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 control systems to be trap free comparable to those now well known in quantum control. In particular, our results encompass end-point control problems for a general class of non-linear control systems of the form of a linear time invariant term with an additional state dependent non-linear term. Trap free landscapes ensure that local optimization methods (such as gradient ascent) can achieve monotonic convergence to effective control schemes in both simulation and practice. Within a large class of non-linear control problems, each of the three conditions is shown to hold for all but a null set of cases. Furthermore, we establish a Lipschitz condition for two of these assumptions; under specific circumstances, we explicitly find the associated Lipschitz constants. A detailed numerical investigation using the D-MOPRH control optimization algorithm is presented for a specific family of systems which meet the conditions for possessing trap free control landscapes. The results obtained confirm the trap free nature of the landscapes of such systems.