Necessary conditions for optimality in relaxed stochastic control problems

In this paper, we are concerned with optimal control problems where the system is driven by a stochastic differential equation of the Ito type. We study the relaxed model for which an optimal solution exists. This is an extension of the initial control problem, where admissible controls are measure valued processes. Using Ekeland's variational principle and some stability properties of the corresponding state equation and adjoint processes, we establish necessary conditions for optimality satisfied by an optimal relaxed control. This is the first version of the stochastic maximum principle that covers relaxed controls.     

A note on the reflection principle for the biharmonic equation and the Stokes system

In this note, we discuss the reflection principle of the Stokes system in a half space for the threedimensional case, and of the biharmonic equation. Admitting different boundary conditions, we use the reflection principle to prove uniqueness of solutions of the Stokes system or the biharmonic equation in weightedLq-spaces     

The balancing principle in solving semi-discrete inverse problems in Sobolev scales by Tikhonov method

In this article we discuss a regularization of semi-discrete ill-posed problem appearing as a result of application of a collocation method to Fredholm integral equation of the first kind. In this context we analyse Tikhonov regularization in Sobolev scales and prove error bounds under general source conditions. Moreover, we study an a posteriori regularization parameter choice by means of the balancing principle.     

Cell polarisation model: The 1D case

We study the dynamics of a one-dimensional non-linear and non-local drift-diffusion equation set in the half-line, with the coupling involving the trace value on the boundary. The initial mass M of the density determines the behaviour of the equation: attraction to self-similar profile, to a steady state of finite time, blow-up for supercritical mass. Using the logarithmic Sobolev and the HWI inequalities we obtain a rate of convergence for the sub-critical and critical mass cases. Moreover, we prove a comparison principle on the equation obtained after space integration. This concentration-comparison principle allows proving blow-up of solutions for large initial data without any monotonicity assumption on the initial data.     

Explicit decay bounds in some quasilinear one‐dimensional parabolic problems

In the first part of this paper we study a thermal diffusion process described by a semilinear parabolic problem and we introduce a new maximum principle in order to obtain explicit decay bounds for the temperature and its gradient. In the second part we find analogous bounds for the so‐called ground water equation in a more general form. Copyright © 1999 John Wiley & Sons, Ltd.     

Oscillation behavior of solution of impulsive fractional differential equation

In this paper, we study the oscillation of impulsive Caputo fractional differential equation. Sufficient conditions for the asymptotic and oscillation of the equation are obtained by using the inequality principle and Bihari Lemma. An example is given to illustrate the results. This is the first time to study the oscillation of impulsive fractional differential equation with Caputo derivative.     

Padé iteration method for regularization

In this study we present iterative regularization methods using rational approximations, in particular, Padé approximants, which work well for ill-posed problems. We prove that the (k, j)-Padé method is a convergent and order optimal iterative regularization method in using the discrepancy principle of Morozov. Furthermore, we present a hybrid Padé method, compare it with other well-known methods and found that it is faster than the Landweber method. It is worth mentioning that this study is a completion of the paper [A. Kirsche, C. Böckmann, Rational approximations for ill-conditioned equation systems, Appl. Math. Comput. 171 (2005) 385–397] where this method was treated to solve ill-conditioned equation systems.     

关于二阶半线性椭圆方程的Dirichlet边值问题一个注记(英文)

本文研究二阶半线性椭圆方程的Dirichlet边值问题.利用山路引理和最小作用原理,获得了在新条件下具有Dirichlet边值问题的二阶半线性椭圆方程的弱解的存在性的结果.     

Positivity and time behavior of a linear reaction–diffusion system,non-local in space and time

We consider a general linear reaction–diffusion system in three dimensions and time, containing diffusion (local interaction), jumps (nonlocal interaction) and memory effects. We prove a maximum principle and positivity of the solution and investigate its asymptotic behavior. Moreover, we give an explicit expression of the limit of the solution for large times. In order to obtain these results, we use the following method: We construct a Riemannian manifold with complicated microstructure depending on a small parameter. We study the asymptotic behavior of the solution to a simple diffusion equation on this manifold as the small parameter tends to zero. It turns out that the homogenized system coincides with the original reaction–diffusion system. Using this and the facts that the diffusion equation on manifolds satisfies the maximum principle and its solution converges to a easily calculated constant, we can obtain analogous properties for the original system. Copyright © 2008 John Wiley & Sons, Ltd.     

