# Optimal Control of Stochastic Partial Differential Equations and Partial (Noisy) Observation Control

• Bernt Øksendal
• Agnès Sulem
Chapter
Part of the Universitext book series (UTX)

## Abstract

Suppose the density Y(tx) of a fish population at time $$t \in [0,T]$$ and at the point $$x \in D \subset \mathbb {R}^n$$ (where D is a given open set) is modeled by a stochastic partial differential equation (SPDE for short) of the form
\begin{aligned} \begin{aligned} {\mathrm {d}}Y(t, x) =&\left[ \frac{1}{2} \Delta Y(t,x) + \alpha Y(t,x) - u(t,x) \right] {\mathrm {d}}t \\&+ \beta Y(t,x) {\mathrm {d}}B(t) + Y(t^-,x) \int _{\mathbb {R}} \zeta \tilde{N} ({\mathrm {d}}t, {\mathrm {d}}\zeta ) ; \ (t, x) \in (0,T) \times D, \end{aligned} \end{aligned}
where we assume that $$\zeta \ge -1 + \varepsilon$$ a.s. $$\nu ({\mathrm {d}}\zeta )$$ for some constant $$\varepsilon > 0$$. The boundary conditions are:
\begin{aligned} Y(0,x) = \xi (x) ; \ x \in D \end{aligned}
\begin{aligned} Y(t,x) = \eta (t,x) ; \ (t, x) \in [0,T) \times \partial D. \end{aligned}