  The widespread adoption of nonlinear Receding Horizon Control (RHC)
strategies by industry has led to more than 30 years of intense research
efforts to provide stability guarantees for these methods. However, current
theoretical guarantees require that each (generally nonconvex) planning problem
can be solved to (approximate) global optimality, which is an unrealistic
requirement for the derivative-based local optimization methods generally used
in practical implementations of RHC. This paper takes the first step towards
understanding stability guarantees for nonlinear RHC when the inner planning
problem is solved to first-order stationary points, but not necessarily global
optima. Special attention is given to feedback linearizable systems, and a
mixture of positive and negative results are provided. We establish that, under
certain strong conditions, first-order solutions to RHC exponentially stabilize
linearizable systems. Crucially, this guarantee requires that state costs
applied to the planning problems are in a certain sense `compatible' with the
global geometry of the system, and a simple counter-example demonstrates the
necessity of this condition. These results highlight the need to rethink the
role of global geometry in the context of optimization-based control.
