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Laplace transforms
Published in C.W. Evans, Engineering Mathematics, 2019
exists. Considerations of this kind are rather advanced but suffice it to say that if f(t) is piecewise continuous on the closed interval [0, ∞] then we shall have a fighting chance of settling the matter. A function is piecewise continuous on an interval if it is possible to partition the interval into a finite number of subintervals in such a way that (1) the function is continuous on each subinterval and (2) the function has a finite right-hand and left-hand limit at each point of discontinuity. However we shall not always restrict our attention to these functions but, occasionally in a somewhat cavalier fashion, we shall allow ourselves more freedom; the only proviso being that the Laplace transform will be presumed to exist.
Fourier Series
Published in Christian Constanda, Solution Techniques for Elementary Partial Differential Equations, 2018
2.3 Definition. A function f is said to be piecewise continuous on an interval [a, b] if it is continuous at all but finitely many points in [a, b], where it has jump discontinuities; that is, at any discontinuity point x, the function has distinct right-hand side and left-hand side (finite) limits f(x+) and f(x−).
Design of globally stabilising switching control law for a type of nonlinear systems
Published in International Journal of Systems Science, 2020
Suppose that a switched system generated by the above family (6) is described by where is a piecewise constant function, which has a finite number of discontinuities (namely, switching times) on every bounded time interval and takes a constant value on every interval between two consecutive switching times. Regarding the stability of the above switched system (7), the following result is presented in Liberzon (2003).