Numerical analysis of time discretization of optimal control problems

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Symplectic schemes 0 I Consider the 2n×2n matrix J := . −I 0 Given H smooth: Rn ×Rn → R, the associated Hamiltonian system is can be written as (note that J −1 = −J) ˙p = −Hq(p,q); ˙q = Hp(p,q) (4) d dt p q = J −1 DH(p,q), (5) and the variational equation (linearization) may be written as d dt Zp Zq = J −1 D 2 H(p,q) Zp Zq . (6) 47
Definition 1. (i) A linear mapping A : R 2n → R 2n is called symplectic if it satisfies A ⊤ JA = J. (ii) A differentiable function ϕ : R 2n → R 2n is called symplectic at (p,q) ∈ R 2n , if the Jacobian matrix is symplectic, i.e., if ϕ ′ (p,q)Jϕ ′ (p,q) = J. (7) We say that ϕ is symplectic if it is symplectic at all points. 48
Page 1 and 2: Numerical analysis of time discreti
Page 3 and 4: This class of problems is quite lar
Page 5 and 6: The Euler discretization • N: num
Page 7 and 8: The simplest optimal time problem I
Page 9 and 10: Fuller’s problem I (work with J.
Page 11 and 12: Robbins state constrained problem I
Page 13 and 14: Robbins state constrained problem I
Page 15 and 16: Beam problem: a second-order state
Page 17 and 18: Discretization by Euler’s method
Page 19 and 20: Reduction by elimination of the con
Page 21 and 22: Shooting problem: discretized probl
Page 23 and 24: Error analysis • F: set of C 1 ma
Page 25 and 26: Second-order optimality conditions
Page 27 and 28: Link with the shooting formulation
Page 29 and 30: A priori parameterized control •
Page 31 and 32: Example: mid-point rule • If uk c
Page 33 and 34: Order computation “by hand” Den
Page 35 and 36: Order of a RK scheme I • We forma
Page 37 and 38: Exercice: order 2 yk +hyk+1,1+ 1 2h
Page 39 and 40: s Lagrangian (contracting yk +hk j=
Page 41 and 42: Partitionned Runge-Kutta (PRK) sche
Page 43 and 44: Conditions for order 1 to 3: dj :=
Page 45 and 46: Conditions for order 5 Table 5: Ord
Page 47: Number of conditions for each order
Page 51 and 52: Back to the mid-point rule • The
Page 53 and 54: Synthesis of part II: unconstrained
Page 55 and 56: III: CONTROL CONSTRAINTS (Hager, Do
Page 57 and 58: Active constraints Denote the set o
Page 59 and 60: Minimization of the Hamiltonian •
Page 65 and 66: Euler discretization ⎧ Min φ(yN)
Page 67 and 68: Geometrical hypotheses: junction po
Page 69 and 70: Junction points for the discretized
Page 71 and 72: Connection with stability analysis
Page 73 and 74: Connection with stability analysis
Page 75 and 76: High-order RK schemes and control c
Page 77 and 78: Abstract stability result (in view
Page 79 and 80: Since F1 is strongly coercive, it f
Page 81 and 82: Examples: high orders are natural
Page 83 and 84: Academic example: first-order state
Page 85 and 86: Academic example: first boundary ar
Page 87 and 88: Academic example: three boundary ar
Page 89 and 90: First-order state constraint: Min
Page 91 and 92: Third-order state constraint: Min
Page 93 and 94: In the sequel: we discuss only firs
Page 95 and 96: Condition at junction point Case of
Page 97 and 98: Homotopy path For θ ∈ [0,1]: Min
Page 99 and 100:
Proof. We have that 0 = pθ k+1fu(u
Page 101 and 102:
and similarly 0 ≥ gi(y θ k−1)
Page 103 and 104:
For ε > 0 small enough, we deduce
Page 105 and 106:
OPEN PROBLEMS • First-order state
Page 107 and 108:
Sources • J.F. Bonnans, J. Lauren
Page 109:
The End ! 108
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Numerical analysis of time discretization of optimal control problems

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