Optimal algorithms for a problem of optimal control

Abstract

This paper deals with optimal algorithms for the approximate solution of a problem of optimal control. The control problem in question is the minimization of a quadratic energy functional, which is equivalent to the solution of a mildly nonlinear two-point second-order elliptic boundary-value problem. The only restriction on the algorithms considered is that they can use only a finite amount of information about the problem element f appearing in the definition of the energy functional. An algorithm having error ϵ is said to be optimal if its cost is minimal among all algorithms that solve the problem to within ϵ. We first suppose that the information available about f consists of a finite set of linear functionals of f, that is, we allow arbitrary linear information. We then show that there is a finite element method (whose degree depends on the smoothness of f) which is an optimal algorithm for the optimal control problem. Note that this finite element method requires the evaluation of the inner products of f with finite element basis functions. These inner products are not usually available in practice; often, only “standard information” is available (meaning that we can evaluate f at a finite set of points). So, we next consider the case where the only information that is available is standard information. We then find that there is a “finite element method with quadrature” which is an optimal algorithm among all algorithms using this standard information. Moreover, we find that standard information is weaker than inner-product information. The asymptotic penalty for using standard information instead of inner-product information is unbounded as ϵ tends to 0.