TWO-GRID WEAK GALERKIN METHOD FOR SEMILINEAR ELLIPTIC DIFFERENTIAL EQUATIONS

In this paper, we investigate a two-grid weak Galerkin method for semilinear elliptic differential equations. The method mainly contains two steps. First, we solve the semi-linear elliptic equation on the coarse mesh with mesh size H, then, we use the coarse mesh solution as a initial guess to linearize the semilinear equation on the fine mesh, i.e., on the fine mesh (with mesh size $h$), we only need to solve a linearized system. Theoretical analysis shows that when the exact solution u has sufficient regularity and $h=H^2$, the two-grid weak Galerkin method achieves the same convergence accuracy as weak Galerkin method. Several examples are given to verify the theoretical results.

Foundations of Engineering Mathematics Applied for Fluid Flows

Based on a brief historical excursion, a list of principles is formulated which substantiates the choice of axioms and methods for studying nature. The axiomatics of fluid flows are based on conservation laws in the frames of engineering mathematics and technical physics. In the theory of fluid flows within the continuous medium model, a key role for the total energy is distinguished. To describe a fluid flow, a system of fundamental equations is chosen, supplemented by the equations of the state for the Gibbs potential and the medium density. The system is supplemented by the physically based initial and boundary conditions and analyzed, taking into account the compatibility condition. The complete solutions constructed describe both the structure and dynamics of non-stationary flows. The classification of structural components, including waves, ligaments, and vortices, is given on the basis of the complete solutions of the linearized system. The results of compatible theoretical and experimental studies are compared for the cases of potential and actual homogeneous and stratified fluid flow past an arbitrarily oriented plate. The importance of studying the transfer and transformation processes of energy components is illustrated by the description of the fine structures of flows formed by a free-falling drop coalescing with a target fluid at rest.

Krasovskii Passivity and μ-Synthesis Controller Design for Quasi-Linear Affine Systems

This paper presents an end-to-end method to design passivity-based controllers (PBC) for a class of input-affine nonlinear systems, named quasi-linear affine. The approach is developed using Krasovskii’s method to design a Lyapunov function for studying the asymptotic stability, and a sufficient condition to construct a storage function is given, along with a supply-rate function. The linear fractional transformation interconnection between the nonlinear system and the Krasovskii PBC (K-PBC) results in a system which manages to follow the provided input trajectory. However, given that the input and output of the closed-loop system do not have the same physical significance, a path planning is mandatory. For the path planning component, we propose a robust controller designed using the μ-synthesis mixed-sensitivity loop-shaping for the linearized system around a desired equilibrium point. As a case study, we present the proposed methodology for DC-DC converters in a unified manner, giving sufficient conditions for such systems to be Krasovskii passive in terms of Linear Matrix Inequalities (LMIs), along with the possibility to compute both the K-PBC and robust controller alike.

Influence of Wind Turbine Design Parameters on Linearized Physics-Based Models in OpenFAST

While most physics involved in wind energy are nonlinear, linearization of the underlying nonlinear wind-system equations is often important for understanding the system response and exploiting well-established methods and tools for analyzing linear systems. Linearized models are important for, e.g., eigenanalysis (to derive structural natural frequencies, damping ratios, and mode shapes) and controls design (based on linear state-space models). In controls co-design (CCD), whose methods often rely on linearized time-domain models of the physics, the physical structure (often called the plant) and controller are designed and optimized concurrently, so, it is important to understand how changes to the physical design affect the linearized system. This work summarizes efforts done to understand the impact of design parameter variations in the physical system (mass, stiffness, geometry, etc.) on the linearized system using OpenFAST.

Long time behaviour and turnpike solutions in mildly non-monotone mean field games

We consider mean field game systems in time-horizon (0,T), where the individual cost depends locally on the density distribution of the agents, and the Hamiltonian is locally uniformly convex. We show that, even if the coupling cost functions are mildly non-monotone, then the system is still well posed due to the effect of individual noise. The rate of anti-monotonicity (the aggregation rate of the cost function) which can be afforded depends on the intensity of the diffusion and on global bounds of solutions. We give applications to either globally Lipschitz Hamiltonians or quadratic Hamiltonians and couplings having mild growth. Under similar conditions, we give a complete description of the ergodic and long time properties of the system. In particular we prove: (i) the turnpike property of solutions in the finite (long) horizon (0,T), (ii) the convergence of the system from (0,T) towards (0,\infty), (iii) the vanishing discount limit of the infinite horizon problem and the long time convergence towards the ergodic stationary solution. We extend previous results which were known only for the case of monotone and smoothing couplings; our approach is self-contained and does not need the use of the linearized system or of the master equation.

