Conclusion and Future Work

It is clear that, even with an imperfect fitness measure, genetic programming is capable of finding functions that at least satisfy the Lyapunov properties over most of their domain. I expect that the method, with a better measure of fitness, will be able to find true Lyapunov functions without false hits. How well this method will scale up to higher-dimensional and more complex system remains to be seen.

Some possible future work, besides testing it on more complex systems, is:

• The measure of fitness has to improve. Perhaps there is a way to replace the approximation with an exact measure. One possibility is interval arithmetic, a feature of Mathematica. Although it produces intervals too conservative (a true Lyapunov function might return a interval with negative values), it warrants more study.

• Sometimes a Lyapunov function with vastly disparate scale would be produced. In some directions it would rise to $10^{20}$; in others it would be only in the tens. It could be a problem when the Lyapunov function is used in the control law: in one direction, the slightest error could saturate the controls; in another direction, the control law could be almost completely numb to large error.

  As a remedy, the Lyapunov function could be subject to more stringent conditions, such as being bound above and below by some positive definite function.

• Incorporating genetic programming into robust nonlinear control design. Genetic programming could search for a control law at the same time as Lyapunov functions. This would require a more advanced genetic programming structure, where a single individual contains multiple trees: one for the Lyapunov function, and others for the controls.