How do I find $u : [0,\infty) \to \mathbb{R}^m$ that minimizes \begin{equation} J(u(\cdot)) = \lim_{t \to \infty} \frac{1}{t} \int_0^t L(x(\tau),u(\tau)) d \tau, \end{equation} subject to \begin{equation} \dot{x}(t) = f(x(t),u(t)) \end{equation} ?
My fist impression is that I cannot use optimal control theory/calculus of variations since the performance measure is not \begin{equation} J(u(\cdot)) = \int_0^{\infty} L(x(\tau),u(\tau)) d \tau. \end{equation}
Thanks