If A and B are known, then the right hand side is known, and must of necessity be symmetric positive semi-definite. Therefore, there is a solution C to the stated equation, and this is the (lower triangular) Cholesky factor of $ AA^T + BB^T$ . Cholesky factorization is a standard routine in any linear algeabra library. MATLAB's chol computes the upper triangular Cholesky factor, so using that, C would be chol(A * A' + B * B')' .
I'll leave it to others to determine whether there is a "closed form" solution in terms of X, Y, and c, but nothing jumps out at me.
Edit: Sorry, did not mean for this to be deleted.