Appl Math Optim 51:201–250 (2005)
2004 Springer Science+Business Media, Inc.
Semilinear Kolmogorov Equations and Applications to
Stochastic Optimal Control
Dipartimento di Matematica, Politecnico di Milano,
Piazza Leonardo da Vinci 32, 20133 Milano, Italy
Abstract. Semilinear parabolic differential equations are solved in a mild sense
in an inﬁnite-dimensional Hilbert space. Applications to stochastic optimal control
problems are studied by solving the associated Hamilton–Jacobi–Bellman equation.
These results are applied to some controlled stochastic partial differential equations.
Key Words. Stochastic optimal control, Hamilton–Jacobi–Bellman equation,
Inﬁnite-dimensional stochastic processes.
AMS Classiﬁcation. 60M30, 93E20.
In this work we study a class of semilinear parabolic differential equations in Hilbert
spaces, in particular, some Hamilton–Jacobi–Bellman equations, and give applications to
stochastic optimal control problems in inﬁnite-dimensions. Our results are then applied
to the optimal control of some stochastic partial differential equations, in particular, a
controlled wave equation.
In the ﬁrst part we study a class of semilinear parabolic differential equations in a
separable Hilbert space H of the form
u(t, x)+ψ(t, x, u(t, x), ∇
u(t, x)), t ∈ [0, T ], x ∈ H,
u(T , x)= ϕ(x).
We often refer to this equation as the semilinear backward Kolmogorov equation. The
This researach was partially supported by the European Community’s Human Potential Programme
under Contract HPRN-CT-2002-00281, Evolution Equations.