Teoremi di Frobenius e Chow per campi vettoriali lipschitziani

Control theory has found several applications in Physics in the last decades, from Statistical and Classical Mechanics to Chaos Theory. This thesis is focused upon a specific topic of this mathematical theory, known as Controllability. We give the two main results in this field, the Frobenius and Chow-Rashevsky theorems, first in a differentiable environment and then in a less smooth one, with the last chapter focused on some examples in Classical Mechanics. Precisely speaking: we’ll start giving the definition of small time locally controllable system; then, basic Differential geometry notions are given (tangent bundle, vector fields and their fluxes, Lie brackets) and we state the Frobenius and Chow Theorems in this differentiable context; we then examine the non-smooth case, particularly focusing on re-defining Lipschitz vector fields and introducing objects like Set-valued maps and Generalized Differential Quotients with their properties, all of which are necessary to give a new precise definition of Lie brackets in this non-differentiable context. This will allow an immediate re-statement of the Frobenius Theorem; an exact formula for the composition of the fluxes of vector fields, then, will let a generalization of the Chow’s theorem in the non-smooth case. The last chapter shows some applications of the two theorems in Classical Mechanics, as well as an interesting connection between Lie and Poisson brackets, which allows the use of this work in Hamiltonian mechanics