Zermelo's navigation problem

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richard
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Zermelo's navigation problem

Post by richard » 2 weeks ago

Hi,
I am triyng to calculate the route wich takes less time to go from point A to point B in the presence of a constant flow (I. E. a simple version of Zermelo's navigation problem) using the GAMS software. However, if I put both points on a straight line and make the constant flow equal to zero, I don't get the results that I should.

Code: Select all

    $set n 10
    set j /0*%n%/;
    sets
    jlast(j)
    jnotlast(j);
    jlast(j)$(ord(j)=card(j))=yes;
    jnotlast(j)=not jlast(j);
    scalar
    
    v aeroplane velocity /10/
    
    u wind velocity in the x axis /0/
    
    n number of intervals /%n%/
    
    density density /1.225/
    variable
    gamma(j),
    x(j),
    y(j),
    time,
    objective;
    
    positive variable
    t(j)
    
    step;
    
    equation
    diffx(j),
    diffy(j),
    obj;
    
    * diffx[j]$(jnotlast(j)).. x[j+1]-x[j] =e=0.5*(t(j+1)-t(j))*(v*cos(gamma(j+1)) + v*cos(gamma(j)) );
    * diffy[j]$(jnotlast(j)).. y[j+1]-y[j] =e=0.5*(t(j+1)-t(j))*(v*sin(gamma(j+1)) + v*sin(gamma(j)) );
    
    diffx[j]$(jnotlast(j)).. x[j+1]-x[j] =e=0.5*step*(v*cos(gamma(j+1))-u + v*cos(gamma(j))-u );
    diffy[j]$(jnotlast(j)).. y[j+1]-y[j] =e=0.5*step*(v*sin(gamma(j+1)) + v*sin(gamma(j)) );
    
    obj.. time =e= n*step;
      
    x.fx('0') = 1.0e-12;
    x.fx('%n%') = 1.0e-12;
    y.fx('0') = 1.0e-12;
    y.fx('%n%') = 10;
    t.fx('0')= 1.0e-12;
    
    y.up(j) = 10;
    y.lo(j) = 1.0e-12;
    t.lo(j)=1.0e-12;
    gamma.up(j)=pi;
    gamma.lo(j)=0;
   
    model brahstron1 /all/;
    
    option
    nlp=ipopt;
    solve brahstron1 using nlp minimize time;
In this code I have defined two diferential equations, for x'=dx/dt and y'=dy/dt using the trapezoid rule. Here, gamma is our control variable and that's why it's free. Step is the difference between the times j and j+1, that is why time is n (number of intervals)*step(size of interval) and is what I seek to reduce.

Below this one can find the restrictions, these include the initial and final points for x and y, also the initial t=0.

I would spect as a result that the angle gamma became pi/2 and y going between 0 and 10, however, I find weird results, as Y being 5 on all the steps.

I have looked at this code for a while and I cannot find where my mistake is.

Can somebody please tell me what am I doing wrong?
Thank you very much for reading.

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