Optimal control of a stochastic heat equation with boundary-noise and boundary-control. (English) Zbl 1123.60052
The authors of the present paper study a controlled state equation of parabolic type with Neumann boundary condition over the interval \([0,\pi]\). In the boundary condition at \(0\) and at \(\pi\) the derivative of the solution equals to the sum of a control and of a white noise in time; the two white noises intervening at the both interval end points are assumed to be independent. Such control problems have been largely studied in the deterministic case and addressed in the stochastic case as well. However, in these works the noise is also always a forcing term of the equation which, in the case of a strictly non degenerate diffusion facilitates the studies, while in the present paper the more complicate situation of a noise acting only at the boundary is studied. For this the authors reformulate the state equation as infinite dimensional evolution equation and prove the existence and uniqueness of a mild solution to the associated Hamilton-Jacobi-Bellman equation. It’s \(C^1\)-regularity is used to construct the optimal feedback control for the control problem. For overcoming the difficulties related with the degeneracy of the second order derivative and the presence of unbounded terms in the Hamilton-Jacobi-Bellman equation, this equation is studied with the help of a suitable forward-backward stochastic differential equation.
Reviewer: Rainer Buckdahn (Brest)
MSC:
| 60H30 | Applications of stochastic analysis (to PDEs, etc.) |
| 49L20 | Dynamic programming in optimal control and differential games |
| 93E20 | Optimal stochastic control |
| 35K20 | Initial-boundary value problems for second-order parabolic equations |
Keywords:
stochastic heat equation; Neuman boundary condition; boundary control; Hamilton-Jacobi-Bellman equation; forward-backward stochastic differential equationReferences:
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