P. Tabuada, Verification and control of hybrid systems: a symbolic approach, 2009.

C. Belta, B. Yordanov, and E. Gol, Formal methods for discrete-time dynamical systems, 2017.

A. Girard, G. Gössler, and S. Mouelhi, Safety controller synthesis for incrementally stable switched systems using multiscale symbolic models, IEEE Transactions on Automatic Control, vol.61, issue.6, pp.1537-1549, 2016.
URL : https://hal.archives-ouvertes.fr/hal-01197426

K. Hsu, R. Majumdar, K. Mallik, and A. Schmuck, Multi-layered abstraction-based controller synthesis for continuous-time systems, International Conference on Hybrid Systems: Computation and Control, pp.120-129, 2018.

A. Weber, M. Rungger, and G. Reissig, Optimized state space grids for abstractions, IEEE Transactions on Automatic Control, vol.62, issue.11, pp.5816-5821, 2017.

A. Saoud and A. Girard, Optimal multirate sampling in symbolic models for incrementally stable switched systems, Automatica, vol.98, pp.58-65, 2018.
URL : https://hal.archives-ouvertes.fr/hal-01860113

J. Camara, A. Girard, and G. Gössler, Safety controller synthesis for switched systems using multi-scale symbolic models, IEEE Conference on Decision and Control and European Control Conference, pp.520-525, 2011.
URL : https://hal.archives-ouvertes.fr/hal-00665226

K. Hsu, R. Majumdar, K. Mallik, and A. Schmuck, Lazy abstraction-based control for safety specifications, IEEE Conference on Decision and Control, pp.4902-4907, 2018.

O. Hussien and P. Tabuada, Lazy controller synthesis using threevalued abstractions for safety and reachability specifications, IEEE Conference on Decision and Control, pp.3567-3572, 2018.

A. Kader, A. Saoud, and A. Girard, Safety controller design for incrementally stable switched systems using event-based symbolic models, European Control Conference, pp.1269-1274, 2019.
URL : https://hal.archives-ouvertes.fr/hal-02054930

A. Swikir, A. Girard, and M. Zamani, From dissipativity theory to compositional synthesis of symbolic models, Indian Control Conference, pp.30-35, 2018.
URL : https://hal.archives-ouvertes.fr/hal-01620525

E. S. Kim, M. Arcak, and M. Zamani, Constructing control system abstractions from modular components, International Conference on Hybrid Systems: Computation and Control, pp.137-146, 2018.

P. Meyer, A. Girard, and E. Witrant, Safety control with performance guarantees of cooperative systems using compositional abstractions, IFAC Conference on Analysis and Design of Hyrbid Systems, pp.317-322, 2015.
URL : https://hal.archives-ouvertes.fr/hal-01180975

A. Saoud, P. Jagtap, M. Zamani, and A. Girard, Compositional abstraction-based synthesis for cascade discrete-time control systems, IFAC Conference on Analysis and Design of Hybrid Systems, pp.13-18, 2018.
URL : https://hal.archives-ouvertes.fr/hal-01761180

E. S. Kim, M. Arcak, and S. A. Seshia, Symbolic control design for monotone systems with directed specifications, Automatica, vol.83, pp.10-19, 2017.

D. Angeli and E. D. Sontag, Monotone control systems, IEEE Transactions on automatic control, vol.48, issue.10, pp.1684-1698, 2003.

D. Zonetti, A. Saoud, A. Girard, and L. Fribourg, A symbolic approach to voltage stability and power sharing in time-varying DC microgrids, European Control Conference, pp.903-909, 2019.
URL : https://hal.archives-ouvertes.fr/hal-02070070

S. Sadraddini and C. Belta, Safety control of monotone systems with bounded uncertainties, IEEE Conference on Decision and Control, pp.4874-4879, 2016.

A. Finkel and P. Schnoebelen, Well-structured transition systems everywhere!, Theoretical Computer Science, vol.256, issue.1-2, pp.63-92, 2001.

G. Higman, Ordering by divisibility in abstract algebras, Proceedings of the London Mathematical Society, vol.3, issue.1, pp.326-336, 1952.

P. Ioannou and C. Chien, Autonomous intelligent cruise control, IEEE Transactions on Vehicular Technology, vol.42, issue.4, pp.657-672, 1993.

A. Saoud, A. Girard, and L. Fribourg, Contract based design of symbolic controllers for interconnected multiperiodic sampled-data systems, IEEE Conference on Decision and Control, pp.773-779, 2018.
URL : https://hal.archives-ouvertes.fr/hal-01857389

, ) ?? ?(x 2 , u 2 ) iff for any x 1 ? ?(x 1 , u 1 ), there exists x 2 ? ?(x 2 , u 2 ) with x 1 ? X x 2 . Hence, (i) ?? (ii). (ii) =? (iii): Let x ? X, u ? U , x 1 ? (? x) and u 1 ? (? u), Proof: (i) ?? (ii): Let x 1 , x 2 ? X and u 1 , u 2 ? U Lemma 1, we have that ?

, Hence, from (iii), we have that ?

, To prove the second inclusion, it is sufficient to show that C is a safety controller for the transition system S and the safety specification X S . We have that dom(C) =? dom(C * ) ?? X S = X S , where the first equality comes from (i), the second inclusion comes from the fact that C * is a safety controller and the last equality comes from the lower closedness of X S . Hence, the first condition of Definition 3 is satisfied. Now let x ? dom(C) and u ? C(x). From construction of the controller C, we have the existence of x ? X such that x ? X x , x ? dom(C * ), From construction of the controller C, it follows immediately that ? dom(C * ) = dom(C)

, Then, ? dom(C * ) =? dom(C) = dom(C * ), where the last equality comes from (i) in Lemma 2. Hence, dom(C * ) is a lower closed set. (ii) Let x 1 , x 2 ? X with x 1 ? X x 2, Proof: (i) We have from (ii) in Lemma 2 that dom(C * ) = dom(C)

C. , where the first inclusion comes from the fact that S is a monotone transition system and the last equality comes from (i). Hence, by maximality of C * , we have that u ? C * (x 1 ). Then, C * (x 2 ) ? C * (x 1 ). (iii) Let x ? X, u ? C * (x), Then, ?(x 2 , u) ? dom(C * ). Hence, we have that ?(x 1 , u) ?? ?(x 2 , u) ?? dom(C * ) = dom(C * )