Given the robot dynamics of the form $$M(\theta)\ddot{\theta}+C(\theta, \dot{\theta})\dot{\theta}+G(\theta)=u$$ where the notations have the standard meaning ($M$ is the inertia matrix, $C$ is Coriolis-centrifugal matrix, and $G$ is the gravity). Let the desired configuration of the robot be $\theta_d$. To track this desired configuration, we apply the following control law with a gravity compensation: $$u=-K_p(\theta-\theta_d)-K_d\dot{\theta}+G(\theta)$$ The gain matrices $K_p, K_d$ can be assumed to be diagonal with the diagonal elements being $k_p, k_d$ respectively.
Now I need to show 1. Stability, 2. Asymptotic Stability, 3. Exponential Stability. Where the definitions of the quantities of the above are as follows: Let $V$ be a non-negative function with derivative ¤ defined along the trajectories/solution of the system.
Stability: If $V$ is locally pdf(positive definite) and $\dot{V} \leq 0$ locally, then equilibrium point $x^* = 0$ is locally stable.
Asymptotic stability: $V$ is locally pdf and locally decresant, $-\dot{V}$ locally pdf, then $x^*=0$ is locally asymptotically stable. $V$ is pdf and radially unbounded and decrescant, $-\dot{V}$ is pdf, then $x^*=0$ is globally asymptotically stable.
Exponential stability:: $x^*=0$ is exponentially stable if there exists a $V$, that satifies the following: $$\alpha_1||x||^2 \leq V(x) \leq \alpha_2||x||^2$$ $$\dot{V}\leq -\alpha_3 ||x||^2$$
I am not sure how to proceed with the given problem. Any help/relevant resource is helpful.