Abstract
The behavior of the run of an impulse differential inclusion, and, in particular, of a hybrid control system, is “summarized” by the “initialization map” associating with each initial condition the set of new initialized conditions and more generally, by its “substratum”, that is a set-valued map associating with a cadence and a state the next reinitialized state. These maps are characterized in several ways, and in particular, as “set-valued” solutions of a system of Hamilton-Jacobi partial differential inclusions, that play the same role than usual Hamilton-Jacobi-Bellman equations in optimal control.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
AUBIN J.-P. (1991) Viability Theory Birkhäuser, Boston, Basel, Berlin
AUBIN J.-P. (1999) Impulse Differential Inclusions and Hybrid Systems:A Viability Approach, Lecture Notes, University of California at Berkeley
AUBIN J.-P. (2000) Lyapunov Functions for Impulse and Hybrid Control Systems, Proceedings of the CDC 2000 Conference
AUBIN Jôle optimal et EDP: Innovations et Applications, IOS Press
AUBIN J.-P. (2000) Boundary-Value Problems for Systems of First-Order Partial Differential Inclusions, NoDEA, 7, 61–84
AUBIN J.-P. (to appear) Viability Kernels and Capture Basins of Sets under Differential Inclusions, SIAM J. Control
AUBIN J.-P. (submitted) Boundary-Value Problems for Systems of Hamilton-Jacobi-Bellman Inclusions with Constraints, SIAM J. Control
AUBIN J.-P. & FRANKOWSKA H. (1990) Set-Valued Analysis, Birkhäuser, Boston, Basel, Berlin
AUBIN J.-P. & FRANKOWSKA H. (1992) Hyperbolic systems of partial differential inclusions, Annali Scuola Normale di Pisa, 18, 541–562
AUBIN J.-P. & HADDAD G. (to appear) Cadenced runs of impulse and hybrid control systems, International Journal Robust and Nonlinear Control
AUBIN J.-P., LYGEROS J., QUINCAMPOIX M., SASTRY S. & SEUBE N. (to appear) Impulse Differential Inclusions: A Viability Approach to Hybrid Systems
BENSOUSSAN A. & MENALDI (1997) Hybrid Control and Dynamic Programming, Dynamics of Continuous, Discrete and Impulse Systems, 3, 395–442
BRANICKY M.S., BORKAR V.S. & MITTER S. (1998) A unified framework for hybrid control: Background, model and theory, IEEE Trans. Autom. Control, 43, 31–45
BRETTE R. (2000) Rotation numbers of orientation preserving circle maps, submitted to Nonlinearity
CARDALIAGUET P., QUINCAMPOIX M. & SAINT-PIERRE P. (1994) Temps optimaux pour des problèmes avec contraintes et sans contrôlabilité locale Comptes-Rendus de l’Académie des Sciences, Série 1, Paris, 318, 607–612
DAY R.H. (1995) Multiple-phase economic dynamics, in Multiple-phase economic dynamics, T. Maruyama & W. Takahashi, Eds., Springer-Verlag, 25–45
MATVEEV A.S., SAVKIN A.V. (2000) Qualitative Theory of Hybrid Dynamical Systems, Birkhäuser
MATVEEV A.S., SAVKIN A.V. (2001) Hybrid dynamical systems: Controller and sensor switching problems, Birkhäuser
ROCKAFELLAR R.T. & WETS R. (1997) Variational Analysis, Springer-Verlag
SHAFT (van der) A. & SCHUMACHER H. (1999) An introduction to hybrid dynamical systems, Springer-Verlag, Lecture Notes in Control, 251
TAVERNI L. (1987) Differential autamata and their discrete simulators, Nonlinear Analysis, TMA, 11, 665–683
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2001 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Aubin, JP. (2001). The Substratum of Impulse and Hybrid Control Systems. In: Di Benedetto, M.D., Sangiovanni-Vincentelli, A. (eds) Hybrid Systems: Computation and Control. HSCC 2001. Lecture Notes in Computer Science, vol 2034. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-45351-2_12
Download citation
DOI: https://doi.org/10.1007/3-540-45351-2_12
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-41866-5
Online ISBN: 978-3-540-45351-2
eBook Packages: Springer Book Archive