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

TL;DR

This paper identifies three practical conditions under which the control landscapes of certain non-linear systems are free of traps, enabling reliable optimization methods like gradient ascent to find effective controls.

Contribution

It provides the first set of tractable, sufficient conditions ensuring trap-free control landscapes for a broad class of non-linear control systems, extending quantum control insights.

Findings

Conditions hold for almost all cases within the class of systems studied.

Lipschitz constants are explicitly derived for two conditions.

Numerical results confirm the trap-free landscape property for systems meeting the conditions.

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.