Forecast horizon in dynamic family of one-dimensional control problems

Ryszarda Rempała

GDML_Books, (1991), p.

Résumé

The forecast horizon is defined as a property of a class of functions. Some general existence conditions are derived. The results are applied to the process x(·) described by the differential equationẋ(t) = e(t,u(t)) - f(t,x(t)), $x(0)=x_{0}$,where e, f are nonnegative and increasing in the second variable, and u(·) denotes a control variable.A cost functional is associated with the process and the control. The cost is characterized by three functions: g(t,u), h(t,x), k(x), and a time interval. A class of functions (e, f, g, h, k) for which the forecast horizons can be explicitly obtained is described. Some applications to economic problems are given.In the second part of the paper a discrete-time, stochastic, linear control problem is considered. The problem is described by means of a sequence of Markov transition functions and two deterministic sequences. For some classes of sequences the forecast horizons are explicitly obtained. Optimal solutions are determined. An economic application of the problem is given.

CONTENTS1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52. Sufficient conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .73. Definitions and hypotheses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94. Properties of arcs and trajectories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .105. Dynamic family of optimal controls . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116. The maximum principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .157. Horizon theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .168. Remarks to horizon theorems. An economic application . . . . . . . . . . . . . . . . . . . . 279. Discrete-time linear systems with stochastic parameters. Horizon theorems . . . . . 2910. Proof of horizon theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3211. Final remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .38  References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

1991 Mathematics Subject Classification: Primary 90C39; Secondary 93C75, 90B05, 90B30, 90A16, 90A05.

Détail

Zbl 0754.90063

EUDML-ID : urn:eudml:doc:219352