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

  • Bernt ØksendalEmail author
  • Agnès Sulem
Part of the Universitext book series (UTX)


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:
$$\begin{aligned} Y(0,x) = \xi (x) ; \ x \in D \end{aligned}$$
$$\begin{aligned} Y(t,x) = \eta (t,x) ; \ (t, x) \in [0,T) \times \partial D. \end{aligned}$$

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© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of OsloOsloNorway
  2. 2.Inria Research Center of ParisParisFrance

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