Nonlinear signal-correction observer and application to UAV navigation

A nonlinear signal-correction observer (NSCO) is presented for signals correction and estimation, which not only can reject the position measurement error, but also the unknown velocity can be estimated, in spite of the existence of large position measurement error and intense stochastic non-Gaussian noise. For this method, the position signal is not required to be bounded. The NSCO is developed for position/acceleration integration, and it is applied to an unmanned aerial vehicle (UAV) navigation: Based on the NSCO, the position and flying velocity of quadrotor UAV are estimated. An experiment is conducted to demonstrate the effectiveness of the proposed method

Universal Time Scale for Thermalization in Two-dimensional Systems

The Fermi-Pasta-Ulam-Tsingou problem, i.e., the problem of energy equipartition among normal modes in a weakly nonlinear lattice, is here studied in two types of two-dimensional (2D) lattices, more precisely in lattices with square cell and triangular cell. We apply the wave-turbulence approach to describe the dynamics and find multi-wave resonances play a major role in the transfer of energy among the normal modes. We show that, in general, the thermalization time in 2D systems is inversely proportional to the squared perturbation strength in the thermodynamic limit. Numerical simulations confirm that the results are consistent with the theoretical prediction no matter systems are translation-invariant or not. It leads to the conclusion that such systems can always be thermalized by arbitrarily weak many-body interactions. Moreover, the validity for disordered lattices implies that the localized states are unstable.Comment: 6 pages, 4 figure

A Liouville theorem for the Euler equations in a disk

We present a symmetry result regarding stationary solutions of the 2D Euler equations in a disk. We prove that a stationary solution with only one stagnation point in a disk must be a circular flow, which confirms a conjecture proposed by F. Hamel and N. Nadirashvili in [J. Eur. Math. Soc., 25 (2023), no. 1, 323-368]. The key ingredient of the proof is to use `local' symmetry properties for the non-negative solutions of semi-linear elliptic equations with a continuous nonlinearity in a ball, which can be established by a rearrangement technique called continuous Steiner symmetrization