1. Consider the one-form ω = (1+2x1+x2)dx1 + x1dx2. Is it exact ? If yes, compute α(x1, x2) such that ω =dα(x1, x2).
2. Consider the one-form ω = x1dx2 + x2dx3 + x3dx4 . Is it integrable ? Compute dω ∧ dω ∧ω.
Do there exist two functions ϕ1and ϕ2 such that ω∈ span{dϕ1,dϕ2} ? If yes, compute ϕ1and ϕ2 .
3. The nonholonomic integrator
Check the accessibility of the following nonholonomic integrator:
x&1 =u1 x&2 =u2
x&3 =x1u 2 −x2u1
Is it fully (means input-state) linearizable by static state feedback ?
Is it fully linearizable by dynamic state feedback ? à Hint: try to find 2 output functions with relative degree 1 and such that the decoupling matrix has rank 1 only !
