1. 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)

2. 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

3. 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

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