6.8 Study of abnormality for particular cases
6.8.3 Underactuated by one input
Now consider the control–affine system with n − 1 input vector fields, ˙
144 6.8. Study of abnormality for particular cases
where the dimension of Q is n.
The constraint submanifolds for abnormality are given by N0[0] = n (Λ, u) ∈ T∗(T Q) × U | HYV s = 0 , s = 1, . . . , n − 1 o , N1[0] = n(Λ, u) ∈ N0[0]| H[Z,YV s ]= 0 , s = 1, . . . , n − 1 o , N2[0] = n (Λ, u) ∈ N1[0]| H[Z,[Z,YV s ]]+ urHhYr: YsiV = 0 , s = 1, . . . , n − 1 o . As the codimension of the distribution of the input vector fields is 1, in a neighbourhood of every (Λ, u) ∈ N1[0] there exists a vector field Y on Q complementary to the subspace spanned by the control vector fields. Then, for (Λ, u) ∈ N1[0] ⊆ annYV × U ,
HhYr: YsiV(Λ) = crsHYV(Λ), where hYr: Ysi = n−1 X i=1 cirsYi+ crsY and cirs, crs∈ C∞(Q).
At present, keeping in mind that the matrix crsis a function on Q, the cases to be considered
are:
1. The matrix crs has maximum rank on N1[0] and HYV(Λ) 6= 0, so that the controls are
completely determined on N2[0]and the algorithm stops.
2. The matrix crshas maximum rank on N1[0] and HYV(Λ) = 0, so that the controls cannot
be determined on N2[0], but the momenta corresponding with the velocities are zero be- cause of the rank ofY and the choice of Y . Due to Hamilton’s equations (6.5.11), the momentum is zero. Thus there are no abnormal extremals.
3. The matrix crshas nonzero rank, the rank is not maximum on N [0]
1 and HYV(Λ) 6= 0, so
that some controls are determined but some conditions must still be stabilized.
4. The matrix crs has nonzero rank, the rank is not maximum on N1[0] and HYV(Λ) = 0,
so that the controls cannot be determined, but the momentum corresponding with the velocities is zero. Due to Hamilton’s equations (6.5.11) the momentum is zero. Thus there are no abnormal extremals.
5. The matrix crsis identically zero on N1[0], so that hY : Y i ⊆ Y . The controls remain un-
determined on N2[0]and the algorithm must go on with the stabilization of H[Z,[Z,YV s ]]= 0
with s = 1, . . . , n − 1.
6. The matrix crsis identically zero on a submanifold S of N1[0], so that hY : Y i ⊆ Y . The
controls remain undetermined on N2[0]and the algorithm must go on with the stabilization of H[Z,[Z,YV
s ]] = 0 with s = 1, . . . , n − 1 and also with the stabilization of the constraints
that determine implicitly the submanifold S.
The associated vector–valued quadratic form for every x ∈ Q is Bx:Yx×Yx −→ TxQ/Yx
(w1, w2) 7−→ πYx(hW1: W2i)
as defined in (6.6.18). Its associated matrix is a (n−1)×(n−1)–matrix given by (crs), because
dim TxQ/Yx= 1 and there are n − 1 input vector fields. For any λ ∈ (TxQ/Yx)∗ ' annYx,
we have the real quadratic form
(λB)x:Yx×Yx −→ R
(w1, w2) 7−→ hλ, hW1: W2i(x)i = Brsar1as2,
where Wi = ariYrand Brs= hλ, hYr: Ysii = crshλ, Y i.
The above different cases in terms of the vector–valued quadratic forms are:
1. If the matrix crs has maximum rank on N1[0] and HYV(Λ) 6= 0, then Bx is definite or
indefinite on N1[0].
2. If the matrix crshas maximum rank on N1[0]and HYV(Λ) = 0, then Bxcan be strongly
semidefinite or essentially indefinite, having in mind Remark 6.8.4. The zero momen- tum makes Bx essentially indefinite and a nonzero momentum can make Bx strongly
semidefinite, although this momentum does not satisfy the hypotheses in this case. 3. If the matrix crshas nonzero rank, the rank is not maximum on N1[0]and HYV(Λ) 6= 0,
then Bxis either strongly semidefinite or essentially indefinite on N1[0], depending on the
rank of the matrix crsand the sign of the eigenvalues.
