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As central topics in systems and control theory, the study of stability, robustness, and sensitivity to disturbances for nonlinear systems has gained with the Input-to-State Stability (ISS) paradigm one of its most powerful tools. Standard formulations of ISS and several related notions characterize stability properties in a global setting with respect to a single equilibrium in the origin and for systems defined in Euclidean space. This classical requirements make the basic theory not applicable for a global analysis of many dynamical behaviours of interest for applications, such as: multistability, periodic oscillations, and different class of attractors. The aim of this work is to extend and further characterize the variety of notions available in standard ISS literature to multistable systems evolving on Riemannian manifolds. The definition of global multistability in consideration is based on the existence of a finite number of compact, globally attractive, invariant sets satisfying a specific condition of acyclicity. Such definition allows for Lyapunov characterizations in hybrid systems and is used to generalize ISS, integral ISS, and Output-to-State Stability (OSS) to multistable systems on manifolds. In different modalities, the novel properties of ISS and integral ISS - respectively denoted as Input-to-State Multistability (ISM) and integral ISM - are shown to be preserved in cascade interconnection and with input shifts. In particular, under suitable conditions, we can establish semi-global practical ISM in perturbed flows and in singular perturbation models.

This thesis covers the design of slow wave structures for travelling wave tubes, with a specific focus on those that could be used for operation at millimetre or shorter wavelengths. Serpentine and a coupled cavity photonic crystal structure are covered in detail, together with the interaction between the electromagnetic waves they support and the electron gun and magnetic beam focusing systems needed for a travelling wave interaction. In Chapter 2, the existing small-signal theory of the travelling wave interaction is introduced and applied to a serpentine travelling wave tube. A set of synthesis equations for the serpentine structure are then derived from the analysis and verified with simulation. In Chapter 3, possible improvements to the serpentine structure for high frequency operation, and operation on harmonics other than the fundamental (for both the phase and the interaction impedance) are considered. From the investigation it can be concluded that higher harmonics allow a larger beam current than the fundamental. In Chapter 4, slow wave structures based on photonic crystals are proposed for use in travelling wave tubes. A specific photonic crystal arrangement – the coupled resonator optical waveguide (CROW) - that does not appear to have been studied previously in this application is then investigated. The conclusion is that a CROW is suitable for use in a travelling wave tube and is significantly more manufacturable than existing approaches. In Chapter 5, the design of a full electron beam system for use with both the original and the improved slow wave structures is presented. The design of an electron gun, cathode and collimating magnet using an immersed flow insertion are all covered in detail. In Chapter 6, conclusions are drawn and avenues for possible future work are presented.

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