Applications of Driftless Control Affine Systems to a Problem of Inventory and Production

Abstract

In this paper, the framework of driftless control affine systems has been used for the study of inventory and production problems. A mathematical model for an economic problem was built by using some optimal control techniques. For the controllability issue of the optimal control system, which involves restrictions on the final quantities required, the Lie geometric methods have been applied. In order to find the optimal solution, the Pontryagin Maximum Principle has been used at the level of a Lie algebroid, which in this case consists in an integrable distribution of the tangent space. Finally, a numerical example is presented.

Lie Geometric Methods in the Study of Driftless Control Affine Systems with Holonomic Distribution and Economic Applications

“…In addition, ẋ3 = u 1 x 2 + u 2 x 1 ≥ 0, and using x 3 (0) = 0, we obtain x 3 (t) ≥ 0. Some different mathematical models can be found in the papers [10,[12][13][14][15]. It results that the system ( 17) is a driftless control affine system on R 3 + , written in the following form:…”

“…In addition, a new approach to maximizing the profit in a stockdependent demand inventory model is presented in [9]. Otherwise, there are a multitude of papers that study the optimization of production and storage costs with various restrictions (see form instance, [10][11][12][13][14][15]).…”

“…In addition, some facts and methods of control theory treated from the geometric point of view are presented in [29]. The geometry of Lie algebroids is used in the study of distributional systems (driftless control afine system) in the papers [12,14,[30][31][32][33][34][35][36].…”

Applications of Optimal Control to Production Planning

“…Agrachev and Sachkov [1] presented some methods of the mathematical control theory treated from the geometric point of view. LaValle [17] gave an unified treatment of control theory including control affine systems and Popescu [22] used the framework of Lie algebroids in the study of driftless control affine systems. One of the motivations for this work is the study of Lagrangian systems with some external constraints.…”

Optimal Control Applications in the Study of Production Management

“…Ortega & Lin [19] gave a review on control theory applications to the production and inventory problems. Popescu [21] used the framework of a driftless control affine system in the study of inventory and production problems. The present paper is organized as follows.…”