Geometric Numerical Integration of Hamiltonian systems: application - - PowerPoint PPT Presentation

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Geometric Numerical Integration of Hamiltonian systems: application to some

• ptimal control problems

Philippe Chartier1

1IPSO

INRIA-Rennes

Optimal Control : Algorithms and Applications, May 30-June 1st 2007

logo First examples Hamiltonian problems Geometric B-series Modified equations Application to control problems

Outline

1

First examples Harmonic oscillator 2-D Kepler Problem

2

Hamiltonian problems Main properties of Hamiltonian systems Symplectic maps Application to Hamiltonian systems

3

Geometric B-series B(utcher)-series Algebraic characterization of geometric properties

4

Modified equations Backward error analysis for ordinary differential equations Geometric properties of the modified equation

5

Application to control problems An optimal control problem without constraints Runge-Kutta discretization of optimality conditions Modified optimal control problem

logo First examples Hamiltonian problems Geometric B-series Modified equations Application to control problems

Outline

1

First examples Harmonic oscillator 2-D Kepler Problem

2

Hamiltonian problems Main properties of Hamiltonian systems Symplectic maps Application to Hamiltonian systems

3

Geometric B-series B(utcher)-series Algebraic characterization of geometric properties

4

Modified equations Backward error analysis for ordinary differential equations Geometric properties of the modified equation

5

Application to control problems An optimal control problem without constraints Runge-Kutta discretization of optimality conditions Modified optimal control problem

logo First examples Hamiltonian problems Geometric B-series Modified equations Application to control problems Harmonic oscillator

Equations and solution

Consider the following Hamiltonian H(p, q) = 1 2(p2 + ω2q2) and the corresponding Hamiltonian system

• ˙

p = − ∂H

∂q

= −ω2q ˙ q =

∂H ∂p

= p . The exact trajectory is known to be an ellipse in the phase-space (p, q) depending on initial conditions (p0, q0).

logo First examples Hamiltonian problems Geometric B-series Modified equations Application to control problems Harmonic oscillator

Three elementary methods

Explicit Euler pn+1 = pn + h(−ω2qn) = pn + (−hω2)qn qn+1 = qn + h(pn) = hpn + qn Implicit Euler

• pn+1

= pn + h(−ω2qn+1) =

1 1+h2ω2 pn

+

−hω2 1+h2ω2 qn

qn+1 = qn + h(pn+1) =

h 1+h2ω2 pn

+

1 1+h2ω2 qn

logo First examples Hamiltonian problems Geometric B-series Modified equations Application to control problems Harmonic oscillator

Three elementary methods

Midpoint rule pn+1 = pn + h(−ω2 qn+1+qn

2

) qn+1 = qn + h(pn+1+pn

2

) i.e.    pn+1 =

1−h2ω2/4 1+h2ω2/4pn

+

−hω2 1+h2ω2/4qn

qn+1 =

h 1+h2ω2/4pn

+

1−h2ω2/4 1+h2ω2/4qn

logo First examples Hamiltonian problems Geometric B-series Modified equations Application to control problems Harmonic oscillator

Computed trajectories for the three methods

−3 −2 −1 1 2 3 −3 −2 −1 1 2 3 u v Explicit Euler (green), Midpoint Rule (red), Implicit Euler (blue)

logo First examples Hamiltonian problems Geometric B-series Modified equations Application to control problems Harmonic oscillator

Theoretical explanation of the different behaviors

All previous schemes can written as a linear recurrence pn+1 qn+1

• = M(hω)

pn qn

• with, for the Explicit Euler method

M(hω) = 1 (−hω2) 1 1

• and eigenvalues λ1,2 = (1 ± ihω). Hence, the energy grows like

(1 + h2ω2)n/2.

logo First examples Hamiltonian problems Geometric B-series Modified equations Application to control problems Harmonic oscillator

Theoretical explanation of the different behaviors

All previous schemes can written as a linear recurrence pn+1 qn+1

• = M(hω)

pn qn

• with, for the Implicit Euler method

M(hω) =

• 1

1+h2ω2 −hω2 1+h2ω2 h 1+h2ω2 1 1+h2ω2

• and eigenvalues λ1,2 = (1 ± ihω)−1. Hence, the energy

decreases like (1 + h2ω2)−n/2.

