A Newton method for the computation of time-optimal boundary controls of one-dimensional vibrating systems

We consider the problem of time-optimal boundary control of a one-dimensional vibrating system subject to a control constraint that prescribes an upper bound for the L2-norm of the image of the control function under a Volterra operator. For the solution of this problem, we propose to use Newtonʹs method to compute the zero of the optimal value function of certain parametric auxiliary problems, where the steering time is the parameter. The formulation of the auxiliary problems, which are problems of norm-minimal control, is based on the method of moments. For a fixed parameter, these problems have a simple structure. We present convergence results with respect to the discretization parameters, where the discretization is done by truncating the system of moment equations. We prove that the optimal value function of the discretized parametric auxiliary problem is differentiable and show how the derivative can be computed, so that Newtonʹs method can be used. We present numerical examples for the problem of time-optimal control of the rotation of an Euler–Bernoulli beam that illustrate the fast convergence of the algorithm with respect to the time-parameter.