Marwan Abukhaled

A computational method for solving optimal control of a system of parallel beams using Legendre wavelets

The optimal control of transverse vibration of two Euler-Bernoulli beams coupled in parallel by discrete springs is considered. An index of performance is formulated which consists of a modified energy functional of two coupled structures at a specified time and penalty functions involving the point control forces. The minimization of the performance index over these forces is subject to the equation of motion governing the structural vibrations, the imposed initial condition as well as the boundary conditions. By use of the modal space technique, the optimal control of distributed parameter systems is simplified into the optimal control of a linear time-invariant lumped-parameter systems. A computationally attractive method based on Legendre wavelets in time domain for solving the optimal control of the lumped parameter systems for any finite interval is proposed. Legendre wavelet integral operational matrix and the properties of a Kronecker product are used to find the approximated optimal trajectory and optimal law of the linear systems with respect to a quadratic cost function by only solving a linear system of algebraic equations. This method provides a straightforward and convenient approach for digital computation. A numerical example is provided to demonstrate the applicability and effectiveness of the proposed method.

Optimal Control Systems by Time-Dependent Coefficients Using CAS Wavelets

This paper considers the problem of controlling the solution of an initial boundary-value problem for a wave equation with time-dependent sound speed. The control problem is to determine the optimal sound speed function which damps the vibration of the system by minimizing a given energy performance measure. The minimization of the energy performance measure over sound speed is subjected to the equation of motion of the system with imposed initial and boundary conditions. Using the modal space technique, the optimal control of distributed parameter systems is simplified into the optimal control of bilinear time-invariant lumped-parameter systems. A wavelet-based method for evaluating the modal optimal control and trajectory of the bilinear system is proposed. The method employs finite CAS wavelets to approximate modal control and state variables. Numerical examples are presented to demonstrate the effectiveness of the method in reducing the energy of the system.

Optimal pointwise control for a parallel system of Euler-Bernoulli beams

Abstract The optimal control of a distributed system consisting of two Euler–Bernoulli beams coupled in parallel with pointwise controllers is considered. An index of performance is formulated which consists of a modified energy functional of two coupled structures at a specified time and penalty functions involving the point control forces. The minimization of the performance index over these forces is subject to the equation of motion governing the structural vibrations, the imposed initial condition as well as the boundary conditions. A maximum principle is derived for optimal point controls of one-dimensional coupled structures undergoing transverse vibrations. The optimal control law is obtained using a maximum principle and the applicability of the results is demonstrated. A method of solution for such a type of structure is suggested by using the eigenfunction expansion and the maximum principle. The solution involves reducing the original problem to a system of ordinary differential equations. The effectiveness of this approach is illustrated numerically by comparing the behavior of the controlled and uncontrolled problem.

Variational Iteration Method for Nonlinear Singular Two-Point Boundary Value Problems Arising in Human Physiology

The variational iteration method is applied to solve a class of nonlinear singular boundary value problems that arise in physiology. The process of the method, which produces solutions in terms of convergent series, is explained. The Lagrange multipliers needed to construct the correctional functional are found in terms of the exponential integral and Whittaker functions. The method easily overcomes the obstacle of singularities. Examples will be presented to test the method and compare it to other existing methods in order to confirm fast convergence and significant accuracy.

Wavelets approach for the optimal control of vibrating plates by piezoelectric patches

The optimal control for vibration suppression of a plate by distributed piezoelectric actuators is considered. A performance index in the form of a weighted quadratic functional of the dynamic response of a rectangular simply supported plate will be minimized within a prescribed time duration using piezoelectric patches (voltages). The minimization of the performance index over these voltages is subject to the equation of motion governing the plates structural vibration and a set of initial and boundary conditions. The solution method is a combination of modal space expansion and direct state parameterization. Modal space expansion will transform the optimal control of a distributed parameter system into the optimal control of a lumped parameter system. Using Legendre wavelets, the quadratic optimization problem is transformed into a mathematical programming problem, where the objective is to minimize a set of unknown coefficients to obtain the optimal trajectory and the optimal control. Numerical examples will be provided to illustrate the effectiveness and the efficiency of the proposed method.

Wavelets approach for optimal boundary control of cellular uptake in tissue engineering

The control of the uptake of growth factors in tissue engineering is mathematically modelled by a partial differential equation subject to boundary and initial conditions. The main objective is to regulate cellular processes for the growth or regeneration of a tissue within an assigned terminal time. The techniques of basis function expansion and direct state parameterization were employed to yield efficient computational methods for this problem. Using Legendre and Chebyshev wavelets, the optimal control of the lumped-parameter system was transformed into a system of algebraic equations. The computational efficiency and effectiveness of the proposed method are illustrated by numerical examples.

Testing pulse density distribution for terrestrial gamma ray flashes

Maximum likelihood fits for the time profiles of 51 terrestrial gamma ray flashes (from Compton Gamma Ray Observatory/Burst and Transient Source Experiment and Fermi Gamma-Ray Space Telescope/Gamma-Ray Burst Monitor) were calculated for five proposed probability densities. A lognormal distribution, which had been used by other researchers, was compared with piecewise Gaussian, piecewise exponential, inverse Gaussian, and Ornstein-Uhlenbeck probability density functions. The piecewise Gaussian and piecewise exponential distributions are justified physically through assuming exponential growth and decay of the electron avalanches which result in the gamma ray bursts and are therefore highly relevant in this context. However, identifying the electron avalanche phenomenon as a form of stochastic exit time process, the inverse Gaussian and Ornstein-Uhlenbeck are reasonable alternatives. Results of the maximum likelihood calculations indicate that the five probability densities fit the gamma ray pulse data equally well. By this comparison, our aim is to indicate to terrestrial gamma ray flash researchers these other at least equally valid distribution functions, which may give insights into the physical processes that the electrons (and the positrons) undergo in the gamma ray flashes.

CAS Wavelets Approach for Optimal Control of Cellular Uptake in Tissue Engineering

The control of the uptake of growth factors in tissue engineering is mathematically modeled via a partial differential equation subject to boundary and initial conditions. The main objective is to regulate cellular processes for the growth or regineration of a tissue within an assigned terminal time. CAS wavelets approach is proposed to solve the optimal control problem.

A Galerkin-Parameterization Method for the Optimal Control of Smart Microbeams

A proposed computational method is applied to damp out the excess vibrations in smart microbeams, where the control action is implemented using piezoceramic actuators. From a mathematical point of view, we wish to determine the optimal boundary actuators that minimize a given energy-based performance measure. The minimization of the performance measure over the actuators is subjected to the full motion of the structural vibrations of the micro-beams. A direct state-control parametrization approach is proposed where the shifted Legendre polynomials are employed to solve the optimization problem. Legendre operational matrix and the properties of Kronecker product are utilized to find the approximated optimal trajectory and optimal control law of the lumped parameter systems with respect to the quadratic cost function by solving linear algebraic equations. Numerical examples are provided to demonstrate the applicability and efficiency of the proposed approach.