logo First examples Hamiltonian problems Geometric B-series Modified equations Application to control problems Harmonic oscillator

Theoretical explanation of the different behaviors

All previous schemes can written as a linear recurrence pn+1 qn+1

• = M(hω)

pn qn

• with, for the Midpoint rule

M(hω) =  

1−h2ω2/4 1+h2ω2/4 −hω2 1+h2ω2/4 h 1+h2ω2/4 1−h2ω2/4 1+h2ω2/4

  and eigenvalues λ1,2 of modulus one. Hence, the energy is constant.

logo First examples Hamiltonian problems Geometric B-series Modified equations Application to control problems 2-D Kepler Problem

Equations

Consider the follwoing Hamiltonian H(p1, p2, q1, q2) = 1 2[(p1)2 + (p2)2] − 1

• (q1)2 + (q2)2 ,

= T(p) + V(q). and the corresponding System

• ˙

p = − ∂H

∂q

= −V ′(q) ˙ q =

∂H ∂p

= p ⇐ ⇒ ¨ q = −V ′(q) The exact trajectory is known to be an ellipse in the phase-space (p, q) depending on initial conditions.

logo First examples Hamiltonian problems Geometric B-series Modified equations Application to control problems 2-D Kepler Problem

Computed trajectories and energies

−3 −2.5 −2 −1.5 −1 −0.5 0.5 1 1.5 2 −3 −2 −1 1 2 3 q1 q2 Euler explicit/implicit

logo First examples Hamiltonian problems Geometric B-series Modified equations Application to control problems 2-D Kepler Problem

Computed trajectories and energies

−2 −1.5 −1 −0.5 0.5 1 1.5 2 −2 −1.5 −1 −0.5 0.5 1 1.5 2 q1 q2 Midpoint Rule

logo First examples Hamiltonian problems Geometric B-series Modified equations Application to control problems 2-D Kepler Problem

Computed trajectories and energies

50 100 150 200 250 300 350 400 450 −8 −7 −6 −5 −4 −3 −2 −1 Time H Energy

logo First examples Hamiltonian problems Geometric B-series Modified equations Application to control problems 2-D Kepler Problem

Motivation for further investigations

Observation Nothing as simple as a linear analysis can sustain the observed superior behavior of the midpoint rule on Kepler problem. Other non-linear problems corroborate these observations. Consequence A more elaborated theory is required to understand what is going on.

logo First examples Hamiltonian problems Geometric B-series Modified equations Application to control problems

Outline

1

First examples Harmonic oscillator 2-D Kepler Problem

2

Hamiltonian problems Main properties of Hamiltonian systems Symplectic maps Application to Hamiltonian systems

3

Geometric B-series B(utcher)-series Algebraic characterization of geometric properties

4

Modified equations Backward error analysis for ordinary differential equations Geometric properties of the modified equation

5

Application to control problems An optimal control problem without constraints Runge-Kutta discretization of optimality conditions Modified optimal control problem

logo First examples Hamiltonian problems Geometric B-series Modified equations Application to control problems Main properties of Hamiltonian systems

For p and q in Rd, and H a smooth scalar function, one can define the following Hamiltonian system

• ˙

p = − ∂H

∂q

˙ q =

∂H ∂q

. Denoting y = p q

• , J =
• Id

−Id

• Canonical equations

˙ y = J−1∇H.

logo First examples Hamiltonian problems Geometric B-series Modified equations Application to control problems Main properties of Hamiltonian systems

Conservation of the Hamiltonian by the flow

Definition The flow ϕt is defined as the function which associates at time t the exact solution of ˙ y = f(y) with initial condition y(0) = y0. Theorem The flow of an Hamiltonian system preserves the value of the Hamiltonian. Proof : Since J is skew-symmetric, along any exact trajectory

• ne has:

d dt H(ϕt(y)) = ∂H ∂y dy dt = (∇H)TJ−1∇H = 0.

logo First examples Hamiltonian problems Geometric B-series Modified equations Application to control problems Main properties of Hamiltonian systems

Conservation of volume

Theorem For a system of the form ˙ y = f(y), with div f = 0, one has Vol(ϕt(A)) = Vol(A) for any compact set A ⊂ Rn. Proof : Ψt(y) = ∂ϕt

