Abstract
A chaos maximizing optimal control problem is formulated and applied to Duffing's equation to maximize the largest Lyapunov exponent. The resulting bang-bang optimal controller yields a positive value of the largest Lyapunov exponent, indicating chaotic behavior. In fact, the largest Lyapunov exponent is approximately twice as large as that achieved with simple sinusoidal forcing at the same amplitude bounds. However, the resulting phase portrait of the optimal trajectory is a limit cycle and is not chaotic at all. This paradoxical result contradicts the basic theory that a bounded trajectory with at least one positive Lyapunov exponent must be chaotic. Details concerning the development of a chaos measurement that is viable for current optimal control theory, a method of continuous normalization, the paradoxical chaotic limit cycle, resolution of the paradox, and closed-loop optimal jump condition in an augmented space are presented. In particular, for systems of differential equations with only piecewise differentiable right-hand sides due to a switching control, a jump discontinuity condition must be imposed on the state perturbations in order to compute correct Lyapunov exponents.
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