hw54 - MAE 288A Assignment 4 Due Tuesday, 24 Nov., 2009...

Info iconThis preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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 ...
View Full Document

This note was uploaded on 08/02/2010 for the course MAE 288 taught by Professor Mceneaney,w during the Spring '10 term at UCSD.

Page1 / 2

hw54 - MAE 288A Assignment 4 Due Tuesday, 24 Nov., 2009...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online