Coupled Galerkin and parameterization methods for optimal control of discretely connected parallel beams

Galerkin and wavelet methods for optimal boundary control of a couple of discretely connected parallel beams are proposed. First, the problem with boundary controls is converted into a problem with distributed controls. The problem is, then, reduced by a Galerkin-based approach into determining the optimal control of a linear time-invariant lumped parameter system, which will be solved by a wavelet-based method using Legendre wavelets. The integration-operational matrix and Kronecker product are utilized to significantly simplify the optimization problem into a system of linear equations. A numerical example is presented to demonstrate the applicability and the efficiency of the proposed method.     

Transport equation on semidiscrete domains and Poisson–Bernoulli processes

In this paper, we consider a scalar transport equation with constant coefficients on domains with discrete space and continuous, discrete or general time. We show that on all these underlying domains, solutions of the transport equation can conserve sign and integrals both in time and space. Detailed analysis reveals that, under some initial conditions, the solutions correspond to counting stochastic processes and related probability distributions. Consequently, the transport equation could generate various modifications of these processes and distributions and provide some insights into corresponding convergence questions. Possible applications are suggested and discussed.     

Solvability of the clamped Euler–Bernoulli beam equation

In this study, solvability of the initial boundary value problem for general form Euler–Bernoulli beam equation which includes also moving point-loads is investigated. The complete proof of an existence and uniqueness properties of the weak solution of the considered equation with Dirichlet type boundary conditions is derived. The method used here is based on Galerkin approximation which is the main tool for the weak solution theory of linear evolution equations as well as in derivation of a priori estimate for the approximate solutions. All steps of the proposed technique are explained in detail.     

A collocation method based on Bernoulli operational matrix for numerical solution of generalized pantograph equation

This paper presents a direct solution technique for solving the generalized pantograph equation with variable coefficients subject to initial conditions, using a collocation method based on Bernoulli operational matrix of derivatives. Only small dimension of Bernoulli operational matrix is needed to obtain a satisfactory result. Numerical results with comparisons are given to confirm the reliability of the proposed method for generalized pantograph equations.     

Aeroelastic analysis of cantilever plates using Peters’ aerodynamic model,and the influence of choosing beam or plate theories as the structural model

In this paper, the aeroelastic analyses of a rectangular cantilever plate of varying aspect ratio is presented. The classical plate theory has been selected as the structural model. The main point that distinguishes this study from previously reported research is employing Peters’ theory to model aerodynamic effect which is not straightforward. The Peters’ aerodynamic model was originally developed to provide lift and moment, which is only applicable to the structural model based on the beam theories. In this study, using the basic concept of the Peters’ aerodynamic model in addition to utilizing the Fourier series, the pressure distribution is derived, which makes Peters’ model applicable to structural models based on plate theory. This combination provides a much simpler state–space aeroelastic model for plates in comparison to the prevalent panel methods, which could lead to a significant reduction in computational time. In addition, the aeroelastic response of the plate with respect to changes in the structural model from the beam theory to the plate theory is evaluated. By using data from an experiment carried out at Duke University, the theoretical results are evaluated. Furthermore, the differences in structural models obtained from the plate and beam theories can be divided into two distinct parts, which are responsible for differences in bending and torsional behaviors of the structure, separately. This approach enables us to measure the effects of differences of each behavior separately, which could provide with a new insight into the problem. It has been determined that the flutter speeds obtained from the beam and plate aeroelastic models are little affected by the difference in bending behavior, but rather is mainly caused by the difference in torsional frequencies.     

Stabilization of a viscoelastic rotating Euler‐Bernoulli beam

In this paper, we consider a rotating Euler‐Bernoulli beam. The beam is made of a viscoelastic material, and it is subject to undesirable vibrations. Under a suitable control torque applied at the motor, we prove the arbitrary stabilization of the system for a large class of relaxation functions by using the multiplier method and some ideas introduced by Tatar (J. Math. Phys. 52:013502, 2011).     

Criteria for -continuity

Bernoullicity is the strongest mixing property that a measure-theoretic dynamical system can have. This is known to be intimately connected to the so-called metric on processes, introduced by Ornstein. In this paper, we consider families of measures arising in a number of contexts and give conditions under which the measures depend -continuously on the parameters. At points where there is -continuity, it is often straightforward to establish that the measures have the Bernoulli property.