L10-StochOptControl

Nick jones elements of stochastic optimal control

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Unformatted text preview: his system around in direction i (Rii small). This is natural for systems generically - we might think that it is easier to control a system to something close to where it might have gone anyway [12]. Nick Jones Elements of Stochastic Optimal Control: ICDNS A more general class of system II Dynamics: dx = (b (x , t ) + Bu )dt + d ξ 1 Cost: w (x , u , t ) = 2 u T Ru + Q (x , t ) Final cost: W (x ) = φ(xT ) Constraint: ν = λBR −1 B We thus have a simple control (relatively simple noise) but a complex system we’d like to control with complex costs. We have also introduced a constraint which is strong but natural. I recommend reading Ref. [3] though it is not required (beyond what you need to understand the practical). Nick Jones Elements of Stochastic Optimal Control: ICDNS A more general class of system III Dynamics: dx = (b (x , t ) + Bu )dt + d ξ 1 Cost: w (x , u , t ) = 2 u T Ru + Q (x , t ) Final cost: W (x ) = φ(xT ) Constraint: ν = λBR −1 B We can thus write the stochastic HJB as: −∂t J (t , x ) = minu ( 1 u T Ru + Q (x , t ) + (b + Bu )T ∂x J (t , x ) + 2 1 2 Tr (ν (t , x , u )∂x J (t , x )). 2 Optimizing over u yields: u (x , t ) = −R −1 B ∂x J (t , x ). Plugging this optimal control back in to the stochastic HJB, defining J (x , t ) = −λ log ψ (x , t ) and using the constraint discussed yields 2 an equation linear in ψ : ∂t ψ = V − b T ∂x − 1 Tr (ν (t , x , u )∂x ) ψ . λ 2 This can be solved backwards in time starting with ψ (x , T ) = exp (−φ(x )/λ) (since J (x , T ) = φ(xT )). Nick Jones Elements of Stocha...
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