Optimal Control Theory - Module 3 - Maximum Principle

subject to (tf , x(tf )) ∈ S = [t0,∞)× S1 where S1 is a k dimensional manifold in Rn S1 = {x ∈ R : h1(x) = h2(x) = · · · = hn−k(x) = 0} where hi are C1 functions from Rn to R subject to ẋ = f(x, u), x(t0) = x0 for u ∈ C[t0, T ] and u(t) ∈ U ⊂ Rm with f and L being C1 functions. Let u∗ : [t0, tf ] → R be an optimal control with state trajectory x∗ : [t0, tf ] → Rn and a constant. Then there exists a function p∗ : [t0, tf ]→ Rn and a constant p0 ≤ 0 (not both zero) for all t ∈ [t0, tf ] such that 1. x∗ and p∗ satisfy Hamilton’s canonical equations, ẋ∗ = Hp(x ∗, u∗, p∗, p0) (1) ṗ∗ = −Hx(x, u∗, p∗, p0) (2)