For m ≥ 1 the only critical points of the m-ideal energy on closed planar curves are round multiply-covered circles, and the associated L²-gradient flows converge exponentially to these circles from W^{2,2} initial data below a curvature-oscillation threshold.
[L2] ,From Spiral to Spline: Optimal Techniques in Interactive Curve DesignUCB/EECS-2009- 162(2009)
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On the generalised ideal flow of closed planar curves
For m ≥ 1 the only critical points of the m-ideal energy on closed planar curves are round multiply-covered circles, and the associated L²-gradient flows converge exponentially to these circles from W^{2,2} initial data below a curvature-oscillation threshold.