R. Alur, C. Courcoubetis, T. A. Henzinger, and P. Ho, Hybrid automata: An algorithmic approach to the specification and verification of hybrid systems, Hybrid Systems (Lecture Notes in Computer Science), 1992.
DOI : 10.1007/3-540-57318-6_30

A. D. Ames, A. Abate, and S. Sastry, Sufficient Conditions for the Existence of Zeno Behavior, Proceedings of the 44th IEEE Conference on Decision and Control, pp.696-701, 2005.
DOI : 10.1109/CDC.2005.1582237

F. Bob, J. R. Caviness, and . Johnson, Quantifier Elimination and Cylindrical Algebraic Decomposition, 1998.

X. Chen, E. Ábrahám, and S. Sankaranarayanan, Flow*: An Analyzer for Non-linear Hybrid Systems, Proceedings of the 25th International Conference on Computer Aided Verification (CAV'13, pp.258-263978, 2013.
DOI : 10.1007/978-3-642-39799-8_18

G. E. Collins, Quantifier elimination for real closed fields by cylindrical algebraic decompostion, Lecture Notes in Computer Science, vol.33, pp.134-183, 1975.
DOI : 10.1007/3-540-07407-4_17

C. C. Conley, Isolated invariant sets and the Morse index, Conference Board of the Mathematical Sciences, vol.38, 1978.
DOI : 10.1090/cbms/038

J. Cortés, Discontinuous dynamical systems, IEEE Control Systems Magazine, vol.28, issue.3, pp.36-73919306, 2008.
DOI : 10.1109/MCS.2008.919306

D. Cox, J. Little, and D. O. Shea, Ideals, Varieties, and Algorithms: An Introduction to Computational Algebraic Geometry and Commutative Algebra, 2010.

H. James, J. Davenport, and . Heintz, Real Quantifier Elimination is Doubly Exponential, J. Symb. Comput, vol.588, pp.1-2, 1988.

M. Jennifer, A. Davoren, and . Nerode, Logics for hybrid systems, Proc. IEEE, vol.88, issue.7, pp.985-1010, 2000.

M. Egerstedt, Behavior Based Robotics Using Hybrid Automata, Proceedings of the Third International Workshop on Hybrid Systems: Computation and Control (HSCC '00, pp.103-116, 2000.

G. Frehse, An Introduction to Hybrid Automata, Numerical Simulation and Reachability Analysis, pp.50-81, 2015.
DOI : 10.1007/978-3-658-09994-7_3

G. Frehse, C. Le-guernic, A. Donzé, S. Cotton, R. Ray et al., SpaceEx: Scalable Verification of Hybrid Systems, Proceedings of the 23rd International Conference on Computer Aided Verification, pp.379-395, 2011.
DOI : 10.1007/978-3-642-00768-2_32

URL : https://hal.archives-ouvertes.fr/hal-00769608

N. Fulton, S. Mitsch, J. Quesel, M. Völp, and A. Platzer, KeYmaera??X: An Axiomatic Tactical Theorem Prover for Hybrid Systems, Automated Deduction -CADE-25: 25th International Conference on Automated Deduction Proceedings (Lecutre Notes in Computer Science), pp.978-981, 2015.
DOI : 10.1007/978-3-319-21401-6_36

K. Ghorbal and A. Platzer, Characterizing Algebraic Invariants by Differential Radical Invariants Held as Part of the European Joint Conferences on Theory and Practice of Software, Tools and Algorithms for the Construction and Analysis of Systems -20th International Conference Proceedings. 279?294, pp.978-981, 2014.

K. Ghorbal, A. Sogokon, and A. Platzer, A hierarchy of proof rules for checking positive invariance of algebraic and semi-algebraic sets, Computer Languages, Systems & Structures, vol.47, pp.19-43, 2017.
DOI : 10.1016/j.cl.2015.11.003

URL : https://hal.archives-ouvertes.fr/hal-01232288

R. Goebel, R. G. Sanfelice, and A. R. Teel, Hybrid dynamical systems, IEEE Control Systems, vol.29, issue.2, pp.28-93, 2008.
DOI : 10.1109/MCS.2008.931718

O. Hájek, Discontinuous differential equations, I, Journal of Differential Equations, vol.32, issue.2, pp.149-1700022, 1979.
DOI : 10.1016/0022-0396(79)90056-1

K. Jack, J. P. Hale, and . Lasalle, Differential Equations: Linearity vs, Nonlinearity. SIAM Rev, vol.5, issue.3, pp.249-272, 1963.

P. Hartman, Ordinary Differential Equations, 1964.

A. Thomas and . Henzinger, The Theory of Hybrid Automata, Proceedings of the 11th Annual IEEE Symposium on Logic in Computer Science (LICS '96). IEEE Computer Society, pp.278-292, 1996.

