  We address the motion planning problem for large multi-agent systems,
utilizing Cosserat rod theory to model the dynamic behavior of vehicle
formations. The problem is formulated as an optimal control problem over
partial differential equations (PDEs) that describe the system as a continuum.
This approach ensures scalability with respect to the number of vehicles, as
the problem's complexity remains unaffected by the size of the formation. The
numerical discretization of the governing equations and problem's constraints
is achieved through Bernstein surface polynomials, facilitating the conversion
of the optimal control problem into a nonlinear programming (NLP) problem. This
NLP problem is subsequently solved using off-the-shelf optimization software.
We present several properties and algorithms related to Bernstein surface
polynomials to support the selection of this methodology. Numerical
demonstrations underscore the efficacy of this mathematical framework.
