Fractional optimal control of a 2-dimensional distributed system using eigenfunctions

This paper presents an eigenfunctions expansion based scheme for Fractional Optimal Control (FOC) of a 2-dimensional distributed system. The fractional derivative is defined in the Riemann–Liouville sense. The performance index of a FOC problem is considered as a function of both state and control variables, and the dynamic constraints are expressed by a Partial Fractional Differential Equation (PFDE) containing two space parameters and one time parameter. Eigenfunctions are used to eliminate the terms containing space parameters and to define the problem in terms of a set of generalized state and control variables. For numerical computation Grünwald–Letnikov approximation is used. A direct numerical technique is proposed to obtain the state and the control variables. For a linear case, the numerical technique results into a set of algebraic equations which can be solved using a direct or an iterative scheme. The problem is solved for different number of eigenfunctions and time discretization. Numerical results show that only a few eigenfunctions are sufficient to obtain good results, and the solutions converge as the size of the time step is reduced.     

Improving vortex models via optimal control theory

Low-order inviscid point vortex models have demonstrated success in capturing the qualitative behavior of aerodynamic forces resulting from unsteady lifting surface maneuvers. However, the quantitative agreement is often lacking for separated flows as a result of the ambiguity in the edge conditions in this fundamentally unsteady process. In this work, we develop a model reduction framework in which such models can be systematically improved with empirical results. We consider the low-order impulse matching vortex model in which, in its original form, Kutta conditions are applied at both edges to determine the strengths of single point vortices shed from each edge. Here, we relax the Kutta condition imposed at the plate׳s edges and instead seek the time rate of change of the vortex strengths that minimize the discrepancy between the model-predicted and high-fidelity simulation force histories, while the vortex positions adhere to the dynamics of the low-order model. A constrained minimization problem is constructed within an optimal control framework and solved by means of variational principles. The optimization approach is demonstrated on several unsteady wing maneuvers, including pitch-up and impulsive translation at a fixed angle of attack. Additionally, a stitching technique is introduced for extending the time interval over which the model is optimized.     

Local discontinuous Galerkin method for a nonlocal viscous conservation laws

The purpose of this article is to investigate high‐order numerical approximations of scalar conservation laws with nonlocal viscous term. The viscous term is given in the form of convolution in space variable. With the help of the characteristic of viscous term, we design a semidiscrete local discontinuous Galerkin (LDG) method to solve the nonlocal model. We prove stability and convergence of semidiscrete LDG method in L² norm. The theoretical analysis reveals that the present numerical scheme is stable with optimal convergence order for the linear case, and it is stable with sub‐optimal convergence order for nonlinear case. To demonstrate the validity and accuracy of our scheme, we test the Burgers equation with two typical nonlocal fractional viscous terms. The numerical results show the convergence order accuracy in space for both linear and nonlinear cases. Some numerical simulations are provided to show the robustness and effectiveness of the present numerical scheme.     

3-D kinematical conservation laws (KCL): Evolution of a surface in – in particular propagation of a nonlinear wavefront

3-D KCL are equations of evolution of a propagating surface (or a wavefront) Ω_t in 3-space dimensions and were first derived by Giles, Prasad and Ravindran in 1995 assuming the motion of the surface to be isotropic. Here we discuss various properties of these 3-D KCL. These are the most general equations in conservation form, governing the evolution of Ω_t with singularities which we call kinks and which are curves across which the normal n to Ω_t and amplitude w on Ω_t are discontinuous. From KCL we derive a system of six differential equations and show that the KCL system is equivalent to the ray equations of Ω_t. The six independent equations and an energy transport equation (for small amplitude waves in a polytropic gas) involving an amplitude w (which is related to the normal velocity m of Ω_t) form a completely determined system of seven equations. We have determined eigenvalues of the system by a very novel method and find that the system has two distinct nonzero eigenvalues and five zero eigenvalues and the dimension of the eigenspace associated with the multiple eigenvalue 0 is only 4. For an appropriately defined m, the two nonzero eigenvalues are real when m>1 and pure imaginary when m<1. Finally we give some examples of evolution of weakly nonlinear wavefronts.     

