Paper Info
Title
Controlled Stochastic Differential Equations under Constraints in Infinite Dimensional Spaces
Abstract
In this paper we study the compatibility (or viability) of a given state constraint $K$ with respect to a controlled stochastic evolution equation in a real Hilbert space $H$. We allow the noise to be a cylindrical Wiener process and admit an unbounded linear operator in the state equation. Our assumptions cover, for instance, controlled heat equations with space-time white noise. Our main result is to prove that if $K$ is $\varepsilon$-viable, then the square of the distance from $K$: $d_K^2(x):= \inf_{y\in K}|x-y|^2$ is a viscosity supersolution of a suitable class of fully nonlinear Hamilton-Jacobi-Bellman equations in $H$. This extends already obtained results into the finite dimensional case. We use the definition of viscosity supersolutions for “unbounded” elliptic equations in infinite variables that have been recently introduced by Święch and Kelome. We discuss several cases where the above necessary condition is also sufficient.
Year
DOI
Venue
2008
10.1137/060674284
SIAM J. Control and Optimization
Keywords
Field
DocType
controlled stochastic differential equations,state equation,state constraint,nonlinear hamilton-jacobi-bellman equation,viscosity supersolutions,elliptic equation,viscosity supersolution,controlled stochastic evolution equation,infinite dimensional spaces,unbounded linear operator,space-time white noise,controlled heat equation,stochastic differential equation,stochastic control
Hilbert space,Differential equation,Nonlinear system,Mathematical analysis,Hamilton–Jacobi equation,Stochastic differential equation,Heat equation,Viscosity solution,Mathematics,Stochastic control
Journal
Volume
Issue
ISSN
47
1
0363-0129
Citations 
PageRank 
References 
2
0.61
0
Authors
3
Name
Order
Citations
PageRank
Rainer Buckdahn16218.36
M. Quincampoix246350.08
Gianmario Tessitore33513.55