Considering the simple dynamical system as follows
where and are sparse matrices, is a input and is incident matrix which poses in the system.
In my thought, the explicit methods work if there is not on the left hand side,
e.g. solve it by forward Euler method and ignore the input for simplicity (), then
, where is the discretized time step.
However, when this is not true, the explicit method still need to solve linear system, because
or say it requires inversion of matrix