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.

Dynamic stability related to parametric resonance is a very important factor in structural dynamics. For example, instability caused by parametric resonance is believed to be the reason for the famous Tacoma bridge collapse in 1940 [

Control in the coefficients is known to be a very effective method in structures governed by elliptic equations [

The model considered in this work is motivated by recent developments in the area of smart materials [

A structure made with such a material is considered where control consists of eliminating a transient disturbance in the structure by varying the material properties in the response to the deformation. The modal dynamic of a structure is governed by a scalar wave equation, where the control variable is the sound speed in the medium. The basic bilinear optimal control problem becomes the minimization of the energy function of the system in a given period of time with a minimum sound speed. Using modal expansion, the optimal control of the distributed parameter system is reduced to the optimal control of a bilinear time-varying lumped parameter system. The parameterization approach is used to approximate the state-variable and each component of the control variable using finite-term wavelets with unknown coefficients. Therefore, the quadratic problem is transformed into a mathematical programming problem with the objective of minimizing the unknown coefficients to give suboptimal solution of the problem. A necessary condition for the optimality of the unknown coefficients is derived as a system of linear algebraic equations for which the solution is used to obtain the optimal control sound speed and optimal state function.

The bilinear system is a kind of nonlinear system where some related problems such as optimal control are much more difficult to solve than those of linear systems. In literature, many authors [

Let

Let the admissible control set be

The optimal control problem is stated as follows:

We pose the problem at hand as a control problem for an finite system of ordinary differential equations by using modal space expansion. Let

Wavelets have been used by many scientists and engineers to solve several problems in areas such as signal and image processing, control problems, and stochastic problems. Wavelets are mathematical functions that are constructed using dilation and translation of a single function called the mother wavelet denoted by

The CAS wavelets employed in this paper are defined as

The integration of the function

To redefine the wavelet functions over the interval

Using the expansion in (

Using (

Multiplying (

Consider the wave equation

For the sake of illustration, the following parameters were assumed:

Comparison between uncontrolled and controlled performance indices.

Controller | |
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It is observed that the proposed control is effective in significantly reducing the performance index of the problem.

A control for a wave equation where the control is a time dependent coefficient is considered. A modal space technique simplifies the optimal control of a distributed parameter system into the optimal control of a bilinear time-invariant lumped-parameter system. A Galerkin CAS wavelet-based method was developed to solve this bilinear optimal control problem. The main aspect of the proposed approach resides in converting the optimization problem into a mathematical programming problem where the necessary conditions of optimality are derived as a system of algebraic equations . A test example, which includes a variable coefficient and one-dimensional hyperbolic equation, demonstrates the capability of the proposed Galerkin-Wavelet approach for solving optimal control problems governed by bilinear systems. Moreover, the numerical simulations show that the optimal control procedure led to a substantial damping in the bilinear system energy.

This method may be extended to treat a more general setting where the coefficients are