We provide an example of a convex infinite horizon problem with a linear objective functional where the different interpretations of the improper integral ∞ 0 f (t, x(t), u(t)) dt in either Lebesgue or Riemann sense lead to different but finite optimal values.
In this paper we treat a resource allocation model defined on an infinite interval. We show that the solution of the corresponding problem with finite horizon cannot be extended to a solution of the infinite horizon problem, since the resource allocation problem in the unmodified setting does not have a solution on an unbounded interval. To change this situation we bring an additional state constraint into the model which contains a weight function. The new problem, called now the adapted resource allocation problem, has an optimal solution which has been identified by means of the duality concept of Klötzler.
We consider a class of linear-quadratic infinite horizon optimal control problems in Lagrange form involving the Lebesgue integral in the objective. The key idea is to introduce weighted Sobolev spaces W 1 2 (IR + , μ) as state spaces and weighted Lebesgue spaces L 2 (IR + , μ) as control spaces into the problem setting. Then, the problem becomes an optimization problem in Hilbert spaces. We use the weight functions μ(t) = e ρt , ρ = 0 in our consideration. This problem setting gives us the possibility to extend the admissible set and simultaneously to be sure that the adjoint variable belongs to a Hilbert space too. For the class of problems proposed, existence results as well as a Pontryagin-type Maximum Principle, as necessary and sufficient optimality condition, can be shown. Based on this principle we develop a Galerkin method, coupled with the Gauss-Laguerre quadrature formulas as discretization scheme, to solve the problem numerically. Results are presented for the introduced model and different weight functions.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.