Applications of Liapounov's Second Method in Reactor Dynamics

Abstract

Introduction. In his classical treatise! Liapounov establishes certain criteria for the stability of solutions of ordinary nonlinear differential equations which do not involve the consideration of the (linear) equations of first variation. These criteria, restated below, constitute Liapounov's Second Method. This method, although well known, has been brought into prominence only recently by The Russian School, notably I. G. Malkin. While these criteria are not particularly easy to apply even to particular linear differential equations (notably with variable coefficients, see Ref. 2), it is of interest that certain nonlinear problems of considerable physical interest, which are treated below, can be handled rather simply in this manner.Ergen and Weinberg 3 and Lipkin' have recently investigated from the physical point of view certain aspects of nonlinear reactor dynamics of homogeneous and heterogeneous reactors with regard to stability by constructing certain positive definite functions of the reactor variables and parameters. These functions will be shown to satisfy the conditions of Theorems 1 and 2 below. Because of the mechanical analogies developed in Ref. 3, these positive definite functions are called Hamiltonians.The purpose of this paper is twofold. First, the theory in References 3 and 4 of interest here is set on a firm mathematical basis by Liapounov's Second Method. Second, still other types of heterogeneous reactors, namely those with heat generated in each medium, are studied with regard to stability thus generalizing the work of Lipkin'.The study of the both physically and mathematically interesting reactor with infinitely many media (i.e. a one dimensional continuous medium) with regard to stability is not complete at this time and is therefore not included. A formal solution of the stability problem for this case has been obtained. However, a rigorous treatment appears to be quite difficult, primarily because the dynamic equations consist of a nonhomogeneous diffusion equation; the two equations are linked through the nonhomogeneous term. Separation of variables together with the use of appropriate boundary and initial conditions achieves a reduction to an infinite system of differential equations. The method used to treat stability in the cases considered in this paper then applies formally equally well in this more complicated case. However, because of convergence problems it appears that the usual variational approach rather than the Second Method will prove advantageous here.

Mathematical Description of Feedback

“…1) can be performed using their definitions in (5.1, Eqs. [2][3][4][5][6][7][8][9]. This calculation reduces to a numerical integration once the steady-state distribution is known.…”

“…In order to obtain the feedback kernel as a sum of expo nentials, we shall first cast (43) into a diagonal form. For this purpose, we define the following matrices [3]:…”

“…Certainly, for small enough Ax, the temperature variation inside the increment Ax will not vary rapidly; hence, we may define the average temperature inside in terms of the endpoint values 2f * = T k+1 + T k (3) The transit time through the region is given by t 0 == h\v = (NAx)/v (4) f We break the axial direction into a number of regions or "lumps;" hence, lumping.…”

Analyzing stability of neutron point kinetics equations with nine photo-neutron groups using Lyapunov exponent method

“…There are several methods for stability analysis of nuclear reactor having been studied before [3,10,11], such as Bode, Nyquist, Routh Hurwitz and Lyapunov second method [3,12,13]. Fu [9], Chen [14], Ergen [15] and etc studied stability analysis using Lyapunov second method. Recently, Della et al [13] has developed a theoretical model to study the stability of Ghana Research Reactor (GHARR-1) for a single group of delayed neutrons taking into consideration thermal hydraulics.…”

A Class of Nonlinear Delay Differential Equations

“…Physically the stability criteria for the identically zero solution of (0.3) describe conditions under which the equilibrium state of the reactor (with unit average power) is stable. Since Ergen, Lipkin and N ohel [6] have been able to give rigorous stability criteria for other and in a sense simpler reactors, the present paper can also be regarded as an extension of that work. It should also be noted that Brownell and Ergen [7] examined the question of existence of periodic solutions of (0.3), but did not consider questions of stability.…”