The motor is weaker than gravity, so the pendulum cannot be lifted; it has to be pumped, like a swing. Watch the textbook do it, then draw the swing you want, angle against time. A tracking controller follows your line as well as the motor allows, and a linear controller takes over the moment the pendulum enters the region it can hold.
The torque limit is 4 N·m against a gravity torque of 9.8 N·m at the horizontal, so energy has to be pumped in over several swings. Cost is the time until the pendulum is caught and held, in seconds, plus a fiftieth of the control energy; a run that never catches scores eighteen plus energy. The catch fires when the state enters V(x) < c, the Lyapunov ellipse from Stable is Still; c = 3.5 was chosen by simulation, not proved. The textbook reference is an energy-shaping controller, three lines of code that swing it up every time; it plays on the left until you run. Today's joint friction is hidden, and it moves the best swing period.
The motor holds the pendulum about twenty-four degrees from the bottom and no further; beyond that gravity wins. So the pendulum is swung up the way a child pumps a swing: push with the motion and every pass adds energy, push against it and you undo your own work. The period lengthens as the amplitude grows, so the rhythm that works at the start is wrong by the end, and today's friction moves it again. Your line is a plan for that rhythm. The textbook does not plan at all; it watches one number, the energy, and it loses to a good line because it cannot see the clock.
The motor's power into the pendulum is torque times angular velocity. Push with the motion and the energy rises; push against it and it falls. Steering one scalar toward its target needs no model of where the pendulum is, only of how much energy it has, and the torque limit is respected by clipping, since a smaller push with the right sign still moves the energy the right way.
It converges to an orbit, not to a point: at the target energy the pendulum reaches the top with zero velocity but does not stay, so a second controller must catch it. It cannot hurry, because it has no clock; it wastes torque near the turning points where θ̇ ≈ 0; and with friction the target must be overshot a little, by an amount today's hidden b decides.
For the double pendulum on a cart the motor acts on the cart alone, and the power it delivers is force times cart velocity. A force against the cart's velocity, scaled by the energy still stored, makes the stored energy a Lyapunov function that only descends. Nothing about the links appears in the rule: two chaotic links are drained by a law that never looked at them.
It stalls where ẋ = 0 and does nothing about the last small swings, which is why the lab hands over to a linear damper designed around the hanging rest once the links are nearly still. A second Lyapunov argument closes the gap the first one leaves.
Near the top the pendulum is nearly linear, an LQR holds it, and a sublevel set of that LQR's V is the natural place to switch: if the set is invariant, entering it settles the matter. Far from the top a nonlinear controller, here your drawn line and a tracker, does the swinging. This is hybrid control, most real swing-ups work this way, and so does a gymnast approaching a handstand.
Honesty: c = 3.5 was chosen by simulation with the 4 N·m limit, and nothing here proves the set invariant under saturation; the game switches once and does not switch back. For the double pendulum, the lab and Kick replace the ellipse with a two-second forward simulation under the balancing gains, which is a test rather than a hope. Real hybrid loops also guard the boundary against chattering with hysteresis.