Summary: We study the isothermal Euler equations with friction and consider non-stationary solutions locally around a stationary subcritical state on a finite time interval. The considered control system is a quasilinear hyperbolic system with a source term. For the corresponding initial-boundary value problem we prove the existence of a continuously differentiable solution and present a method of boundary feedback stabilization. We introduce a Lyapunov function which is a weighted and squared \(H^1\)-norm of the difference between the non-stationary and the stationary state. We develop boundary feedback conditions which guarantee that the Lyapunov function and the \(H^1\)-norm of the difference between the non-stationary and the stationary state decay exponentially with time. This allows us also to prove exponential estimates for the \(C^0\)- and \(C^1\)-norm.