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Partial differential equations (PDEs) describe various physical phenomena such as magnetism, heat transfer, fluid flow, or vibrations in elastic structures, etc. A control system may be a mechanical system being affected by various inputs available to the operator of the system. The dynamics of the system are given by a set of states which satisfy PDEs. The inputs changing the dynamics of the system are generally referred to control variables and the measurable quantities to the operator of the system are called observed variables.
A feedback control system is one of the earliest control systems in the history. It is a control system that regulates the system based on the feedback of actual observed states of the system. A cruise-control (known as speed-control) system used in modern vehicles is a very good example of a feedback control system. The cruise control allows the car to maintain its speeds at a preset value. This is accomplished by measuring the vehicle speed, comparing it to the desired or reference speed, and automatically adjusting the throttle according to a control law. Other examples of feedback control systems include thermostat and anti-lock braking system.
One of the most challenging problems in engineering applications is the vibration control of elastic structures. The vibration in the system can be can be written as a system of partial differential equations, and therefore, the states are infinitely many to describe the system dynamics. This is in deep contrast to the systems written by the ordinary differential equations which only have finitely many states. From the engineering point of view, this may be seen obscure since only the first finite number of equations (modes) are generally taken into account in practice and the others are ignored (and fingers are crossed!). An interesting question is “How can we design a finite dimensional feedback controller for an infinite dimensional system?" This is one of the tough problems in the theory of PDEs and requires deep mathematical challenges.