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 InputOutput Finite-Time Stability (IO-FTS). Their aim is to provide quantitative bounds, during a finite time interval, on the state and output trajectories, respectively.