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5Russian Journal of Nonlinear Dynamics, 2018, vol. 14, no. 4, pp. 583-593. Full-texts are available at http://nd.ics.org.ru DOI: 10.20537/nd180411
MATHEMATICAL PROBLEMS OF NONLINEARITY
MSC 2010: 49K30
Optimal Bang-Bang Trajectories in Sub-Finsler Problem on the Cartan Group
Yu. Sachkov
The Cartan group is the free nilpotent Lie group of step 3, with 2 generators. This paper studies the Cartan group endowed with the left-invariant sub-Finsler norm. We adopt the viewpoint of time-optimal control theory. By Pontryagin maximum principle, all sub-Finsler length minimizers belong to one of the following types: abnormal, bang-bang, singular, and mixed. Bang-bang controls are piecewise controls with values in the vertices of the set of control parameter.
In a previous work, it was shown that bang-bang trajectories have a finite number of patterns determined by values of the Casimir functions on the dual of the Cartan algebra. In this paper we consider, case by case, all patterns of bang-bang trajectories, and obtain detailed upper bounds on the number of switchings of optimal control.
For bang-bang trajectories with low values of the energy integral, we show optimality for arbitrarily large times.
The bang-bang trajectories with high values of the energy integral are studied via a second order necessary optimality condition due to A. Agrachev and R. Gamkrelidze. This optimality condition provides a quadratic form, whose sign-definiteness is related to optimality of bangbang trajectories. For each pattern of these trajectories, we compute the maximum number of switchings of optimal control. We show that optimal bang-bang controls may have not more than 11 switchings. For particular patterns of bang-bang controls, we obtain better bounds. In such a way we improve the bounds obtained in previous works.
On the basis of results of this work we can start to study the cut time along bang-bang trajectories, i.e., the time when these trajectories lose their optimality. This question will be considered in subsequent works.
Keywords: sub-Finsler geometry, optimal control, switchings, bang-bang trajectories
Received October 24, 2018 Accepted December 03, 2018
The research leading to these results has received funding from the Ministry of Education and Science of the Russian Federation in the framework of the project RFMEFI60716X0153.
Yurii Sachkov yusachkov@gmail.com
A. K. Ailamazyan Program Systems Institute of RAS
ul. Petra I 4a, Veskovo, Pereslavl district, Yaroslavl region, 152021 Russia
1. Introduction
Sub-Finsler geometry on Lie groups has received considerable attention during last years due to its applications, especially in geometric group theory and in harmonic analysis, see articles [4, 6, 10] and introductions of [16, 18] for a broad explanation of the reasons and for several references of the state-of-the-art. To our knowledge the term sub-Finsler appears for the first time in paper [11].
In the case of step two nilpotent Lie groups and homogeneous spaces there is a good understanding of sub-Finsler structures (Heisenberg group, flat Martinet case, Grushin plane) after work [16]. On the other hand, a detailed study of the left-invariant sub-Finsler structure on the free nilpotent Lie group of step 3 with 2 generators (called the Cartan group) began in works [18, 19]. This paper continues those works.
We adopt the viewpoint of time-optimal control theory. Pontryagin maximum principle [13] implies that sub-Finsler length minimizers are of one of the following types: abnormal, bangbang, singular, or mixed (concatenations of finite number of bang and singular arcs). In this work we study optimality of bang-bang trajectories. There is a finite number of patterns of these trajectories described in [18, 19], and for each pattern we prove an upper bound on the number of switchings of bang-bang optimal control. The main tool here is a second order necessary optimality condition due to A. Agrachev and R. Gamkrelidze [15].
This work has the following structure. In Section 2 we recall the problem statement and some previously obtained results from [18, 19]. In Section 3 the second order optimality condition by Agrachev - Gamkrelidze [15] is stated. In Section 4 we prove the main results of this paper: we consider all patterns of bang-bang trajectories, and obtain upper bounds on the number of switchings of the optimal control. Results of Section 4 improve Theorem 6 [18] by giving detailed bounds on the number of switchings for all patterns of bang-bang optimal control. Finally, some concluding remarks are given in Section 5.
2. Problem statement and previous results
Consider the 5-dimensional free nilpotent Lie algebra with 2 generators, of step 3. There exists a basis L = span(X1,..., X5) in which the product rule in L takes the form
[Xi, X2] = X3, [Xi, X3] = X4, [X2X3] = X5, ad X4 = ad X5 = 0.