The Runge–Kutta control volume discontinuous finite element method for systems of hyperbolic conservation laws

In this paper, a new high‐order and high‐resolution method called the Runge–Kutta control volume discontinuous finite element method (RKCVDFEM) was proposed to solve 1D and 2D systems of hyperbolic conservation laws. Its main advantage lies in the local conservation, and it is simpler than the Runge–Kutta discontinuous Galerkin finite element method (RKDGM). The theoretical analysis showed that the RKCVDFEM has formally an optimal convergence order for 1D systems. Based on logically rectangular grids of irregular quadrilaterals, a scheme for 2D systems was constructed. Some classical problems were simulated and the validity of the method was presented. Copyright © 2010 John Wiley & Sons, Ltd.     

THE CONSERVATION LAW OF NONHOLONOMIC SYSTEM OF SECOND-ORDER NON-CHETAVE’S TYPE IN EVENT SPACE

The conservation law of nonholonomic system of second-order non-Chataev’s type in event space is studied. The Jourdain’s principle in event space is presented. The invariant condition of the Jourdain’s principle under infinitesimal transformation is given by introducing Jourdain’s generators in event space. Then the conservation law of the system in event space is obtained under certain conditions. Finally a calculating example is given.     

An optimal control approach to the analysis of inelastic structures under cyclic loading

Optimal control theory is used to study the asymptotic behaviour of elasto-viscoplastic structures under cyclic loading. With this approach, the asymptotic state is found as the solution of a minimization problem. General properties of this method are established. A simple thermomechanical problem is studied to illustrate and validate this approach. An interest of the proposed method lies in its capacity to handle other nonlinearities than plasticity. To illustrate this point, the approach is extended to the coupled viscoplasticity/frictionless contact problem. Some numerical results are given for an elasto-viscoplastic half-plane under cyclic frictionless indentation.     

Symmetry classification of quasi-linear PDE’s containing arbitrary functions

We consider the problem of performing the preliminary “symmetry classification” of a class of quasi-linear PDE’s containing one or more arbitrary functions: we provide an easy condition involving these functions in order that nontrivial Lie point symmetries be admitted, and a “geometrical” characterization of the relevant system of equations determining these symmetries. Two detailed examples will elucidate the idea and the procedure: the first one concerns a nonlinear Laplace-type equation, the second a generalization of an equation (the Grad–Schlüter–Shafranov equation) which is used in magnetohydrodynamics.     

Attitude control for part actuator failure of agile small satellite

The stability and singularity problem of agile small satellite （ASS） with actuator failure is discussed in this paper. Firstly, the three-axis stabilized controller of an ASS is designed, where micro control moment gyros （MCMG＇s） in pyramid configuration （PC） is used as the actuator. By using the same controller and steering law, the control results before and after one gyro fails are compared by simulation. The variation of singular momentum envelope before and after one gyro fails is also compared. The simulation results show that the failure intensively decreases the capacity of output torque, which leads to the emergence of more singular points and the rapid saturation of MCMG＇s. Finally, the parameters of system controller are changed to compare the control effect.     

Gradient method of optimal control applied to the operation of a dam water gate

An extension of the authors' previous methods is presented for the optimal control of flood propagation via a dam gate, based on a combination of the finite element and gradient methods. It is assumed in previous papers that the control duration is the same as the duration of the flood. However, the duration of the control does not necessarily coincide with that of the flood flow. To overcome this difficulty, the gradient method is applied to solve the free terminal time-fixed terminal condition problem. It is shown that the water elevation can be controlled exactly the same as with the previously presented method. It is also shown that the computation can be terminated at a far shorter time than the terminal time of the flood.     

Singularly perturbed methods in the theory of optimal control of systems governed by partial differential equations

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Optimal control theory and Newton–Euler formalism for Cosserat beam theory

A Newton–Euler formalism is derived for Cosserat beam theory in a purely deductive manner, thanks to an analogy with optimal control theory. The method relies upon joint use of Gauss least constraint principle, Appell's equations and optimal control theory, that was used successfully in a previous work for the classical case of discrete Newton–Euler backward and forward recursions of multibody systems. To cite this article: G. Le Vey, C. R. Mecanique 334 (2006).     

