Path-Dependent Hamilton–Jacobi Equations In Infinite Dimensions

Keywords

Minimax solutions; Nonlinear evolution equations; Optimal control; Path-dependent PDEs

Abstract

We propose notions of minimax and viscosity solutions for a class of fully nonlinear path-dependent PDEs with nonlinear, monotone, and coercive operators on Hilbert space. Our main result is well-posedness (existence, uniqueness, and stability) for minimax solutions. A particular novelty is a suitable combination of minimax and viscosity solution techniques in the proof of the comparison principle. One of the main difficulties, the lack of compactness in infinite-dimensional Hilbert spaces, is circumvented by working with suitable compact subsets of our path space. As an application, our theory makes it possible to employ the dynamic programming approach to study optimal control problems for a fairly general class of (delay) evolution equations in the variational framework. Furthermore, differential games associated to such evolution equations can be investigated following the Krasovskiĭ–Subbotin approach similarly as in finite dimensions.

Publication Date

10-15-2018

Publication Title

Journal of Functional Analysis

Volume

275

Issue

8

Number of Pages

2096-2161

Document Type

Article

Personal Identifier

scopus

DOI Link

https://doi.org/10.1016/j.jfa.2018.07.010

Socpus ID

85050665046 (Scopus)

Source API URL

https://api.elsevier.com/content/abstract/scopus_id/85050665046

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