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

Abstract

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: <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\tdot{e} + a_{2}\ddot{e} + a_{1}\dot{e} + a_{0}e + f(e)=0</tex> In this case, the stability of the nonlinear system may be inferred by considering an associated linear system in which the nonlinear function <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">f(e)</tex> is replaced by <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">ke</tex> . If the linear system is asymptotically stable for <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">k_{1} &lt; k &lt; k_{2}</tex> , then the nonlinear system will be asymptotically stable in-the-large for any <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">f(e)</tex> for which <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">k_{1} &lt; \frac{f(e)}{e} &lt; k_{2}.</tex> 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 <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">k</tex> .