On universal deformations in analysis of Signorini’s nonlinear theory of hyperelastic media

It is shown that uniform compression/tension and simple shear as universal deformations are quite useful in studying Signorini’s nonlinear theory of hyperelastic materials. They make it possible to formulate restrictions for the elastic constants of the theory and to explain the Poynting effect __________ Translated from Prikladnaya Mekhanika, Vol. 43, No. 12, pp. 54–60, December 2007.     

Dynamic Analysis of an Infinite Cylindrical Hole in a Saturated Poroelastic Medium

In this paper, the dynamic response of an infinite cylindrical hole embedded in a porous medium and subjected to an axisymmetric ring load is investigated. Two scalar potentials and two vector potentials are introduced to decouple the governing equations of Biot’s theory. By taking a Fourier transform with respect to time and the axial coordinate, we derive general solutions for the potentials, displacements, stresses and pore pressures in the frequency-wave-number domain. Using the general solutions and a set of boundary conditions applied at the hole surface, the frequency-wave-number domain solutions for the proposed problem are determined. Numerical inversion of the Fourier transform with respect to the axial wave number yields the frequency domain solutions, while a double inverse Fourier transform with respect to frequency as well as the axial wave number generates the time-space domain solution. The numerical results of this paper indicate that the dynamic response of a porous medium surrounding an infinite hole is dependant upon many factors including the parameters of the porous media, the location of receivers, the boundary conditions along the hole surface as well as the load characteristics.     

One-dimensional unsteady inertial flow in phreatic aquifers induced by a sudden change of the boundary head

We are examining the classical problem of unsteady flow in a phreatic semi-infinite aquifer, induced by sudden rise or drawdown of the boundary head, by taking into account the influence of the inertial effects. We demonstrate that for short times the inertial effects are dominant and the equation system describing the flow behavior can be reduced to a single ordinary differential equation. This equation is solved both numerically by the Runge-Kutta method and analytically by the Adomian’s decomposition approach and an adequate polynomial-exponential approximation as well. The influence of the viscous term, occurring for longer times, is also taken into account by solving the full Forchheimer equation by a finite difference approach. It is also demonstrated that as for the Darcian flow, for the case of small fluctuations of the water table, the computation procedure can be simplified by using a linearized form of the mass balance equation. Compact analytical expressions for the computation of the water stored or extracted from an aquifer, including viscous corrections are also developed.     

FLEXURAL WAVE PROPAGATION IN NARROW MINDLIN＇S PLATE

Appling Mindlin＇s theory of thick plates and Hamilton system to propagation of elastic waves under free boundary condition, a solution of the problem was given. Dispersion equations of propagation mode of strip plates were deduced from eigenfunction expansion method. It was compared with the dispersion relation that was gained through solution of thick plate theory proposed by Mindlin. Based on the two kinds of theories, the dispersion curves show great difference in the region of short waves, and the cutoff frequencies are higher in Hamiltonian systems. However, the dispersion curves are almost the same in the region of long waves.     

A new method for solving Biot＇s consolidation of a finite soil layer in the cylindrical coordinate system

A new method is developed to solve Biot＇s consolidation of a finite soil layer in the cylindrical coordinate system. Based on the governing equations of Biot＇s consolidation and the technique of Laplace transform, Fourier expansions and Hankel transform with respect to time t, coordinate θ and coordinate r, respectively, a relationship of displacements, stresses, excess pore water pressure and flux is established between the ground surface （z = 0） and an arbitrary depth z in the Laplace and Hankel transform domain. By referring to proper boundary conditions of the finite soil layer, the solutions for displacements, stresses, excess pore water pressure and flux of any point in the transform domain can be obtained. The actual solutions in the physical domain can be acquired by inverting the Laplace and the Hankel transforms.