Control landscapes for a class of non-linear dynamical systems: sufficient conditions for the absence of traps

Abstract

We establish three tractable, jointly sufficient conditions for the control landscapes of non-linear dynamical systems to be trap free. Trap free landscapes ensure that local optimization methods (such as gradient ascent) can achieve monotonic convergence to the objective in both simulations and in practical circumstances. These results extend prior research primarily regarding the Schrödinger equation to a broader class of non-linear control problems encompassing both quantum and other dynamical systems. This outcome elucidates that these previous conclusions on quantum control landscapes were not specifically contingent upon any features unique to quantum dynamics. As an illustration of the new general results we demonstrate that they encompass end-point objectives for a general class of non-linear control systems having the form of a linear time invariant term with an additional state dependent non-linear term. Within this large class of non-linear control problems, each of the three sufficient conditions is shown to hold for all but a null set of cases. We establish a Lipschitz condition for two of these sufficient conditions and under specific circumstances we explicitly find the associated Lipschitz constants. A detailed numerical investigation using the D-MOPRH gradient control optimization algorithm is presented for a particular example amongst this family of systems. The numerical results confirm the trap free nature of the landscapes of such systems.