Periodic diagonal controlled-diffusion HJB Cauchy problem

Let $d\ge 1$, $T>0$, and let $A$ be a nonempty finite ordered control set. For each $a\in A$ and $j=1,\dots ,d$, let ${\alpha }_j^a,b_j^a,f^a:{\mathbb{T}}^d\times [0,T]\to \mathbb{R}$ be bounded and periodic, jointly continuous in $(x,t)$, and uniformly Lipschitz in $(x,t)$. Assume a common constant ${\alpha }_{\ast }>0$ satisfies ${\alpha }_j^a(x,t)\ge {\alpha }_{\ast }\text{for every }a,j,x,t\mathrm{.}$ The diagonal controlled-diffusion HJB Cauchy problem is $u_t-{\max }_{a\in A}\left\{{\sum }_{j=1}^d\right({\alpha }_j^a(x,t)u_{x_j{x}_j}+b_j^a(x,t)u_{x_j})+f^a(x,t)\}=0\text{on }{\mathbb{T}}^d\times (0,T),$ with continuous periodic initial data $u(x,0)=u_0(x)$. In the convention $u_t+F(x,t,Du,D^{2}u)=0$, this has $F(x,t,p,X)=-{\max }_{a\in A}\{{\sum }_j({\alpha }_j^a{X}_{jj}+b_j^a{p}_j)+f^a\},$ which is nonincreasing in each diagonal Hessian entry. No off-diagonal diffusion or arbitrary anisotropic central stencil is covered.

The lower bound ${\alpha }_{\ast }>0$ is part of this selected coefficient branch; the monotonicity proof below only needs the separate mesh diffusion-dominance and time-step inequalities. The definition itself asserts no solution existence or uniqueness. Local draft items now prove comparison for the full stated coefficient class and use it to prove unconditional convergence and existence along an admissible mesh family. Those new arguments remain subject to the native independent reviews and gates and have not received workflow acceptance.