Symplectic integrators very nearly conserve the total energy and are particularly useful when treating long times. We demonstrate some of the properties of these integrators by exploring the structure of first-, second-, and fourth-order symplectic integrators and apply them to the simple harmonic oscillator. We consider numeric, geometric, and analytic aspects of the integrators with particular attention to the computed energies.

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We consider the iterative learning control problem from an adaptive control viewpoint. It is shown that some standard Lyapunov adaptive designs can be modi ed in a straightforward manner to give a solution to either the feedback or feedforward ILC problem. Some of the common assumptions of nonlinear iterative learning control are relaxed: eg. we relax the common linear growth asssumption on the nonlinearities and handle systems of arbitrary relative degree. It is shown that generally a linear rate of convergence of the MSE can be achieved, and a simple robustness analysis is given. For linear plants we show that a linear rate of MSE convergence can be achieved for non-minimum phase plants.

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