Suppose we have a dynamic optimization problem, such as to minimize
$$J(x_0,u) = \int_0^\infty \bigg(L(x,\dot{x}) +u^2 \bigg)dt$$ where $$\dot{x} = f(x) + g(x)u, \quad x_0 = x(t=0)$$
To be clear, $x$ is a vector and $f$, $g$ are vector fields.
but $u(x) = u(a,x)$ where a is a constant scalar. For example, $u = ax_1x_2$.
I am stuck while trying to use methods like writing the Hamilton-Jacobi-Bellman equation because it appears to leave $u$ a general function, but I want to specify it to be only of the parameter $a$, while holding $a$ constant. Is there a way to augment the equation such that $\dot{a} = 0$ or otherwise constrain this to solve the problem? Thanks in advance.