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Variational principles are derived for multiwalled carbon nanotubes undergoing linear vibrations using the semi-inverse method with the governing equations based on nonlocal Timoshenko beam theory which takes small scale effects and shear deformation into account. Physical models based on the nonlocal theory approximate the nanoscale phenomenon more accurately than the local theories by taking small scale phenomenon into account. Variational formulation is used to derive the natural and geometric boundary conditions which give a set of coupled boundary conditions in the case of free boundaries which become uncoupled in the case of the local theory. Hamilton's principle applicable to this case is also given. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

Abstract: 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. [Copyright &y& Elsevier]

Abstract: Pointwise control of the viscous Burgers equation in one spatial dimension is studied with the objective of minimizing the distance between the final state function and target profile along with the energy of the control. An efficient computational method is proposed for solving such problems, which is based on special orthonormal functions that satisfy the associated boundary conditions. Employing these orthonormal functions as a basis of a modal expansion method, the solution space is limited to the smallest lower subspace that is sufficient to describe the original problem. Consequently, the Burgers equation is reduced to a set of a minimal number of ordinary nonlinear differential equations. Thus, by the modal expansion method, the optimal control of a distributed parameter system described by the Burgers equation is converted to the optimal control of lumped parameter dynamical systems in finite dimension. The time-variant control is approximated by a finite term of the Fourier series whose unknown coefficients and frequencies giving an optimal solution are sought, thereby converting the optimal control problem into a mathematical programming problem. The solution space obtained is based on control parameterization by using the Runge–Kutta method. The efficiency of the proposed method is examined using a numerical example for various target functions. [Copyright &y& Elsevier]