Verification of Aizerman's conjecture for a class of third-order systems

The second method of Lyapunov is used to validate Aizerman's conjecture for the class of third-order nonlinear control systems described by the following differential equation: \tdot{e} + a_{2}\ddot{e} + a_{1}\dot{e} + a_{0}e + f(e)=0 In this case, the stability of the nonlinear system may be inferred by considering an associated linear system in which the nonlinear function f(e) is replaced by ke . If the linear system is asymptotically stable for k_{1} , then the nonlinear system will be asymptotically stable in-the-large for any f(e) for which k_{1} The Lyapunov function used to prove this result is determined in a straightforward manner by considering the physical behavior of the system at the extreme points of the allowable range of k .