Sub- and Super-optimality Principles and Construction of Almost Optimal Strategies for Differential Games in Hilbert Spaces

Święch, Andrzej

doi:10.1007/978-0-8176-8089-3_8

Annals of the International Society of Dynamic Games

2010

DOI: 10.1007/978-0-8176-8089-3_8

|View full text |Cite

Sub- and Super-optimality Principles and Construction of Almost Optimal Strategies for Differential Games in Hilbert Spaces

Andrzej Święch

Abstract: We prove sub-and superoptimality principles of dynamic programming and show how to use the theory of viscosity solutions to construct almost optimal strategies for two-player, zero-sum differential games driven by abstract evolution equations in Hilbert spaces.

Help me understand this report

Search citation statements

Order By: Relevance

Paper Sections

Select...

Citation Types

Supporting

Mentioning

Contrasting

Year Published

2024

Publication Types

Select...

Article2

Relationship

Self Cite0

Independent2

Authors

Journals

Cited by 2 publications

References 24 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

Minimax solutions of Hamiltoni-Jacobi equations in dynamical optimization problems for hereditary systems

Gomoyunov,

Lukoyanov

2024

Uspekhi Mat. Nauk

View full text Add to dashboard Cite

Настоящая статья содержит обзор результатов, касающихся развития теории уравнений Гамильтона-Якоби для наследственных динамических систем. Особенность этих систем состоит в том, что скорость изменения их состояния зависит не только от текущего положения, как в классическом случае, но и от всего пройденного пути - истории движения. Большая часть статьи посвящена динамическим системам, движение которых описывается при помощи функционально-дифференциальных уравнений запаздывающего типа. Кроме того, затрагиваются и более общие системы, описываемые функционально-дифференциальными уравнениями нейтрального типа, а также тесно связанные с ними системы, описываемые дифференциальными уравнениями с производными дробного порядка. Рассматриваются так называемые наследственные уравнения Гамильтона-Якоби, которые для указанных классов систем играют роль, аналогичную роли классических уравнений Гамильтона-Якоби в задачах динамической оптимизации обыкновенных дифференциальных систем. В контексте приложений к задачам управления основное внимание уделяется минимаксному подходу к понятию обобщенного решения рассматриваемых уравнений Гамильтона-Якоби, а также его связи с вязкостным подходом. Приводятся опирающиеся на обсуждаемые конструкции методы построения оптимальных стратегий управления по принципу обратной связи с памятью истории движения. Библиография: 183 названия.

show abstract

Minimax solutions of Hamiltoni-Jacobi equations in dynamical optimization problems for hereditary systems

Gomoyunov,

Lukoyanov

2024

Uspekhi Mat. Nauk

View full text Add to dashboard Cite

show abstract

Minimax solutions of Hamilton-Jacobi equations in dynamic optimization problems for hereditary systems

Gomoyunov,

Lukoyanov

2024

Russian Math. Surveys

View full text Add to dashboard Cite

A survey of the results concerning the development of the theory of Hamilton-Jacobi equations for hereditary dynamical systems is presented. One feature of these systems is that the rate of change of their state depends not only on the current position, like in the classical case, but also on the full path travelled, that is, the history of the motion. Most of the paper is devoted to dynamical systems whose motion is described by functional differential equations of retarded type. In addition, more general systems described by functional differential equations of neutral type and closely related systems described by differential equations with fractional derivatives are considered. So-called path-dependent Hamilton-Jacobi equations are treated, which play for the above classes of systems a role similar to that of the classical Hamilton-Jacobi equations in dynamic optimization problems for ordinary differential systems. In the context of applications to control problems, the main attention is paid to the minimax approach to the concept of a generalized solution of the Hamilton-Jacobi equations under consideration and also to its relationship with the viscosity approach. Methods for designing optimal feedback control strategies with memory of motion history which are based on the constructions discussed are presented. Bibliography: 183 titles.

show abstract

scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.

Contact Info

hi@scite.ai

10624 S. Eastern Ave., Ste. A-614

Henderson, NV 89052, USA

Blog Terms and Conditions API Terms Privacy Policy Contact Cookie Preferences Do Not Sell or Share My Personal Information

Made with 💙 for researchers

Part of the Research Solutions Family.

Sub- and Super-optimality Principles and Construction of Almost Optimal Strategies for Differential Games in Hilbert Spaces

Abstract: We prove sub-and superoptimality principles of dynamic programming and show how to use the theory of viscosity solutions to construct almost optimal strategies for two-player, zero-sum differential games driven by abstract evolution equations in Hilbert spaces.

Cited by 2 publications

References 24 publications

Minimax solutions of Hamiltoni-Jacobi equations in dynamical optimization problems for hereditary systems

Minimax solutions of Hamiltoni-Jacobi equations in dynamical optimization problems for hereditary systems

Minimax solutions of Hamilton-Jacobi equations in dynamic optimization problems for hereditary systems

Contact Info

Product

Resources

About