Control landscapes for a class of nonlinear dynamical systems: sufficient conditions for the absence of traps
Abstract
We establish three tractable, jointly sufficient conditions for the control landscapes of nonlinear 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 nonlinear 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 endpoint objectives for a general class of nonlinear control systems having the form of a linear time invariant term with an additional state dependent nonlinear term. Within this large class of nonlinear 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 DMOPRH 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.
 Publication:

Journal of Physics A Mathematical General
 Pub Date:
 August 2018
 DOI:
 10.1088/17518121/aacc85
 arXiv:
 arXiv:1710.03226
 Bibcode:
 2018JPhA...51G5103R
 Keywords:

 Mathematics  Optimization and Control;
 Mathematical Physics
 EPrint:
 6 Figures