Prompt Title: Adaptive Control Convergence.

To show that both e and θ converge to zero using the LaSalle invariance principle, we first need to choose an appropriate Lyapunov function candidate V(e, θ). A common choice for adaptive control systems is:

V(e, θ) = 0.5 * e^2 + 0.5 * θ^2

Now, we'll compute the time derivative of V along the trajectories of the closed-loop error dynamics:

Vdot(e, θ) = ∂V/∂e * edot + ∂V/∂θ * θdot

From the given closed-loop error dynamics, we have:

edot = -e + θw θdot = -ew

Now, computing the partial derivatives of V:

∂V/∂e = e ∂V/∂θ = θ

Substituting these into the expression for Vdot, we get:

Vdot(e, θ) = e(-e + θw) + θ(-ew) Vdot(e, θ) = -e^2 + eθw - eθw Vdot(e, θ) = -e^2

We observe that Vdot(e, θ) is negative semi-definite. According to the LaSalle invariance principle, the largest invariant set in the set {e, θ | Vdot(e, θ) = 0} is the set of points to which all the trajectories converge.

In this case, Vdot(e, θ) = 0 if and only if e = 0. Therefore, the largest invariant set is {e, θ | e = 0}.

Now, let's analyze the dynamics of θ on this set:

θdot = -ew(t)

Since w(t) is a non-zero constant, θdot is non-zero only if e ≠ 0. But on the invariant set, e = 0, and thus θdot = 0. This implies that θ must remain constant on the invariant set. Furthermore, since V(e, θ) is positive definite, V(0, θ) = 0.5 * θ^2, and V(0, θ) = 0 if and only if θ = 0. This means that both e and θ converge to zero.

By the LaSalle invariance principle, we have shown that both e and θ converge to zero for the given adaptive control system.