This thesis is concerned with input-output stability theory. Within this framework, it is of interest how inputs map to outputs through an operator that represents a system to be controlled or the controller itself. The Small Gain, Passivity, and Conic Sector Stability Theorems can be used to assess the stability of a negative feedback interconnection involving two systems that each have specific input-output properties.

Our first contribution concerns characterization of the input-output properties of linear timevarying (LTV) systems. We present various theorems that ensure that a LTV system has finite gain, is passive, or is conic. We also consider the stability of various negative feedback interconnections.

Motivated by the robust nature of passivity-based control, we consider how to overcome passivity violations. This investigation leads to the hybrid conic systems framework whereby systems are described in terms of multiple conic bounds over different operating ranges. A special case relevant to systems that experience a passivity violation is the hybrid passive/finite gain framework. Sufficient conditions are derived that ensure the negative feedback interconnection of two hybrid conic systems is stable.

The input-output properties of gain-scheduled systems are also investigated. We show that a gain-scheduled system composed of conic subsystems has conic bounds as well. Using the conic bounds of the subsystems along with the scheduling signal properties, the overall conic bounds of the gain-scheduled system can be calculated. We also show that when hybrid very strictly passive/finite gain (VSP/finite gain) subsystems are gain-scheduled, the overall map is also hybrid VSP/finite gain.

Being concerned with the control of aerospace systems, we use the theory developed in this thesis to control two interesting plants. We consider passivity-based control of a spacecraft endowed with magnetic torque rods and reaction wheels. In particular, we synthesize a LTV input strictly passive controller. Using hybrid theory we control single- and two-link flexible manipulators. We present two controller synthesis schemes, each of which employs numerical optimization techniques and attempts to have the hybrid VSP/finite gain controllers mimic a H₂ controller. One of our synthesis methods uses the Generalized Kalman-Yakubovich-Popov Lemma, thus realizing a convex optimization problem