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In this paper a class of linear-quadratic infinite horizon optimal control problems is considered. Problems of this type are not only of practical interest. They also appear as an approximation of nonlinear problems. The key idea is to introduce weighted Sobolev spaces as state space and weighted Lebesgue spaces as control spaces into the problem setting. We investigate the question of existence of an optimal solution in these spaces and establish a Pontryagin type Maximum Principle as a necessary optimality condition including transversality conditions.

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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.