In the article, we prove that the function x ? (1-x)pK(?x) is logarithmically concave on (0,1) if and only if p ? 7/32, the function x ? K(?x)/log(1+4/?1-x) is convex on (0,1) and the function x ? d2/dx2 [K(?x)- log (1+4/?1-x) is absolutely monotonic on (0,1), where K(x) = ??/20 (1-x2 sin2t)-1/2 dt (0 < x < 1) is the complete elliptic integral of the first kind.

In this article, we extend the adaptive tracking control to more general nonlinear systems with multiple uncertainties, including output constraint, input delay, unknown parameter, and external disturbances for the first time. Without any growth assumptions, the adaptive backstepping technique is combined with the parameter separation technique to solve the parametric nonlinearities while the related results need to apply restrictive assumptions or use an approximation-based scheme to deal with them. To alleviate the serious uncertainties caused by output constraint and input delay, the barrier Lyapunov function (BLF) and the Pade approximation method are employed in a unified framework when the control coefficient is known. Under the case of the unknown control coefficient, the Nussbaum gain technique is further combined to compensate it. Then, under both cases, universal adaptive state-feedback control strategies merged with rigorous stability analysis are proposed, respectively, which guarantees all the signals are uniformly ultimately bounded. In addition, the reference signal tracks the system output into a compact set of the origin and the constraint of output is not violated. Finally, the proposed controllers are applied to an inverted pendulum system, which demonstrates the designed controller is effective.

In this paper, we focus to design intelligence numerical computing through artificial neural networks (ANNs) which are backpropagated with Levenberg–Marquard technique (NN-BLMT) for the theoretical approach of physical aspects of heat generation in second-grade fluid (PA-HG-SGF) due to Riga plate. The Riga plate (RP) is known as the physical interaction stimulus, which consists of electrodes and enduring electromagnets that are located on plane surface. The purpose of NN-BLMT is to learn the weight of neural networks on the basis of optimization of the fitness value based on mean-square error between the proposed result and reference numerical solution. The innovation and reliability of NN-BLMT used will be greatly better as compared to traditional numerical techniques that are used to solve commercial and industrial problems. NN-BLMT is fast and easy to apply on nonlinear problems and get best results. The original model PA-HG-SGF in term of PDEs is first converted into system of nonlinear ODEs through suitable transformation and then numerically solved. Through Adam numerical technique (AMT) in Mathematica software, a dataset for PA-HG-SGF is attained for different scenarios of PA-HG-SGF by variation of second-grade parameter, heat-generation parameter, Hartman number, thermal lamination parameter, thermal composure parameter and Prandtl number. Expected solutions are described for PA-HG-SGF through the NN-BLMT testing, training, and validation process. In addition, NN-BLMT relative studies and performance analysis are validated by histogram studies, regression analysis and MSE and then analyzed PA-HG-SGF.