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Unformatted text preview: orbed (by the
V (x , t ) ﬁeld – I’ll explain) we weight it to zero.
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J (x , t ) −λ log N N unabsorbed exp(−φ(yi )/λ).
i∈
N
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ψ (x , t ) = N i ∈unabsorbed exp(−φ(yi )/λ) Nick Jones Elements of Stochastic Optimal Control: ICDNS Using diﬀusions to solve a large class of control problems II
∂t ρ = − V − ∂x (b ρ) +
λ 1
2 2 ij νij ∂ x∂∂ xj ρ .
i J (x , t ) = −λlog dy ρ(y , T x , t )exp (−φ(y )/λ).
1
J (x , t ) −λ log N N unabsorbed exp(−φ(yi )/λ).
i∈
1
ψ (x , t ) N N unabsorbed exp(−φ(yi )/λ)
i∈
We can then deduce our optimal controls from our optimal
costtogo:
u (x , t ) = −R −1 B ∂x J (t , x )
In the case where the optimal control is unique we can
approximate u (x , t ) directly through the form:
1
u (x , tj ) ψ(x ,t ) N unabsorbed exp(−φ(yi )/λ)ξj where we are
i∈
considering the discrete time of our simulation and ξj is the
perturbation at time j . This is particularly cute since it tells us that
we can interpret our noise perturbations which are successful at
steering the particle as controls. We are now set for the practical.
Nick Jones Elements of Stochastic Optimal Control: ICDNS Remark
I’ve just taken us through what is sometimes called PathIntegral
Control [3]. It relied on this constraint (inversely) connecting my
the costs of control in a direction to whether the noise was strong
in that direction (scaled by a parameter which can be interpreted
as a temperature).
It turns out th...
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 Spring '14
 NickJones
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