∂y (y) is solution of

d dt Ψt(y) = ∂f ∂y (ϕt(y))Ψt(y), Ψ0(y) = IRn. Hence d dt det(Ψt(y)) = det(Ψt) Tr(Ψ−1

t

• ∂yf(ϕt(y))
• Ψt) = 0,

and

• ϕt(A)

dz =

• A

det(Ψt(y))dy =

• A

det(Ψ0(y))dy =

• A

dy.

logo First examples Hamiltonian problems Geometric B-series Modified equations Application to control problems Main properties of Hamiltonian systems

Conservation of volume

Theorem The flow of an Hamiltonian system preserves the volume. Proof : For an Hamiltonian system f = J−1∇H divf = Tr( ∂f ∂y ) = Tr(J−1∇2H) = Tr(∇2HJ−T) = −Tr(J−1∇2H) = −divf = 0.

logo First examples Hamiltonian problems Geometric B-series Modified equations Application to control problems Symplectic maps

Dimension d = 1

The oriented area of the Parallelogram P = {tξ + sη | 0 ≤ t, s ≤ 1}, generated by the two vectors ξ = ξp ξq

• and η =

ηp ηq

• is of the form:

Oriented area

• riented.area(P) =
• ξp

ηp ξq ηq

• = ξpηq − ξqηp.

logo First examples Hamiltonian problems Geometric B-series Modified equations Application to control problems Symplectic maps

Dimension d > 1

In dimension d > 1, one replaces this expression by the sum ω(ξ, η) of the oriented areas of the projections of P over the (pi, qi)-planes: Skew-symmetric bilinear form ω ω(ξ, η) =

d

• i=1
• ξp

i

ηp

i

ξq

i

ηq

i

• =

d

• i=1

(ξp

i ηq i − ξq i ηp i ) = ξTJη.

logo First examples Hamiltonian problems Geometric B-series Modified equations Application to control problems Symplectic maps

Dimension d > 1

pI qI R2d−2 ξ η (ξp

i ηq i − ξq i ηp i )

Figure: The map ω

logo First examples Hamiltonian problems Geometric B-series Modified equations Application to control problems Symplectic maps

Linear symplectic maps

Definition A linear map A : R2d → R2d is said to be symplectic iff: ATJA = J, i.e., equivalently, iff: ∀(ξ, η) ∈ R2d × R2d, ω(Aξ, Aη) = ω(ξ, η). In dimension 1, the symplecticity of A is nothing else but the the preservation of areas.

logo First examples Hamiltonian problems Geometric B-series Modified equations Application to control problems Symplectic maps

Linear symplectic maps

Definition A linear map A : R2d → R2d is said to be symplectic iff: ATJA = J, i.e., equivalently, iff: ∀(ξ, η) ∈ R2d × R2d, ω(Aξ, Aη) = ω(ξ, η). In dimension d > 1, it accounts for the preservation of the sum

• f the oriented areas of the projection over the (pi, qi)-planes.

logo First examples Hamiltonian problems Geometric B-series Modified equations Application to control problems Symplectic maps

General situation

Definition A smooth map g from U, an open subset of R2d, into R2d, is said to be symplectic if its Jacobian matrix g′(p, q) is symplectic for all (p, q) in U, i.e. iff: ∀(p, q) ∈ U,

• g′(p, q)

T Jg′(p, q) = J,

• r equivalently iff:

∀(p, q) ∈ U, ∀(ξ, η) ∈ R2d × R2d, ω(g′(p, q)ξ, g′(p, q)η) = ω(ξ, η).

logo First examples Hamiltonian problems Geometric B-series Modified equations Application to control problems Symplectic maps

An integral quantity preserved by symplectic maps

Let M = ψ(K) be a 2-D submanifold of U with ψ(s, t) a smooth

• map. M can be seen as a union of small parallelograms

generated by the vectors ∂ψ ∂s ds and ∂ψ ∂t dt, and we can write the sum of oriented areas of the projections

• ver the (pi, qi)-planes of all these parallelograms as

The integral form of ω Ω(M) =

• K

ω(∂ψ ∂s (s, t), ∂ψ ∂t (s, t))dsdt.

logo First examples Hamiltonian problems Geometric B-series Modified equations Application to control problems Symplectic maps

Theorem Let g : U → R2d be smooth and symplectic on U. Then, g preserves Ω(M), that is to say: Ω(g(M)) = Ω(M).

