Numerical analysis of time discretization of optimal control problems

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Optimality systems (original and discretized problem) For the original problem: ⎧ ⎪⎨ ⎪⎩ ˙yt = f(ut,yt), −˙¯pt = pfy(ut,yt), t ∈ (0,T); 0 = pfu(ut,yt), t ∈ (0,T); y0 = y 0 ; pT = φ ′ (¯yT), and for the discretized problem ⎧ yk+1−yk ⎪⎨ hk pk −pk+1 = f(uk,yk), = pk+1fy(uk,yk), k = 0,...,N −1, ⎪⎩ hk 0 = pk+1fu(uk,yk), k = 0,...,N −1, y0 = y 0 ; pN = φ ′ (yN). 17
Reduction by elimination of the control: continuous problem Reduction hypothesis: ū continous, and Huu[¯pt](ūt,¯yt) = ¯ptfuu(ūt,¯yt) uniformly invertible. Then by the IFT (Implicit function theorem), “locally in time” Reduced optimality system: ⎧ ⎨ ⎩ Hu[p](u,y) = 0 iff u = Υ(p,y). ˙yt = f(Υ(yt,pt),yt), −˙¯pt = pfy(Υ(yt,pt),yt), t ∈ (0,T); y0 = y 0 ; pT = φ ′ (¯yT), 18
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: Discretization by Euler’s method
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 and 48: Number of conditions for each order
Page 49 and 50: Definition 1. (i) A linear mapping
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
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High-order RK schemes and control c
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Abstract stability result (in view
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Since F1 is strongly coercive, it f
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Examples: high orders are natural
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Academic example: first-order state
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Academic example: first boundary ar
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Academic example: three boundary ar
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First-order state constraint: Min
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Third-order state constraint: Min
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In the sequel: we discuss only firs
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Condition at junction point Case of
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Homotopy path For θ ∈ [0,1]: Min
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Proof. We have that 0 = pθ k+1fu(u
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and similarly 0 ≥ gi(y θ k−1)
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For ε > 0 small enough, we deduce
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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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