Lp-Optimal Boundary Control for the Wave Equation

We study problems of boundary controllability with minimal L p-norm (p ∈ [2, ∞]) for the one-dimensional wave equation, where the state is controlled at both boundaries through Dirichlet or Neumann conditions. The problem is to reach a given terminal state and velocity in a given finite time, while minimizing the L p-norm of the controls. We give necessary and sufficient conditions for the solvability of this problem. We show as follows how this infinite-dimensional optimization problem can be transformed into a problem which is much simpler: The feasible set of the transformed problem is described by a finite number of simple pointwise equality constraints for the control function in the Dirichlet case while, in the Neumann case, an additional integral equality constraint appears. We provide explicit complete solutions of the problems for all p ∈ [2, ∞] in the Dirichlet case and solutions for some typical examples in the Neumann case. 1. Introduction. In this paper, we discuss two-sided Dirchlet or Neumann controls for the one-dimensional wave equation for p between 2 and ∞. We consider the problem of exact control; that is, starting from the zero position we want to reach a given terminal state in a given finite time. Our aim is to find control functions with minimal L p-norm that steer the system to the target. For certain typical cases, we present explicit representations of such optimal control functions in terms of the given target state. It is well known that, in the L 2-case, the optimal control functions can be characterized as the L 2-norm minimal solutions of a trigonometric moment problem, which has been analyzed in depth (see [4], [19], [15]). For the L p-case (p > 2) there are only a few publications on the subject and even the question of existence of solutions, which is equivalent to the question of L p-controllability, has not been solved completely. In the present paper, we give a complete analysis of this problem for the boundary control of the one-dimensional wave equation. The problem can be reduced to the case of the minimal time interval, where controllability is possible. This allows an answer to be given to the question of solvability of the problem of L p-controllability in terms of the properties of the target states. We use the control function for the minimal time interval to transform the infinite-dimensional problem into a problem, which is …