(p0, q0) qI qI g

∂ψ ∂s ds ∂ψ ∂t dt ∂g◦ψ ∂s

ds

∂g◦ψ ∂t

dt pI R2d−2 g(p0, q0) pI

Figure: Image of M by g

logo First examples Hamiltonian problems Geometric B-series Modified equations Application to control problems Symplectic maps

Theorem Let g : U → R2d be smooth and symplectic on U. Then, g preserves Ω(M), that is to say: Ω(g(M)) = Ω(M). Proof : g(M) can be parametrized by g ◦ ψ on K so that Ω(g(M)) =

• K ω(∂g◦ψ

∂s (s, t), ∂g◦ψ ∂t (s, t))dsdt

=

• K ω(g′(ψ(s, t))∂ψ

∂s (s, t), g′(ψ(s, t))∂ψ ∂t (s, t))dsdt

=

• K
• ∂ψ

∂s (s, t)

T g′(ψ(s, t)) T Jg′(ψ(s, t))

• =J

∂ψ ∂t (s, t)dsdt

= Ω(M).

logo First examples Hamiltonian problems Geometric B-series Modified equations Application to control problems Application to Hamiltonian systems

Characterization of Hamiltonian systems (I)

Theorem (Poincar´ e 1899) Let H(p, q) be a twice continuously differentiable function from an open subset U ⊂ R2d into R2d. Then, for all t such that ϕt exists, ϕt is symplectic. Proof: For all t such that ϕt exists, Ψt = ∂ϕt

∂y0 satisfies

˙ Ψt = J−1∇2H(ϕt(y0))Ψ, Ψ0 = I2d. Given that ∇2H(ϕt(y0)) is symmetric, we have:

d dt

• ΨT

t JΨt

• =
• ∂ϕt

∂y0

T ∇2H J−TJ

−J−1J=−I

Ψt + ΨT

t J−1J I

∇2HΨt = 0. The conclusion now follows from (for t = 0) ΨT

t JΨt = ITJI = J.

logo First examples Hamiltonian problems Geometric B-series Modified equations Application to control problems Application to Hamiltonian systems

Characterization of Hamiltonian systems (II)

Theorem Let U be a simply connected open subset and f a smooth on U. If ϕt(y) is symplectic for small t and y in U, then ˙ y = f(y) is Hamiltonian on U, i.e there exists a smooth H defined on U such that ∀y ∈ U, f(y) = J−1∇H(y). Proof : For all t such that ϕt exists, ∂ϕt

∂y0 is the solution of

˙ Ψt = f ′(ϕt(y0))Ψt, Ψ0 = I2d. Upon differentiating the symplecticity relation, we obtain: 0 = d

dt

• ΨT

t JΨt = ΨT t

(f ′(ϕt(y0)))TJ + JTf ′(ϕt(y0))

• Ψt.

For t = 0, we get (f ′(ϕt(y0)))TJ + JTf ′(ϕt(y0)) = 0, and (Jf ′(ϕt(y0)))T = Jf ′(ϕt(y0)). Pfaff’s Lemma allows to conclude.

logo First examples Hamiltonian problems Geometric B-series Modified equations Application to control problems

Outline

1

First examples Harmonic oscillator 2-D Kepler Problem

2

Hamiltonian problems Main properties of Hamiltonian systems Symplectic maps Application to Hamiltonian systems

3

Geometric B-series B(utcher)-series Algebraic characterization of geometric properties

4

Modified equations Backward error analysis for ordinary differential equations Geometric properties of the modified equation

5

Application to control problems An optimal control problem without constraints Runge-Kutta discretization of optimality conditions Modified optimal control problem

logo First examples Hamiltonian problems Geometric B-series Modified equations Application to control problems B(utcher)-series

Definition

Given a n-dimensional system of differential equations y′(x) = f(y(x)) a B-series B(a, y) is a formal expression of the form B(a) = idRn +

• t ∈ T

h|t| σ(t) a(t) F(t) = idRn + ha( )f(·) + h2a( )(f ′f)(·) + · · · the index set T = { , , , , · · · } is the set of trees, |t| and σ(t) are fixed positive integers, F(t) is a map from Rn to Rn obtained from f and its derivatives, a is a function defined on T which characterizes B(a, y).

