Necessary and sufficient stability condition

In this chapter we study the stability with respect to the initial data and the right-hand side of two-layer and three-layer difference schemes that are treated as operator-difference schemes with operators in Hilbert space. Necessary and sufficient stability conditions are discovered and then the corresponding a priori estimates are obtained through such an analysis by means of the energy inequality method. A regularization method for the further development of various difference schemes of a desired quality (in accuracy and economy) in the class of stability schemes is well-established. Numerous concrete schemes for equations of parabolic and hyperbolic types are available as possible applications, bring out the indisputable merit of these methods and unveil their potential. 

Prigoginc and Defay (1954) have shown the validity of Equations 46 or 47 to cau.se the validity of Equations 39 and 40, henr , the diffusional stability condition (any of Equations 46, 47, 53-56) is a necessary and sufficient stability condition of the ono-phase state of multicomponent systems (including the metastable state). 

If (a) above is satisfied, then the necessary and sufficient condition for stability is either... 

Theorem 4 The stability with respect to the initial data of scheme (25) with constant operators is necessary and sufficient for the stability with respect to the right-hand side, provided condition (24) of the norm of concordance holds. Moreover, in that case a priori estimate (20) is valid. 

This condition is necessary and sufficient for the stability of the weighted scheme, which interests us. 

Remark If the operators A and B are commuting, then condition (14) is necessary and sufficient for the stability of scheme (la) in the space Hq ... 

Therefore, the method of estimation of the transition operator norm permits us to prove that condition (14) is necessary and sufficient for the stability of scheme (1) with respect to the initial data in the space Ha (for B B ) and in the space Hb (for B = B > 0) with constant M = 1. 

The first analysis is connected with the case when A is a constant self-adjoint positive operator A = A > 0. As we have shown in Section 2, a necessary and sufficient condition for the stability of the weighted scheme (47) with respect to the initial data is... 

Let now // be a complex space and S be a non-self-adjoint operator. Then a necessary and sufficient condition for the stability in the space Ha with respect to the initial data of the scheme... 

Along these lines, it is worth noting that we are still in the framework of the general stability theory outlined in Chapter 6, Section 2 for difference schemes like (22) asserting that a necessary and sufficient condition for the estimate < pWllkWo valid for any 0 < / < 1 and any operator... 

If the characteristic polynomial passes the coefficient test, we then construct the Routh array to find the necessary and sufficient conditions for stability. This is one of the few classical techniques that we do not emphasize and the general formula is omitted. The array construction up to a fourth order polynomial is used to illustrate the concept. 

There is, however, another statement of the necessary and sufficient condition of thermodynamic stability of the multicomponent system in relation to mutual diffusion and phase separation that is less stringent than equation (3.20) because it may be fulfilled not for every component of the multicomponent system. For example, in the case of the ternary system biopolymeri + biopolymer2 + solvent, it appears enough to fulfil only two of the inequalities (Prigogine and Defay, 1954)... 

Condition 1 The necessary and sufficient condition for stability of a single steady state is B < B. ... 

Linearized or asymptotic stability analysis examines the stability of a steady state to small perturbations from that state. For example, when heat generation is greater than heat removal (as at points A— and B+ in Fig. 19-4), the temperature will rise until the next stable steady-state temperature is reached (for A— it is A, for B+ it is C). In contrast, when heat generation is less than heat removal (as at points A+ and B— in Fig. 19-4), the temperature will fall to the next-lower stable steady-state temperature (for A+ and B— it is A). A similar analysis can be done around steady-state C, and the result indicates that A and C are stable steady states since small perturbations from the vicinity of these return the system to the corresponding stable points. Point B is an unstable steady state, since a small perturbation moves the system away to either A or C, depending on the direction of the perturbation. Similarly, at conditions where a unique steady state exists, this steady state is always stable for the adiabatic CSTR. Hence, for the adiabatic CSTR considered in Fig. 19-4, the slope condition dQH/dT > dQG/dT is a necessary and sufficient condition for asymptotic stability of a steady state. In general (e.g., for an externally cooled CSTR), however, the slope condition is a necessary but not a sufficient condition for stability i.e., violation of this condition leads to asymptotic instability, but its satisfaction does not ensure asymptotic stability. For example, in select reactor systems even... 

Local asymptotic stability criteria may be obtained by first solving the steady-state equations to obtain steady states and then linearizing the transient mass and energy balance equations in terms of deviations of variables around each steady state. The determinant (or slope) and trace conditions derived from the matrix A in the set of equations obtained are necessary and sufficient for asymptotic stability. 

The stationary point of the system (5.2) is by a definition stable one, if all the roots of its characteristic equation (5.11) have the negative real parts. The Routh-Hurwitz criteria presented in Ref. [206] permits escaping the calculations of these roots to establish the simple relations between the coefficients ock, which allow to point out simple stability conditions. For instance, in the case of terpolymerization the positivity of both coefficients oq and oc2 is regarded to be a criteria of such stability, and as for four-component copolymerization the following non-equality a3 < oq < ot2 has also to be hold. At arbitrary number m of the components the positivity of all ak is regarded to be necessary (but not sufficient) stability condition. For the stability of the boundary SP of m-component system located inside the certain boundary 1-subsimplex of monomers Mk, M2,. .., M, the stability of the above SP in such subsimplex and negativity of all values of X, Xl+1,..., vm x (5.13) are needed. 

The Gibbs stability theory provides necessary and sufficient conditions to investigate stability problems with well-defined boundary conditions in equilibrium state. Some examples of this are... 

Table 12.1 Necessary and sufficient conditions for the stability of equilibrium state...

Recall that the occurrence of the positively defined Lyapunov function is the necessary and sufficient condition of the uniqueness and stability of the stationary state (see Section 3.4). 

Are there NBMOs in a molecule In some cases the information on the stability and reactivity of a hydrocarbon can be assessed by determining whether the compound has singly populated NBMOs. In terms of the graph theory the presence of NBMOs is determined by establishing the necessary and sufficient conditions for the existence of zeros in the graph s spectrum. Below we cite a number of important theorems allowing the presence of NBMOs in a given AS to be established. 

The determinant of the - adjacency matrix A. It was observed that this determinant is often equal to zero and this is a necessary and sufficient condition for the presence of non-bonding molecular orbitals in Hiickel theory. The actual numerical value of det A is correlated to the thermodynamic stability of the molecule [Graovac and Gutman, 1978 Trinajstic, 1992]. 

The complete set of necessary and sufficient conditions for stability, as first given by Amundson in 1955, is derived in a rather different way. The basic idea is to focus attention on small perturbations away from a given steady state. If they are sufficiently small, they can be described by linear equations and we shall be able to see just how they grow or die away. It can be proved that this establishes local stability, in the sense that sufficiently small perturbations will certainly die out. It does not say anything... 

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