Applied Mathematics and Optimization最新文献

We study a system described by a type of initial and boundary value problem of the Hirota equation with nonhomogeneous boundary conditions posed on a bounded interval. Firstly, we prove the local well-posedness of the system in the space (H^s(0,1)) by using an explicit solution formula and contraction mapping principle for any (sge 1). Secondly, we obtain the global well-posedness in (H^1(0,1)) and (H^2(0,1)) by the norm estimation. Especially, the main difficulty is that the characteristic equation corresponding to Hirota equation needs to be solved by construction and that nonlinear terms are taken into consideration. In addition, the norm estimate for global well-posedness of solution in (H^1(0,1)) and (H^2(0,1)) are complicated.

This paper investigates a linear-quadratic (LQ) Stackelberg game for mean-field type backward stochastic system, in which the cost functional is also mean-field type. In our model, the leader first announces the terminal goal satisfying pointwise and affine constraints and open-loop dynamic decisions at the initial time which takes into account the best response of the follower. Then two interrelated optimization problems are sequentially solved by the follower (a backward LQ problem) and the leader (a backward-forward LQ problem). The open-loop Stackelberg equilibrium is obtained by virtue of duality theory and represented by some fully coupled mean-field backward-forward stochastic differential equations with mixed initial-terminal conditions, whose global solvability is discussed in some nontrivial cases by Riccati decoupling method and discounting method. As an application, we address a product pricing problem.

We show how the sticky dynamics for the one-dimensional pressureless Euler-alignment system can be obtained as an (L^2)-gradient flow of a convex functional. This is analogous to the Lagrangian evolution introduced by Natile and Savaré for the pressureless Euler system, and by Brenier et al. for the corresponding system with a self-interacting force field. Our Lagrangian evolution can be seen as the limit of sticky particle Cucker–Smale dynamics, similar to the solutions obtained by Leslie and Tan from a corresponding scalar balance law, and provides us with a uniquely determined distributional solution of the original system in the space of probability measures with quadratic moments and corresponding square-integrable velocities. Moreover, we show that the gradient flow also provides an entropy solution to the balance law of Leslie and Tan, and how their results on cluster formation follow naturally from (non-)monotonicity properties of the so-called natural velocity of the flow.

Discrete time stochastic optimal control problems and Markov decision processes (MDPs), respectively, serve as fundamental models for problems that involve sequential decision making under uncertainty and as such constitute the theoretical foundation of reinforcement learning. In this article we study the numerical approximation of MDPs with infinite time horizon, finite control set, and general state spaces. Our set-up in particular covers infinite-horizon optimal stopping problems of discrete time Markov processes. A key tool to solve MDPs are Bellman equations which characterize the value functions of the MDPs and determine the optimal control strategies. By combining ideas from the full-history recursive multilevel Picard approximation method, which was recently introduced to solve certain nonlinear partial differential equations, and ideas from Q-learning we introduce a class of suitable nonlinear Monte Carlo methods and prove that the proposed methods do not suffer from the curse of dimensionality in the numerical approximation of the solutions of Bellman equations and the associated discrete time stochastic optimal control problems.

In this article, the following controlled convective Brinkman-Forchheimer extended Darcy (CBFeD) system is considered in a d-dimensional torus:

where (din {2,3}), (mu ,alpha ,beta >0), (gamma in {mathbb {R}}), (r,qin [1,infty )) with (r>qge 1). We prove the exponential stabilization of CBFeD system by finite- and infinite-dimensional feedback controllers. The solvability of the controlled problem is achieved by using the abstract theory of m-accretive operators and density arguments. As an application of the above solvability result, by using infinite-dimensional feedback controllers, we demonstrate exponential stability results such that the solution preserves an invariance condition for a given closed and convex set. By utilizing the unique continuation property of controllability for finite-dimensional systems, we construct a finite-dimensional feedback controller which exponentially stabilizes CBFeD system locally, where the control is localized in a smaller subdomain. Furthermore, we establish the local exponential stability of CBFeD system via proportional controllers.

We consider the steady Navier–Stokes equations with mixed boundary conditions, where a regularized directional do-nothing (RDDN) condition is defined on the Neumann boundary portion. An auxiliary Stokes reference flow, which also works as a lifting of the inhomogeneous Dirichlet boundary values, is used to define the RDDN condition. Our aim is to study the minimization of a velocity tracking cost functional with controls localized on a part of the boundary. We prove the existence of a solution for this optimal control problem and derive the corresponding first order necessary optimality conditions in terms of dual variables. All results are obtained under appropriate assumptions on the size of the data and the controls, which, however, are less restrictive when compared with the case of the classical do-nothing outflow condition. This is further confirmed by the numerical examples presented, which include scenarios where only noisy data is available.

This paper is devoted to solving a class of second order Hamilton–Jacobi–Bellman (HJB) equations in the Wasserstein space, associated with mean field control problems involving common noise. The well-posedness of viscosity solution to the HJB equation under a new notion is established under general assumptions on the coefficients. Our approach adopts the smooth metric developed by Bayraktar et al. (Proc Am Math Soc 151(09):4089–4098, 2023) as our gauge function for the purpose of smooth variational principle used in the proof of comparison theorem. Further estimates and regularity of the metric, including a novel second order derivative estimate with respect to the measure variable, are derived in order to ensure the uniqueness and existence.

The purpose of this paper is to develop Pareto optimality conditions and constraint qualifications (CQs) for Multiobjective Programs with Cardinality Constraints (MOPCaC). In general, such problems are difficult to solve, not only because they involve a cardinality constraint that is neither continuous nor convex, but also because there may be a potential conflict between the various objective functions. Thus, we reformulate the MOPCaC based on the problem with continuous variables, namely the relaxed problem. Furthermore, we consider different notions of optimality (weak/strong Pareto optimal solutions). Thereby, we define new stationarity conditions that extend the classical Karush-Kuhn-Tucker (KKT) conditions of the scalar case. Moreover, we also introduce new CQs, based on the recently defined multiobjective normal cone, to ensure compliance with such stationarity conditions. Important statements are illustrated by examples.

This paper studies the controllability for a Keller–Segel type chemotaxis model with singular sensitivity. Based on the Hopf–Cole transformation, a nonlinear parabolic system, which has first-order couplings, and the coupling coefficients are functions that depend on both time and space variables, is derived. Then, the controllability result is proved by a new global Carleman estimate for general coupled parabolic equations allowed to contain a convective term. Also, the global existence of nonnegative solution for the chemotaxis system is discussed.

In this work, we use the integral definition of the fractional Laplace operator and study a sparse optimal control problem involving a fractional, semilinear, and elliptic partial differential equation as state equation; control constraints are also considered. We establish the existence of optimal solutions and first and second order optimality conditions. We also analyze regularity properties for optimal variables. We propose and analyze two finite element strategies of discretization: a fully discrete scheme, where the control variable is discretized with piecewise constant functions, and a semidiscrete scheme, where the control variable is not discretized. For both discretization schemes, we analyze convergence properties and a priori error bounds.