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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.

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The answer is no. As a former student of Davis who works in control theory, I can tell you that (1) comes from examining the "energy" of the system: i.e. by examining the (in this case delta) derivative of the scalar quantity ||x(t)||^2 . Take a look at Jackson's and Dacuhna's dissertations from Baylor and you will see where these matrix inequalities come from as they both present the derivations and calculations. Jackson did a follow up paper where he examined the analogous results in the nabla case as well.

There are times in the times scales process where making educated "guesses" can get you pretty far based upon what is known in the classical differential and difference equations case, but often times the generalization to the more arbitrary case can be quite surprising. For example, Bohner argues that the appropriate "generalized" version of the logistic equation actually requires a shift to get the same types of dynamics that you get in the continuous case. His generalization actually yields the standard Beverton-Holt model in the difference equation case as you would expect, but he plays a bit of a magic trick to introduce the shift that is hiding there for the general case to guarantee two equilibria: one unstable at 0 and one stable at some some positive value. So long story short, it can be a quite a bit of an art form when trying to generalize to arbitrary systems: yes there is some "educated guesswork" involved, but usually the generalizations involve thinking about the connections and disconnections that we see between the two classic extremes.

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