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 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.

A pseudospectral method for budget-constrained infinite horizon optimal control problems

“…We have presented the extension of the indirect pseudospectral method for infinite horizon optimal control problems which arises in a natural way from the method developed in [17] through incorporating an additional isoperimetric, or also called budget-constraint, into the discretization scheme applied to the system of necessary optimality conditions in form of a Pontryagin Type maximum Principle, established in [13]. The quality and the convergence rate of the method presented here is comparable with those of the method in [17] and this fact gives us a reason to presume similar quality of convergence for the whole class of budget-constrained control problems, although we do not present any rigorous convergence proof here. Besides the convergence proof, one of the most important focuses of upcoming research will be the generalization of the method to nonlinear infinite horizon optimal control problems.…”

“…We now intend to describe an indirect numerical solution method, i.e. it acts according to the scheme "first optimize, then discretize", and represents an extended version of the indirect pseudospectral method presented in [17], for a class of budget-constrained infinite horizon optimal control problems. For incorporating the method into existing numerical methods compare scheme in Fig.…”

“…In the last decades, developing numerical solution methods for infinite horizon optimal control problems has emerged a lot of attention and a plenty of new results were obtained, cf. [5], [7], [9], [17]. Among them, both direct and indirect solution methods were established.…”

“…Among them, both direct and indirect solution methods were established. In the present paper, we extend the indirect pseudospectral method, introduced in [17], onto the class of budget-constrained infinite horizon optimal control problems. The plausibility of investigating this class of problems can be seen e.g.…”