Geometric Numerical Integration of Hamiltonian systems: application to some optimal control problems Philippe Chartier 1 1 IPSO 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 First examples 1 Harmonic oscillator 2-D Kepler Problem Hamiltonian problems 2 Main properties of Hamiltonian systems Symplectic maps Application to Hamiltonian systems Geometric B-series 3 B(utcher)-series Algebraic characterization of geometric properties Modified equations 4 Backward error analysis for ordinary differential equations Geometric properties of the modified equation Application to control problems 5 An optimal control problem without constraints logo Runge-Kutta discretization of optimality conditions Modified optimal control problem
First examples Hamiltonian problems Geometric B-series Modified equations Application to control problems Outline First examples 1 Harmonic oscillator 2-D Kepler Problem Hamiltonian problems 2 Main properties of Hamiltonian systems Symplectic maps Application to Hamiltonian systems Geometric B-series 3 B(utcher)-series Algebraic characterization of geometric properties Modified equations 4 Backward error analysis for ordinary differential equations Geometric properties of the modified equation Application to control problems 5 An optimal control problem without constraints logo Runge-Kutta discretization of optimality conditions Modified optimal control problem
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 ( p 2 + ω 2 q 2 ) and the corresponding Hamiltonian system � − ∂ H − ω 2 q ˙ p = = ∂ q . ∂ H ˙ q = = p ∂ p The exact trajectory is known to be an ellipse in the logo phase-space ( p , q ) depending on initial conditions ( p 0 , q 0 ) .
First examples Hamiltonian problems Geometric B-series Modified equations Application to control problems Harmonic oscillator Three elementary methods Explicit Euler � p n + 1 p n + h ( − ω 2 q n ) ( − h ω 2 ) q n = = p n + q n + 1 = q n + h ( p n ) = hp n + q n Implicit Euler � − h ω 2 1 p n + h ( − ω 2 q n + 1 ) p n + 1 = = 1 + h 2 ω 2 p n + 1 + h 2 ω 2 q n h 1 q n + 1 = q n + h ( p n + 1 ) = 1 + h 2 ω 2 p n + 1 + h 2 ω 2 q n logo
First examples Hamiltonian problems Geometric B-series Modified equations Application to control problems Harmonic oscillator Three elementary methods Midpoint rule � p n + 1 p n + h ( − ω 2 q n + 1 + q n = ) 2 q n + h ( p n + 1 + p n q n + 1 = ) 2 i.e. 1 − h 2 ω 2 / 4 − h ω 2 p n + 1 = 1 + h 2 ω 2 / 4 p n + 1 + h 2 ω 2 / 4 q n 1 − h 2 ω 2 / 4 h q n + 1 = 1 + h 2 ω 2 / 4 p n + 1 + h 2 ω 2 / 4 q n logo
First examples Hamiltonian problems Geometric B-series Modified equations Application to control problems Harmonic oscillator Computed trajectories for the three methods Explicit Euler (green), Midpoint Rule (red), Implicit Euler (blue) 3 2 1 0 v −1 −2 logo −3 −3 −2 −1 0 1 2 3 u
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 � p n + 1 � � p n � = M ( h ω ) q n + 1 q n with, for the Explicit Euler method � 1 � ( − h ω 2 ) M ( h ω ) = 1 1 and eigenvalues λ 1 , 2 = ( 1 ± ih ω ) . Hence, the energy grows like ( 1 + h 2 ω 2 ) 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 � p n + 1 � � p n � = M ( h ω ) q n + 1 q n with, for the Implicit Euler method � � − h ω 2 1 1 + h 2 ω 2 1 + h 2 ω 2 M ( h ω ) = h 1 1 + h 2 ω 2 1 + h 2 ω 2 and eigenvalues λ 1 , 2 = ( 1 ± ih ω ) − 1 . Hence, the energy logo decreases like ( 1 + h 2 ω 2 ) − n / 2 .
