An indirect pseudospectral method for the solution of linear-quadratic optimal control problems with infinite horizon

Abstract

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 as state spaces and weighted Lebesgue spaces as control spaces into the problem setting. Then, the problem becomes an optimization problem in Hilbert spaces. We use the weight functions 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.