4. If the matrix crshas nonzero rank, the rank is not maximum on N1[0]and HYV(Λ) = 0,
then Bxis strongly semidefinite or essentially indefinite.
5. If hY : Y i ⊆ Y , then Bxis essentially indefinite on N1[0]. To be more precise, it is zero.
6. If the matrix crsis zero on a submanifold S of N1[0], Bx is essentially indefinite on S,
and on N1[0]− S the previous cases arise again.
Whenever Bx is essentially indefinite, every (λ B)x is indefinite. The property (i) in Lemma
C.1.2 makes us believe that those abnormal extremals will not be optimal. The idea is to estab- lish a connection between the tangent perturbation vectors and the image of the vector–valued quadratic form. Then, due to (i) in Lemma C.1.2 the necessary separation condition for op- timality will not be satisfied if Bx is essentially indefinite. This result has not been proved
yet.
Conjecture 6.8.5. Let ΣF = (Q, ∇,Y , U, F, I) be an optimal control problem. If (Υ, u): I →
T Q × U is an abnormal optimal solution, then the vector–valued quadratic form B(τQ◦Υ)(t)is
either zero or semidefinite at everyt ∈ I.
This conjecture gives a necessary condition for having abnormal extremals. It does not discard the existence of normal lifts associated to these extremals.
146 6.8. Study of abnormality for particular cases
This conjecture makes sense to Proposition 4.5.2 and results in [Hirschorn and Lewis 2002]. There, among other hypotheses if the vector–valued quadratic form is indefinite, then the con- trol–affine system is STLC. In Proposition 4.5.2, a necessary condition for abnormality is not to be STLC.
Here we have studied carefully the constraint algorithm, twice applying the tangency con- ditions. The final submanifolds cannot be always obtained at this point. That is why a new research has already been started to connect the constraint algorithm with the high–order Max- imum Principle [Krener 1977], in order to give a geometric version of the results stated by Krener [1977] through the constructions of mappings related to the vector–valued quadratic form at each step of the algorithm.
Strict abnormal extremals in
nonholonomic and kinematic control
systems
W
e continue the study of abnormality in optimal control problems for mechanical control systems. The approach considered in this chapter consists of taking advantage of particular nonholonomic control mechanical systems, which are equivalent to kinematic control system. For more details in that equivalence see, for instance [Bloch 2003, Bullo and Lewis 2005a;c, Mu˜noz-Lecanda and Y´aniz-Fern´andez 2008].The control system in subRiemannian geometry [Montgomery 1995; 2002] is control–linear, as with kinematic systems. As mentioned by Liu and Sussmann [1994b; 1995] and Mont- gomery [1994] there exist local strict abnormal minimizers for the problem of the shortest paths. As the kinematic control systems can be equivalent to nonholonomic control systems, we are going to use the strict abnormal minimizers in subRiemannian geometry to character- ize the abnormal extremals for certain mechanical control systems. Dealing with a kinematic system is by far easier than dealing with a mechanical control system, which is either con- trol–affine or nonlinear, because there is no drift and because more information is known about the control–linear systems [Liu and Sussmann 1995, Montgomery 1994; 1995; 2002].
Firstly, we investigate whether it is feasible to obtain any connection between the optimal control problems associated to the two control systems. Then, Pontryagin’s Maximum Principle will be used to connect the abnormal extremals of both optimal control problems [Barbero- Li˜n´an and Mu˜noz Lecanda 2008c].
This chapter is organized as follows: In §7.1 the different definitions and results associated with the optimal control problems for nonholonomic and kinematic systems are described, in particular, the possible equivalence between both problems. The Hamiltonian problems for both control problems are stated in §7.2 so as to apply the Maximum Principle. Definition 7.2.1 about the different kinds of extremals for the mechanical case is especially important as it gives a justification of the study made in [Bullo and Lewis 2005b] and reviewed in §6.4. In §7.2.4 it is shown how to use the strict abnormal minimizers in subRiemannian geometry to characterize the extremals for the corresponding optimal control problem with nonholonomic mechanical system by means of an example where there exists a local strict abnormal minimizer for the time–optimal control problem for the mechanical system.
148 7.1. Optimal control problem: nonholonomic versus kinematic
7.1
Optimal control problem with nonholonomic mechanical sys-
tems versus kinematic systems
First, we study the nonholonomic and kinematic control systems from the viewpoint of control theory. Then, we study them from the approach of optimal control theory in §7.1.2.