This paper focuses on optimization-based control of multi-aircraft systems that have several mission objectives. Signal Temporal Logic (STL) is used to express the mission specifications that combine temporal and logical constraints. A methodology is presented to construct an optimization problem in the form of Mixed-Integer Linear Programming (MILP) by using the differential flatness property of a nonlinear dynamical system and STL specifications to generate feasible trajectories. Contrary to general implementations of Temporal Logic to discrete-time systems, the proposed method deals with continuous-time systems. It can be used to find optimal control strategies to achieve the assigned tasks for nonlinear dynamical systems without discretizing the system dynamics. As an illustration, we present an air traffic control example. The nonlinear dynamical model for the aircraft is represented as a partially differentially flat system, and the presented method is applied to manage approach control and to solve the arrival sequencing problem. The method is also applied with a quadrotor fleet to show that the method can be used with different multi-agent systems.