What is G and what is H? What is the difference in the way they are defined?

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Both G(s)\, and H(s)\, are notations for transfer functions. Sometimes we use G(i\omega )\, (or H(i\omega )\,) to refer to the "frequency response" if the system is asymptotically stable; it can be obtained by evaluating G(s)\, at s=i\omega \,.

--Shuo