given that we want to minimize a performance
$$\min \int_{t_{0}}^{t_{1}} L\big[x(t),u(t)\big ]\ \mathrm dt$$ subject to $$\dot{x} = f(x,u)$$ $$x(0) = x_{0}$$
when constructing the Hamiltonian would we first multiply L by negative one to make it a maximisation probelm? so $$H(x,u,\lambda) = -L + \lambda^{T}f(x,u)$$
I understand that when doing time minimization problems($L = 1$), you don't multiply by $-1$ cause time cannot be negative. Am I correct in doing this, multiplying $L$ by negative $1$ then using PMP?