Periodic first-order controlled Hamilton–Jacobi Cauchy problem

Let $d,q\ge 1$, $T>0$, and let $A$ be a nonempty compact subset of ${\mathbb{R}}^q$ (a finite action set is included). For each $a\in A$, let $b_a:{\mathbb{T}}^d\times [0,T]\to {\mathbb{R}}^d$ and ${\mathrm{\ell }}_a:{\mathbb{T}}^d\times [0,T]\to \mathbb{R}$ be bounded and jointly continuous in $(x,t,a)$. Write $b_a,{\mathrm{\ell }}_a$ also for their periodic lifts to ${\mathbb{R}}^d\times [0,T]$. Assume there are constants $B,L<\mathrm{\infty }$ such that $\mathrm{\mid }{b}_a(x,t)\mathrm{\mid }\le B,\mathrm{\mid }{\mathrm{\ell }}_a(x,t)\mathrm{\mid }\le L,$ and for all $a\in A$, $x,y\in {\mathbb{R}}^d$, and $t,s\in [0,T]$, $\mathrm{\mid }{b}_a(x,t)-b_a(y,s)\mathrm{\mid }+\mathrm{\mid }{\mathrm{\ell }}_a(x,t)-{\mathrm{\ell }}_a(y,s)\mathrm{\mid }\le L(\mathrm{\mid }x-y\mathrm{\mid }+\mathrm{\mid }t-s\mathrm{\mid })\mathrm{.}$ Given continuous periodic initial data $u_0:{\mathbb{T}}^d\to \mathbb{R}$, define $H(x,t,p)={\max }_{a\in A}\{-b_a(x,t)\cdot p-{\mathrm{\ell }}_a(x,t)\}\mathrm{.}$ The periodic first-order controlled Hamilton–Jacobi Cauchy problem is $u_t+H(x,t,Du)=0\text{on }{\mathbb{T}}^d\times (0,T),u(x,0)=u_0(x)\mathrm{.}$ The signs are fixed by $u_t={\min }_{a\in A}\{b_a\cdot Du+{\mathrm{\ell }}_a\}$. Whenever the controls are discretized, use a sequence of finite nonempty sets $A_k\subseteq A$ and define $H_k(x,t,p):={\max }_{a\in A_k}(-b_a\cdot p-{\mathrm{\ell }}_a)\mathrm{.}$ Define the Hamiltonian error by ${\eta }_k(R):={\sup }_{x,t,\mathrm{\mid }p\mathrm{\mid }\le R}\mathrm{\mid }{H}_k(x,t,p)-H(x,t,p)\mathrm{\mid }\mathrm{.}$ The sequence is admissible when ${\eta }_k(R)\to 0$ for every $R<\mathrm{\infty }$. Exact minimization over $A$ corresponds to ${\eta }_k=0$. Scheme convergence below uses this explicit control-grid error whenever the controls are discretized.

The compact action set and continuity make each displayed maximum attained. The coefficient assumptions imply the exact first-order comparison bounds on the periodic extension to ${\mathbb{R}}^d$: $\mathrm{\mid }H(x,t,p)-H(y,s,p)\mathrm{\mid }\le L(1+\mathrm{\mid }p\mathrm{\mid })(\mathrm{\mid }x-y\mathrm{\mid }+\mathrm{\mid }t-s\mathrm{\mid }),\mathrm{\mid }H(x,t,p)-H(x,t,p^{\mathrm{\prime }})\mathrm{\mid }\le B\mathrm{\mid }p-p^{\mathrm{\prime }}\mathrm{\mid }\mathrm{.}$ Thus the published whole-space comparison theorem applies once bounded upper semicontinuous subsolutions and bounded lower semicontinuous supersolutions with the same pointwise initial ordering have been established.