I have been reading "Partial Differential Equation Models in Macroeconomics" (Achdou, 2014). One of the problems
$$\max_{\{c_{t}\}} \mathbb{E}_{0} \int_0^{\infty} e^{-\rho t} u(c_t)\mathrm{d} t$$
Subjected to
$$\mathrm{d} a_{t}=\left(z_{t}+r(t) a_{t}-c_{t}\right) \mathrm{d} t$$ $$\mathrm{~d} z_{t}=\mu\left(z_{t}\right) \mathrm{d} t+\sigma\left(z_{t}\right) \mathrm{d} W_{t}$$ $$a_{t} \geq \underline{a}$$
The authors then state that the state constraint $a_t \geq \underline{a}$ impose the following value condition
$$z+r a+\partial_{p} H\left(\partial_{a} v\right) \geq 0, \quad a=\underline{a}$$
My question is what is the general form of this condition. That is, what is the value condition when the state variable has multiple dimensions. I have been looking for a while at the literature but the notation is so convoluted that I cannot find an answer.
This will largely depend on the problem but it will be pretty much the same as you have here but for each of the component of the state for which you have a lower bound.
Consider for instance the deterministic system $\dot{x}=f(x,u)$ and we have the lower bound $\underline x\in\mathbb{R}^n$ for the state $x\in\mathbb{R}^n$, that it we want to have $x(t)\ge\underline{x}$ (component-wise) for all $t\ge0$.
This means that the derivative of the $i$-th state should be nonnegative when $x_i(t)=\underline{x}_i$. This translate into
$$\left.f_i(x,u)\right|_{x_i=\underline{x}_i}\ge0$$
for all $i=1,\ldots,n$ and all $x\ge\underline{x}$. This will impose some conditions on the control inputs that may be quite complex depending on the geometry of the problem.
In the case of optimal control, just substitute $u$ by the optimal control.