From what I have read from the timescale, most results of continuous time and discrete time systems can be generalized to arbitrary timescales by considering the generalized derivative operator instead of the forward difference operator or the standard derivative. In particular, the concepts of the time scale can be applied to the literature of control theory and to linear matrix inequalities to address discrete and continuous systems through a unified approach.

I was wondering if, instead of using the generalized derivative operator, one would use a generalized formula for the simultaneous treatment of both continuous and discrete time. Instead, one could instead use a heuristic to derive the equations associated with a time-discrete result by directly perturbing the equations associated with the continuous-time counterpart.

**example**:

To let $ M prec 0 $ call that $ M $ is a symmetrically negatively defined matrix and $ Delta $ the generalized derivative operator. Suppose that $ mathbb {T} $ is a top unlimited time scale with limited grain size and associated step size, which is given by $ mu $,

in the [1] it is proved that there is a symmetrically positive defined matrix $ P $ satisfying

$$ tag {1} label {1} A ^ T (t) P + PA (t) + mu (t) A ^ T (t) PA (t) prec 0 $$

for all $ t $ in the time scale $ mathbb {T} $, then $ x = 0 $ is an asymptotic stable equilibrium point of the dynamic linear system $ x ^ { Delta} = A (t) x $,

Is there a known heuristic that "guesses" eqref {1} by only recognizing the continuous time?

Lyapunov equation is $ A ^ T (t) P + PA (t) prec 0 $?

[1]: Davis, John M. et al. "Algebraic and dynamic Lyapunov equations on time scales." 42. Southeastern Symposium on Systems Theory (SSST). IEEE, 2010.