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This paper proposes a data-driven optimal tracking control scheme for unknown general nonlinear systems using neural networks. First, a new neural networks structure is established to reconstruct the unknown system dynamics of the form ˙ x(t) = f (x(t))+g(x(t))u(t). Two networks in parallel are designed to approximate the functions f (x) and g(x). Then the obtained data-driven models are used to build the optimal tracking control. The developed control consists of two parts, the feed-forward control and the optimal feedback control. The optimal feedback control is developed by approximating the solution of the Hamilton-Jacobi-Bellman equation with neural networks. Unlike other studies, the Hamilton-Jacobi-Bellman solution is found by estimating the value function derivative using neural networks. Finally, the proposed control scheme is tested on a delta robot. Two trajectory tracking examples are provided to verify the effectiveness of the proposed optimal control approach. 

Abstract A key challenge in aerosol pollution studies and climate change assessment is to understand how atmospheric aerosol particles are initially formed^1,2. Although new particle formation (NPF) mechanisms have been described at specific sites^3–6, in most regions, such mechanisms remain uncertain to a large extent because of the limited ability of atmospheric models to simulate critical NPF processes^1,7. Here we synthesize molecular-level experiments to develop comprehensive representations of 11 NPF mechanisms and the complex chemical transformation of precursor gases in a fully coupled global climate model. Combined simulations and observations show that the dominant NPF mechanisms are distinct worldwide and vary with region and altitude. Previously neglected or underrepresented mechanisms involving organics, amines, iodine oxoacids and HNO₃probably dominate NPF in most regions with high concentrations of aerosols or large aerosol radiative forcing; such regions include oceanic and human-polluted continental boundary layers, as well as the upper troposphere over rainforests and Asian monsoon regions. These underrepresented mechanisms also play notable roles in other areas, such as the upper troposphere of the Pacific and Atlantic oceans. Accordingly, NPF accounts for different fractions (10–80%) of the nuclei on which cloud forms at 0.5% supersaturation over various regions in the lower troposphere. The comprehensive simulation of global NPF mechanisms can help improve estimation and source attribution of the climate effects of aerosols. 

Earth’s inner core is predominantly composed of solid iron (Fe) and displays intriguing properties such as strong shear softening and an ultrahigh Poisson’s ratio. Insofar, physical mechanisms to explain these features coherently remain highly debated. Here, we have studied longitudinal and shear wave velocities of hcp-Fe (hexagonal close-packed iron) at relevant pressure–temperature conditions of the inner core using in situ shock experiments and machine learning molecular dynamics (MLMD) simulations. Our results demonstrate that the shear wave velocity of hcp-Fe along the Hugoniot in the premelting condition, defined asT/T_m(T_m: melting temperature of iron) above 0.96, is significantly reduced by ~30%, while Poisson’s ratio jumps to approximately 0.44. MLMD simulations at 230 to 330 GPa indicate that collective motion with fast diffusive atomic migration occurs in premelting hcp-Fe primarily along [100] or [010] crystallographic direction, contributing to its elastic softening and enhanced Poisson’s ratio. Our study reveals that hcp-Fe atoms can diffusively migrate to neighboring positions, forming open-loop and close-loop clusters in the inner core conditions. Hcp-Fe with collective motion at the inner core conditions is thus not an ideal solid previously believed. The premelting hcp-Fe with collective motion behaves like an extremely soft solid with an ultralow shear modulus and an ultrahigh Poisson’s ratio that are consistent with seismic observations of the region. Our findings indicate that premelting hcp-Fe with fast diffusive motion represents the underlying physical mechanism to help explain the unique seismic and geodynamic features of the inner core.