Relaxation and linear programs in a hybrid control model

Héctor Jasso-Fuentes ¹   ; Jose-Luis Menaldi ²

¹ Departamento de Matemáticas CINVESTAV-IPN A. Postal 14-740 Ciudad de México, 07000, México
² Department of Mathematics Wayne State University Detroit, MI 48202, U.S.A.

Applicationes Mathematicae, Tome 46 (2019) no. 2, pp. 191-227

Résumé

Some optimality results for hybrid control problems are presented. The hybrid model under study consists of two subdynamics, one of a standard type governed by an ordinary differential equation, and the other of a special type having a discrete evolution. We focus on the case when the interaction between the subdynamics takes place only when the state of the system reaches a given fixed region of the state space. The controller is able to apply two controls, each applied to one of the two subdynamics, whereas the state follows a composite evolution, of continuous type and discrete type. By the relaxation technique, we prove the existence of a pair of controls that minimizes an incurred (discounted) cost. We conclude the analysis by introducing an auxiliary infinite-dimensional linear program to show the equivalence between the initial control problem and its associated relaxed counterpart.

Détail

DOI : 10.4064/am2387-6-2019

Keywords: optimality results hybrid control problems presented hybrid model under study consists subdynamics standard type governed ordinary differential equation other special type having discrete evolution focus interaction between subdynamics takes place only state system reaches given fixed region state space controller able apply controls each applied subdynamics whereas state follows composite evolution continuous type discrete type relaxation technique prove existence pair controls minimizes incurred discounted cost conclude analysis introducing auxiliary infinite dimensional linear program equivalence between initial control problem its associated relaxed counterpart