Applied Mathematics and Optimization最新文献

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.

Based on probabilistic methods, we discuss the relationship between viscosity and distribution solutions for semi-linear partial differential equations (PDEs) with Neumann boundary conditions. We also extend the research to a type of nonlinear PDEs, which is completed through the well-posedness and continuity results of solutions to the corresponding forward-backward SDE.

This paper is concerned with a semilinear Rao–Nakra sandwich beam under the action of three nonlinear localized frictional damping terms in which the core viscoelastic layer is constrained by the pure elasticity or piezoelectric outer layers. The main goal is to prove its asymptotic behavior by applying minimal amount of support to the damping. We firstly prove that the system is global well-posedness by the theory of monotone operators. For asymptotic behavior of solutions, we obtain uniform decay rate results of the system and the energy decay rates are determined by a nonlinear first-order ODE. The existence of a smooth global attractor with finite fractal dimension and generalized exponential attractors are finally obtained.

We derive new necessary and sufficient conditions for strict efficiency in vector optimization problems for non-smooth mappings. Unlike other approaches, our conditions are described in terms of a suitable directional curvature functional that allows us to derive no-gap second-order optimality conditions in an abstract setting. Our approach allows us to apply our results even when classical assumptions such as the second-order regularity conditions to the feasible set fail, extending the applicability of our approach. As applications to mathematical programming, we provide new primal and dual Karush-Kuhn-Tucker (KKT) second-order necessary and sufficient conditions. We provide some examples to illustrate our findings.

We present an alternative view for the study of optimal control of partially observed Markov Decision Processes (POMDPs). We first revisit the traditional (and by now standard) separated-design method of reducing the problem to fully observed MDPs (belief-MDPs), and present conditions for the existence of optimal policies. Then, rather than working with this standard method, we define a Markov chain taking values in an infinite dimensional product space with the history process serving as the controlled state process and a further refinement in which the control actions and the state process are causally conditionally independent given the measurement/information process. We provide new sufficient conditions for the existence of optimal control policies under the discounted cost and average cost infinite horizon criteria. In particular, while in the belief-MDP reduction of POMDPs, weak Feller condition requirement imposes total variation continuity on either the system kernel or the measurement kernel, with the approach of this paper only weak continuity of both the transition kernel and the measurement kernel is needed (and total variation continuity is not) together with regularity conditions related to filter stability. For the discounted cost setup, we establish near optimality of finite window policies via a direct argument involving near optimality of quantized approximations for MDPs under weak Feller continuity, where finite truncations of memory can be viewed as quantizations of infinite memory with a uniform diameter in each finite window restriction under the product metric. For the average cost setup, we provide new existence conditions and also a general approach on how to initialize the randomness which we show to establish convergence to optimal cost. In the control-free case, our analysis leads to new and weak conditions for the existence and uniqueness of invariant probability measures for nonlinear filter processes, where we show that unique ergodicity of the measurement process and a measurability condition related to filter stability leads to unique ergodicity.

In this paper we establish necessary optimality conditions for minimax multiprocess problems: these are optimal control problems in which we have a family of control systems coupled by endpoint constraints and a minimax cost functional to minimize.