logo First examples Hamiltonian problems Geometric B-series Modified equations Application to control problems B(utcher)-series

Taylor series of the Euler approximation

In trying to get the Taylor expansion of the Implicit Euler approximation y1 = y0 + hf(y1)

• ne gets successively

y1 = y0 + h f

• =y′

+O(h2), y1 = y0 + h f

• =y′

+h2 f ′f

• =y′′

+O(h3), y1 = y0 + h f

• =y′

+h2 f ′f

• =y′′

+h3 f ′f ′f + 1 2f ′′ f, f

• =y(3)=f ′f ′f+f ′′

f,f

• + O(h4).

logo First examples Hamiltonian problems Geometric B-series Modified equations Application to control problems B(utcher)-series

Rooted trees

Definition (Rooted trees) The set of rooted trees is recursively defined by:

1

∈ T , σ( ) = 1.

2

if (t1, . . . , tn) ∈ T n are distinct trees then t = [t1, . . . , t1

• r1

, . . . , tn, . . . , tn

• rn

] ∈ T and σ(t) = n

i=1 ri!σ(ti))ri.

The order of a tree |t| is its number of vertices. Example Tree t Order |t| 1 2 3 3 4 4 4 4 Symetry σ(t) 1 1 2 1 6 1 2 1

logo First examples Hamiltonian problems Geometric B-series Modified equations Application to control problems B(utcher)-series

Elementary differentials

Definition For each t ∈ T , the elementary differential F(t) associated with t is the mapping from Rn to Rn, defined recursively by:

1

F( )(y) = f(y),

2

F([t1, . . . , tn])(y) = f (n)(y)

• F(t1)(y), . . . , F(tn)(y)
• .

Example F( )(y) = f ′(y)f(y), F( )(y) = f ′(y)f ′(y)f(y), F( )(y) = f (3)(y)

• f(y), f(y), f(y)
• .

logo First examples Hamiltonian problems Geometric B-series Modified equations Application to control problems B(utcher)-series

B-series integrators

B-series expansions of some integrators

1

Exact solution: y(h) = y + hf() + h2

2.1(f ′f)(y) + h3 3.2(f ′′(f, f))(y) + h3 6.1(f ′f ′f))(y) + . . . = B(1/γ, y0) with

γ([t1, . . . , tn] = |t|γ(t1) · · · γ(tn).

logo First examples Hamiltonian problems Geometric B-series Modified equations Application to control problems B(utcher)-series

B-series integrators

B-series expansions of some integrators

1

Exact solution: y(h) = y + hf() + h2

2.1(f ′f)(y) + h3 3.2(f ′′(f, f))(y) + h3 6.1(f ′f ′f))(y) + . . . = B(1/γ, y0) with

γ([t1, . . . , tn] = |t|γ(t1) · · · γ(tn).

2

Explicit Euler: y + hf(y) = B(a, y) with a( ) = 1 and a(t) = 0 for all t = .

logo First examples Hamiltonian problems Geometric B-series Modified equations Application to control problems B(utcher)-series

B-series integrators

B-series expansions of some integrators

1

Exact solution: y(h) = y + hf() + h2

2.1(f ′f)(y) + h3 3.2(f ′′(f, f))(y) + h3 6.1(f ′f ′f))(y) + . . . = B(1/γ, y0) with

γ([t1, . . . , tn] = |t|γ(t1) · · · γ(tn).

2

Explicit Euler: y + hf(y) = B(a, y) with a( ) = 1 and a(t) = 0 for all t = .

3

Implicit Euler: Y = y + hf(Y) and y + hf(Y) = B(a, y) with a(t) = 1 for all t ∈ T .

logo First examples Hamiltonian problems Geometric B-series Modified equations Application to control problems B(utcher)-series

B-series integrators

B-series expansions of some integrators

1

Exact solution: y(h) = y + hf() + h2

2.1(f ′f)(y) + h3 3.2(f ′′(f, f))(y) + h3 6.1(f ′f ′f))(y) + . . . = B(1/γ, y0) with

γ([t1, . . . , tn] = |t|γ(t1) · · · γ(tn).

2

Explicit Euler: y + hf(y) = B(a, y) with a( ) = 1 and a(t) = 0 for all t = .

