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Unformatted text preview: MAE 288A Assignment 4 Due Tuesday, 24 Nov., 2009 Note: You may ﬁnd it helpful to write code in order to solve some of these problems. You may work in Matlab, C, C++, Java, Fortran, Maple, or Mathematica. Please include copies of any codes used. It is possible that not all problems will be graded. 1. (10) Consider the optimal control problem with dynamics ˙ ξ = Aξt + But , ξs = x ∈ I n R and payoﬀ J (s, x; u· ) =
1 s 1 (ξ 2t 1T − x)T C (ξt − x) + 1 uT Dut + 2 ξ1 Qξ1 . ˆ ˆ 2t (Note that x is a ﬁxed vector – not time-dependent.) Let u· be a ˆ minimizing control, and let the value function be denoted by V (s, x) for all s ∈ [0, 1]. Let the control take values in I m . You may assume R all matrices are as nice as needed. In particular, let C , D , and Q be positive deﬁnite, symmetric matrices. Suppose V (s, x) takes the ¯ ¯ form V (s, x) = 1 (x − xs )T Rs (x − xs ) + rs for all s, x. Find diﬀerential 2 equations for Rs , xs and rs . ¯ 2. (5) Consider the special case A= 01 , 30 B= Q= 0 , 1 C= x= ˆ 10 , 01 1 . 0 D = 1, 40 , 04 Find V (0, x). You can use Matlab or similar software. 1 3. (10) Consider the HJB PDE and boundary condition |∇V |2 − 1 = 0 (x, y ) ∈ G V (x, y ) = 0 (x, y ) ∈ ∂G where G = (x, y ) ∈ I 2 R x2 + y2 = 1 . 4 Obtain the characteristic equations and their initial conditions. Compute the value at the point √ √ x 2 / √2 1/√2 = . − (1/4) y 2/ 2 1/ 2 (The point is given in a form that should make it easy to ﬁnd the correct characteristic.) What would you do if given a generic point in G? Can you write a system of equations that could be solved to yield the value there? 2 ...
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- Spring '10
- Equations, Boundary value problem, Generic property, MAE 288A Assignment