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We consider a gradient approximation scheme that is based on applying needleshaped inputs. By using ideas known from the classic proof of the PontryaginMaximum Principle we derive an approximation that reveals that the consideredsystem moves along a weighted averaged gradient. Moreover, based on the sameideas, we give similar results for arbitrary periodic inputs. We also present anew gradient-based optimization algorithm that is motivated by our calculationsand that can be interpreted as a combination of the heavy ball method andNesterov's method.

Gradient approximation and extremum seeking via needle variations