A class of weighted Hardy type inequalities in $$\mathbb {R}^N$$

In the paper we prove the weighted Hardy type inequality $$\begin{aligned} \int _{{{\mathbb {R}}}^N}V\varphi ^2 \mu (x)dx\le \int _{\mathbb {R}^N}|\nabla \varphi |^2\mu (x)dx +K\int _{\mathbb {R}^N}\varphi ^2\mu (x)dx, \end{aligned}$$ ∫ R N V φ 2 μ ( x ) d x ≤ ∫ R N | ∇ φ | 2 μ ( x ) d x + K ∫ R N φ 2 μ ( x ) d x , for functions $$\varphi $$ φ in a weighted Sobolev space $$H^1_\mu $$ H μ 1 , for a wider class of potentials V than inverse square potentials and for weight functions $$\mu $$ μ of a quite general type. The case $$\mu =1$$ μ = 1 is included. To get the result we introduce a generalized vector field method. The estimates apply to evolution problems with Kolmogorov operators $$\begin{aligned} Lu=\varDelta u+\frac{\nabla \mu }{\mu }\cdot \nabla u \end{aligned}$$ L u = Δ u + ∇ μ μ · ∇ u perturbed by singular potentials.

Modified Vector Field Path-Following Control System for an Underactuated Autonomous Surface Ship Modelin the Presence of Static Obstacles

A modified path-following control system using the vector field method for an underactuated autonomous surface ship model is proposed in the presence of static obstacles. With this integrated system, autonomous ships are capable of following the predefined path, while avoiding the obstacles automatically. It is different from the methods in most published papers, which usually study path-following and obstacle collision avoidance, separately. This paper considers the coupled path following and collision avoidance task as a whole. Meanwhile, the paper also shows the heading control design method in the presence of static obstacles. To obtain a strong stability property, a nonlinear autopilot is designed based on the manoeuvring tests of the free-running ship model. The equilibrium point of the controller is globally exponentially stable. For the guidance system, a novel vector field method was proposed, and the proof shows the coupled guidance and control system is uniform semi-global exponentially stable (USGES). To prevent the obstacles near the predefined path, the proposed guidance law is augmented by integrating the repelling field of obstacles so that it can control the ship travel toward the predefined path through the obstacles safely. The repelling field function is given considering the obstacle shape and collision risk using the velocity obstacle (VO) algorithm. The simulations and ship model test were performed to validate the integrated system of autonomous ships.

EVALUATION OF UAV AUTONOMOUS FLIGHT ACCURACY WHEN CLASSICAL NAVIGATION ALGORITHM IS USED

This article examines and shows mathematical results of classical algorithm, which is used for small Unmanned Aerial Vehicle (UAV) navigation. The research is done with mathematical UAV model, which eliminates aerodynamics while the chosen flight path is followed by using vector field method. Lots of attention is dedicated to show possible flight path error values with representation of modelled flight path trajectories and deviations from the flight mission path. All of the modelled flight missions are done in two-dimensional space and all of the collected data with flight path error values are evaluated statistically. The most possible theoretical flight path error values are found and the general flight path error tendencies are predicted.