1 Introduction --  1.1 Motivation and History --  1.2 Summary and Organization --  1.3 Summary --  2 Literature Review --  2.1 Nerode et al Approach to Viability of Hybrid Systems [50],[71] --  2.2 Aubin et al Approach to Viability of Hybrid Systems [15] --  2.3 Deshpande{Varaiya Approach to Viability of Hybrid Systems [35] --  2.4 Related Literature --  2.5 Conclusion --  3 Hybrid Model --  3.1 Hybrid Phenomena and Hybrid Model --  3.2 Hybrid Trajectories and their Ordering --  3.3 Continuity, Fixed Points, and Correct Finite Control Automaton --  3.4 Uncertainty in Hybrid Systems --  3.5 The Three-Tank Problem --  3.6 Nerode{Kohn Formalism for Hybrid Systems --  3.7 Conclusion --  4 Viability --  4.1 Background --  4.2 Time{Independent Viability Set --  4.3 Fixed Point Approximation --  4.4 Computation of TIC{COFPAA{I for Three Admissible Control Law Classes --  4.4.1 Piecewise Constant Control --  4.4.2 Piecewise Constant with Finite Switching --  4.4.3 Piecewise Constant with Polynomial Control --  4.5 Time{Dependent Viability Set --  4.5.1 Piecewise Constant Control --  4.6 Examples --  4.6.1 Time{Independent Constraints --  4.6.2 Time{Dependent Constraints --  4.7 Conclusion --  5 Robust Viability --  5.1 Uncertainty and Robustness --  5.2 Ordering of the Controllability Operator under Uncertainty --  5.3 The Uncertain Controllability Operator and the Uncertainty Operator --  5.4 Robust Viability --  5.5 Robust Viability Control Design --  5.6 Examples --  5.7 Conclusion --  6 Viability in Practice --  6.1 Reachable Set Computation of the Controllability Operator --  6.2 Viable Cascade Control and Application to a Batch Polymerization Process [55][56] --  6.2.1 Batch Polymerization Process Model --  6.2.2 Hybrid Model --  6.2.3 Viable Cascade Control --  6.2.4 Batch Polymerization Control --  6.2.5 Discussion and Conclusions --  6.2.6 Appendix --  6.3 Conclusion --  7 An Operator Approach to Viable Attainability of Hybrid Systems [60] --  7.1 Introduction --  7.2 Attainability and the Attainability Operator --  7.3 Viable Attainability and the Viable Attainability Operator --  7.4 Simulation Examples --  7.5 Conclusion --  8 Some Topics Related to the Controllability Operator --  8.1 Topological Continuity Arising from Fixed Point Approximation Algorithm --  8.2 The Lattice over Control Laws of the Controllability Operator --  8.3 Homotopic Approximation under PWC_ -- k --  PWCPC_ -- k --  8.4 Conclusion --  9 Conclusions --  References.

The problem of viability of hybrid systems is considered in this work. A model for a hybrid system is developed including a means of including three forms of uncertainty: transition dynamics, structural uncertainty, and parametric uncertainty. A computational basis for viability of hybrid systems is developed and applied to three control law classes. An approach is developed for robust viability based on two extensions of the controllability operator. The three-tank example is examined for both the viability problem and robust viability problem. The theory is applied through simulation to an active magnetic bearing system and to a batch polymerization process showing that viability can be satisfied in practice. The problem of viable attainability is examined based on the controllability operator approach introduced by Nerode and colleagues. Lastly, properties of the controllability operator are presented.