Control of boundary impedance in two-dimensional material-body cloaking by the wave flow method

Control problems are considered for a two-dimensional electromagnetic field model describing electromagnetic wave scattering in a unbounded homogeneous medium containing an anisotropic permeable inclusion with a partially covered (cloaked) boundary. The control is a function involved in the impedance boundary condition on the covered part of the boundary. The solvability of the original mixed transmission problem for the two-dimensional Helmholtz equation and of the control problems is proved. Optimality systems describing necessary extremum conditions are derived. The uniqueness and stability of optimal solutions with respect to certain perturbations of the cost functional and the incident wave are established.     

Numerical solution of the Goursat problem on a triangular domain with mixed boundary conditions

For the Goursat problem, we consider a triangular domain with mixed Dirichlet and impedance boundary conditions imposed on it. We develop an algorithm for its numerical solution mainly based on Runge-Kutta method and trapezoidal formula. Iterative techniques are constructed to compute some data for the nonlinear part of the differential equation and the impedance boundary condition. Error estimates are derived. Examples are presented to illustrate the effectiveness of the method.     

Solvability of Control Problems for Stationary Equations of Magnetohydrodynamics of a Viscous Fluid

We consider the boundary-value problem for the stationary equations of magnetohydrodynamics of a viscous incompressible fluid with nonhomogeneous boundary conditions for the velocity and electromagnetic field. We study global solvability of this problem and establish some sufficient conditions for uniqueness of its solution. We state control problems for the model of magnetohydrodynamics under consideration, study their solvability, give and examine optimality systems for both arbitrary and particular quality functionals.     

Two-parameter extremum problems of boundary control for stationary thermal convection equations

Two-parameter extremum problems of boundary control are formulated for the stationary thermal convection equations with Dirichlet boundary conditions for velocity and with mixed boundary conditions for temperature. The cost functional is defined as the root mean square integral deviation of the desired velocity (vorticity, or pressure) field from one given in some part of the flow region. Controls are the boundary functions involved in the Dirichlet condition for velocity on the boundary of the flow region and in the Neumann condition for temperature on part of the boundary. The uniqueness of the extremum problems is analyzed, and the stability of solutions with respect to certain perturbations in the cost functional and one of the functional parameters of the original model is estimated. Numerical results for a control problem associated with the minimization of the vorticity norm aimed at drag reduction are discussed.     

Coefficient inverse extremum problems for stationary heat and mass transfer equations

A technique is developed for analyzing coefficient inverse extremum problems for a stationary model of heat and mass transfer. The model consists of the Navier-Stokes equations and the convection-diffusion equations for temperature and the pollutant concentration that are nonlinearly related via buoyancy in the Boussinesq approximation and via convective heat and mass transfer. The inverse problems are stated as the minimization of certain cost functionals at weak solutions to the original boundary value problem. Their solvability is proved, and optimality systems describing the necessary optimality conditions are derived. An analysis of the latter is used to establish sufficient conditions ensuring the local uniqueness and stability of solutions to the inverse extremum problems for particular cost functionals.     

Robust control problems for the multilayer quasi-geostrophic system

In this article, we study some robust control problems associated with the multilayer quasi-geostrophic equations of the ocean and related to data assimilation in oceanography. We consider higher norms (compared to [T. Tachim Medjo, Robust control problems associated with the multilayer quasi-geostrophic equations of the ocean, Appl. Math. Optim. 51(3) (2005) 333–360]) in the definition of the cost functionals. We prove the existence and uniqueness of solutions. The result relies on better a priori estimates on the solutions to the multilayer quasi-geostrophic system obtained using a new formulation that we introduce for the multilayer quasi-geostrophic equation of the ocean. The new formulation replaces the non-homogenous boundary conditions (and the non-local constraint) on the stream-function by a simple homogenous Dirichlet boundary condition.     

