Higher-order numerical scheme for linear quadratic problems with bang–bang controls

This paper considers a linear-quadratic optimal control problem where the control function appears linearly and takes values in a hypercube. It is assumed that the optimal controls are of purely bang-bang type and that the switching function, associated with the problem, exhibits a suitable growth around its zeros. The authors introduce a scheme for the discretization of the problem that doubles the rate of convergence of the Euler's scheme. The proof of the accuracy estimate employs some recently obtained results concerning the stability of the optimal solutions with respect to disturbances

On some open problems in optimal control

Several open problems are presented concerning regularity properties of solutions of optimal control problems with constraints

Rationally Risking Addiction: A Two-Stage Approach

We extend the Becker-Murphy rational addiction model to account for a period before the onset of addiction. While during the first stage of recreational consumption of the addictive good does not imply negative effects, the second stage is analogous to the classical Becker-Murphy model. In line with neurological research, the onset of addiction is a random event positively related to the past consumption of the addictive good. The resulting multistage optimal control model with random switching time is analyzed by way of a transformation into an age-structured deterministic optimal control model. This enables us to analyze in detail the anticipation of the second stage, including the possible emergence of a Skiba point. A numerical example demonstrates that it is optimal to stop consuming the addictive good in case of an early onset (i.e. at a low level of cumulative consumption) of addiction. A late onset tends to lead into long-run addiction

Optimal Cyclic Exploitation of Renewable Resources

The paper contributes to the topic of optimal utilization of spatially distributed renewable resource. Namely, a problem of "sustainable" optimal cyclic exploitation of a renewable resource with logistic law of recovery is investigated. The resource is distributed on a circle and is collected by a single harvester moving along the circle. The recovering and harvesting rates are position dependent, and the latter depends also on the velocity of the harvester, which is considered as a control. The existnce of an optimal solution is proved, as well as necessary optimality conditions for the velocity of the harvester. On this base, a numerical approach is proposed, and some qualitative properties of the optimal solutions are established. The results are illustrated by numerical examples, which reveal some economically meaningful features of the optimal harvesting