Numerical analysis of time discretization of optimal control problems

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I: ORIENTATION We consider the problem of minimizing the cost function T 0 as well as ℓ(ut,yt)dt+φ(y0,yT) subject to: ˙yt = f(ut,yt), t ∈ (0,T), Control constraints: c(ut) ≤ 0, t ∈ (0,T) State constraints: g(yt) ≤ 0, t ∈ (0,T) Mixed state and control constraints: c(ut,yt) ≤ 0, t ∈ (0,T) Initial-final equality and inequality constraints: Φi(y0,yT) = 0, i = 1,...,r1, Φi(y0,yT) ≤ 0, i = r1+1,...,r. 1
This class of problems is quite large: • It includes the case of design parameters = states with zero derivative Special case: variable horizon ˙yt = Tf(ut,yt), t ∈ (0,1) • If data depend smoothly on time, we may set time as a state with derivative equal to 1 • Multiphases systems (separation of stage rockets) enter in this framework by setting “phases in parallel” • We may skip the integral cost, adding ˙yn+1 = ℓ(ut,yt). The new cost is yn+1,T +φ(y0,yT) • It includes some delay systems (H. Maurer) 2
Page 1: Numerical analysis of time discreti
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 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
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Active constraints Denote the set o
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Minimization of the Hamiltonian •
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Second-order optimality conditions
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Second-order optimality conditions
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Euler discretization ⎧ Min φ(yN)
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Geometrical hypotheses: junction po
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Junction points for the discretized
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Connection with stability analysis
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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
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Sources • J.F. Bonnans, J. Lauren
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The End ! 108
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Numerical analysis of time discretization of optimal control problems

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