A Product Formula Approach to a Nonhomogeneous Boundary Optimal Control Problem Governed by Nonlinear Phase-field Transition System

In this paper we prove the convergence of an iterative scheme of fractional steps type for a non-homogeneous Cauchy-Neumann boundary optimal control problem governed by non-linear phase-field system, when the boundary control is dependent both on time and spatial variables. Moreover, necessary optimality conditions are established for the approximating process. The advantage of such approach leads to a numerical algorithm in order to approximate the original optimal control problem.     

An Optimal Control Problem of a Coupled Nonlinear Parabolic Population System

An optimal control problem for a coupled nonlinear parabolic population system is considered. The existence and uniqueness of the positive solution for the system is shown by the method of upper and lower solutions. An explicit prior bound of solutions to the system is given by considering an auxiliary coupled linear system. The existence of the optimal control is proved and the characterization of the optimal control is established.     

An Optimal Control Problem for a Singular System of Solid–Liquid Phase Transition

In this article, we deal with a control problem for a singular system regarding a phase-field model which describes a solid–liquid transition by the Ginzburg–Landau theory. The purpose is to control the system by the means of the heat supply r able to guide it into a certain state with a solid (or liquid) part in a prescribed subset Ω₀ of the space domain Ω, and maintain it in this state during a period of time. The transition is described by a nonlinear differential system of two equations for the phase field and temperature. The control problem is set for some expressions of the cost functional which might reveal cases of physical interest. An approximating control problem is introduced and the existence of at least an optimal pair is proved. The first-order optimality conditions for the approximating problem are determined and a convergence result is given.     

State-Constrained Optimal Control Governed by Elliptic Differential Equations in Unbounded Domains

This work concerns the maximum principle for optimal control problems governed by elliptic differential equations in unbounded domains. Some state constraints are considered. This work was supported by National Natural Science Foundation of China, Grants 10401041 and 10471053.     

Boundary Control Problem for a Nonlinear Convection–Diffusion–Reaction Equation

Computational Mathematics and Mathematical Physics - The solvability of boundary-value and extremum problems for a nonlinear convection–diffusion–reaction equation with mixed boundary...     

Stability of a Turnpike Phenomenon for a Discrete-Time Optimal Control System

We study the structure of solutions of a discrete-time control system with a compact metric space of states X which arises in economic dynamics. This control system is described by a nonempty closed set Ω⊂X×X which determines a class of admissible trajectories (programs) and by a bounded upper semicontinuous objective function v:Ω→R ¹ which determines an optimality criterion. We are interested in turnpike properties of the approximate solutions which are independent of the length of the interval, for all sufficiently large intervals. In the present paper, we show that these turnpike properties are stable under perturbations of the objective function v.     

On a Problem of Optimal Stochastic Control with Incomplete Information

We study a problem of Bayesian adaptive control arising in mathematical finance. For this particular problem, we obtain explicit formulas for the value function and the optimal control policy.     

Stability Analysis of an Optimal Control Problem for a Hyperbolic Equation

In this paper, we derive some results concerning the continuous dependence on parameters of optimal solutions to an optimal control problem that involves a quasilinear hyperbolic differential equation with a boundary condition and a nonlinear integral functional of action; continuous dependence of such kind is sometimes referred to as stability or sensitivity. To present a sufficient condition for the continuous dependence, we use the Kuratowski–Painlevé upper limit of a sequence of sets. Also offered is a technical interpretation of the results.     

Overlaying Classifiers: A Practical Approach to Optimal Scoring

The Receiver Operating Characteristic (ROC) curve is one of the most widely used visual tools to evaluate performance of scoring functions regarding their capacities to discriminate between two populations. It is the goal of this paper to propose a statistical learning method for constructing a scoring function with nearly optimal ROC curve. In this bipartite setup, the target is known to be the regression function up to an increasing transform, and solving the optimization problem boils down to recovering the collection of level sets of the latter, which we interpret here as a continuum of imbricated classification problems. We propose a discretization approach, consisting of building a finite sequence of N classifiers by constrained empirical risk minimization and then constructing a piecewise constant scoring function s _N (x) by overlaying the resulting classifiers. Given the functional nature of the ROC criterion, the accuracy of the ranking induced by s _N (x) can be conceived in a variety of ways, depending on the distance chosen for measuring closeness to the optimal curve in the ROC space. By relating the ROC curve of the resulting scoring function to piecewise linear approximates of the optimal ROC curve, we establish the consistency of the method as well as rate bounds to control its generalization ability in sup -norm. Eventually, we also highlight the fact that, as a byproduct, the algorithm proposed provides an accurate estimate of the optimal ROC curve.     

