The price of control

Every point of the plane is a controller: the lean times k₁ plus its rate times k₂ is how hard the stick is pushed back. Drag the gold dot: with k₁ below 1, or k₂ below 0, the stick falls; inside the corner fenced off by the coral dashes, both poles in the small plane at top right are left of the line, and after a push the stick comes back upright. Count the lean and the push both by their squares, multiply the push's share by the price r, and add it all up along the curves after a push: that is what the controller costs. Drawn for every controller, the costs make a bowl, and the small ring is its bottom, the cheapest. The bottom-right corner adds the cost up along the curves and also computes it from Lyapunov's equation; drag the dot into the ring and the cost comes down to the least the Riccati equation gives, while the two poles land on the hyperbola. Now drag the price r: the cheaper the push, the further the bottom slides out along the gold parabola; drag the dot into the new ring and it pushes harder and recovers faster, its poles moved but still on the same hyperbola.