Necessary and sufficient conditions for stability on finite time horizon

Control theory is often interested in studying stability and stabilization of dynamical systems in an infinite time horizon. However, in many practical situations, focusing on the system behavior in a finite time interval is more important than requiring the system to reach a given equilibrium. Analysis on finite time horizon can be useful, for example, to study state and output transients due to disturbances or to examine the effects of sudden changes of the state variable due to external or internal system perturbations. Moreover, some systems naturally evolve in a finite time interval, as for example earthquakes or animal and vegetal systems. Furthermore, only qualitative behavior of dynamical systems are usually taken into account, as in Lyapunov Asymptotic Stability and classic Input-Output Lp-Stability. In many applications though, it is necessary to provide specific bounds to the system state and/or output variables. When dealing with linear models obtained linearizing nonlinear systems around an equilibrium, for example, it is important to keep the state trajectory close to the equilibrium point to avoid the effects of nonlinearity; the presence of variables physical constraints, like actuators saturations, is another example. Two different concepts of stability over a finite time horizon are dealt with in this thesis, namely Finite-Time Stability (FTS) and Input-Output Finite-Time Stability (IO-FTS). Their aim is to provide quantitative bounds, during a finite time interval, on the state and output trajectories, respectively. These quantitative bounds can be very useful for hybrid systems, e.g. Switching Linear Systems (SLSs), which present jumps in the state space in particular instants of time called resetting times. Hybrid systems exhibit both continuous-time and discrete-time dynamics and are used to model systems such as a thermostat turning the heat on and off, a server switching between buffers in a queueing networks, the gear shift control in a car. This thesis extends the FTS results already present in the previous literature, i.e. necessary and sufficient condition to check finite-time stability, to a larger class of hybrid systems, namely SLS. It also copes with the very useful case of uncertainty on the resetting times. In the context of IO-FTS, this thesis provides necessary and sufficient conditions to check the stability of Linear Time Varying (LTV) systems and sufficient conditions to check the stability of SLSs, also in the case of resetting times not a priori known. For both FTS and IO-FTS, dfferent stabilization problems are solved for SLSs and LTV systems. The new concept of Structured IO-FTS is introduced, which makes possible to limit the effort on the actuators by introducing quantitative bounds on the control inputs. Some applications are considered to demonstrate the effectiveness of the developed control techniques.