Stabilization of a class of underactuated Euler Lagrange system using an approximate model

The energy shaping method, Controlled Lagrangian, is a well-known approach to stabilize the underactuated Euler Lagrange (EL) systems. In this approach, to construct a control rule, some nonlinear and nonhomogeneous partial differential equations (PDEs), which are called matching conditions, must be solved. In this paper, a method is proposed to obtain an approximate solution of these matching conditions for a class of underactuated EL systems. To develop this method, the potential energy matching condition is transformed to a set of linear PDEs using an approximation of inertia matrices. Hence, the assignable potential energy function and the controlled inertia matrix both are constructed as a common solution of these PDEs. Subsequently, the gyroscopic and dissipative forces are determined as the solution for kinetic energy matching condition. Conclusively, the control rule is constructed by adding energy shaping rule and additional dissipation injection to provide asymptotic stability. The stability analysis of the closed-loop system which used the control rule derived with the proposed method is also provided. To demonstrate the success of the proposed method, the stability problem of the inverted pendulum on a cart is considered.

Vectorial variational problems in L ∞ constrained by the Navier–Stokes equations*

We study a minimisation problem in L p and L ∞ for certain cost functionals, where the class of admissible mappings is constrained by the Navier–Stokes equations. Problems of this type are motivated by variational data assimilation for atmospheric flows arising in weather forecasting. Herein we establish the existence of PDE-constrained minimisers for all p, and also that L p minimisers converge to L ∞ minimisers as p → ∞. We further show that L p minimisers solve an Euler–Lagrange system. Finally, all special L ∞ minimisers constructed via approximation by L p minimisers are shown to solve a divergence PDE system involving measure coefficients, which is a divergence-form counterpart of the corresponding non-divergence Aronsson–Euler system.

Quantum Control for Trapped Particles at Matter Surface

In this theoretic work, trapped particles at matter surface to be considered as target system of quantum control. At the framework of variational method in Hilbert space, it would be quite interesting for us to explore particles which is trapped via optical lattice or other kinds of constraints at a matter surface (metal, crystal). The aim of this task is to survey theoretical control for quantum particles as they are appeared and trapped at matter surface (cf. [1]). The physical background of this work is laying on the specified particles motion or reaction under a certain chemical surface. As is well known, one can move a particle at surface smoothly through a point force above it or according to a proper angle, such quantum mechanical motion had already been achieved by the IBM team several years ago. At the viewpoint of quantum control, what is theoretic support? Can we make these control theoretically, computationally or experimentally? In fact, free trapped particles had been considered by scientists and researchers at worldwide scale. The most exciting things in this study is to take particle as target as it constrained on a surface. Theoretically, this work is to describe quantum control system consisting of time-varying Schrodinger equation at physical constraints condition. Then to apply control theory to quantum system of trapped particles, find and characterize optimal quantum control. Further, to compose optimality system (Euler-Lagrange system). Comprising of control free trapped particle, this work is focusing on control taking place at matter surface (on it particle is trapped), that is, try to discuss the external force constrain (e.g. optical lattice) and surface constrain are acting at particle together. Amazing result is desired in control of different multi-forces as control inputs, what would be happened as a particle changing its position, displacement or status under trapped situation? can we make a trapped chemical quantum well, or a physical optical lattice which worked using external force? what is extension of such kind of works at a variety of fields? whether the general quantum control is worked in this case? It is the purpose to solve these mysteries in this work, and report the initial conclusion of theoretic aspect for trapped particle at matter surface.

Quantum Control for Trapped Particles at Matter Surface

In this theoretic work, trapped particles at matter surface to be considered as target system of quantum control. At the framework of variational method in Hilbert space, it would be quite interesting for us to explore particles which is trapped via optical lattice or other kinds of constraints at a matter surface (metal, crystal). The aim of this task is to survey theoretical control for quantum particles as they are appeared and trapped at matter surface (cf. [1]). The physical background of this work is laying on the specified particles motion or reaction under a certain chemical surface. As is well known, one can move a particle at surface smoothly through a point force above it or according to a proper angle, such quantum mechanical motion had already been achieved by the IBM team several years ago. At the viewpoint of quantum control, what is theoretic support? Can we make these control theoretically, computationally or experimentally? In fact, free trapped particles had been considered by scientists and researchers at worldwide scale. The most exciting things in this study is to take particle as target as it constrained on a surface. Theoretically, this work is to describe quantum control system consisting of time-varying Schrodinger equation at physical constraints condition. Then to apply control theory to quantum system of trapped particles, find and characterize optimal quantum control. Further, to compose optimality system (Euler-Lagrange system). Comprising of control free trapped particle, this work is focusing on control taking place at matter surface (on it particle is trapped), that is, try to discuss the external force constrain (e.g. optical lattice) and surface constrain are acting at particle together. Amazing result is desired in control of different multi-forces as control inputs, what would be happened as a particle changing its position, displacement or status under trapped situation? can we make a trapped chemical quantum well, or a physical optical lattice which worked using external force? what is extension of such kind of works at a variety of fields? whether the general quantum control is worked in this case? It is the purpose to solve these mysteries in this work, and report the initial conclusion of theoretic aspect for trapped particle at matter surface.

Analytical and experimental nonzero-sum differential game-based control of a 7-DOF robotic manipulator

We formulate a Nash-based feedback control law for an Euler–Lagrange system to yield a solution to noncooperative differential game. The robot manipulators are broadly used in industrial units on the account of their reliable, fast, and precise motions, while consuming a significant lumped amount of energy. Therefore, an optimal control strategy needs to be implemented in addressing efficiency issues, while delivering accuracy obligation. As a case study, we here focus on a 7-DOF robot manipulator through formulating a two-player feedback nonzero-sum differential game. First, coupled Euler–Lagrangian dynamic equations of the manipulator are briefly presented. Then, we formulate the feedback Nash equilibrium solution to achieve perfect trajectory tracking. Finally, the performance of the Nash-based feedback controller is analytically and experimentally examined. Simulation and experimental results reveal that the control law yields almost perfect tracking and achieves closed-loop stability.

Quantitative analysis of a system of integral equations with weight on the upper half space

<p style='text-indent:20px;'>In this paper we analyzed the integrability and asymptotic behavior of the positive solutions to the Euler-Lagrange system associated with a class of weighted Hardy-Littlewood-Sobolev inequality on the upper half space <inline-formula><tex-math id="M1">\begin{document}$ \mathbb{R}_+^n. $\end{document}</tex-math></inline-formula> We first obtained the optimal integrability for the solutions by the regularity lifting theorem. And then, with this integrability, we investigated the growth rate of the solutions around the origin and the decay rate near infinity.</p>