Convexity conditions are identified under which optimal controls in the class of strict controls exist for a large class of stochastic processes under a long-term average criterion in the presence of hard and/or soft constraints. The result adapts a similar result obtained by Haussmann and Lepeltier (1990) for a controlled diffusion under a mixed optimal- stopping/finite-horizon/first-exit criterion. The approach taken in this paper is to utilize an equivalent linear programming formulation of the control problem. These results apply to controlled processes such as diffusions, Markov chains, sim- ple Markov jump processes, diffusions with jumps, regime- switching diffusions and solutions to L ́ evy stochastic differential equations.