflow/outflow conditions. The structure is modeled by a stationary 1D beam equation with a load density involving the force exerted by the fluid and, thereby, may vary its position. In a smallness regime, we prove the existence and uniqueness of the solution to the PDE-ODE coupled system.

In this paper, we study the social optima for a kind of time-inconsistent linear-quadratic (LQ) mean-field problem. We first propose the concept of (varepsilon)-social equilibrium strategy for our problem within the game theoretic framework. Next, an auxiliary time-inconsistent control problem is constructed by the scheme of person-by-person optimality and a set of linear feedback equilibrium controls for this auxiliary problem is presented explicitly via several ordinary differential equations (ODEs). Then we can derive a new kind of consistency condition (CC) system consisting of a forward stochastic differential equation (SDE) and three flows of backward stochastic differential equations (BSDEs), and obtain a set of decentralized admissible controls for the original problem. The wellposedness of this kind of CC system is also discussed in this paper. We prove that the set of admissible controls given above is indeed an (varepsilon)-social equilibrium strategy for our original problem. Finally, a numerical example is presented to illustrate our theoretic results visually.