The optimal control of a distributed parameter system

This paper deals with the optimal control of a perturbed rod which is fixed at one end while the control force is applied at the other end. By using the variational method, a Fredholm integral equation of the second kind, which is a necessary and sufficient condition for the optimal control function, is obtained. A sequence of approximate solutions for the integral equation is derived by using the fact that the kernel can be approximated by a suitable sequence of degenerate kernels. The condition for the convergence of the sequence of the approximate solutions is stated. This method is net limited to the problem discussed in this paper, but can be applied to the solution of integral equations whenever the kernel can be approximated by a sequence of degenerate kernels. The paper concludes with a numerical example.