Optimal control of a stochastic heat equation with boundary-noise and boundary-control
Optimal control of a stochastic heat equation with boundary-noise and boundary-control
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DOI：
10.1051/cocv:2007001
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发表时间：
2007-01-01
影响因子：
1.4
通讯作者：
Tessitore, Gianmario
中科院分区：
文献类型：
--
作者：
Debussche, Arnaud;Fuhrman, Marco;Tessitore, Gianmario
研究了有界真实的区间上具有Neumann边界条件的非线性随机热方程的最优控制问题。这里的特殊性在于控制和噪声都作用于边界。我们首先将状态方程重新表述为一个无限维的随机演化方程。本文的第一个主要结果是证明了相应的Hamilton-Jacobi-Bellman（HJB）方程的温和解的存在性和唯一性。这样的解决方案的C1正则性，然后用于构造控制问题的最优反馈。为了克服二阶算子的退化和无界项的存在所引起的困难，我们引入一个合适的正倒向随机微分方程组来研究HJB方程，如[14，27]中分别对有限维和有限维半线性抛物方程提出的方法。
We are concerned with the optimal control of a nonlinear stochastic heat equation on a bounded real interval with Neumann boundary conditions. The specificity here is that both the control and the noise act on the boundary. We start by reformulating the state equation as an infinite dimensional stochastic evolution equation. The first main result of the paper is the proof of existence and uniqueness of a mild solution for the corresponding Hamilton-Jacobi-Bellman (HJB) equation. The C 1 regularity of such a solution is then used to construct the optimal feedback for the control problem. In order to overcome the difficulties arising from the degeneracy of the second order operator and from the presence of unbounded terms we study the HJB equation by introducing a suitable forward-backward system of stochastic differential equations as in the appraoch proposed in [ 14,27] for finite dimensional and in finite dimensional semilinear parabolic equations respectively.