The goal of this paper is the formulation of the volumetric modulated arc radiation therapy (VMAT) delivery problem to moving tumors treated on medical linear accelerators. The main aim of the study is to determine and understand the interdependence between several parameters (dose rate, gantry speed and MLC speed) in the delivery of VMAT treatment plan to tumors that move. The secondary aim is to prove local delivery optimality (minimal beam ON time). The problem of VMAT delivery to moving target is formulated as a control problem with constraints. The relationships between parameters of arc therapy delivery are derived under constraint of treatment plan invariance and limitations on delivery parameters. The non-uniqueness of arc therapy irradiation is revealed from these relationships. The most efficient delivery of arc therapy is formulated as optimal control problem and solved by geometrical methods. A computer program is developed to find numerical solutions for delivery of specific VMAT plan. Explicit examples of VMAT delivery plans for 3-dimensional target motion are computed and illustrated with graphical representations of the variability of delivery parameters.