Do FIR filter have any application in control theory?
FIR filters are fairly common in some areas of control theory. As they usually incur a lot of added phase/time-delay, they are not really usable in the feedback path of regular control systems, but they are useful when the added phase/time-delay is not affecting the system in an adverse way, or when the particular phase response and time-delay is desired.

Examples:


Feed-forward control. FIR filters are useful for producing filters that approximate arbitrary frequency responses, hence they can be used to shape a reference signal. A typical example is to use an FIR filter with the inverse frequency response of the plant -- trying to counteract the dynamics of the plant in order to get a desired output. Phase/time-delay is not interfering with the stability or performance since the computation can be done offline. FIR filters can often produce higher performance than IIR filters, especially where there are non-minimum phase zeros.
Learning systems (iterative learning control [ILC] and repetitive control [RC]). In these systems, again, FIR filters are often used to generate an inverse response filter of the plant. This is done to improve the performance of the learning, as this enables a higher gain to be used for a larger bandwidth. The time-delay is here desired (RC), or it does not matter, as the computations are done offline (ILC). Low-pass FIR filters are also used here for their linear-phase properties, as the linear-phase property makes the filter not interfere with the phase of other filters, and this improves performance.