Laplace transform[1]. Starting with a signal in the time domain, you can take the Fourier transform to map it into the frequency domain and then the Laplace transform (Z-transform[2]) to map it into discrete time domain. Discrete time is what digital computers do. At this point implementing the control loop on a digital computer is remarkably straight forward.
Taking your system's closed loop transfer function[3], you can use the above Laplace and Z-transforms to produce the desired closed loop control algorithm to "optimally" control your system. (I put "optimally" in quotes because it is optimum for the system as defined; if your understanding / definition of your system is wrong, it will be optimally controlling for a different system and be suboptiomal for your actual system.)
The Fourier transform and its computational offshoot, the Fast Fourier Transform. We wouldn't have any of modern electronics and telecommunications without the mathematical insight that the FT brought. The fact that you can go from the time domain to the frequency domain and back again is nothing less than astounding. Optics was also benefitted by the FT because you can use is to filter unwanted optical frequencies and diffraction effects from your images using Fourier transformation.
Taking your system's closed loop transfer function[3], you can use the above Laplace and Z-transforms to produce the desired closed loop control algorithm to "optimally" control your system. (I put "optimally" in quotes because it is optimum for the system as defined; if your understanding / definition of your system is wrong, it will be optimally controlling for a different system and be suboptiomal for your actual system.)
[1] https://en.wikipedia.org/wiki/Laplace_transform
[2] https://en.wikipedia.org/wiki/Z-transform
[3] https://en.wikipedia.org/wiki/Control_theory#Closed-loop_tra...