Weighted functional spaces approach in infinite horizon optimal control problems: A systematic analysis of hidden opportunities and advantages

This paper constitutes the advantages of using the weighted Sobolev and weighted Lebesgue spaces in optimal control problems defined on an infinite time interval. Based on numerous examples, it demonstrates how the introduction of certain weight or density functions into the problem statement may influence the existence of an optimal solution and the optimality of a chosen candidate. The positive effects of the proper relation between the state and the co-state functional spaces for the necessary optimality conditions as well as for the development of numerical schemes are discussed. This paper is based on over one decade of research in this field and summarizes the main findings concerning the use of weight functions in optimal control problems.