3

Implicit Euler: Y = y + hf(Y) and y + hf(Y) = B(a, y) with a(t) = 1 for all t ∈ T .

4

Midpoint rule: Y = y + h

2f(Y) and y + hf(Y) = B(a, y) with

a(t) = (1/2)|t|−1 for all t ∈ T .

logo First examples Hamiltonian problems Geometric B-series Modified equations Application to control problems Algebraic characterization of geometric properties

ODEs with an invariant

Assume I(ϕt(y)) = I(y) for all (t, y). The integrator B(a, y) preserves I iff for all y, I

• B(a, y)
• = I(y).

Theorem The integrator B(a, y) preserves I that for all couples (f, I) of a vector field f and a first integral I, iff a( ) = 1 and a satisfies ∀m ≥ 2, ∀(t1, . . . , tm) ∈ T m, a(t1) · · · a(tm) =

m

• j=1

a(tj ◦

• i=j

ti). The notation s ◦

j tj is used here to denote the tree obtained

by connecting the roots of all trees tj to the root of s.

logo First examples Hamiltonian problems Geometric B-series Modified equations Application to control problems Algebraic characterization of geometric properties

Algebraic conditions for geometric properties

Theorem Consider a B-series integrator B(a, y). Then, it can not preserve general invariants for all problems (f, I). it preserves quadratic invariants for all problems (f, I) with I(y) = yTCy iff ∀(t1, t2) ∈ T 2, a(t1)a(t2) = a(t1 ◦ t2) + a(t2 ◦ t1). (1) it is symplectic for all problems f = J−1∇H iff (1) holds true. it preserves H for all problems f = J−1∇H only if it is conjugate to a symplectic B-series integrator. if it is symplectic, it preserves a modified Hamiltonian for all problems f = J−1∇H. it can not be symplectic and preserve H for all problems

logo First examples Hamiltonian problems Geometric B-series Modified equations Application to control problems

Outline

1

First examples Harmonic oscillator 2-D Kepler Problem

2

Hamiltonian problems Main properties of Hamiltonian systems Symplectic maps Application to Hamiltonian systems

3

Geometric B-series B(utcher)-series Algebraic characterization of geometric properties

4

Modified equations Backward error analysis for ordinary differential equations Geometric properties of the modified equation

5

Application to control problems An optimal control problem without constraints Runge-Kutta discretization of optimality conditions Modified optimal control problem

logo First examples Hamiltonian problems Geometric B-series Modified equations Application to control problems Backward error analysis for ordinary differential equations

Concept of backward error analysis Given a differential equation ˙ y = f(y), y(0) = y0 and a numerical one-step method (say a B-series integrator) yn+1 = B(a, y) the idea of backward analysis is to find a modified differential equation ˙

• y =

fh( y) = f( y) + hf2( y) + h2f3( y) + h3f4( y) + . . . ,

• y(0) = y0

such that for tn = nh yn = y(tn)

logo First examples Hamiltonian problems Geometric B-series Modified equations Application to control problems Backward error analysis for ordinary differential equations

BACKWARD ERROR ANALYSIS ˙ y = f(y) ˙ z = fh(z) z(0), z(h), z(2h), . . . = y0, y1, y2, y3, . . . n u m e r i c a l m e t h

• d

e x a c t s

• l

u t i

• n

logo First examples Hamiltonian problems Geometric B-series Modified equations Application to control problems Backward error analysis for ordinary differential equations

Illustration with a toy-problem

Example Lotka-Volterra in normal form ˙ u = ev − 2 ˙ v = 1 − eu i.e. y′ = f(y) with f(y) = (ev − 2, 1 − eu)T. Elementary differentials F( ) = f ′f =

• ev(1 − eu)

−eu(ev − 2)

• , F(

) = f ′f ′f = −eu+vf, F( ) = f ′′(f, f) =

• ev(1 − eu)2

−eu(ev − 2)2

• .

logo First examples Hamiltonian problems Geometric B-series Modified equations Application to control problems Backward error analysis for ordinary differential equations

Modified vector field ˜ fh = f + h

• a(

) − 1 2

• f ′f + h2

2

• a(

) − a( ) + 1 6

• f ′′(f, f)

+h2

• a(

) − a( ) + 1 3

• f ′f ′f + O(h3).