Integral equation methods for scattering by infinite rough surfaces

In this paper, we consider the Dirichlet and impedance boundary value problems for the Helmholtz equation in a non‐locally perturbed half‐plane. These boundary value problems arise in a study of time‐harmonic acoustic scattering of an incident field by a sound‐soft, infinite rough surface where the total field vanishes (the Dirichlet problem) or by an infinite, impedance rough surface where the total field satisfies a homogeneous impedance condition (the impedance problem). We propose a new boundary integral equation formulation for the Dirichlet problem, utilizing a combined double‐ and single‐layer potential and a Dirichlet half‐plane Green's function. For the impedance problem we propose two boundary integral equation formulations, both using a half‐plane impedance Green's function, the first derived from Green's representation theorem, and the second arising from seeking the solution as a single‐layer potential. We show that all the integral equations proposed are uniquely solvable in the space of bounded and continuous functions for all wavenumbers. As an important corollary we prove that, for a variety of incident fields including an incident plane wave, the impedance boundary value problem for the scattered field has a unique solution under certain constraints on the boundary impedance. Copyright © 2003 John Wiley & Sons, Ltd.     

Stability estimates in identification problems for the convection-diffusion-reaction equation

Identification problems for the stationary convection-diffusion-reaction equation in a bounded domain with a Dirichlet condition imposed on the boundary of the domain are studied. By applying an optimization method, these problems are reduced to inverse extremum problems in which the variable diffusivity and the volume density of substance sources are used as control functions. Their solvability is proved for an arbitrary weakly lower semicontinuous cost functional and particular cost functionals. An analysis of the optimality system is used to establish sufficient conditions on the input data under which the solutions of particular extremum problems are unique and stable with respect to small perturbations in the cost functional and in one of the functions involved in the boundary value problem.     

Stability estimates for solutions of control problems for the Maxwell equations with mixed boundary conditions

We consider extremal problems for the time-harmonic Maxwell equations with mixed boundary conditions for the electric field. Namely, the tangential component of the electric field is given on one part of the boundary, and an impedance boundary condition is posed on the other part. We prove the solvability of the original mixed boundary value problem and the extremal problem. We obtain sufficient conditions on the input data ensuring the stability of solutions of specific extremal problems under certain perturbations of both the performance functional and some functions occurring in the boundary value problem.     

The Solution of a Free Boundary Problem Related to Environmental Management Systems

Abstract

Explicit solutions of free boundary problems are notoriously difficult to find. In this article, we consider two log-normal diffusions. One represents the level of pollution, or degradation, in some environmental area. The second models the social, political, or financial cost of the pollution. A single control parameter is considered that reduces the rate of pollution. The optimal time to implement the change in the parameter is found by explicitly solving a free boundary problem. The novelty is that the smooth pasting conditions, which are difficult to justify, are not used in the derivation.     

Three positive solutions for the one-dimensional p-Laplacian

In this paper, nonlinear two point boundary value problems with p-Laplacian operators subject to Dirichlet boundary condition and nonlinear boundary conditions are studied. We show the existence of three positive solutions by the five functionals fixed point theorem.     

On the optimization of the initial boundary value problem for a conservation law

In the present note, the theory of shift differentiability for the Cauchy problem is extended to the case of an initial boundary value problem for a conservation law. This result allows to exhibit an Euler-Lagrange equation to be satisfied by the extrema of integral functionals defined on the solutions of initial boundary value problems of this kind.     

Mixed boundary value problems with a shift for a pair of metaanalytic and analytic functions

In this article, we reconsider the mixed boundary value problem on the unit circle for a pair of metaanalytic and analytic functions as in Du and Wang (2008) [9]. By adopting appropriate transformations, we convert the problem into two independent boundary value problems for analytic functions. We then obtain expressions of solution and condition of solvability for the mixed boundary value problem. The forms of the solutions and the condition of solvability here are rather dissimilar to those in Du and Wang (2008) [9]. But the equivalence is established at the end of this article.     

