minimize the fuel needed for a hohmann's transfer from earth to saturn in matlab

main.m
% Define initial state of spacecraft at beginning of transfer
r0_Earth = [x0_Earth; y0_Earth; z0_Earth];
v0_Earth = [vx0_Earth; vy0_Earth; vz0_Earth];
t0 = 0;

% Define mathematical model of spacecraft's motion through space
% This can be based on classical orbital mechanics
% Define the optimization variables for times and magnitudes of engine burns
% In this example, we use 3 burns: departure burn, mid-course correction, capture burn
% The optimization variables are the durations (t1,t2,t3) and magnitudes (m1,m2,m3) of each burn
x0 = [t1, t2, t3, m1, m2, m3];

% Define the optimization problem
fun = @(x) hohmannFuel(x, r0_Earth, v0_Earth, t0, r_Saturn, v_Saturn);
options = optimoptions(@fmincon,'Algorithm','sqp','MaxIterations',10000,'MaxFunctionEvaluations',100000);
lb = [0,0,0,0,0,0]; % lower bounds for optimization variables
ub = [1e6,1e6,1e6,1e6,1e6,1e6]; % upper bounds for optimization variables
[x_opt, f_min] = fmincon(fun,x0,[],[],[],[],lb,ub,[],options);

% Solve the optimization problem
% This will give the optimal sequence of engine burns that minimizes the total fuel used during the transfer.
% Note that this function must return the total fuel used as the objective function
function f = hohmannFuel(x, r0_Earth, v0_Earth, t0, r_Saturn, v_Saturn)
   t1 = x(1);
   t2 = x(2);
   t3 = x(3);
   m1 = x(4);
   m2 = x(5);
   m3 = x(6);

   % simulate spacecraft's motion through space using optimized engine burns
   % this can be done using the mathematical model of spacecraft's motion
   
   % evaluate the success of the Hohmann transfer by comparing the final position and velocity of the spacecraft with the desired values at Saturn
   % adjustments to the engine burns can be made if necessary
   
   % return the total fuel used as the objective function
   f = m1 + m2 + m3;
end
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