The boundary control of the wave equation

This thesis is devoted to the solution of optimal control problems governed by linear and nonlinear wave equations and the estimation of the errors in approximating these solutions. First, the boundary control of a linear wave equation with an integral performance criterion and fixed final states is considered. This problem is modified into the one consisting of the minimization of a linear functional over a set of positive Radon measures, the optimal measure is then approximated by a finite combination of atomic measures and so the problem is converted to a finite-dimensional linear programming problem. The solution of this problem is used to construct a piecewise-constant control. In estimating the integral performance criterion and fixed final states from the mentioned finite-dimensional linear program, some errors occur. We have established some general results concerning these errors, and estimate them in term of the number of linear constraints appeared in the finite-dimensional linear program. Finally, the existence and numerical estimation of the distributed control of a nonlinear wave equation with an integral performance criterion and fixed final states is considered. Again by means of the well-known process of embedding, the problem is replaced by another one in which the minimum of a linear form is sought over a subset of pairs of positive Radon measures defined by linear equalities. The minimization in the new problem is global, and it can be approximated by the solution of a finite-dimensional linear program. However, the final states in this case are only reached asymptotically, that is, as the number of constraints being considered tends to infinity.