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Viscosity solution
About: Viscosity solution is a research topic. Over the lifetime, 2392 publications have been published within this topic receiving 69103 citations.
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TL;DR: The notion of viscosity solutions of scalar fully nonlinear partial differential equations of second order provides a framework in which startling comparison and uniqueness theorems, existence theorem, and continuous dependence may now be proved by very efficient and striking arguments as discussed by the authors.
Abstract: The notion of viscosity solutions of scalar fully nonlinear partial differential equations of second order provides a framework in which startling comparison and uniqueness theorems, existence theorems, and theorems about continuous dependence may now be proved by very efficient and striking arguments. The range of important applications of these results is enormous. This article is a self-contained exposition of the basic theory of viscosity solutions
5,267 citations
Book•
18 Dec 1992TL;DR: In this paper, an introduction to optimal stochastic control for continuous time Markov processes and to the theory of viscosity solutions is given, as well as a concise introduction to two-controller, zero-sum differential games.
Abstract: This book is intended as an introduction to optimal stochastic control for continuous time Markov processes and to the theory of viscosity solutions. The authors approach stochastic control problems by the method of dynamic programming. The text provides an introduction to dynamic programming for deterministic optimal control problems, as well as to the corresponding theory of viscosity solutions. A new Chapter X gives an introduction to the role of stochastic optimal control in portfolio optimization and in pricing derivatives in incomplete markets. Chapter VI of the First Edition has been completely rewritten, to emphasize the relationships between logarithmic transformations and risk sensitivity. A new Chapter XI gives a concise introduction to two-controller, zero-sum differential games. Also covered are controlled Markov diffusions and viscosity solutions of Hamilton-Jacobi-Bellman equations. The authors have tried, through illustrative examples and selective material, to connect stochastic control theory with other mathematical areas (e.g. large deviations theory) and with applications to engineering, physics, management, and finance. In this Second Edition, new material on applications to mathematical finance has been added. Concise introductions to risk-sensitive control theory, nonlinear H-infinity control and differential games are also included.
3,885 citations
Book•
18 Dec 1997
TL;DR: In this paper, the main ideas on a model problem with continuous viscosity solutions of Hamilton-Jacobi equations are discussed. But the main idea of the main solutions is not discussed.
Abstract: Preface.- Basic notations.- Outline of the main ideas on a model problem.- Continuous viscosity solutions of Hamilton-Jacobi equations.- Optimal control problems with continuous value functions: unrestricted state space.- Optimal control problems with continuous value functions: restricted state space.- Discontinuous viscosity solutions and applications.- Approximation and perturbation problems.- Asymptotic problems.- Differential Games.- Numerical solution of Dynamic Programming.- Nonlinear H-infinity control by Pierpaolo Soravia.- Bibliography.- Index
2,747 citations
TL;DR: In this article, the authors examined viscosity solutions of Hamilton-Jacobi equations, and proved the existence assertions by expanding on the arguments in the introduction concerning the relationship of the vanishing-viscosity method and the notion of viscoity solutions.
Abstract: Publisher Summary This chapter examines viscosity solutions of Hamilton–Jacobi equations. The ability to formulate an existence and uniqueness result for generality requires the ability to discuss non differential solutions of the equation, and this has not been possible before. However, the existence assertions can be proved by expanding on the arguments in the introduction concerning the relationship of the vanishing viscosity method and the notion of viscosity solutions, so users can adapt known methods here. The uniqueness is then the main new point.
2,407 citations
TL;DR: In this paper, the previous equation of Buddenberg and the author has been modified to give a general equation for viscosity as a function of molecular weights and viscosities of the pure components of the mixture.
Abstract: By application of the kinetic theory, with several simplifying assumptions, the previous equation of Buddenberg and the author has been modified to give a general equation for viscosity as a function of molecular weights and viscosities of the pure components of the mixture. Agreement of the equation with experimental data is demonstrated for a number of highly irregular binary gas systems and mixtures of three to seven components.
2,183 citations