I consider the following function:
$H_k(x,t,u,\psi_k,\psi_{0_k})=\langle f(x,u),\psi_k\rangle+\psi_{0_k}e^{-rt}g(x,u)-\psi_{0_k}e^{-(r+1)t}\dfrac{\Vert u-z_k(t) \Vert^2}{1+\sigma_k}$
Futhermore i know that following equality holds almost everywhere:
$\dfrac{d}{dt}H_k(x_k(t),t,u_k(t),\psi_k(t),\psi_{0_k})= \dfrac{\partial H_k}{\partial t}(x_k(t),t,u_k(t),\psi_k(t),\psi_{0_k})$
Then i found in literature the derivation regarding time of the function like:
$\dfrac{d}{dt}H_k(x_k(t),t,u_k(t),\psi_k(t),\psi_{0_k})= \dfrac{\partial H_k}{\partial t}(x_k(t),t,u_k(t),\psi_k(t),\psi_{0_k})= -\psi_{0_k}re^{-rt}\Big [ g(x_k(t),u_k(t))+(r+1)e^{-(r+1)t}\dfrac{\Vert u_k(t)-z_k(t)\Vert^2}{1+\sigma_k}\Big ] +2\psi_{0_k}e^{-(r+1)t}\dfrac{\langle u_k(t)-z_k(t),\dot z_k(t)\rangle}{1+\sigma_k}$
Can someone verify this expression? I dont understand why there is used a big bracket. Does the last term stands for the derivation of the norm-term?