Introduction Definitions Regularity Second-order analysis Shooting algorithm Remarks & conclusion No-gap second-order optimality conditions for optimal control problems with state constraints Application to the shooting algorithm Audrey Hermant CMAP ´ Ecole Polytechnique and INRIA Futurs, France 13th Czech-French-German Conference on Optimization Heidelberg, September 19, 2007 Joined work with J. Fr´ ed´ eric Bonnans CFG 07 A. Hermant Second-order optimality conditions for optimal control problems with state constraints 1/25
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No-gap second-order optimality conditions for …hermant/HermantCFG07.pdfCf Kawasaki-P´ales-Zeidan (necessary cond., additional term), Malanowski-Maurer (SSC). CFG 07 A. Hermant Second-order
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Problem 2: Well-posedness of the shooting algorithm
I Useful algorithm to obtain solutions with a high precision andlow complexity.
I Principle: reduce the problem to a multi-points boundaryvalue problem and solve the finite-dimensional shootingequation using a Newton method.
I Theoretical difficulties due to pure state constraints:reformulation of the optimality conditions, the algorithm takesinto account only a part of the optimality conditions.
Is this algorithm well-posed? (Jacobian of the shooting mappinginvertible).
CFG 07 A. Hermant Second-order optimality conditions for optimal control problems with state constraints 5/25
The order of the state constraint gi , denoted by qi , is the smallestnumber of derivation of t → gi (y(t)), when y satisfies y = f (u, y),to have an explicit dependence in u.
I More precisely, the time derivatives of gi satisfy:
g(j)i (u, y) = g
(j−1)i ,y (y)f (u, y) = g
(j)i (y), j = 1, . . . , qi − 1
g(qi )i (u, y) = g
(qi−1)i ,y (y)f (u, y), g
(qi )i ,u 6≡ 0.
I For mixed control-state constraints, we set
qi := 0, g(qi )i (u, y) := ci (u, y), i = r + 1, . . . , r + s.
CFG 07 A. Hermant Second-order optimality conditions for optimal control problems with state constraints 7/25
Assume that (LIC) holds and u is continuous. Then there exists alocal change of variables z = Φ(y), v = Ψ(u, y) such that, in thenew coordinates, the dynamics and the constraints write locally{
z(qi )i = vi , i = 1, . . . , r
˙z = f (v , z)
zi ≤ 0, i = 1, . . . , r ,vi ≤ 0, i = r + 1, . . . , r + s.
CFG 07 A. Hermant Second-order optimality conditions for optimal control problems with state constraints 8/25
(A5) • For all entry/exit times τ ∈ Ti of state constraints:if qi is odd (resp. even), the derivative of order 2qi (resp.2qi − 1) of t → gi (y(t)) is discontinuous at τ .
• All essential touch points τ ∈ T essi (qi ≥ 2) are reducible,
i.e.g
(2)i (u(τ), y(τ)) < 0.
(A6) Strict complementarity on boundary arcs for state constraints:
dηi
dt> 0 a.e. on int I (gi (y)).
CFG 07 A. Hermant Second-order optimality conditions for optimal control problems with state constraints 14/25
I By (A1), Hu(u(t), y(t), p(t)) = 0 iff u(t) = Υ(y(t), p(t)).I The first-order optimality condition writes (two-points
boundary value problem):
y = f (Υ(y , p), y), y(0) = y0
−p = Hy (Υ(y , p), y , p), p(T ) = φy (y(T )).
I Shooting algorithm: Find a zero of the shooting mappingp0 → p(T )− φy (y(T )) with
y = f (Υ(y , p), y), y(0) = y0
−p = Hy (Υ(y , p), y , p), p(0) = p0.
I Constrainted case: when the structure of the trajectory isknown, introduce junction times as unknown of the shootingmapping, as well as jump parameters of the costate for stateconstraints (Bryson et al. 63).
CFG 07 A. Hermant Second-order optimality conditions for optimal control problems with state constraints 20/25
I By (A1), Hu(u(t), y(t), p(t)) = 0 iff u(t) = Υ(y(t), p(t)).I The first-order optimality condition writes (two-points
boundary value problem):
y = f (Υ(y , p), y), y(0) = y0
−p = Hy (Υ(y , p), y , p), p(T ) = φy (y(T )).
I Shooting algorithm: Find a zero of the shooting mappingp0 → p(T )− φy (y(T )) with
y = f (Υ(y , p), y), y(0) = y0
−p = Hy (Υ(y , p), y , p), p(0) = p0.
I Constrainted case: when the structure of the trajectory isknown, introduce junction times as unknown of the shootingmapping, as well as jump parameters of the costate for stateconstraints (Bryson et al. 63).
CFG 07 A. Hermant Second-order optimality conditions for optimal control problems with state constraints 20/25
Let (u, y) be a local solution of (P) satisfying (A1)-(A8). Thenthe shooting algorithm is well-posed in the neighborhood of (u, y)(invertible Jacobian of the shooting mapping), iff:(i) State constraints of order qi ≥ 3 have no boundary arc;(ii) The no-gap sufficient second-order condition
Q(v)−r∑
i=1
∑τ∈T ess
i
[ηi (τ)](g
(1)i ,y (y(τ))zv (τ))2
g(2)i (u(τ), y(τ))
> 0, ∀v ∈ C (u) \ {0}
holds, i.e. (u, y) satisfies the quadratic growth condition.
CFG 07 A. Hermant Second-order optimality conditions for optimal control problems with state constraints 22/25
I We obtain no-gap second-order optimality conditions for purestate constraints of arbitrary orders and mixed control-stateconstraints, and a characterization of the well-posedness ofthe shooting algorithm.
I Outlook: Numerical applications of the shooting algorithm,using homotopy/continuation methods to automaticallydetect the structure of the trajectory and initialize some of theshooting parameters.
Reference of this talk: J.F. Bonnans, A.H., Second-order analysis for
optimal control problems with pure and mixed state constraints, INRIA
Research Report 6199 (2007), submitted.
http://hal.inria.fr/inria-00148946
CFG 07 A. Hermant Second-order optimality conditions for optimal control problems with state constraints 25/25