Analysis of Basic PMP Solution Find control u(t) which minimizes the function: 𝐽 𝑥,𝑢 = 0 𝑇 𝐿 𝑥,𝑢 𝑑𝑡 + 𝑉 𝑥 𝑇 x(t=0) = x0, T fixed, 𝑥 =𝑓(𝑥,𝑢) To find a necessary condition to minimize J(x,u), we augment the cost function with the dynamics constraints 𝐽 𝑥,𝑢,𝜆 = 𝐽 𝑥,𝑢 + 0 𝑇 𝜆 𝑇 (𝑓 𝑥,𝑢 − 𝑥 )𝑑𝑡 = 0 𝑇 𝐻 𝑥,𝑢 −𝜆 𝑇 𝑥 𝑑𝑡 +𝑉(𝑥 𝑇 ) Extremize the augmented cost w.r.t. variations dx(t); du(t); dl (t); 𝛿 𝐽 = 0 𝑇 𝜕𝐻 𝜕𝑥 +𝜆 𝑇 𝑥 𝛿𝑥 + 𝜕𝐻 𝜕𝑢 𝛿𝑢+ 𝜕𝐻 𝜕𝜆 − 𝑥 𝑇 𝛿𝜆 dt + 𝜕𝑉 𝜕𝑥 − 𝜆 𝑇 𝑇 𝛿𝑥 𝑇 + 𝜆 𝑇 0 𝛿𝑥 0 𝐻 𝑥,𝑢 ≡𝐿 𝑥,𝑢 +𝜆 𝑇 𝑓(𝑥,𝑢)

Dimension of PMP Solution Unknowns n state variables x(t) n co-state variables 𝜆(t) m control variables u(t) Optimal control satisfies: 𝑥 = 𝜕𝐻 𝜕λ 𝑇 =𝑓(𝑥,𝑢) 𝑥 0 = 𝑥 0 n o.d.e.s with I.C.s 2 point B.V. problem −λ = 𝜕𝐻 𝜕𝑥 𝑇 λ 𝑇 = 𝜕𝑉 𝜕𝑥 𝑥 𝑇 n o.d.e.s with T.C.s 𝜕𝐻 𝜕𝑢 =0 ; m algebraic equations

Additional Constraints? Example: what if final state constraints fi(x(T))=0, i=1,…,p are desired: Add constraints to cost using additional Lagrange multipliers Multipliers 𝐽𝑎 𝑥,𝑢 = 0 𝑇 𝐿 𝑥,𝑢 𝑑𝑡 + 𝑉 𝑥 𝑇 + 𝑣 1 𝜑 1 𝑥 𝑇 + …+ 𝑣 𝑝 𝜑 𝑝 𝑥 𝑇 Cost augmentation Extremizing the constrained, augmented the cost function yields 𝑥 = 𝜕𝐻 𝜕λ 𝑇 =𝑓(𝑥,𝑢) 𝑥 0 = 𝑥 0 n o.d.e.s with I.C.s −λ = 𝜕𝐻 𝜕𝑥 𝑇 λ 𝑇 = 𝜕𝑉 𝜕𝑥 (𝑥 𝑇 )+ 𝜕𝜑 𝜕𝑥 (𝑥 𝑇 ) 𝑣 n o.d.e.s with T.C.s 𝜕𝐻 𝜕𝑢 =0 ; m algebraic equations 𝜑 1 𝑥 𝑇 =0; …; 𝜑 𝑝 𝑥 𝑇 =0 p algebraic equations

Additional Constraints? Example: what if final time is undetermined? Final time T is an additional variable H(T) = 0 gives one additional constraint equation!