Affine Non Autonomous State Space system
For starters, can you tell the dimensions of each matrix ? It will make it easier to think further.

Assuming that u(t) and c are both of same dimensions. If they are, you can just make it :
der(x) = F*x(t)+G*[u(t) + c] , because u(t) is an input ( usually the one we control ) and c is also an input, only it's an external one. u(t) suggests it's time varying input, but that's just a general case, it doesn't have to be.

Edit after your comment:

Now that you have explained that u(t) is a 1x1 scalar, I see a solution, that may be too simple to work. You said you are using lsim command in Matlab, so I suppose you pass state space (ss) system model to it. In that case, since according to your comment u(t) is a scalar, G*u(t) is a constant 6x1 vector. Therefore, you can make a substition and say [G*u + c] is your new G matrix. For now I assume you are familiar with this part of using Matlab, but just in case you need a kickstart:

Matlab documentation on this website: 
LINK
Says you can pass systems to lsim in this fashion : 

[y,t] = lsim(sys,u,t)  
[y,t,x] = lsim(sys,u,t)        % for state-space models only 
[y,t,x] = lsim(sys,u,t,x0)    % with initial state