Necessary Conditions for Control Effort Minimization of Euler-Lagrange Systems

The L1 norm of the control vector is a suitable measure for the effort needed to control a vehicle, since, in several cases of practical interest, it can provide accurate estimate of the fuel consumption. In this paper, we address the problem of minimizing the weighted sum of the L1 norms of the control vectors of N vehicles moving in formation. Specifically, modeling each agent as a six degrees-of-freedom rigid body subject to external forces and moments, and holonomic and nonholonomic constraints, we give necessary conditions for minimizing the formation’s control effort. In addition, we provide necessary conditions for the existence of singular controls for the abnormal and the normal optimal control problems. Two of our main results show that singular controls have order of singularity equal to one and are analytical in the junction between singular and non-singular arcs. In order to highlight the framework presented in this paper, we provide a numerical example concerning a formation of F-16 performing an Immelmann turn.