Journal of Optimization Theory and Applications

, Volume 71, Issue 1, pp 67–83

Impulsive control systems with commutative vector fields

  • A. BressanJr.
  • F. Rampazzo
Contributed Papers

DOI: 10.1007/BF00940040

Cite this article as:
Bressan, A. & Rampazzo, F. J Optim Theory Appl (1991) 71: 67. doi:10.1007/BF00940040


We consider variational problems with control laws given by systems of ordinary differential equations whose vector fields depend linearly on the time derivativeu=(u1,...,um) of the controlu=(u1,...,um). The presence of the derivativeu, which is motivated by recent applications in Lagrangian mechanics, causes an impulsive dynamics: at any jump of the control, one expects a jump of the state.

The main assumption of this paper is the commutativity of the vector fields that multiply theuα. This hypothesis allows us to associate our impulsive systems and the corresponding adjoint systems to suitable nonimpulsive control systems, to which standard techniques can be applied. In particular, we prove a maximum principle, which extends Pontryagin's maximum principle to impulsive commutative systems.

Key Words

Commutativity of vector fieldsadjoint systemsmeasurable functionsoptimal controlsmaximum principle

Copyright information

© Plenum Publishing Corporation 1991

Authors and Affiliations

  • A. BressanJr.
    • 1
  • F. Rampazzo
    • 2
  1. 1.S.I.S.S.A.TriesteItaly
  2. 2.Department of Pure and Applied MathematicsUniversity of PadovaPadovaItaly