This work presents a novel method to solve analysis and design problems for nonlinear
control systems, the classical solutions of which rely on the solvability, or on the solution
itself, of partial differential equations or inequalities. The first part of the thesis is dedicated
to the analysis of nonlinear systems. The notion of Dynamic Lyapunov function
is introduced. These functions allow to study stability properties of equilibrium points,
similarly to standard Lyapunov functions. In the former, however, a positive definite
function is combined with a dynamical system that render Dynamic Lyapunov functions
easier to construct than Lyapunov functions. These ideas are then extended to characterize
Dynamic Controllability and Observability functions, which are exploited in the model
reduction problem for nonlinear systems. Constructive solutions to the L2-disturbance
attenuation and the optimal control problems are proposed in the second part of the thesis.
The key aspect of these solutions is the definition of Dynamic Value functions that,
generalizing Dynamic Lyapunov functions, consist of a dynamical feedback and a positive
definite function. In the last part of the thesis a similar approach is utilized to simplify
the observer design problem via the Immersion and Invariance technique. Finally, the
effectiveness of the methodologies is illustrated by means of several applications, including
range estimation and the optimal robust control of mechanical systems, combustion engine
test benches and the air path of a diesel engine.