Complexity of the minimum-time damping of a pendulum

The problem of minimum-time damping of a pendulum is a classical problem of control theory. In the linear case, described by the equation ẍ + x = u, |u| ≤ 1, its solution is stated in [1]. The optimal control is of bang-bang type, i.e. it takes values u = ±1, and the switching curve which separates the domain of the phase plane, where u = −1 from the domain u = +1 consists of unit semicircles centered at points of the form (2k +1, 0), where k is an integer. The real physical pendulum controlled by a torque in the joint is governed by the equation ẍ + sin x = εu, |u| ≤ 1, where x is the vertical angle, and ε is the maximal amplitude of the control torque. The parameter ε is arbitrary: it might be large, small, of order 1. We are interested most in the case of a small ε. The maximum principle says that the optimal control has the form u = sign ψ, where the “adjoint” variable satisfies the equation ψ̈ + (cos x)ψ = 0. Thus, the control is still of the bang-bang type, but the time instants of switchings are roots of a rather nontrivial function, a solution of the general Sturm–Liouville/Schrödinger equation. The complexity of a control is characterized mainly by the switching number. In the linear case this number for a trajectory connecting the initial point (x, ẋ) with (0, 0) is Tπ + O(1), where T is the duration of the motion. In its turn, T = π √ E 2 + O(1), where E = 1 2 ẋ 2 + 12x 2 is the energy. Thus, each trajectory possesses a finite number of switches, but if the initial energy is large