Stochastic control on Hilbert space for linear evolution equations with random operator-valued coefficients
Stochastic control on Hilbert space for linear evolution equations with random operator-valued coefficients
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DOI：
10.1137/0319023
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发表时间：
1980-06
影响因子：
1.2
通讯作者：
N. Ahmed
中科院分区：
文献类型：
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作者：
N. Ahmed
考虑可分Hilbert空间上随机发展方程dxi=(A(T)xi+B(T)u)dt+sigma(T)dw$的最优控制问题，其中A(T)，B(T)，sigma(T)，tgeqq0是渐进可测的算子值随机过程，A一般无界.证明了发展方程(弱解)的存在唯一性。然后，我们给出了二次(随机)代价函数最优控制的存在性和最优性的必要条件。对于最优反馈控制，我们求解了一个随机算子Riccati方程和一个倒向随机发展方程。通过转置从正向发展方程产生的随机同构来求解后向方程。最优反馈控制通过状态的随机仿射变换给出。文中还给出了一些算例，说明了结果的有效性。这项工作是对Bismut[SIAM J.Control Opti.，14(1976)，pp.419-444；15(1977)，pp.1-4]和B…
We consider a problem of optimal control of the stochastic evolution equation $d\xi = (A(t)\xi + B(t)u)dt + \sigma (t)dw$, on a separable Hilbert space, where $\{ A(t),B(t),\sigma (t),t \geqq 0\} $ are progressively measurable operator-valued random processes with A generally unbounded. We prove the existence and uniqueness of (weak) solutions of the evolution equation. Then we present the existence of optimal controls and necessary conditions of optimality for a quadratic (random) cost function. For optimal feedback controls we solve a random operator Riccati equation and a backward stochastic evolution equation. The backward equation is solved by transposing a random isomorphism generated from a forward evolution equation. The optimal feedback control is given by a random affine transformation of the state. Some examples are presented to indicate usefulness of the results. This work is a partial extension of the results of Bismut [SIAM J. Control Optim., 14 (1976), pp. 419–444; 15 (1977), pp. 1–4] and B...