I have previously done this question on math stack, and I think that maybe here is better.
I am studying the theory of discrete-time systems, comparing them with continuous ones in the control theory field.In particular, I have focused on the advantages and disadvantages to prefer sometimes the first over the latter.
Some papers say that nowadays, most control systems are implemented through digital devices (as DSP, etc.). So sometimes, it is preferable to design directly a discrete-time controller rather than a continuous one. Indeed the engineers prefer this choice.
I have also read that designing a continuous-time controller is not entirely avoided; indeed, many mathematicians prefer this choice. In that case, the idea is that a continuous-time is approximately a discrete-time system, with an arbitrarily small sampling time.
My question is: what are the disadvantages of choosing a too small sampling time above all on the control effort?
Note: the control effort is the amount of energy or power necessary for the controller to perform its duty.