Invariant manifolds of the optimal control problem

The optimal control problem is that of balancing an inverted planar pendulum on a cart that moves in the plane of the pendulum with an applied horizontal force u, subject to a quadratic cost function. We only consider the model equations associated with the pendulum, so this system is two-dimensional. The problem can be reformulated as a four-dimensional Hamiltonian system with Hamiltonian

H(x1, x2, p1, p2) = Q(x1, x2, u*(x1, x2, p1, p2)) + p1 x2 + p2 f(x1, x2) + p2 c(x1, x2) u*(x1, x2, p1, p2),

where the first two coordinates (x₁, x₂) represent state space and the second, dual pair of coordinates (p₁, p₂) are related to the control. The functions f, c, Q, and u^* are defined as

The parameters are g = 9.8, l = 0.5, m_r = 0.2, m = 2, ₁ = 0.1, ₂ = 0.05, and _u = 0.01. The control u^*(x₁, x₂, p₁, p₂) is (locally) optimal if (x₁, x₂, p₁, p₂) lies on the stable manifold of the origin.

	Four different projections of the stable manifold of the origin.
	The animation shows how the stable manifold grows in four projections at the same time (2.4MB).