The text below concerns the proof derived by Evans and Souganidis in their 1984 paper that related the “weaker” (or viscosity) solution to the Cauchy problem of HJ equations’ lower value, viz.,

Evans, L. C., & Souganidis, P. E. (1984). Differential games and representation formulas for solutions of Hamilton-Jacobi-Isaacs equations. Indiana Univ. Math. J, 33(5), 773–797.

Specifically, equation 3.4 under Theorem 3.1 should read,

\begin{align} W(t,x) \ge \sup_{y \in M(t)} { \int_t^{t+\sigma} h \left(s, x(s), y(s), \delta [y](s) \right) ds + V\left(t+\sigma, x(t+\sigma) \right) } - \epsilon \end{align}

and not

\begin{align} W(t,x) \ge \sup_{y \in M(t)} { \int_t^{t+\sigma} h \left(s, x(s), \delta [y](s) \right) ds + V\left(t+\sigma, x(t+\sigma) \right) } - \epsilon \end{align}

The authors omitted the control term $y(s) \in Y \vert y$ is measurable.

In a similar vein, $W(t,x)$ on page 778 should have the supremum before the definite integral i.e.,

\begin{align} W(t,x) \ge \sup_{y \in M(t)} { \int_t^T h \left(s, x(s), y(s), \beta [y](s) \right) ds +g(x(T)) } - 2 \epsilon \end{align}

A similar logic would apply to the proof of the upper value of the Cauchy HJ equation i.e., (3.2).