  This article introduces and provides the mathematical foundation of the open
string Floer theory of Landau-Ginzburg model viaWitten equation. We introduce
the concept of regular tame exact Landau-Ginzburg system on a noncompact
Kaehler manifold, and define the notion of Landau-Ginzburg branes, as the
objects of the Fukaya category. The study on Witten equation in our context
provides the construction of the Fukaya category of Landau-Ginzburg model which
was conjectured to be existed in Gaiotto-Moore-Witten's work [22] and
Kapranov-Kontsevich-Soibelman's work [30]. This is the first paper in the
series of studies on Landau-Ginzburg models in the contexts of the mirror
symmetry and other topics.
