Initial and final value theorems

Frequently, it is required to determine the initial or final value of the system response to some forcing function. It is possible to evaluate this information without inverting the appropriate transform into the time domain. 

The value of the response at t = 0 is given by the initial value theorem which states that  

The latter theorem enables the system steady-state gain to be determined as the relation between the input and output as t - oo after a step change of unit magnitude h s been applied to the system. 

Determine the steady-state gain of the U-tube manometer described in Fig. 7.21. 

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For a more elaborate treatment of Laplace transform and the initial and final value theorems, the reader is referred to the many textbooks on process control. 

The initial and final value theorem for the first-order system become ... 

Apply the initial and final value theorems to the transform derived in Example 3.1 ... 

In spite of the complex dependence of v(t) on the viscoelastic functions, the limit values of the Poisson ratio can easily be obtained. Thus the theorem of the initial and final values establishes that if a function/(t) has a limit, the following relationships hold ... 

The value of the work is therefore dependent on the shape of the whole of the curve between the initial and final volumes, and hence, in determining the work done in tracing out an element of the curve we must know the direction of that element, or in other words the elasticity of the fluid. This of course is merely a special case of the theorem that Bk is in general an imperfect differential. 

We now present two theorems which can be used to find the values of the time-domain function at two extremes, t = 0 and t = °°, without having to do the inverse transform. In control, we use the final value theorem quite often. The initial value theorem is less useful. As we have seen from our very first example in Section 2.1, the problems that we solve are defined to have exclusively zero initial conditions. 

The symmetry of an isolated atom is that of the full rotation group R+ (3), whose irreducible representations (IRs) are D where j is an integer or half an odd integer. An application of the fundamental matrix element theorem [22] tells that the matrix element (5.1) is non-zero only if the IR DW of Wi is included in the direct product x of the IRs of ra and < f. The components of the electric dipole transform like the components of a polar vector, under the IR l)(V) of R+(3). Thus, when the initial and final atomic states are characterized by angular momenta Ji and J2, respectively, the electric dipole matrix element (5.1) is non-zero only if D(Jl) is contained in Dx D(j 2 ) = D(J2+1) + T)(J2) + )(J2-i) for j2 > 1 This condition is met for = J2 + 1, J2, or J2 — 1. However, it can be seen that a transition between two states with the same value of J is allowed only for J 0 as DW x D= D( D(°) is the unit IR of R+(3)). For a hydrogen-like centre, when an atomic state is defined by an orbital quantum number , this can be reduced to the Laporte selection rule A = 1. This is of course formal, as it will be shown that an impurity state is the weighted sum of different atomic-like states with different values of but with the same parity P = ( —1) These states are represented by an atomic spectroscopy notation, with lower case letters for the values of (0, 1, 2, 3, 4, 5, etc. correspond to s, p, d, f, g, h, etc.). The impurity states with P = 1 and -1 are called even- and odd-parity states, respectively. For the one-valley EM donor states, this quasi-atomic selection rule determines that the parity-allowed transitions from Is states are towards np (n > 2), n/ (n > 4), nh (n > 6), or nj (n > 8) states. For the acceptor states in cubic semiconductors, the even- and odd-parity states labelled by the double IRs T of Oh or Td are indexed by + or respectively, and the parity-allowed transition take place between Ti+ and... 

The measured E value is directly proportional to the difference Eb(IE) = Ef — E,. The final state in PES consists of an ion and the outgoing photoelectron. The electronic structure of material is often described by approximate, one-electron wavefunctions (MO theory). MO approximation neglects electron correlation in both the initial and final states, but fortunately this often leads to a cancellation of errors when Ef, is calculated. A related problem arises when one tries to use the same wavefunctions to describe 4q and I f. This frozen orbital approximation is embedded in the Koopmans approximation (or the Koopmans theorem as it is most inappropriately called), equation 4,... 

The Correspondence Theorem is a formal expression of the requirement that the radiation calculated hy quantum theory must agree in frequency, intensity and polarization with that calculated by classical theory when the difference between the initial and final quantum states tends asymptotically to zero. This condition is met when the difference between the two values of a particular quantum number is small compared with the absolute value. 

Theorem 1 If dF is an exact differential, then the line integral J dF depends only on the initial and final points and not on the choice of curve joining these points. Further, the line integral equals the value of the function F at the final point minus the value of the function at the beginning point. We say that the line integral of an exact differential is path-independent... 

When H has reached its minimum value this is the well known Maxwell-Boltzmann distribution for a gas in thermal equilibrium with a uniform motion u. So, argues Boltzmann, solutions of his equation for an isolated system approach an equilibrium state, just as real gases seem to do. Up to a negative factor -k, in fact), differences in H are the same as differences in the thermodynamic entropy between initial and final equilibrium states. Boltzmann thought that his //-theorem gave a foundation of the increase in entropy as a result of the collision integral, whose derivation was based on the Stosszahlansatz. 

Back transformation from the transfer function to a differential equation in the time-domain, can be achieved by substitution of 5 -x(5 ) = dx(t)/dt. The variable s is independent of the time and indicates more or less the rate of change. This can be explained by the final-value theorem and the initial value theorem apphed to a variation in the input variable of a system. 

The above derivation shows that Jarzynski s identity is an immediate consequence of the Feynman-Kac theorem. This connection has not only theoretical value, but is also useful in practice. First, it forms the basis for an equilibrium thermodynamic analysis of nonequilibrium pulling experiments [3, 15]. Second, it helps in deriving a Jarzynski identity for dynamics using thermostats. Moreover, this derivation clarifies an important aspect trajectories can be thought of as mapping initial conditions (I = 0) to trajectory endpoints, and the Boltzmann factor of the accumulated work reweights that map to give the desired Boltzmann distribution. Finally, it can be easily extended to transformations between steady states [17] in which non-Boltzmann distributions are stationary. 

If the absorption coefficients in a quasi-linear photoreaction are not known, the rank of the Jacobi matrix, the eigenvalues of the reactions and the final absorbances can be calculated. Furthermore the initial values of the absorbances and their derivatives with respect to time can be determined. Any additional statements require additional information since for linear systems the following theorem is valid [15] ... 

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