Mathematical model of the dynamics of a hydrofoil vessel as a complex technical system

Определена структура единой среды моделирования, состоящая из трёх блоков: блок, где задаются или формируются значения исследуемых параметров, влияющие на выходные показатели судна, как объекта моделирования, блок представляющий собой ядро единой среды моделирования и блок, где формируется совокупность тех или иных показателей, подлежащих анализу. Определена математическая модель динамики возмущенного движения СПК, при этом использованы следующие системы координат: земная прямоугольная горизонтальная правая, связанная с судном прямоугольная правая и вспомогательная нецентральная прямоугольная правая. Определены основные допущения математической модели. Представлены уравнения динамики судна на подводных крыльях в общем виде и определены силы и моменты, действующие на судно на подводных крыльях в крыльевом режиме движения. Гидродинамические силы и моменты, возникающие на каждом из крыльевых устройств, определены расчетным путем. Работа движителей моделируется заданием среднего упора, направленного по оси вала движителя и параллельного диаметральной плоскости судна. В модели динамики предусмотрена возможность задания аэродинамических сил и моментов, действующие на СПК в крыльевом режиме. Разработана математическая модель электрогидравлического привода, состоящая из суммирующего устройства, электрогидроусилителя и силового интегрирующего привода, охваченных общей обратной связью по положению и скорости перемещения, а также модель системы управления движением, которая является одной из важнейших подсистем СПК, формирующей алгоритмы управления, поступающие на входы ЭГП соответствующих ИО, расположенных на несущих поверхностях КУ. При решении некоторых задач, связанных с проектированием СПК и его технических систем, особенно для получения оценочных значений фазовых координат судна на начальных этапах проектирования или решения специальных задач, разработана линеаризованная система дифференциальных уравнений объекта. The structure of a unified modeling environment has been determined, which consists of three blocks: a block where the values of the studied parameters are set or formed, which affect the output indicators of the vessel as an object of modeling, a block that is the core of a unified modeling environment and a block where a set of certain indicators is formed. analysis. A mathematical model of the dynamics of the disturbed motion of the SPK was determined, with the following coordinate systems used: earth rectangular horizontal right, rectangular right connected to the ship and auxiliary off-center rectangular right. The basic assumptions of the mathematical model are determined. The equations of the dynamics of a hydrofoil ship in general form are presented and the forces and moments acting on a hydrofoil ship in the wing mode of motion are determined. The hydrodynamic forces and moments arising on each of the wing devices are determined by calculation. The operation of the propellers is modeled by setting the middle stop directed along the axis of the propeller shaft and parallel to the diametral plane of the vessel. The dynamics model provides for the possibility of setting aerodynamic forces and moments acting on the HFV in the wing mode. A mathematical model of an electrohydraulic drive has been developed, consisting of a summing device, an electrohydraulic amplifier and a power integrating drive, covered by a general feedback on the JJposition and speed of movement, as well as a model of a motion control system, which is one of the most important subsystems of the SPC that forms control algorithms entering the EGP inputs of the corresponding EUT located on the bearing surfaces of the KU. When solving some problems related to the design of the HFV and its technical systems, especially for obtaining the estimated values of the phase coordinates of the vessel at the initial stages of design or solving special problems, a linearized system of differential equations of the object was developed.

On the convexity of the reachable set with respect to a part of coordinates at small time intervals

We investigate the convexity of the reachable sets for some of the coordinates of nonlinear systems with integral constraints on the control at small time intervals. We have proved sufficient convexity conditions in the form of constraints on the asymptotics of the eigenvalues of the Gramian of the controllability of a linearized system for some of the coordinates. There are two nonlinear third-order systems under study as examples. The system linearized along a trajectory generated by zero control is uncontrollable, and the system in the other example is completely controllable. We investigate the sufficient conditions for convexity of projection of reachable sets. Numerical modeling has been carried out, demonstrating the non-convexity of some projections even for small time intervals.

A stability theorem for equilibria of delay differential equations in a critical case with application to a model of cell evolution

In this paper the stability of the zero equilibrium of a system with time delay is studied. The critical case of a multiple zero root of the characteristic equation of the linearized system is treated by applying a Malkin type theorem and using a complete Lyapunov-Krasovskii functional. An application to a model for malaria under treatment considering the action of the immune system is presented.