Explicit Euler method ˜ f eE

h

= f − 1 2hf ′f + h2 12f ′′(f, f) + h2 3 f ′f ′f + O(h3),

logo First examples Hamiltonian problems Geometric B-series Modified equations Application to control problems Backward error analysis for ordinary differential equations

Modified vector field ˜ fh = f + h

• a(

) − 1 2

• f ′f + h2

2

• a(

) − a( ) + 1 6

• f ′′(f, f)

+h2

• a(

) − a( ) + 1 3

• f ′f ′f + O(h3).

Implicit Euler method ˜ f iE

h = f + 1

2hf ′f + h2 12f ′′(f, f) + h2 3 f ′f ′f + O(h3),

logo First examples Hamiltonian problems Geometric B-series Modified equations Application to control problems Backward error analysis for ordinary differential equations

Modified vector field ˜ fh = f + h

• a(

) − 1 2

• f ′f + h2

2

• a(

) − a( ) + 1 6

• f ′′(f, f)

+h2

• a(

) − a( ) + 1 3

• f ′f ′f + O(h3).

Midpoint Rule ˜ f mr

h

= f − h2 24f ′′(f, f) + h2 12f ′f ′f + O(h3).

logo First examples Hamiltonian problems Geometric B-series Modified equations Application to control problems Backward error analysis for ordinary differential equations

−20 −15 −10 −5 5 −14 −12 −10 −8 −6 −4 −2 2 4 u v Explicit Euler

Figure: Exact solutions of modified equations (red) versus Explicit Euler (green).

logo First examples Hamiltonian problems Geometric B-series Modified equations Application to control problems Backward error analysis for ordinary differential equations

−7 −6 −5 −4 −3 −2 −1 1 2 3 −4 −3 −2 −1 1 2 3 u v Implicit Euler

Figure: Exact solutions of modified equations (red) versus Implicit Euler (green).

logo First examples Hamiltonian problems Geometric B-series Modified equations Application to control problems Backward error analysis for ordinary differential equations

−8 −6 −4 −2 2 4 −4 −3 −2 −1 1 2 3 u v Midpoint Rule

Figure: Exact solutions of modified equations (red) versus Midpoint Rule (green).

logo First examples Hamiltonian problems Geometric B-series Modified equations Application to control problems Geometric properties of the modified equation

Theorem The modified vector field ˜ fh of an integrator B(a, y) satisfies: If Φf,h has order p, then

• fh(y) = f(y) + hpfp+1(y) + hp+1fp+2(y) + . . . .

If B(a, y) is symmetric, then f2j = 0 for all j. If B(a, y) preserves a first integral I(y), then I(y) is a first integral of the modified differential equation. If B(a, y) is symplectic for f(y) = J−1∇H(y), then ˜ fh is Hamiltonian If B(a, y) is volume-preserving for a divergence free f, then ˜ fh is divergence free.

logo First examples Hamiltonian problems Geometric B-series Modified equations Application to control problems

Outline

1

First examples Harmonic oscillator 2-D Kepler Problem

2

Hamiltonian problems Main properties of Hamiltonian systems Symplectic maps Application to Hamiltonian systems

3

Geometric B-series B(utcher)-series Algebraic characterization of geometric properties

4

Modified equations Backward error analysis for ordinary differential equations Geometric properties of the modified equation

5

Application to control problems An optimal control problem without constraints Runge-Kutta discretization of optimality conditions Modified optimal control problem

logo First examples Hamiltonian problems Geometric B-series Modified equations Application to control problems An optimal control problem without constraints

Original formulation (P)        Find (y, u) satisfying min Φ(y(1)), ˙ y(t) = f(y(t), u(t)), t ∈ (0, 1), y(0) = y0. Pontryagin formulation (H(y, p, u) := pTf(y, u)) (OC)            min Φ(y(1)), ˙ y(t) = Hp(y(t), p(t), u(t)) ˙ p(t) = −Hy(y(t), p(t), u(t)) H(y(t), p(t), u(t)) = minα H(y(t), p(t), α) y(0) = y0, p(1) = Φ′(y(1)).

logo First examples Hamiltonian problems Geometric B-series Modified equations Application to control problems An optimal control problem without constraints

