This basically adds one more ingredient to partial differential equations: a function that we can select.
And then the question becomes: if this function has such and such limitation, can we make the solution of the differential equation have such and such property?
Control theory also takes into consideration possible discretization of the domain, which allows using numerical methods to solve partial differential equations, as well as digital, rather than analogue control methods.
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Control theory studies dynamical input-output systems, feedback, stability, and response.
Control theory is an interdisciplinary branch of engineering and mathematics that deals with the behavior of dynamic systems. It focuses on the design and implementation of controllers that manage the system's behavior to achieve desired objectives. Control theory is used in various applications, including aerospace, automotive, robotics, manufacturing, and process control. At its core, control theory strives to develop mathematical models of systems and to analyze their behavior over time.