The Lie algebra L is called the Cartan algebra, and the corresponding connected simply connected Lie group M is called the Cartan group. We will use the following model:
M = r5
M Rx,y,z,v,w,
with the Lie algebra L modeled by left-invariant vector fields on R5
v d Vd x2 + V2 d
^ 1 — — — 77 TT~ —
dx 2 dz 2 dw''
x d_ + xd_ + + y2 d
dy 2 dz 2 dv'
d d d
3 dz X dv ^ dw'
X4 — tP")
dv
X - d A5 — "o •
dw
The product rule in the Cartan group M in this model is given in [12].
Left-invariant sub-Finsler problem on the Cartan group is stated as the following timeoptimal problem:
q = U1X1 + U2X2, q e M, u e U = {u e R2 | \\u\\^ < 1}, (2.1)
\\u\\^ = max(|ui|, |u21),
q(0) = qo = Id = (0,...,0), q(T) = qi, (2.2)
T — min . (2.3)
Problem (2.1)-(2.3) was considered first in papers [18, 19]. We recall some results of those papers.
Existence of optimal controls follows from Rashevsky-Chow and Filippov theorem [13]. Pontryagin Maximum Principle implies that optimal abnormal controls are constant. Introduce linear-on-fibers Hamiltonians hi(X) = (X,Xi), X e T*M, i = 1,..., 5. A normal extremal arc Xt, t e I = (a, 3) C [0, T] is called:
• a bang-bang arc if
card{t e I | h1h2(Xt) = 0} < to,
• a singular arc if one of the condition holds:
h1(Xt) = 0, t e I (h1-singular arc), or h2(Xt) = 0, t e I (h2-singular arc),
• a mixed arc if it consists of a finite number of bang-bang and singular arcs.
Singular controls have one of components constantly equal to 1 or —1, thus they are optimal. The fix-time attainable set along singular trajectories was explicitly described and was shown to be semi-algebraic.
Bang-bang extremal trajectories satisfy the Hamiltonian system with the Hamiltonian function H = |hi| + |h21:
'hi = —S2h3, h2 = sih3,
h3 = sih4 + S2h5, (2.4)
h4 = h5 = 0, q = siXi + S2X2.
The dual of the Lie algebra L* = T^M has Casimir functions h.4, h.5, E = + h1/7.5 — h^h^,
14
thus Hamiltonian system (2.4) has integrals h4, h5, E, and H.
The mapping (X,q) — (kX,q),k > 0, preserves extremal trajectories, thus we can consider only the reduced case
H (A) = 1.
With the use of the coordinate 9 e S1 = R/2nZ:
h\ = sgn(cos 9) cos2 9, h2 = sgn(sin 9) sin2 9, the vertical part of Hamiltonian system (2.4) reduces to the following system:
9 =
hs
| sin 291 h 3 = sh + S2h5,
(2.5)
s1 = sgn cos 9, s2 = sgn sin 9. Consider the cylinder
c = t*0 m n{H = 1}.
In work [18] it was shown that bang-bang trajectories can be represented as images of an exponential mapping: {q(t)} = Exp(A, t), X G C, t > 0. The exponential mapping is single-valued for generic X G C, and is multi-valued for certain special subsets of C, see [18].
System (2.5) is preserved by the group of symmetries of the square {(h1,h2) G R2 | |h1| + + |h2| = 1}. Thus in the study of system (2.5) we can restrict ourselves by the case h4 ^ h5 ^ 0. This group of symmetries reduces the cylinder C to the fundamental domain of the group {X G C | h4 ^ h5 ^ 0}. Further, this fundamental domain admits a stratification by invariant subsets of the Hamiltonian system (2.5):
{X G C | h4 ^ h5 ^ 0} = U4=1Ci,
C1 = {X G C | h4 >h5 > 0},
C2 = {X G C | h4 >h5 = 0},
C3 = {X G C | h4 = h5 > 0},
C4 = {X G C | h4 = h5 = 0}.
