In this paper, the functional dynamic equation of second order is studied on an arbitrary time scale under milder restrictions without the assumed conditions in the recent literature. The Nehari, Hille, and Ohriska type oscillation criteria of the equation are investigated. The presented results confirm that the study of the equation in this formula is superior to other previous studies. Some examples are addressed to demonstrate the finding.

In this paper, the sharp Hille-type oscillation criteria are proposed for a class of second-order nonlinear functional dynamic equations on an arbitrary time scale, by using the technique of Riccati transformation and integral averaging method. The obtained results demonstrate an improvement in Hille-type compared with the results reported in the literature. Some examples are provided to illustrate the significance of the obtained results.

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This paper aims to improve Hille oscillation criteria for the third-order functional dynamic equation p2(ξ)ϕγ2p1ξϕγ1yΔ(ξ)ΔΔ+a(ξ)ϕγy(g(ξ))=0, on an above-unbounded time scale T. The obtained results improve related contributions reported in the literature without restrictive conditions on the time scales. To demonstrate the essential results, an example is presented.

Due to the expected increase in the electric vehicles (EVs) sales and hence the increase of the price of rare-earth permanent magnets, the switched reluctance motors (SRMs) are gaining increasing research interest currently and in the future. The SRMs offer numerous advantages regarding their structure and converter topologies. However, they suffer from the high torque ripple and complex control algorithms. This paper presents an improved direct instantaneous torque control (DITC) strategy of SRMs for EVs. The improved DITC can fulfill the vehicle requirements. It involves a simple online torque estimator and a torque error compensator. The turn-on angle is defined analytically to achieve wide speed operation and maximum torque per ampere (MTPA) production. Moreover, the turn-off angles are optimized for minimum torque ripples and the highest efficiency. In addition, this paper provides a detailed comparison between the proposed DITC and the most applicable torque control techniques of SRMs for EVs, including indirect instantaneous torque control (IITC), using torque sharing function (TSF) strategy and average torque control (ATC). The results show the superior performance of the proposed DITC because it has the lowest torque ripples, the highest torque tor current ratio, and the best efficiency over the low and medium speed ranges. Moreover, the comparison shows the advantages of each control technique over the range of speed control. It provides a very clear overview to develop a universal control technique of SRM for EVs by merging two or more control techniques.