  In this paper, we consider stochastic control problems on the Sierpinski
gasket. An order comparison lemma is derived using heat kernel estimate for
Brownian motion on the gasket. Using the order comparison lemma and techniques
of BSDEs, we establish a Pontryagin stochastic maximum principle for these
control problems. It turns out that the stochastic maximum principle on the
Sierpinski gasket involves two necessity equations in contrast to its
counterpart on Euclidean spaces. This effect is due to singularity between the
Hausdorff measure and the energy dominant measure on the gasket, which is a
common feature shared by many fractal spaces. The linear regulator problems on
the gasket is also considered as an example.
