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 $\epsilon$-viable, then the square of the distance from $K$: $d_K^2(x)\coloneq\inf_{y\in K}|x-y|^2$ 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 Świ\c ech and Kelome. We discuss several cases where the above necessary condition is also sufficient.''

Buckdahn, R., Quincampoix, M., Tessitore, G. (2008). Controlled stochastic differential equations under constraints in infinite dimensional spaces. SIAM JOURNAL ON CONTROL AND OPTIMIZATION, 47(1), 218-250 [10.1137/060674284].

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 $\epsilon$-viable, then the square of the distance from $K$: $d_K^2(x)\coloneq\inf_{y\in K}|x-y|^2$ 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 Świ\c ech and Kelome. We discuss several cases where the above necessary condition is also sufficient.''
Scheda breve Scheda completa Scheda completa (DC)
Articolo in rivista - Articolo scientifico
stochastic differential equations, constraints, Hilbert spaces
English
218
250
Buckdahn, R., Quincampoix, M., Tessitore, G. (2008). Controlled stochastic differential equations under constraints in infinite dimensional spaces. SIAM JOURNAL ON CONTROL AND OPTIMIZATION, 47(1), 218-250 [10.1137/060674284].
Buckdahn, R; Quincampoix, M; Tessitore, G
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/10281/5507
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