Comparison of convex optimization-based approaches to solve nonconvex optimal control problems

Nonconvex optimal control problems are generally difficult to solve due to their non-convexity. The main non-convexity in many problems may come from the nonlinearity of system dynamics. This paper presents two approaches, namely the direct linearization approach and the nonlinearity-kept & linearization approach, on how to transform the nonlinear dynamics into linear dynamics, with the aim to efficiently solve the original problems by convex optimization (which requires all equality constraints to be linear). The first approach directly linearizes the nonlinear dynamics, while the second approach first attempts to keep some nonlinearity in the nonlinear dynamics and then partially linearizes an obtained control-affine system to get linear dynamics. Details on these convex optimization-based approaches and issues on feasibility and convergence will be discussed. Then, two atmospheric flight problems with highly nonlinear dynamics, which are optimal flight of aerodynamically controlled aircrafts and fuel-optimal precise landing of rockets, are selected to test the performance of these two approaches. Our preliminary results show that the nonlinearity-kept & linearization approach achieves better performance in convergence than the direct linearization approach, and it is crucial to preserve certain nonlinearity inherent in the original problems in the process of convexification.