K. H. Johansson, M. Egerstedt, J. Lygeros, and S. Sastry, On the regularization of Zeno hybrid automata, Systems & Control Letters, vol.38, issue.3, pp.141-150, 1999.
DOI : 10.1016/S0167-6911(99)00059-6

S. Kong, S. Gao, W. Chen, and E. M. Clarke, dReach: ??-Reachability Analysis for Hybrid Systems, Tools and Algorithms for the Construction and Analysis of Systems -21st International Conference, TACAS 2015, Held as Part of the European Joint Conferences on Theory and Practice of Software, ETAPS 2015, pp.200-205978, 2015.
DOI : 10.1007/978-3-662-46681-0_15

J. Liu, N. Zhan, and H. Zhao, Computing semi-algebraic invariants for polynomial dynamical systems, Proceedings of the ninth ACM international conference on Embedded software, EMSOFT '11, pp.97-106, 2011.
DOI : 10.1145/2038642.2038659

J. Lygeros, K. H. Johansson, S. Sastry, and M. Egerstedt, On the existence of executions of hybrid automata, Proceedings of the 38th IEEE Conference on Decision and Control (Cat. No.99CH36304), pp.2249-2254, 1999.
DOI : 10.1109/CDC.1999.831255

J. Lygeros, K. H. Johansson, S. N. Simi?, J. Zhang, S. Shankar et al., Dynamical properties of hybrid automata, IEEE Transactions on Automatic Control, vol.48, issue.1, pp.2-17806650, 2002.
DOI : 10.1109/TAC.2002.806650

B. Mishra, Algorithmic Algebra, pp.978-979, 1993.
DOI : 10.1007/978-1-4612-4344-1

P. J. Mosterman, Mode Transition Behavior in Hybrid Dynamc Systems, Proc. of the 2003 Winter Simulation Conference, pp.623-631, 2003.

J. Pieter, G. Mosterman, and . Biswas, A theory of discontinuities in physical system models, Journal of the Franklin Institute, vol.335, issue.396, pp.401-439, 1998.

P. J. Mosterman, F. Zhao, and G. Biswas, An Ontology for Transitions in Physical Dynamic Systems, Proceedings of the Fifteenth National Conference on Artificial Intelligence and Tenth Innovative Applications of Artificial Intelligence Conference, AAAI 98, IAAI 98, pp.219-224, 1998.

M. Eva, R. Navarro-lópez, and . Carter, Hybrid automata: an insight into the discrete abstraction of discontinuous systems, International Journal of Systems Science, vol.42, issue.11, pp.1883-1898, 2011.

D. Novikov and S. Yakovenko, Trajectories of polynomial vector fields and ascending chains of polynomial ideals, Annales de l'institut Fourier, pp.563-609, 1999.
DOI : 10.5802/aif.1683

A. Platzer, Differential Dynamic Logic for Hybrid Systems, Journal of Automated Reasoning, vol.30, issue.1, pp.143-189, 2008.
DOI : 10.1007/978-1-4612-0601-9

A. Platzer, Logical Analysis of Hybrid Systems: Proving Theorems for Complex Dynamics, pp.978-981, 2010.

R. G. Sanfelice, R. Goebel, and A. R. Teel, Generalized solutions to hybrid dynamical systems, ESAIM: Control, Optimisation and Calculus of Variations, vol.11, issue.4, pp.699-724, 2008.
DOI : 10.1109/TAC.1966.1098336

A. Seidenberg, A New Decision Method for Elementary Algebra, The Annals of Mathematics, vol.60, issue.2, pp.365-374, 1954.
DOI : 10.2307/1969640

A. Tarski, A decision method for elementary algebra and geometry. Bull. Amer, Math. Soc, vol.59, pp.978-981, 1951.

G. Teschl, Ordinary Differential Equations and Dynamical Systems, Graduate Studies in Mathematics, vol.140, 2012.
DOI : 10.1090/gsm/140

A. Tiwari, Abstractions for hybrid systems, Formal Methods in System Design, vol.92, issue.3, pp.57-83, 2008.
DOI : 10.1007/3-540-58179-0_45

I. Vadim and . Utkin, Sliding Modes in Control and Optimization, pp.978-981, 1992.

J. Arjan, . Van-der, H. Schaft, and . Schumacher, An introduction to hybrid dynamical systems, Lecture Notes in Control and Information Sciences, vol.251, 2000.

S. Wang, N. Zhan, and L. Zou, An Improved HHL Prover: An Interactive Theorem Prover for Hybrid Systems, pp.382-399, 2015.
DOI : 10.1109/EMSOFT.2013.6658587

S. Hans and . Witsenhausen, A class of hybrid-state continuous-time dynamic systems, IEEE Trans. Automat. Control, vol.11, issue.2, pp.161-167, 1966.

F. Zhao and V. I. Utkin, Adaptive simulation and control of variable-structure control systems in sliding regimes, Automatica, vol.32, issue.7, pp.1037-10420005, 1996.
DOI : 10.1016/0005-1098(96)00036-2