# Can time-scale calculus be used to derive a counterpart theorem of discrete-time dynamic systems directly from continuos-time dynamics systems?

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

What I was wondering is if rather than using the generalized derivative operator to obtain a generalized formula to deal with both continuous and discrete time simultaneously, one could use instead a heuristic to derive the equations associated to a discrete-time result by directly perturbing the equations associated to the continuous-time counterpart.

Example: Let $$M \prec 0$$ denote that $$M$$ is a symmetric negative-definite matrix and $$\Delta$$ the generalized derivative operator. Suppose that $$\mathbb{T}$$ is a time-scale unbounded above, with bounded graininess and associated step-size given by $$\mu$$.

In [1] it is proved that if there exists a symmetric positive-definite 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 a asymptotic stable equilibrium point of the dynamic linear system $$x^{\Delta} = A(t)x$$.

Is there any known heuristic to "guess" \eqref{1} by only knowing that 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." 2010 42nd Southeastern Symposium on System Theory (SSST). IEEE, 2010.