Characterization of the Darboux point for particular classes of problems

A minimal sufficient condition for global optimality involving the Darboux point, analogous to the minimal sufficient condition of local optimality involving the conjugate point, is presented. The Darboux point is then characterized for optimal control problems with linear dynamics, cost functionals with a general terminal state term and an integrand quadratic in the state and control, and general terminal conditions. The Darboux point is shown to be the supremum of a sequence of conjugate points. If the terminal state term is quadratic, along with a scalar quadratic boundary condition, then the Darboux point is also the time at which the Riccati matrix becomes unbounded, giving a characterization of the unboundedness of the Riccati matrix at points which are not in general conjugate points.This research was supported by the National Science Foundation under Grant No. GK-30115.This is Definition 2.1 of Ref. 1.     

Stability estimates for the solutions of control problems for the stationary magnetohydrodynamic equations

We study control problems for the stationary magnetohydrodynamic equations. In these problems, one has to find an unknown vector function occurring in the boundary condition for the magnetic field and the solution of the boundary value problem in question by minimizing a performance functional depending on the velocity and pressure. We derive new a priori estimates for the solutions of the original boundary value problem and the extremal problem and prove theorems on the local uniqueness and stability of solutions for specific performance functionals.     

A boundary control problem for the blood flow in venous insufficiency. The general case

The purpose of this article is to introduce and study an optimal control problem with medical applications. When a vein loses its elasticity, phenomena such as stagnation and recirculation of the blood may appear; these phenomena produce medical complications. We propose an optimization model in order to diminish the negative consequences of the lack of vein elasticity. We extend a previous model involving the interaction between a viscous fluid and an elastic boundary to the case when both the fluid and the elastic medium occupy three dimensional domains. After establishing the existence and uniqueness of the solution for the coupled problem, we present a boundary control problem in order to determine an exterior compression that realizes a blood flow without recirculation. Since it is not possible to find such a compression directly, we consider a sequence of cost functionals and we study the corresponding optimal control problems. The existence and uniqueness of the optimal controls are proved and the optimality conditions that characterize the optimal controls are derived. Finally, we establish the relation between the control problem with physical meaning and the sequence of optimal controls already constructed.     

利用远场模式的完全与不完全数据反演声波阻尼区域

提出一种方法,利用远场模式的完全数据与不完全数据反演声波阻尼区域,证明了方法的收敛性,并给出若干数值例子.     

Efficient solutions in V-KT-pseudoinvex multiobjective control problems: A characterization

In this paper, we introduce a new condition on functionals involved in a multiobjective control problem, for which we define the V-KT-pseudoinvex control problem. We prove that a V-KT-pseudoinvex control problem is characterized so that a Kuhn–Tucker point is an efficient solution. We generalize recently obtained optimality results of known mathematical programming problems and control problems. We illustrate these results with an example.     

Optimal Boundary Control of Steady-State Flow of a Viscous Inhomogeneous Incompressible Fluid

We study the problem of optimal boundary control of two-dimensional steady-state flow of a viscous inhomogeneous incompressible fluid. The role of control is played by the values of the velocity on a part of the boundary of the domain considered. On the remaining part of the boundary, the vector of flow velocity and the fluid density are given. We seek the fluid density as a scalar function (determined by the initial data) of the stream function, study the solvability of the problem, and obtain necessary optimality conditions.     

Differential quadrature method for pricing American options

In this article, differential quadrature method (DQM), a highly accurate and efficient numerical method for solving nonlinear problems, is used to overcome the difficulty in determining the optimal exercise boundary of American option. The following three parts of the problem in pricing American options are solved. The first part is how to treat the uncertainty of the early exercise boundary, or free boundary in the language of the PDE treatment of the American option, because American options can be exercised before the date of expiration. The second part is how to solve the nonlinear problem, because the problem of pricing American options is nonlinear. And the third part is how to treat the initial value condition with the singularity and the boundary conditions in the DQM. Numerical results for the free boundary of American option obtained by both DQM and finite difference method (FDM) are given and from which it can be seen the computational efficiency is greatly improved by DQM. © 2002 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 18: 711–725, 2002; Published online in Wiley InterScience (www.interscience.wiley.com); DOI 10.1002/num.10028.