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.

Comment on ‘Control landscapes are almost always trap free: a geometric assessment’

“…The authors of Ref. [2] broadly applied the discussed results in their subsequent publications [7,8] to explain a variety of experimental evidences of trap-free landscapes in the areas far beyond the scope of quantum control, such as chemical synthesis, property optimization and evolutionary biology. It is worth stressing that these ex-planations are derived from the argument summarized as theorem 3 in this comment, which we have shown by counterexamples to be incorrect.…”

“…Since classical mechanics is merely a limiting case of quantum mechanics, the reported result has pivotal practical implications. Namely, the authors argue in their subsequent publications [7,8] that optimal control problems in nearly all areas, from chemistry and material science to biological evolution, fundamentally belong to a "simple" category and can be solved in polynomial times using a relatively small number of controls.…”

Variational Quantum Gate Optimization

“…We tried a variety of existing optimizers and found that a gradient-based optimizer L-BFGS-B [61] shows the fastest convergence to an optimal value, even if we take into account the number of measurements necessary for the gradient evaluation. It is known that there exist only a few local traps in quantum gate optimization in general [65]. Thus, we expect local traps do not largely affect the performance of VQGO.…”

The Surprising Ease of Finding Optimal Solutions for Controlling Nonlinear Phenomena in Quantum and Classical Complex Systems

“…Nonlinear phenomena cover vast areas in the sciences and beyond, and it is an open question about which applications will satisfy the same general N-CL principle. A primary question concerning N-CLs 9,11 is whether local optima exist forming traps, as shown in Figure 1, compared to the trap-free situation in Figure 2. These two figures are simply schematics with two control variables, u = [u 1 , u 2 ], along the horizontal axes and with the objective fitness value indicated by the height domains (e.g., directed and natural evolutions) do not share this feature, at least at small scales over the associated landscapes where some degree of roughness can be expected.…”