A mesh-independence principle for operators equations and the Steffensen method

In this study we prove the mesh-independence principle via Steffensen’s method. This principle asserts that when Steffensen’s method is applied to a nonlinear equation between some Banach spaces, as well as to some finite-dimensional discretization of that equation, then the behavior of the discretized process is asymptotically the same as that for the original iteration. Local and semilocal convergence results as well as an error analysis for Steffensen’s method are also provided.     

A Stochastic Recursive Optimal Control Problem Under the G-expectation Framework

In this paper, we study a stochastic recursive optimal control problem in which the objective functional is described by the solution of a backward stochastic differential equation driven by \(G\) -Brownian motion. Under standard assumptions, we establish the dynamic programming principle and the related Hamilton–Jacobi–Bellman (HJB) equation in the framework of \(G\) -expectation. Finally, we show that the value function is the viscosity solution of the obtained HJB equation.     

Three positive solutions for nonlinear elliptic equations in finite strip with hole

In this paper, we study the effect of the domain on the multiplicity of positive solutions for the nonlinear elliptic equation. Using the Ekeland variational principle and the center mass function, we prove that there are at least three positive solutions for the nonlinear elliptic equation in finite strip with holes.     

全空间上带Hardy-Sobolev项的拟线性椭圆方程解的存在性

讨论了全空间上一类带Hardy-Sobolev项的拟线性椭圆问题,利用集中紧原理和适当的实验函数,得到并验证了(PS)_c条件,从而证明了该问题非平凡解的存在性.     

Finite time blow-up and global solutions for a class of semilinear parabolic equations at high energy level

In this paper we study the initial boundary value problem of a class of semilinear parabolic equation. Our main tools are the comparison principle and variational methods. In this paper, we will find both finite time blow-up and global solutions at high energy level.     

A Positive Solution of Mixed Non-Linear Fractional Delay Differential Equations with Integral Boundary Conditions

In this paper, we study mixed non-linear fractional delay differential equations with integral boundary conditions. We obtain an equivalence result between the proposed problem and non-linear Fredholm integral equation of the second kind. Further, we establish existence and uniqueness of positive solutions for the problem using Guo-Krasnoseleskii’s fixed point theorem and Banach contraction principle.     

Stochastic maximum principle for mean-field forward-backward stochastic control system with terminal state constraints

In this paper,we consider an optimal control problem with state constraints,where the control system is described by a mean-field forward-backward stochastic differential equation(MFFBSDE,for short)and the admissible control is mean-field type.Making full use of the backward stochastic differential equation theory,we transform the original control system into an equivalent backward form,i.e.,the equations in the control system are all backward.In addition,Ekeland’s variational principle helps us deal with the state constraints so that we get a stochastic maximum principle which characterizes the necessary condition of the optimal control.We also study a stochastic linear quadratic control problem with state constraints.     

Large deviation principle for the fourth-order stochastic heat equations with fractional noises

In this paper, we shall study a fourth-order stochastic heat equation driven by a fractional noise, which is fractional in time and white in space. We will discuss the existence and uniqueness of the solution to the equation. Furthermore, the regularity of the solution will be obtained. On the other hand, the large deviation principle for the equation with a small perturbation will be established through developing a classical method.     

Large deviation principle for reflected Poisson driven stochastic differential equations in epidemic models

Abstract

We establish a large deviation principle for a reflected Poisson driven stochastic differential equation. Our motivation is to study in a forthcoming paper the problem of exit of such a process from the basin of attraction of a locally stable equilibrium associated with its law of large numbers. Two examples are described in which we verify the assumptions that we make to establish the large deviation principle.     

Stability analysis of time-fractional differential equations with initial data

In this work, we investigate the stability of solution for a class of time-fractional differential equation with initial data by the help of a fractional Duhamel principle; we use it and the superposition principle to obtain the solution of our problem. We present some theoretical results by some scientific theorems and lemmas.     

Complexiton solutions to the Hirota‐Satsuma‐Ito equation

The Hirota bilinear method is a powerful tool for solving nonlinear evolution equations. Together with the linear superposition principle, it can be used to find a special class of explicit solutions that correspond to complex eigenvalues of associated characteristic problems. These solutions are known as complexiton solutions or simply complexitons. In this article, we study complexiton solutions of the the Hirota‐Satsuma‐Ito equation which is a (2 + 1)‐dimensional extension of the Hirota‐Satsuma shallow water wave equation known to describe propagation of unidirectional shallow water waves. We first construct hyperbolic function solutions and consequently derive the so‐called complexitons via the Hirota bilinear method and the linear superposition principle. In particular, we find nonsingular complexiton solutions to the Hirota‐Satsuma‐Ito equation. Finally, we give some illustrative examples and a few concluding remarks.