Final value theorem

In mathematical analysis, the final value theorem (FVT) is one of several similar theorems used to relate frequency domain expressions to the time domain behavior as time approaches infinity. Mathematically, if in continuous time has (unilateral) Laplace transform , then a final value theorem establishes conditions under which Likewise, if in discrete time has (unilateral) Z-transform , then a final value theorem establishes conditions under which An Abelian final value theorem makes assumptions about the time-domain behavior of (or ) to calculate . Conversely, a Tauberian final value theorem makes assumptions about the frequency-domain behaviour of to calculate (or ) (see Abelian and Tauberian theorems for integral transforms). In the following statements, the notation '' means that approaches 0, whereas '' means that approaches 0 through the positive numbers. Suppose that every pole of is either in the open left half plane or at the origin, and that has at most a single pole at the origin. Then as , and . Suppose that and both have Laplace transforms that exist for all . If exists and exists then . Remark Both limits must exist for the theorem to hold. For example, if then does not exist, but Suppose that is bounded and differentiable, and that is also bounded on . If as then . Suppose that every pole of is either in the open left half-plane or at the origin. Then one of the following occurs: as , and . as , and as . as , and as . In particular, if is a multiple pole of then case 2 or 3 applies ( or ). Suppose that is Laplace transformable. Let . If exists and exists then where denotes the Gamma function. Final value theorems for obtaining have applications in establishing the long-term stability of a system. Suppose that is bounded and measurable and . Then exists for all and . Elementary proof Suppose for convenience that on , and let . Let , and choose so that for all . Since , for every we have hence Now for every we have On the other hand, since is fixed it is clear that , and so if is small enough.