We study constrained viscosity solutions with an unbounded growth for a class of first order Hamilton-Jacobi-Bellman equations arising in hybrid control systems. To deal with the boundary constraint and rapid growth of the solutions, we construct a particular set of test functions and under very mild conditions establish a comparison theorem which gives the estimate of distance between the subsolution and the supersolution. The comparison theorem implies uniqueness of the constrained viscosity solution if its existence is ensured; and under some additional assumptions we give an existence result by showing that the value function is a constrained viscosity solution. We then apply the obtained uniqueness results to an optimal scheduling problem and finally to stochastic manufacturing systems.