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I'm confronted to a problem of trajectory optimization. The problem consists in going from point A to point B in the Hill frame (relative navigation) subject to a dynamics described by Clohessy-Wiltshire equations. I want to find the fuel optimal trajectory with three $\Delta$v's: one at $t_0$ one at $t_f$ and one at an arbitrary (but set for the moment) time $t_k$. The evolution of the system is given by

$\mathbf{x}(t+1) = A\mathbf{x}(t) + B\mathbf{u}(t)$

where $\mathbf{x}$ is the state vector and $\mathbf{u}$ is my input (i.e. the $\Delta$v's). Working on this relation one gets

$\mathbf{x}_j = (A_{j-1}...A_1A_0)\mathbf{x}_0 + [A_{j-1}\cdot \cdot \cdot A_1B_0, A_{j-1}\cdot \cdot \cdot A_2B_1, ..., A_{j-1}B_{j-2}, B_{j-1}, \mathbf{0}][\mathbf{u}_0, ..., \mathbf{u}_{j-2}, \mathbf{u}_{j-1},...,\mathbf{u}_{N-1}]^T \equiv C_j\mathbf{x}_0 + D_j\mathbf{u} \quad (1)$ for $j = 1,...N$.

So my optimization problem is as follow: find min of $\sum_{i=1}^N ||\mathbf{u}_i||$ subject to (1) and $\mathbf{x}_0$ and $\mathbf{x}_N$ given (other constraints such as $||\mathbf{x}_i|| > d$ for all $i$ will later be implemented).

If I'm correct, the decision variables are $\mathbf{z} = [\mathbf{x}_0,..., \mathbf{x}_N, \mathbf{u}_0,..., \mathbf{u}_N]$.

I'm not sure of how (and if it's possible) to formulate the constraints as $M\mathbf{z} \leq \mathbf{b}$ ? How to implement it ? (for instance in matlab) Do I need an initial guess on the trajectory ?

Thanks

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    $\begingroup$ This looks like it has the start of a good question, but I'm not really sure what you are asking for. As it is, this looks like a homework problem for physics, and you might get a better answer over there. $\endgroup$ – PearsonArtPhoto Jul 22 '15 at 15:13