Abstract
In the present paper, two optimal control problems are studied using Lie geometric methods and applying the Pontryagin Maximum Principle at the level of a new working space, called Lie algebroid. It is proved that the framework of a Lie algebroid is more suitable than the cotangent bundle in order to find the optimal solutions of some driftless control affine systems with holonomic distributions. Finally, an economic application is given.
Highlights
In the last decades, Lie geometric methods have been applied successfully in different domains of research such as dynamical systems or optimal control theory
In the book [4], the notions of deterministic optimal control systems governed by ordinary differential equations are studied
We solve two optimal control problems and prove that the framework of a Lie algebroid is more suitable than the cotangent space in the study of some driftless control affine systems with holonomic distributions
Summary
Lie geometric methods have been applied successfully in different domains of research such as dynamical systems or optimal control theory. We solve two optimal control problems and prove that the framework of a Lie algebroid is more suitable than the cotangent space in the study of some driftless control affine systems with holonomic distributions. The known results about Lie geometric methods in optimal control theory for control affine systems are presented, including the controllability issues in the case of holonomic and nonholonomic distributions. The fourth section deals with the study of an economic problem of inventory and production using the mathematical model of optimal control and Lie geometric methods for controllability issues. The optimal solution is obtained using the Pontryagin Maximum Principle on a Lie algebroid This approach simplifies the study and shows the connection between the geometry of Lie algebroids and optimal control for distributional systems
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