Further, we have the following stratifications:
C1 = u8 ui= C1
C1 = E- a(-h4), C21 = E-1 (-h4, -h5), C31 = E-1(-h5), C41 = E-1(-h5,h5),
C51 = E- >5 ), C1 = E-1(h5, h4), C1 = E-1(h4), Cg1 = E-1(h4, +œ),
C 2 = U 6 ui= C 2
C2 = E- a(-h4), C22 = E-1 (-h4,0), C32 = E-1(0),
C42 = E- a(0,h4), C52 = E-1(h4 ), C62 = E-1(h4, +œ),
C 3 = U 4 ui= C 3
C3 = E- a(-h4), C23 = E-1 (-h4,0), C33 = E-1(h4), C43 = E-1(h4, +œ),
C 4 = C4 u C24,
c4 = E- -1(0), C24 = E-1(0, +œ).
In paper [19] was obtained the following optimality result for bang-bang trajectories with low energy E.
Theorem 1 ([19, Theorem 2]). If a bang-bang extremal \t, t e [0, satisfies the
inequality
min(-|h41, —|h5|) <E < max(-|h4|, —|h5|) (2.6)
then it is optimal.
3. Theorem by Agrachev — Gamkrelidze
We obtain an upper bound on the number of switchings on optimal bang-bang trajectories via the following theorem due to A. Agrachev and R. Gamkrelidze.
Theorem 2 ([15, 16]). Let (q(-),u(-)) be an extremal pair for problem (2.1)-(2.3) and let X. be an extremal lift of q(-). Assume that X. is the unique extremal lift of q(-), up to multiplication by a positive scalar. Assume that there exist 0 = t0 < ti <t2 < ... <tk < rk+i = T and u°,...,uk e U such that u(-) is constantly equal to uj on (rj,Tj+i) for j = 0,...,k.
Fix j = 1,.. .,k. For i = 0,...,k let Yi = uiXi + u2>X2 and define recursively the operators
Pj = Pj-i = Id Vec(M),
Pi = Pi-i o e(ti-ti-1 )adYi-1, i = j + 1, ...,k, Pi = Pi+i o e-(ti+2-ti+l)adYi+1, i = 0,...,j — 2.
Define the vector fields
Zi = Pi(Yi), i = 0,...,k.
Let Q be the quadratic form
Q(a)= Y, aiai{Xtj, [Zi,Zi](q(tj))),
defined on the space
{k k
a = (ao, ...,ak) e Rk+i ^ ai = 0, ^ aiZi(q(j)) = 0
i=0 i=0
If Q is not negative-semide^finite, then q(-) is not optimal.
4. Bounds on the number of switchings
Now we obtain bounds on the number of switchings for bang-bang optimal trajectories Exp(A,i) with A G uf=1Ccase by case.
4.1. Case A € C1
In the case A G C1 system (2.5) has phase portrait given in Fig. 1.
4.1.1. Low values of integral E
Theorem 1 implies the following statement.
Corollary 1. If A G C1 U C2 U then the trajectory Exp(A, t), t G [0, is optimal.
4.1.2. High values of integral E
We apply Theorem 2 and obtain the following upper bounds on the number of switchings on optimal bang-bang trajectories. An example of detailed computation on the basis of Theorem 2 is given in the proof of Theorem 5 [19].
h3
Fig. 1. Phase portrait of system (2.5) in case A g C1.
Theorem 3. Let A G ui-=4Cl. Then the bang-bang trajectory Exp(A, t) with k switchings is not optimal, where k is given by the following tables:
• A G u C5 ^ Table 1,
• A G C6 ^ Table 2,
• A G C7 ^ Table 3,
• A G C8 ^ Table 4.
Remark 1. We explain now how Tables 1-4 should be read.
Consider Table 1. The first line — Start — gives the values of («i(0), w2(0)) = (sgn ^i(0), sgn h2(0)) and, if necessary, the signs of h3(0) as a lower index. For example, the first column of Table 1 corresponds to (w1(0),w2(0)) = (sgnh1(0),sgnh2(0)) = (+1, +1). The second column of Table 1 corresponds to the initial values (w1(0),w2(0)) = (sgn h1(0), sgn h2(0)) = ( —1, +1) and sgn h3(0) = +1. The second line of Table 1 gives the number of switchings k for the corresponding A g Cj u Cj such that the bang-bang trajectory Exp(A, t) is not optimal. Similar agreement is applied for Tables 2, 4.
