A gas flow control problem (Q1076959)

The problem of maximizing a one-dimensional stationary gas flow through a channel is reduced to the problem \[ \int^{t_ 0}_{0}q(T_ 1,T_ 2)dt\to \max,\quad \int^{t_ 0}_{0}(q(T_ 1,T_ 2)/T_ 2)dt=const\quad (\geq 0), \] where q is a given function, \(T_ 1\) is a wall temperature and \(T_ 2\) is a gas temperature. Funtions \(T_ 1=T_ 1(t)\) and \(T_ 2=T_ 2(t)\) play the role of control. Properties of optimal \(T_{1opt}\) and \(T_{2opt}\) are established. For example, \(T_{2opt}=const\) and \(T_{1opt}\) has only one switching point.