Liapunov's stability on autonomous nuclear reactor dynamical systems

Abstract

Point kinetics model for nuclear reactor physics leads to a stiff system of ordinary differential equations. Stability conditions for linear and nonlinear nuclear reactor point kinetics equations are determined without explicitly solving these systems. Three different techniques, linearization, Liapunov's direct, and variable gradient methods, are applied to investigate the stability of the equilibrium solutions of the proposed systems. Linear system is solved analytically to check the results obtained by Liapunov's direct and variable gradient methods. For the autonomous nonlinear system with Newtonian feedback, linearization technique is used to discuss the stability in a small vicinity of the equilibrium solution, while variable gradient method is applied to establish asymptotic stability. The nonlinear system is close to the linear system, so that it has been proved to be almost linear. Generalized power series method (GPWS) with variable step sizes is used as a numerical method to solve the nonlinear system. Results of numerical calculations are presented in the form of diagrams that confirm the validity of stability conditions obtained by Liapunov's methods. Our results indicate that no trajectories go to infinity as the time tends to infinity. Also, it is proved that the system is conservative by constructing a Liapunov's function, which is valid everywhere without any restrictions, associated with zero Liapunov's derivative function.