It is hard to find the global optimum of general nonlinear and nonconvex optimization problems in a reasonable time. This article presents a method to transfer the receding horizon control approach, where nonlinear, nonconvex optimization problems are considered, into graph-search problems. Specifically, systems with symmetries are considered to transfer system dynamics into a finite-state automaton. In contrast to traditional graph-search approaches where the search continues until the goal vertex is found, the transfer of a receding horizon control approach to graph-search problems presented in this article allows to solve them in real time. We prove that the solutions are recursively feasible by restricting the graph search to end in accepting states of the underlying finite-state automaton. The approach is applied to trajectory planning for multiple networked and autonomous vehicles. We evaluate its effectiveness in simulation and experiments in the Cyber-Physical Mobility Lab, an open-source platform for networked and autonomous vehicles. We show real-time capable trajectory planning with collision avoidance in experiments on off-the-shelf hardware and code in MATLAB for two vehicles.

Classical turnpikes correspond to optimal steady states which are attractors of infinite-horizon optimal control problems. In this paper, motivated by mechanical systems with symmetries, we generalize this concept to manifold turnpikes. Specifically, the necessary optimality conditions projected onto a symmetry-induced manifold coincide with those of a reduced-order problem defined on the manifold under certain conditions. We also propose sufficient conditions for the existence of manifold turnpikes based on a tailored notion of dissipativity with respect to manifolds. Furthermore, we show how the classical Legendre transformation between Euler–Lagrange and Hamilton formalisms can be extended to the adjoint variables. Finally, we draw upon the Kepler problem to illustrate our findings.