Consider $$H(x, p)=sup _{u in U}{-f(x, u) cdot p-f^0(x, u)}$$

This is a classical Hamiltonian coming from an infinite horizon control problem of ODE’s with state equation, i.e.

begin{equation}label{state_eq_finite_dim}

left{begin{array}{l}

y^{prime}(t)=f(y(t), u(t)), t>0 \

y(0)=x in mathbb{R}^n

end{array}right.

end{equation}

where $u in mathcal{U}={u:(0,+infty) rightarrow U$ measurable$}$, $U subset mathbb{R}^n$

and the problem is to minimize

begin{equation}

J(x, u)=int_{0}^{+infty} e^{-lambda t} f^0(y_x(t), u(t)) d t

end{equation}

Assume for simplicity that $f,f^0 in C^1_B$ and $f$ is lipchitz uniformly in $u$

Now in (Bardi, Martino, and Italo Capuzzo-Dolcetta. Optimal control and viscosity solutions of Hamilton-Jacobi-Bellman equations. Springer Science & Business Media, 1997.) it is claimed that for fixed $x$ the function $H(x,p)$ is convex in $p$.

How do you see that?