Fitting curves with several complicated conditions

I have a system of 26 (linear, nonhomogeneous, ordinary) differential equations that looks like this (in matrix form):

X'(t) = A X(t) + J(t)

Depending on whether I use Dirac's delta on J(t) or a constant number, I can either achieve a decay curve or a constant line (after a sufficient time).

The problem is, I actually have a " system of system of differential equations".

What I want to do is to calculate the above system of differential equations so that few of the element of X(t) combined will have certain inclination. However, I want it to also satisfy other conditions that require same equation used above but with different J(t) and slightly different constant used for matrix A.

The reason I am doing this is because I have eight unknown constants (the rest is fixed) in matrix A that I know will be achieved if all the conditions I have mentioned above has been satisfied. However, despite above system being analytically solvable, it cannot be solved symbolically (because it's too complicated), and I must enter a specific value for each constants before running the calculation.

The question is, is there a way to make MATLAB enter random numbers into the eight unknown constants until it converges into the decay curve that satisfies all the conditions mentioned above? Is there like a script to do this? I am not very good at using scripts because I've never experienced programming and stuff.

Sorry for making a long question. I am not a native English speaker so I am not good at expressing what I am trying to say.

Thank you.