Lyapunov function method

The stability of stationary points in cases (al), (a2), (d) is proven in Appendix 45 using the Lyapunov function method. 

Chapter 5 provides some examples of purely analyti( al tools useful for describing CA. It discusses methods of inferring cycle-state structure from global eigenvalue spectra, the enumeration of limit cycles, the use of shift transformations, local structure theory, and Lyapunov functions. Some preliminary research on linking CA behavior with the topological characteristics of the underlying lattice is also described. 

In principle, to study the local stability of a stationary point from a linear approximation is not difficult. Some difficulties are met only in those cases where the real parts of characteristic roots are equal to zero. More complicated is the study of its global stability (in the large) either in a particular preset region or throughout the whole phase space. In most cases the global stability can be proved by using the properly selected Lyapunov function (a so-called second Lyapunov method). Let us consider the function V(c) having first-order partial derivatives dY/dCf. The expression... 

From the various versions of this method we will choose only one. Let V < 0 and, only at the rest point under study c, V - 0. Then let Vhave its minimum, V(c) = at the point c and for some e > Vmin the set specified by the inequality V(c0) < e is finite. Therefor any initial conditions c0 from this set the solution of eqn. (73) is c(t, k, c0) - c at t - oo. V(c) is called a Lyapunov function. The arbitrary function whose derivative is negative because of the system is called a Chetaev or sometimes a dissipative function. Physical examples are free energy, negative entropy, mechanical energy in systems with friction, etc. Studies of the dissipative functions can often provide useful information about a given system. A modern representation for the second Lyapunov method, including a method of Lyapunov vector functions, can be found in ref. 20. 

In the systems that are far from equilibrium, the stratification into sub systems with fast and slow is also possible, with the subsystem with fast internal variables being characterized by the minimum of the relevant Lyapunov function (provided that such a function exists for the particular process scheme). The ways to describe systems that can be stratified in accordance with the timing hierarchy of the processes involved are under intensive study in modem chemical engineering and biophysics. The methods are based on models that take into account mechanistic (deterministic) and statistical degrees of freedom and their contribution to processes of energy transfer and chemical conversions in the systems with a very complicated process hierarchy (for example, catalytic and biological processes). 

Considerations of this type are characteristic of the Lyapunov method of examination of the stability of a stationary point the function W defined by equation (A18) is an example of the so-called Lyapunov function. Note also that to draw the conclusion about an asymptotic stability of the stationary point (0, 0), in addition to inequality (A 19) deriving from properties of the Lyapunov function and properties of the investigated system, inequality (A20) was also required. 

The second and, apparently, the more general idea is the idea of formation of a new scientific discipline - "Model Engineering" (Gorban and Karlin, 2005 Gorban et al., 2007). The subject of the discipline is the choice of an outset statement of the solved problem which is the most suitable (optimal) both for conceptual analysis and for computations. The transfer of kinetic description into the space of thermodynamic variables became a main method for this discipline. In the method the solved problem can be represented as one-criterion problem of search for extremum of the function that has the properties of the Lyapunov functions (monotonously moving to fixed points). 

Dynamic nonlinear analysis techniques (Isidori 1995) are not directly applicable to DAE models but they should be transformed into nonlinear input-affine state-space model form by possibly substimting the algebraic equations into the differential ones. There are two possible approaches for nonlinear stability analysis Lyapunov s direct method (using an appropriate Lyapunov-function candidate) or local asymptotic stability analysis using the linearized system model. 

Following the Morbidelli and Varma criterion, several other methods have been proposed in recent years in order to characterize the highly sensitive behavior of a batch reactor when it reaches the runaway boundaries. Among the most successful approaches, the evidence of a volume expansion in the phase space of the system has been widely exploited to characterize runaway conditions. For example, Strozzi and Zaldivar [9] defined the sensitivity as a function of the sum of the time-dependent Lyapunov exponents of the system and the runaway boundaries as the conditions that maximize or minimize this Lyapunov sensitivity. This has put the basis for the development of a new class of runaway criteria referred to as divergence-based approaches [5,10,18]. These methods usually identify runaway with the occurrence of a positive divergence of the vector field associated with the mathematical model of the reactor. 

The method of Lyapunov s function consists of constructing a function V(x,y) with the following properties ... 

In the case of nonlinear systems that cannot be reduced to a linearized system, stability is much more difficult to assess. Lyapunov s direct method [1] requires a suitable energy function to be found. Often, only numerical time integration gives an indication of the dynamic behaviour and stability that cannot be proven otherwise. 

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