Appl Math Optim 55:285–326 (2007)
2007 Springer Science+Business Media, Inc.
Inﬁnite Horizon Stochastic Optimal Control Problems with
Degenerate Noise and Elliptic Equations in Hilbert Spaces
Dipartimento di Matematica, Politecnico di Milano,
Piazza Leonardo da Vinci 32, 20133 Milano, Italy
Abstract. Semilinear elliptic partial differential equations are solved in a mild
sense in an inﬁnite-dimensional Hilbert space. These results are applied to a stochas-
tic optimal control problem with inﬁnite horizon. Applications to controlled stochas-
tic heat and wave equations are given.
Key Words. Stochastic optimal control, Stationary Hamilton–Jacobi–Bellman
equations, Inﬁnite-dimensional stochastic processes.
AMS Classiﬁcation. 60H30, 93E20.
In this paper we study a class of semilinear partial differential equations of elliptic
type on a Hilbert space. We continue our previous work , when the case of a partial
differential equation of parabolic type was treated. By mean of the solution of this elliptic
equation, we are able to solve a stochastic optimal control problem with inﬁnite horizon.
In the ﬁrst part of the paper we study the following equation in a separable Hilbert
Au(x) − λu(x) = ψ(x, u(x),∇
u(x)), x ∈ H, (1.1)
where ψ is a given map and λ>0. The linear operator A is formally deﬁned by
A f (x) =
f (x)) +Ax,∇ f (x)
+F(x),∇ f (x)
where ∇ f ∈ H and ∇
f ∈ L(H, H) are respectively ﬁrst- and second-order Gateaux
derivatives of f , and ∇
f is the derivative of f in the directions selected by G, called