A Newton method for the computation of time-optimal boundary controls of one-dimensional vibrating systems
The problem of time-optimal boundary control for a one-dimensional vibrating system is considered. Namely, given a system described by \[ y_{tt}(x,t) + Ly(x,t) = Bu(t), \] with initial conditions: \(y(x,0) = y_0\), \(y_t(x,0) = y_1, \;x {\i}(0,1)\), one looks for \(u\) which in the shortest time \(T\) steers the system to the rest: \(y(x,T) = 0, \;y_t(x,T) = 0, \;x \in (0,1)\), and satisfies some inequality constraints and additional moment equations. To solve the problem the author introduces a parametric auxiliary problem with fixed controlling time, where the function from the inequality constraint is taken as the objective function and the moment equations are taken as constraint functions. It appears that the smallest root of the optimal value function for this problem is the minimal controlling time and it can be approximated by zeros of the optimal value functions of some discretized auxiliary problems. Thus, after proving that the latter are differentiable the author uses the Newton method to compute the zeros of these optimal value functions getting the approximations of the minimal controlling time and the time-optimal control as well. Some numerical examples for the time-optimal control of the rotating Euler-Bernoulli beam are also presented.
scientific article; zbMATH DE number 1419812
scientific article