Original formulation (P)        Find (y, u) satisfying min Φ(y(1)), ˙ y(t) = f(y(t), u(t)), t ∈ (0, 1), y(0) = y0. Hamiltonian formulation (H(y, p) := H(y, p, ϕ(y, p)) (OC′)        min Φ(y(1)), ˙ y(t) = Hp(y(t), p(t)) ˙ p(t) = −Hy(y(t), p(t)) y(0) = y0, p(1) = Φ′(y(1)).

logo First examples Hamiltonian problems Geometric B-series Modified equations Application to control problems Runge-Kutta discretization of optimality conditions

The Runge-Kutta discretisation (A, b) for problem (P) (DP)        min Φ(yN), yk+1 = yk + h s

i=1 bif(yki, uki),

k = 0, . . . N − 1, yki = yk + h s

j=1 aijf(ykj, ukj),

i = 1 . . . s, y0 = y0. is equivalent Hager[00] or Bonnans/Laurent-Varin[06] to the symplectic partitionned RK-discretisation for (OC′) (DOC)                min Φ(yN), yk+1 = yk + h s

i=1 biHp(yki, pki),

yki = yk + h s

j=1 aijHp(ykj, pkj),

i = 1 . . . s, pk+1 = pk − h s

i=1 biHy(yki, pki),

pki = pk − h s

j=1 ˆ

aijHy(ykj, pkj), i = 1 . . . s, y0 = y0, pN = Φ′(yN).

logo First examples Hamiltonian problems Geometric B-series Modified equations Application to control problems Modified optimal control problem

Existence of a modified problem

Question Can one interpret the numerical solution obtained as the exact solution of a modified optimal control problem of the form: ( P)    Min Φ(x(1)), ˙ x(t) =

• f(x(t), u(t)),

t ∈ (0, 1), x(0) = x0, where

• f(x, u) = f(x, u) + hf2(x, u) + h2f3(x, u) + . . . ?

logo First examples Hamiltonian problems Geometric B-series Modified equations Application to control problems Modified optimal control problem

Note that if this is possible, then problem ( P) has necessary conditions ( OC) from the Pontryagin principle, ( OC)          ˙ x(t) =

• f(x(t), u(t)),

˙ p(t) = −pT fx(x(t), u(t)), = pT fu(x(t), u(t)) ⇐ ⇒ u(t) = ϕ(x(t), p(t)), x(0) = x0, p(1) = Φ′(x(1)), where

• ϕ(x, p) = ϕ(x, p) + hϕ2(x, p) + h2ϕ3(x, p) + . . . .

is (formally) given by 0 = pT fu(x, u) ⇐ ⇒ u = ϕ(x, p).

logo First examples Hamiltonian problems Geometric B-series Modified equations Application to control problems Modified optimal control problem

The linear case

Equations (x, u ∈ Rn, A, Z, S, B ∈ Rn×n, Z T = Z, ST = S, det(B) = 0)    Min 1

2

1

0 (xTZx + uTSu)dt,

˙ x = Ax + Bu x(0) given

• r equivalently:

Hamiltonian system (H = 1

2(xTZx + uTSu) + pT(Ax + Bu))

˙ y = ˙ x ˙ p

• =
• A

−BS−1BT −Z −AT

• y := My.

logo First examples Hamiltonian problems Geometric B-series Modified equations Application to control problems Modified optimal control problem

Symplectic Runge-Kutta discretization ˙ y = Mhy, where Mh =

• Ah

−Λh −Zh −AT

h

• ,

and Λh = BS−1BT + O(h) > 0 and symmetric for small h. Thus: Λh = BhS−1BT

h .

Modified optimal control problem    Min 1

2

1

0 (xTZhz + uTSu)dt

˙ x = Ahx + Bhu x(0) given . When dimu < dimx, one can add extra control variables and recover the result (though for a stationary problem).

logo First examples Hamiltonian problems Geometric B-series Modified equations Application to control problems Further work

These preliminary results raise several questions:

1

Modified control problem: is is possible to extend to non linear problems, using the formalism of B-series?

2

What is the advantage of symplecticity for optimal control problems? Can this be seen from the modified problem, as this is the case for ordinary differential equations?

3

Conjugate points can be defined for stationary point control

• problems. Computing numerically the conjugate points

with the criteria det ∂xn

∂p0 = 0 (using the variational equation)

yields exactly the conjugate points of the modified control problem... Can symplectic methods be useful for computing conjugate points ?