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We develop a novel optimal control algorithm to change the phase of an oscillator using a minimum energy input, which also minimizes the oscillator’s transversal distance to the uncontrolled periodic orbit. Our algorithm uses a two-dimensional reduction technique based on both isochrons and isostables. We develop a novel method to eliminate cardiac alternans by connecting our control algorithm with the underlying physiological problem. We also describe how the devised algorithm can be used for spike timing control which can potentially help with motor symptoms of essential and parkinsonian tremor, and aid in treating jet lag. To demonstrate the advantages of this algorithm, we compare it with a previously proposed optimal control algorithm based on standard phase reduction for the Hopf bifurcation normal form, and models for cardiac pacemaker cells, thalamic neurons, and circadian gene regulation cycle in the suprachiasmatic nucleus. We show that our control algorithm is effective even when a large phase change is required or when the nontrivial Floquet multiplier is close to unity; in such cases, the previously proposed control algorithm fails. 

In this article, we devise two related control algorithms to change the degree of synchrony of a population of noise-free, identical, uncoupled neural oscillators using a single control input. The algorithms are based on phase reduction, and use a population-level partial differential equation formulation to change the phase distribution of the neurons as desired. Motivated by the pathological neural synchronization hypothesized to be present in patients suffering from essential and parkinsonian tremor, we take our control objective to be the desynchronization of an initially synchronized neural population. Through numerical simulations, we are able to show that our algorithms work for both Type I and Type II neural populations. To demonstrate the versatility of our control algorithms, we also show that they can be applied to synchronize an initially desynchronized neural population as well. For the systems considered in this paper, the control algorithms can be applied to achieve any desired traveling-wave neural phase distribution, as long as the combination of initial and desired phase distributions is non-degenerate.