# Forecast horizon in dynamic family of one-dimensional control problems

• Publisher: Instytut Matematyczny Polskiej Akademi Nauk(Warszawa), 1991

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## Abstract

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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)), $x\left(0\right)={x}_{0}$,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 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 401991 Mathematics Subject Classification: Primary 90C39; Secondary 93C75, 90B05, 90B30, 90A16, 90A05.

## How to cite

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Ryszarda Rempała. Forecast horizon in dynamic family of one-dimensional control problems. Warszawa: Instytut Matematyczny Polskiej Akademi Nauk, 1991. <http://eudml.org/doc/219352>.

@book{RyszardaRempała1991,
abstract = {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 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 401991 Mathematics Subject Classification: Primary 90C39; Secondary 93C75, 90B05, 90B30, 90A16, 90A05.},
author = {Ryszarda Rempała},
keywords = {forecast horizon; optimal control problem; maximum principle; Bellman's equation; economic models; sufficient conditions for the existence},
language = {eng},
location = {Warszawa},
publisher = {Instytut Matematyczny Polskiej Akademi Nauk},
title = {Forecast horizon in dynamic family of one-dimensional control problems},
url = {http://eudml.org/doc/219352},
year = {1991},
}

TY - BOOK
AU - Ryszarda Rempała
TI - Forecast horizon in dynamic family of one-dimensional control problems
PY - 1991
CY - Warszawa
PB - Instytut Matematyczny Polskiej Akademi Nauk
AB - 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 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 401991 Mathematics Subject Classification: Primary 90C39; Secondary 93C75, 90B05, 90B30, 90A16, 90A05.
LA - eng
KW - forecast horizon; optimal control problem; maximum principle; Bellman's equation; economic models; sufficient conditions for the existence
UR - http://eudml.org/doc/219352
ER -

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