Impulsive control systems without commutativity assumptions

  • A. BressanJr.
  • F. Rampazzo
Contributed Papers

DOI: 10.1007/BF02193094

Cite this article as:
Bressan, A. & Rampazzo, F. J Optim Theory Appl (1994) 81: 435. doi:10.1007/BF02193094


This paper is concerned with optimal control problems for an impulsive system of the form
$$\dot x(t) = f(t, x, u) + \sum\limits_{i = 1}^m {g_i } (t, x, u)\dot u_i ,u(t) \in U,$$
where the measurable controlu(·) is possibly discontinuous, so that the trajectories of the system must be interpreted in a generalized sense. We study in particular the case where the vector fieldsgi do not commute. By integrating the distribution generated by all the iterated Lie brackets of the vector fieldsgi, we first construct a local factorizationA1×A2 of the state space. If (x1,x2) are coordinates onA1×A2, we derive from (1) a quotient control system for the single state variablex1, withu, x2 both playing the role of controls. A density result is proved, which clarifies the relationship between the original system (1) and the quotient system. Since the quotient system turns out to be commutative, previous results valid for commutative systems can be applied, yielding existence and necessary conditions for optimal trajectories. In the final sections, two examples of impulsive systems and an application to a mechanical problem are given.

Key Words

Lie brackets local integral manifolds optimal impulsive controls 

Copyright information

© Plenum Publishing Corporation 1994

Authors and Affiliations

  • A. BressanJr.
    • 1
  • F. Rampazzo
    • 2
  1. 1.Scuola Internazionale Superiore di Studi AvanzatiTriesteItaly
  2. 2.Dipartimento di Matematica Pura e ApplicataUniversità di PadovaPadovaItaly

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