Pontryagin’s Principle for Optimal Control Problem Governed by 3D Navier–Stokes Equations

This paper deals with the Pontryagin maximum principle for optimal control problems governed by 3D Navier–Stokes equations with pointwise control constraint. The obtained result is proved by using some results on regularity of solutions of the Navier–Stokes equations and techniques of optimal control theory.     

Viscosity Solutions for a System of Integro-PDEs and Connections to Optimal Switching and Control of Jump-Diffusion Processes

We develop a viscosity solution theory for a system of nonlinear degenerate parabolic integro-partial differential equations (IPDEs) related to stochastic optimal switching and control problems or stochastic games. In the case of stochastic optimal switching and control, we prove via dynamic programming methods that the value function is a viscosity solution of the IPDEs. In our setting the value functions or the solutions of the IPDEs are not smooth, so classical verification theorems do not apply.     

Finite Difference Method for an Optimal Control Problem for a Nonlinear Time-dependent Schrödinger Equation

The finite difference method is applied to an optimal control problem for a system governed by a nonlinear Schrödinger equation with a complex coe?cient. The optimal control problem is discretized by the finite difference method, the error estimate for the finite difference scheme is established and the convergence of difference approximations of the optimal control according to the functional is proved.     

A Free Boundary Problem for an Elliptic–Parabolic System: Application to a Model of Tumor Growth

Abstract

In this article, we study a free boundary problem for a system of two partial differential equations, one parabolic and other elliptic. The system models the growth of a tumor with arbitrary initial shape. We establish the existence and uniqueness of a solution for some time interval. In the special case where we only have the elliptic equation, the problem coincides with the Hele–Shaw problem.     

New Approach to Solving a System of Variational Inequalities and Hierarchical Problems

This paper deals with a viscosity iterative method, in real Hilbert spaces, for solving a system of variational inequalities over the fixed-point sets of possibly discontinuous mappings. Under classical conditions, we prove a strong convergence theorem for our method. The proposed algorithm can be applied for instance to solving variational inequalities in some situations when the projection methods fail. Moreover, the techniques of analysis are novel and provide new tools in designing approximation schemes for combined and bilevel optimization problems.     

Existence and Uniqueness of the Global Smooth Solution to the Periodic Boundary Value Problem of Fractional Nonlinear Schrödinger System

In this paper, we study a class of coupled fractional nonlinear Schrödinger system with periodic boundary condition. Using Galerkin method, the existence of global smooth solution is obtained. We also prove the uniqueness of the solution.     

A New Analytical Approach to Solve Magnetohydrodynamics Flow Over a Nonlinear Porous Stretching Sheet by Laplace Padé Decomposition Method

The aspire of this article is to bring in a new approximate method, that is to say the Laplace Padé decomposition method which is a mixture of Laplace decomposition and Padé approximation to offer an analytical approximate way out to magnetohydrodynamics flow over a nonlinear porous stretching sheet. This new iteration approach provides us with a convenient way to approximate solution. A closed agreement between the obtained solution and some well-known results has been established. The proposed procedure can be applied to handle other nonlinear problems.     

A THEORY FOR MARSHALLING RALLWAY CARS INTO A TRAIN——An Approach to a Multiple Objective Problem

About twefity years ago, on some marshalling yards in China, an empirical method, alled the Tabulation Method, was used to make feasible and effective plans for marshalling railway cars into a train, especially, into a local train. Theoretical studies of this method developed some mathematical models of the nature of of combinatorial optimization. The main part of this paper is a survey of a mathematical theory, called the Theory of Standard Numeration. When there is one loco (there are two locos) in operation on the marshalling yard (one on each side of the yard), this theory says that every feasible plan for making up a local train can be characterized by a finite sequence of binary(ternary) numbers which are called the characterlshc numbers of the cars. If the minimum value of the number of ordered subsequences in a standard partition of the initial train is n (which can easily be obtained by the Tabulation Method)and 2~(m-2)相似文献   

On a New Approach to the Problem of Distribution of Zeros of Hermite—Padé Polynomials for a Nikishin System

A new approach to the problem of the zero distribution of type I Hermite—Padé polynomials for a pair of functions f₁, f₂ forming a Nikishin system is discussed. Unlike the traditional vector approach, we give an answer in terms of a scalar equilibrium problem with harmonic external field which is posed on a two-sheeted Riemann surface.