Viscosity Solutions and Applications

9706760 Swiech拟议的研究围绕粘性解的概念，粘性解是Hamilton-Jacobi-Bellman-Isaacs(HJBI)型方程的弱解的概念。它既有基本的，也有应用的，可以分为两部分：无限维空间中的偏微分方程组及其在随机最优控制中的应用，以及椭圆型和抛物型方程。关于第一部分，研究者计划继续研究Hilbert空间中二阶HJBI方程粘性解(即广义解)的理论发展。基本问题是解的存在唯一性、正则性、收敛性质等。从应用的角度来看，HJBI方程非常重要，因为它们是所谓的动态规划方程，对应于由随机偏微分方程组驱动的动力系统的随机最优控制问题。第二部分主要研究完全非线性、二阶、一致椭圆型和抛物型偏微分方程组理论中的一些基本问题。主要问题是在空间变量中不连续的非线性一致抛物型和椭圆型方程的解的正则性和唯一性，以及在更大范围内发展具有可测成分的完全非线性方程的广义解的理论。尽管最近有一些结果，但唯一性仍然是一个主要的开放问题，即使对于线性方程，它与具有不连续扩散系数的随机微分方程鞅解的唯一性有关。最后，研究者提出了退化椭圆型和抛物型可测偏微分方程弱解的概念。例如，这种类型的方程出现在涉及由平均曲线驱动的运动的问题中。经典的粘性溶液理论在这里不适用。有必要将其扩展到此类案件。粘性解的概念被应用数学家广泛使用，所提出的研究在应用科学中具有潜在的深远影响，特别是它涉及HJBI方程和由随机偏微分方程组驱动的动力系统的控制的部分。在现实生活中，这些系统是由偏微分方程控制的，表现出噪声行为。应用范围从工程到数学金融。人们不得不提到非线性部分观测系统的随机最优控制(所谓的非线性滤波)，由随机偏微分方程建模的系统的分布式和边界控制，工程上广泛应用的无限维系统的极小极大型控制(与无限维微分对策有关)，小噪声系统的风险敏感控制(在自适应跟踪设备的控制中很重要)，以及最近利率动态的市场模型(期权定价)。调查人员相信，拟议的研究将有助于理解上述问题，并将提供一个可供在这些领域工作的人使用的数学框架。

9706760 Swiech The proposed research revolves around the notion of viscosity solution, which is a notion of weak solution for equations of Hamilton-Jacobi-Bellman-Isaacs (HJBI) type. It is both basic and applied and can be divided into two parts: partial differential equations in infinite dimensional spaces and applications to stochastic optimal control, and elliptic and parabolic equations. As regards the first part, the investigator plans to continue research in the development of the theory of viscosity solutions (i.e. generalized solutions) of second order HJBI equations in Hilbert space. The basic issues are the existence and uniqueness of solutions, their regularity, convergence properties, etc. HJBI equations are of great importance from the point of view of applications, since they are so-called dynamic programming equations, corresponding to problems of stochastic optimal control of dynamical systems driven by stochastic partial differential equations. The second part of the proposed research concentrates on some fundamental problems in the theory of fully nonlinear, second order, uniformly elliptic and parabolic partial differential equations. The major issue is the regularity and uniqueness of solutions of nonlinear, uniformly parabolic and elliptic equations which are not continuous in the space variable and, on a larger scale, the development of a theory of generalized solutions for fully nonlinear equations with measurable ingredients. Despite some recent results, uniqueness is in general still a major open problem, even for linear equations, where it is connected to the uniqueness of so-called martingale solutions of stochastic differential equations with discontinuous diffusion coefficient. Finally the investigator proposes to work on a notion of weak solution for degenerate elliptic and parabolic partial differential equations with measurable terms. Equations of this type appear, for instance, in problems involving motion driven by mean curva ture. The classical theory of viscosity solutions does not apply here. Its extension to such cases is needed. The notion of viscosity solution is widely used by applied mathematicians, and the proposed research has a potential of having far-reaching consequences in applied sciences, especially the part of it dealing with HJBI equations and control of dynamical systems driven by stochastic partial differential equations. In real life these are systems governed by partial differential equations that exhibit noisy behavior. Applications range from engineering to mathematical finance. One has to mention stochastic optimal control of nonlinear partially observed systems (so-called nonlinear filtering), distributed and boundary control of systems modelled by stochastic partial differential equations, minimax type control of infinite dimensional systems widely used in engineering (which is connected to infinite dimensional differential games), risk-sensitive control of small noise systems (important in control of adaptive tracking devices), and recent market models of interest rate dynamics (option pricing). The investigator is convinced that the proposed research will help understand the above problems and will provide a mathematical framework that can be used by people working in these areas.