Standard form of the maximum principle The controlled system of - - PDF document

I

Standard form of the maximum principle

The controlled system of diff. equations dxi dt = fi(x, u), x ∈ Rn, u ∈ U, contravariant variables with upper indices, f1, . . . , fn covariant “auxiliary” variables with lower in- dices, ψ1, . . . , ψn, The Hamiltonian of the problem and the cor- responding Hamiltonian system, H(ψ, x, u) = ψαfα(x, u), dxi dt = ∂H ∂ψi , dψi dt = −∂H ∂xi. The maximum condition for eliminating the parameter u, H(ψ(t), x(t), u(t)) = max

v∈U H(ψ(t), x(t), v) ∀t,

.

II

The Pontryagin derivative PX

Extremals of the problem I =

t1

t0

L(q, ˙ q)dt = min, q = (q1, . . . , qn), ˙ q = ( ˙ q1, . . . , ˙ qn), are solutions of the Euler-Lagrange equation ∂2L ∂ ˙ q2 ¨ q + ∂2L ∂q ∂ ˙ q ˙ q − ∂L ∂q = 0. The Euler-Lagrange derivative: q → ∂2L ∂ ˙ q2 ¨ q + ∂2L ∂q ∂ ˙ q ˙ q − ∂L ∂q . Invariant formulation of the time-optimal pr-

• blem:

X = X(x, u) ∈ TxM ⊂ TM, x ∈ M, u ∈ U. The fiberwise linear Hamiltonian HX(ψ, u) of the problem and the Pontryagin derivative PX: HX(ψ, u)

def

= < ψ, X(πψ, u) >, ψ ∈ T ∗

πψM,

iPXω = dHX

Computing HX and PX in canonical coordi- nates q = (q1, . . . , qn), p = (p1, . . . , pn): ω = dqα ∧ dpα, X = Xα ∂ ∂xα, ψ = pαdxα HX =< ψ, X >= pαXα = pX.

    

PX = Q ∂ ∂q + P ∂ ∂p = Qα ∂ ∂qα + P ∂ ∂pα iPX dq ∧ dp = det

• < dq, PX >< dp, PX >

dq dp

• =

Q dp − P dq = dHX = p∂X ∂q dq + X dp = ⇒ Q = X, P = −p∂X ∂q dq, ⇓

                                    

PX = X ∂ ∂q − p∂X ∂q ∂ ∂p ⇓ dq dt = X = ∂HX ∂p , dp dt = −p∂X ∂q = −∂HX ∂q

III

Identification of PX

Pontryagin derivative PX is the unique ex- tension as a vector field on T ∗M of the Lie bracket operator adXY = [X, Y ], initially de- fined as a derivation on the C∞(M)-module V ect M. The C∞(T ∗M)-module

M = M0 ⊕ M1 ⊂ C∞(T ∗M), M0 = π∗C∞(M), M1 = V ect M

and unique extension to V ect T ∗M of deriva- tions on M, which preserve both submodules

M0, M1.

The importance of M in the geometry of T ∗M: A = F(q, p) ∀A ∈ C∞(T ∗M),

X

• M

= Y

• M

= ⇒ X = Y ∀X, Y ∈ V ect T ∗M.

PX = X ∂ ∂q − p∂X ∂q ∂ ∂p ⇓ I) PX · π∗a = π∗ · Xa, II) PXHY = HadXY , ⇓ PX is a Hamiltonian lift over X, i.e. π∗ · PX

• ψ

= Xπψ ∀ψ ∈ T ∗M; II) PXHY = HadXY ⇓ If the flow Gt is generated by PXt, Xt = X(x, u(t)), then Gt is bundle preserving over the flow gt generated by Xt : Gt

• T ∗

xM

: T ∗

xM −

→ T ∗

gtxM

∀x ∈ M.

IV Expression of PX(= adX)through the Lie derivative LX over X, etLX = etX

∗

: TM − → TM. The conjugate to the bundle-preserving flow etLX over the flow etX:

• etLX

∗

• T ∗

xM

def

=

 etLX

• Te−tXxM

 

∗

: T ∗

e−tXxM ←

− T ∗

xM

∀x ∈ M The natural duality et adX =

• etLX

∗−1 =

• e−tLX

∗

The same duality, if etLX, et adX are consid- ered as corresponding pullback flows of auto- morphisms and restricted to Λ1(M), V ect M: etX < θ, Y >=< etLXθ, et adXY > ⇓ X < θ, Y >=< LXθ, Y > + < θ, adXY >

Pontryagin derivative PX is the unique ex- tension as a vector field on T ∗M of the Lie bracket operator adXY = [X, Y ], initially de- fined as a derivation on the C∞(M)-module V ect M. PX preserves the submodules M0, M1 of fiber- wise constant and fiberwise linear functions

• n T ∗M, hence every flow on T ∗M generated

by a nonstationary vector field, PXt, Xt = X(x, u(t)), is bundle-preserving. There exists a natural duality between the Lie and Pontryagin derivatives expressed by the relation etPX =

• etLX

∗−1

=

• e−tLX

∗

• r, in the infinitesimal form,

X < θ, Y >=< LX, Y > + < θ, PX > .

• Note. The operator X → PX is not a con-

nection since applied to aX, a ∈ C∞(M), it differentiates a.