Stability from an operational viewpoint

The concept of stability for single-input/single-output linear time-invariant plants is reformulated within the framework of Mikusinski's generalized functions and then characterized; each plant is given by convolution with a generalized function called the impulse response. The impulse response of a stable plant is the generalized derivative of a function of bounded variation, and is therefore the sum of three uniquely determined functions: the first being an absolutely integrable function, the second being an absolutely convergent sum of at most countably many delays, and the third being an atomless singular measure, a stochastic phenomenon.<
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