Liapunov function

Remark 12. The introduction of the Liapunov function V appears somewhat arbitrary but accords with the axioms proposed by Wei [9] see the discussion at the end of the paper. 

The reader will have noticed that the Liapunov function used in the proof of the theorem was not obvious on either biological or mathematical grounds. Its discovery by Hsu greatly simplified and extended earlier arguments given in [HHW]. This is typical of applications of the LaSalle corollary. Considerable ingenuity, intuition, and perhaps luck are required to find a Liapunov function. 

Even for systems that have nothing to do with mechanics, it is occasionally possible to construct an energy-like function that decreases along trajectories. Such a function is called a Liapunov function. If a Liapunov function exists, then closed orbits are forbidden, by the same reasoning as in Example 7.2.2. 

To be more precise, consider a system x = f(x) with a fixed point at x. Suppose that we can find a Liapunov function, i.e., a continuously differentiable, realvalued function V(x) with the following properties ... 

Unfortunately, there is no systematic way to construct Liapunov functions. Divine inspiration is usually required, although sometimes one can work backwards. Sums of squares occasionally work, as in the following example. 

By constructing a Liapunov function, show that the system x = -x-l-4y, y = -X - y has no closed orbits. 

The proof involves the construction of a Liapunov function, a smooth, positive definite function that decreases along trajectories. As discussed in Section 7.2, a Liapunov function is a generalization of an energy function for a classical mechanical system—in the presence of friction or other dissipation, the energy decreases monotonically. There is no systematic way to concoct Liapunov functions, but often it is wise to try expressions involving sums of squares. 

By defining an appropriate Liapunov function, show that e(t) —> 0 as t —> >. Solution First we write the equations governing the error dynamics. Subtract-... 

Hence E<0, with equality only if e = 0. Therefore is a Liapunov function, and so e = 0 is globally asymptotically stable. ... 

Exponentially fast synchronization) The Liapunov function of Example... 

Liapunov function (Section 6) reaction intermediates (Sections 4 and 5) coefficients in quadratic function dimensionless concentration Frank-Kamenetskii dimensionless heat release rate dimensionless temperature RTm/E... 

The Liapunov definition of stability has the great virtue that it provides a method of analysis that by-passes the arduous task of integrating the reactor equations. Whereas local stability of the equilibrium states (c.,T.) is easily established via equations, use of Liapunov function V(c,T) establishes a region of stability (Figure 13). This is the region enclosed by the contour V c,T) = constant, which touches c,T) = 0 tangentially. 

Within this region all tr ectories [c(t), T(/)] return to the undisturbed state since dK/dt is negative the Liapunov function V defines a region of asyrrgptotic... 

The derivative dVjdt is easily found by ditferentiating equation (40) and using the reactor equations (37) for dc/dr and dr/dt. Different values of coefiBcients Oil. a,i, and Oji will provide different r ons of asymptotic stability. The union of all these (UAS) is a more extensive r on of asymptotic stability than any individual. An investigation of the CSTR using this approach was carried out by Berger and Perlmutter "" typical results are own in Figure 14 in terms of the UAS obtained from three choices of Liapunov functions. 

Stability studies of the steady-state solutions of the TRAM are in general very complex and beyond the scope of this Report. The more successful approaches have involved the use of Liapunov functions, " of collocation methods, or of topological fixed-point methods. The genoation of r ons of stability inevitably involves a considerable amount of computational effort. For a full discussion of these methods the reader is again referred to Perlmutter s book. ... 

Energy methods can also be applied to systems not initially at equilibrium. A Liapunov function must be found which (a) vanishes for the initial state, (b) is positive for all other states, and (c) decreases in value as perturbation amplitude decreases. The initial state is stable if the function s value decreases continuously during system response to all possible perturbations. As with the thermodynamic energy method, the Liapimov method is generally easier to use than perturbation of the governing equations, especially for perturbations of large amplitude. However, a Liapunov function must first be foimd. Further information on this approach may be found in Denn (1975), Dussan (1975), and Joseph (1976). 

Farkas, H. Noszticzius, Z. (1985a). Generalized Lotka-Volterra schemes and the construction of two-dimensional Expodator cores and their Liapunov-function via critical Hopf bifurcations. J. Chem. Soc. Faraday 2, 81, 1487-505. 

Farkas, H. Noszticzius, Z. (1985b) Use of Liapunov—function in dissipative and explosive models. Ber. Bunsenges. Phys. Chem., 89, 604-5. 

Higgins, J. (1968). Some remarks on Shear s Liapunov function for systems of chemical reactions. J. Theor. Biol., 21, 293-304. 

The second technique to be briefly described in this section is that of global Liapunov-functions. Indeed., we have made use of the Liapunov-technique already in Sections 7.2 and 7.4 when defining a Liapunov-function L by... 

Here p is the normalization constant exp(-X ). The formulation given in this chapter has the advantage of the physical interpretation in terms of species-specific thermodynamic driving forces and in terms of Liapunov functions further our formulation is generalizable to autocatalytic systems and many variable systems. 

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