Pontryagin’s maximum principle

Use a fixed final time T, fixed initial state, no state constraints and a free terminal state. Minimize a smooth terminal cost φ(x(T)) plus the integral of L(t,x,u), subject to ẋ=f(t,x,u) and measurable inputs in a compact set U. Assume f and L continuous in their arguments and continuously differentiable in x, with their state derivatives continuous and bounded on compact state sets uniformly in t and u. Assume nearby admissible trajectories exist to T, so the state equation has unique absolutely continuous solutions and well-defined first variations there. Assume an optimum exists. Under these hypotheses, the free-endpoint maximum principle has a normal costate formulation.

Define H=L+pᵀf. The state follows H_p, the costate follows −H_x, and the chosen control minimizes H over U almost everywhere along the optimal state and costate. The terminal rule is p(T)=∇φ(x(T)). A convention H=pᵀf−L instead leads to maximization and a corresponding costate sign change. Mixing the minimizing Hamiltonian with the maximizing control rule gives the wrong command. A derivative condition H_u=0 applies only to an interior smooth minimizing control; constrained minimizers can lie on the boundary and require comparison over the whole allowed set.

The free terminal state permits arbitrary terminal state variations, which produces the gradient transversality rule. A fixed terminal state instead has no arbitrary terminal variation, so p(T)=∇φ is not the correct standalone rule. Terminal manifold constraints introduce endpoint multipliers or normal directions. A free terminal time adds another transversality relation involving the Hamiltonian and explicit time dependence. These are different problems, not interchangeable decorations on the same boundary-value system. State constraints can introduce still more complicated multiplier behavior and are outside the simple formulation above.

In a general constrained-endpoint formulation, write H=λ₀L+pᵀf with λ₀≥0 and a nontrivial multiplier pair. The normal case has λ₀>0 and can be scaled to one; an abnormal extremal has λ₀=0 and cannot be normalized that way. For the basic smooth free-terminal-state problem above, λ₀=0 would force p(T)=0 and then p identically zero by the adjoint equation, contradicting nontriviality. That is why normality is justified there. It should not be assumed automatically after adding endpoint constraints.

The coupled equations usually form a two-point boundary-value problem: x is known at the initial time and p at the final time. Shooting methods guess missing initial costates, integrate forward and correct terminal residuals. Such methods can be sensitive because unstable directions magnify the guessed error. Finding one numerical root establishes only an extremal candidate unless a sufficient condition is also checked. Different extremals can have different costs, and a candidate can fail admissibility through a control bound or an unintended endpoint mismatch.

For convex examples, global optimality can sometimes be verified independently by a convexity inequality or direct lower bound. In more general nonlinear problems, second variations or a verified value function may distinguish an optimum from other extremals. Bang-bang controls arise when the minimizing Hamiltonian is affine in a bounded control and its coefficient does not vanish. A vanishing coefficient over an interval is a singular case requiring more analysis; it is not permission to assign an arbitrary command without checking dynamics and higher conditions.