I have problems with writing the current value Hamiltonian and finding the related necessary and sufficient conditions:
I should max the RHS of : $\rho U_t = log (\chi) + \frac{1}{\rho} log(\lambda)+ M(t)log(\lambda) $
s.t.(resource constraint): $a(\lambda)t + \chi = L$ ; and $M=\dot{\mu}$
According to the book, substitute for $\chi$ from the resource constraint and solve the problem that has $\mu \ge 0$ and $\lambda \ge0 $ as the only constraints
$\frac{log(\lambda)}{\rho}=\frac{a(\lambda)}{La(\lambda)\mu}$ $~~$[1]
$\frac{a'(\lambda)\mu}{La(\lambda)\mu}=\frac{\mu}{\rho \lambda}$ $~~$[2]
Where $\chi$ is the total amount of labor in manufacturing corresponding to the optimal aggregate demand for intermediate goods (since one unit of labor manufactures one unit of intermediate, and we have a unit continuum of industries).
$\lambda$ is the stepsize of the innovation (innovation evolves stochastically and increases in each period by a factor of $\lambda > 1$, with probability $\mu dt$ in a time interval $dt$. $\mu$ is the rate of innovation
M(t) is the expected number of successes in innovation before time $\tau$ : $M(\tau)= \int_{0}^{\tau} \mu(s) d s$.
$L$ is labor and $a(\lambda)$ is an increasing function.
Sorry for this question but what I learned in class was:
FOC:

derivative hamiltonian w.r.t. controls variables=0

derivative hamiltonian w.r.t. state variable = d(costate variable)t

transversality condition
How to find [1],[2]?