When a cost functional is strictly convex?

Question

Let $l_1:H\times X\to \mathbb{R}$, $l_2:V\to \mathbb{R}$ and $L:[0,T]\times H\times H\times X\to \mathbb{R}$. Let $J$ be a cost functional given by $$J(f,g,u,u')=l_1(u(T), u'(T))+l_2(g)+\int_{0}^{T}L(t,u(t),u'(t), f(t))dt.$$ What is more $L(t,v,w,\cdot)$ is convex for all $t\in [0,T]$, $v\in H$, $w\in H$. When the cost functional $J$ is strictly convex??? I think I have to assume that $l_1$ and $l_2$ are convex and $L(t,v,w,\cdot)$ is strixtly convex for all $t\in [0,T]$, $v\in H$, $w\in H$. Am I correct? Can I achive strict convexity of $J$ in a diffrent way?