Results 1 to 1 of 1

Math Help - Help with solution of supposedly simple Optimal Control problem?

  1. #1
    Newbie
    Joined
    Feb 2013
    From
    Göteborg
    Posts
    1

    Help with solution of supposedly simple Optimal Control problem?

    Hi, I'm trying to solve the following optimal control problem. I am sincerely sorry that I am not providing it in TeX format, but I hope it is readable


    minimize over u(t) : [ integral from 0 to tf : [ a1*(x1(t) - x1f)^2 + a2*(x2(t) - x2f)^2 + r*u(t)^2 ] ]

    subject to:
    dx1/dt = x2(t)
    dx2/dt = u(t)
    x1(0) = 0;
    x2(0) = 0;
    x1(tf) = x1f;
    x2(tf) = x2f;

    finding the final time tf is not part of the problem and it can be considered fixed. a1,a2 and r are constants, x1 and x2 the state variables, u the control signal and x1f, x1f the desired state at the final time.

    I have proceded in the usual fashion by setting up the Hamiltonian as

    H =
    a1*(x1(t) - x1f)^2 + a2*(x2(t) - x2f)^2 + r*u(t)^2 + p1(t)*x2(t) + p2(t)*u(t)

    where p1(t) and p2(t) are the costate variables satisfying:

    dp1/dt = -partial derivative of H w.r.t. x1
    dp2/dt = - partial derivative of H w.r.t. x2

    From the necessary optimality conditions dH/du = 0 th optimal control uStar(t) is given as

    uStar(t) = -p2(t)/2r

    After inserting this in the original state equations I figured that since I have four first order differential equations (in x1,x2,p1,p2) and four boundary conditions I should be able to solve this, and tried with MATLAB's symbolic math toolbox. And indeed, I got a solution, only problem is that its to long to be printed (exceedes 25 000 characters :S) and of no practical use... But the question is, I guess, if this i due to the problem it self or just an effect of matlabs symbolics handler. I ofcourse tried simplify(...) but to no use.

    Any Ideas on how to proceed?
    The problem seems simple enough and it should be possible to get a closed expression.

    Thanks!

    Robert




    Last edited by hultr; February 21st 2013 at 02:13 PM. Reason: correction
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Another Optimal Control Question
    Posted in the Advanced Applied Math Forum
    Replies: 0
    Last Post: October 11th 2010, 09:08 PM
  2. Optimal control problem
    Posted in the Calculus Forum
    Replies: 1
    Last Post: September 29th 2010, 05:20 PM
  3. Optimal Control Question
    Posted in the Advanced Applied Math Forum
    Replies: 0
    Last Post: September 28th 2010, 08:01 AM
  4. Optimal Control Theory: About the Riccati Dfiferential Equation
    Posted in the Advanced Applied Math Forum
    Replies: 2
    Last Post: June 28th 2010, 01:04 PM
  5. Integrating factor in optimal control theory
    Posted in the Differential Equations Forum
    Replies: 2
    Last Post: June 5th 2010, 03:39 PM

Search Tags


/mathhelpforum @mathhelpforum