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.

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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.

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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.

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In this paper, we deal with infinite horizon optimal control problems involving affine-linear dynamics and prove the existence of optimal solutions. The innovation of this paper lies in the special setting of the problem, precisely in the choice of weighted Sobolev and weighted Lebesgue spaces as the state and control spaces, respectively, which turns out to be meaningful for various problems. We apply the generalized Weierstraß theorem to prove the existence result. A lower semicontinuity theorem which is needed for that is shown under weakened assumptions.