Table 3 should be read as follows. Consider, e.g., entry N = 10 of Table 3. The sequence of signs (+, —) + +— has the following meaning:
• the signs (+, —) determine the initial control (w1(0),w2(0)) = (sgnh1(0), sgn h2(0)) = (+1, —1),
• the subsequent signs + +— determine the signs of h3(t) between switchings of control, i.e.,
- sgn hs(t) = +1, t g [0,11 ];
- sgnh3(t) = +1, t G [ti,t2];
- sgn h3(t) = —1, t G [t2,T],
where t1, t2 are switching times at which h3(t) vanishes.
The number 9 for entry N = 10 of Table 3 gives the number of switchings of a non-optimal bang-bang control.
The same agreement on reading similar tables is used in subsequent subsections.
The below cases A G4=2 C1 are considered similarly to the above case A G C1.
Table 1. A G C} U C}
Start (-,+)-
k 8 9 9 8
Table 2. A e Cl
Start (+,+)+ (-,+)+ (-,+)- (+,+)-
k 9 10 10 10 9 11
Table 3. A eC}
N
Start
N
Start
k
N
Start
N
Start
+
+
+ ++
+, -
17
-, +
+ ++
25
+ ++
+
+
+ +-
10
+
18
+
+ +-
26
+ +-
+
+
+ -+
11
+
+ -+
11
19
+
+ -+
27a
+ - + -
10
+
+
+--
12
+
+--
20
+
+--
27b
+ - + -
11
+
+
- ++
13
+
- ++
21
+
- ++
10
28
+--
+
+
- +-
14
+
- +-
12
22a
+
- + - +
12
29
- ++
+
+
- -+
10
15a
+
22b
+
- + --
11
30
- +-
+
+
15b
+
-- + -
10
23
+
- -+
10
31
- -+
16
+
24
+
32
Table 4. A e C}
Start
k 8 8 6 6
4.2. Case A € C2
In the case X e C2 system (2.5) has phase portrait given in Fig. 2.
4.2.1. Low values of integral E
Theorem 1 implies the following statement.
Corollary 2. If X e Cf U C| U Cf, then the trajectory Exp(X,t), t e [0, +ro), is optimal.
4.2.2. High values of integral E
Theorem 4. Let X e U2=4C2. Then the bang-bang trajectory Exp(X, t) with k switchings is not optimal, where k is given by the following tables:
• X e C42 ^ Table 5,
• X e C62 ^ Table 6.
k
k
k
1
9
9
8
8
8
2
9
9
9
8
3
8
9
4
8
9
7
5
9
8
6
9
7
9
8
9
8
8
Fig. 2. Phase portrait of system (2.5) in case A g C2. Table 5. A g C|
Start (+,+)+ (-,+)+ (+,+)-
k 8 9 8 8 9 8
Table 6. A e C2
Start
k 7 6 6 7
4.3. Case A € C3
In the case A G C3 system (2.5) has phase portrait given in Fig. 3.
h3
Fig. 3. Phase portrait of system (2.5) in case A g C3.
4.3.1. Low values of integral E
Theorem 1 implies the following statement.
Corollary 3. If A G C'3, then the trajectory Exp(A, t), t G [0, +ro), is optimal.
4.3.2. High values of integral E
Theorem 5. Let X £ u4=2Cf. Then the bang-bang trajectory Exp(A, t) with k switchings is not optimal, where k is given by the following tables:
• X £ C23 U C33 ^ Table 7,
• X £ C43 ^ Table 8.
Table 7. A g Cf u cf
Start (-,+)-
k 7 6 7 6
Table 8. A G Cf
Start
k 7 6 7 7
4.4. Case A € C4
In the case X £ C4 system (2.5) has phase portrait given in Fig. 4.
h3
-^-e-2-
Ci
7T 2 2L * 3-ïï 2 T
Fig. 4. Phase portrait of system (2.5) in case X £ C4.
4.4.1. Low values of integral E
Theorem 1 implies the following statement.
Corollary 4. If X £ C4, then the trajectory Exp(X, t), t £ [0, is optimal.
4.4.2. High values of integral E
Theorem 6. Let X £ C2- Then the bang-bang trajectory Exp(X,t) with k = 7 switchings is not optimal.
5. Conclusion
An obvious next question that arises after the upper bounds on the number of switchings of optimal bang-bang control is the following one: when exactly do the bang-bang trajectories lose their optimality? That is, we would like to describe the cut time along bang-bang trajectories.
We hope that this is possible by (extension of) the symmetry method applied successfully for
description of cut time in several sub-Riemannian and Riemannian problems [20-24]. This
question will be studied in forthcoming papers.
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