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 � p n + 1 � � p n � = M ( h ω ) q n + 1 q n with, for the Midpoint rule 1 − h 2 ω 2 / 4 − h ω 2 1 + h 2 ω 2 / 4 1 + h 2 ω 2 / 4 M ( h ω ) = 1 − h 2 ω 2 / 4 h 1 + h 2 ω 2 / 4 1 + h 2 ω 2 / 4 and eigenvalues λ 1 , 2 of modulus one. Hence, the energy is logo constant.
First examples Hamiltonian problems Geometric B-series Modified equations Application to control problems 2-D Kepler Problem Equations Consider the follwoing Hamiltonian 1 1 2 [( p 1 ) 2 + ( p 2 ) 2 ] − H ( p 1 , p 2 , q 1 , q 2 ) = � ( q 1 ) 2 + ( q 2 ) 2 , = T ( p ) + V ( q ) . and the corresponding System � − ∂ H − V ′ ( q ) ˙ p = = ∂ q q = − V ′ ( q ) ⇒ ¨ ⇐ ∂ H ˙ = = q p ∂ p The exact trajectory is known to be an ellipse in the logo phase -space ( p , q ) depending on initial conditions.
First examples Hamiltonian problems Geometric B-series Modified equations Application to control problems 2-D Kepler Problem Computed trajectories and energies Euler explicit/implicit 3 2 1 q2 0 −1 −2 logo −3 −3 −2.5 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 q1
First examples Hamiltonian problems Geometric B-series Modified equations Application to control problems 2-D Kepler Problem Computed trajectories and energies Midpoint Rule 2 1.5 1 0.5 q2 0 −0.5 −1 −1.5 logo −2 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 q1
First examples Hamiltonian problems Geometric B-series Modified equations Application to control problems 2-D Kepler Problem Computed trajectories and energies Energy 0 −1 −2 −3 −4 H −5 −6 −7 logo −8 0 50 100 150 200 250 300 350 400 450 Time
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 First examples 1 Harmonic oscillator 2 -D Kepler Problem Hamiltonian problems 2 Main properties of Hamiltonian systems Symplectic maps Application to Hamiltonian systems Geometric B-series 3 B(utcher)-series Algebraic characterization of geometric properties Modified equations 4 Backward error analysis for ordinary differential equations Geometric properties of the modified equation Application to control problems 5 An optimal control problem without constraints logo Runge-Kutta discretization of optimality conditions Modified optimal control problem
First examples Hamiltonian problems Geometric B-series Modified equations Application to control problems Main properties of Hamiltonian systems For p and q in R d , and H a smooth scalar function, one can define the following Hamiltonian system � − ∂ H ˙ p = ∂ q . ∂ H ˙ q = ∂ q Denoting � p � � � 0 I d y = , J = q − I d 0 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 ϕ t is defined as the function which associates at time t the exact solution of ˙ y = f ( y ) with initial condition y ( 0 ) = y 0 . Theorem The flow of an Hamiltonian system preserves the value of the Hamiltonian. Proof : Since J is skew -symmetric, along any exact trajectory one has: dt H ( ϕ t ( y )) = ∂ H d dy dt = ( ∇ H ) T J − 1 ∇ H = 0 . ∂ y 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 ( ϕ t ( A )) = Vol ( A ) for any compact set A ⊂ R n . Proof : Ψ t ( y ) = ∂ϕ t ∂ y ( y ) is solution of dt Ψ t ( y ) = ∂ f d ∂ y ( ϕ t ( y ))Ψ t ( y ) , Ψ 0 ( y ) = I R n . � � Hence d dt det (Ψ t ( y )) = det (Ψ t ) Tr (Ψ − 1 ∂ y f ( ϕ t ( y )) Ψ t ) = 0 , t � � � � dz = det (Ψ t ( y )) dy = det (Ψ 0 ( y )) dy = dy . and logo ϕ t ( A ) A A A
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 Tr ( ∂ f div f = ∂ y ) Tr ( J − 1 ∇ 2 H ) = Tr ( ∇ 2 HJ − T ) = − Tr ( J − 1 ∇ 2 H ) = = − div f = 0 . logo
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