  We establish necessary and sufficient conditions for viability of evolution
inclusions with locally monotone operators in the sense of Liu and R\"ockner
[J. Funct. Anal., 259 (2010), pp. 2902-2922]. This allows us to prove
wellposedness of lower semicontinuous solutions of Hamilton-Jacobi-Bellman
equations associated to the optimal control of evolution inclusions. Thereby,
we generalize results in Bayraktar and Keller [J. Funct. Anal., 275 (2018), pp.
2096-2161] on Hamilton-Jacobi equations in infinite dimensions with monotone
operators in several ways. First, we permit locally monotone operators. This
extends the applicability of our theory to a wider class of equations such as
Burgers' equations, reaction-diffusion equations, and 2D Navier-Stokes
equations. Second, we allow state constraints. Third, we have uniqueness of
viscosity solutions. Our results on viability and lower semicontinuous
solutions are new even in the case of monotone operators.
