Averaging in optimal control and applications

For problems of optimal control whose dynamics exhibit fast and slow variables, averaging is a relevant approach [1, 4]. One possibility is to average the extremal flow given by Pontryagin maximum principle, but other approaches are also available. A typical application is control in space mechanics [3, 2]. The motion of a spacecraft can be regarded as a perturbation of the periodic motion around the main attracting body (case of negative energy). Besides the control itself (thrust of the engine), one wants to take into account further perturbations such as the potentials of more distant bodies (e.g. of the Moon and the Sun, in addition to the Earth attraction), atmospheric damping, solar pressure, etc. Treating these effects as small perturbations leads to averaging at least with respect to the fast angle describing the position of the spacecraft on its osculating orbit. Possible costs include minimization of time or consumption.

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Keywords. optimal control, averaging, space mechanics

References

  1. Bensoussan, A.; Kokotovic, P.; Blankenship, G. Singular Perturbations for Deterministic Control Problems. Lect. Notes Contr. Inform. Sci. (1986), 9–171.
  2. Bombrun, A.; Pomet, J.-B. The averaged control system of fast oscillating control systems. SIAM J. Control Optim. 51 (2013), no. 3, 2280–2305.
  3. Bonnard, B.; Caillau, J.-B. Geodesic flow of the averaged controlled Kepler equation. Forum Math. 21 (2009), no. 5, 797–814.
  4. Chaplais, F. Averaging and deterministic optimal control. SIAM J. Control Optim. 25 (1987), no. 3, 767–780.