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5Russian Journal of Nonlinear Dynamics, 2020, vol. 16, no. 2, pp. 355-367. Full-texts are available at http://nd.ics.org.ru DOI: 10.20537/nd200209
MATHEMATICAL PROBLEMS OF NONLINEARITY
MSC 2010: 93C10, 49K30
Optimal Bang-Bang Trajectories in Sub-Finsler Problems on the Engel Group
The Engel group is the four-dimensional nilpotent Lie group of step 3, with 2 generators. We consider a one-parameter family of left-invariant rank 2 sub-Finsler problems on the Engel group with the set of control parameters given by a square centered at the origin and rotated by an arbitrary angle. We adopt the viewpoint of time-optimal control theory. By Pontryagin's 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 parameters.
We describe the phase portrait for bang-bang extremals.
In previous work, it was shown that bang-bang trajectories with low values of the energy integral are optimal for arbitrarily large times. For optimal bang-bang trajectories with high values of the energy integral, a general upper bound on the number of switchings was obtained.
In this paper we improve the bounds on the number of switchings on optimal bang-bang trajectories via a second-order necessary optimality condition due to A. Agrachev and R. Gamkre-lidze. This optimality condition provides a quadratic form, whose sign-definiteness is related to optimality of bang-bang 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 9 switchings. For particular patterns of bang-bang controls, we obtain better bounds. In such a way we improve the bounds obtained in previous work.
On the basis of the 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 work.
Keywords: sub-Finsler problem, Engel group, bang-bang extremal, optimality condition
Received March 23, 2020 Accepted May 15, 2020
This work was carried out with the financial support of the state, represented by the Ministry of Science and Higher Education of the Russian Federation (unique identifier of the project RFMEFI60419X0236).
Yuri Sachkov [email protected]
Control Processes Research Center
A. K. Ailamazyan Program Systems Institute of RAS
Pereslavl-Zalessky, Russia
Yu. Sachkov
1. Introduction
Sub-Finsler geometry is a natural generalization of the sub-Riemannian one. A sub-Rie-mannian geometry on a smooth manifold M is given by a vector distribution A on M and an inner product in A. A sub-Finsler structure is defined by a norm in A.
In recent years there has been a noticeable interest in sub-Finsler geometry in view of its applications in geometric group theory [1], spaces with length metrics [2], and control theory [3]. An important question of both sub-Finsler and sub-Riemannian geometry is the description of length minimizers and spheres, and the natural simplest cases here are nilpotent structures. The left-invariant sub-Finsler problem on the Heisenberg group was studied in [7, 8]. Nilpotent lo sub-Finsler structures in the Martinet and Grushin cases were studied in [4]. Left-invariant sub-Finsler problems on the Engel and Cartan groups were studied via convex trigonometry techniques in [5]. Moreover, these techniques were applied to generalizations of a series of classical optimization problems to the sub-Finsler case [6].
The next natural case is the Engel group, the 4-dimensional nilpotent Lie group of step 3 and rank 2. A study of a one-parameter family of sub-Finsler structures on the Engel group with the set of control parameters given by a square was started in [9]. The sub-Finsler problems were considered as time-optimal control problems. Pontryagin's maximum principle was applied, and extremal trajectories were described. Some upper bounds on the number of smooth pieces of optimal bang-bang and mixed trajectories were presented.
In this note we continue that work. We describe the phase portrait for bang-bang extremals and present detailed optimality conditions which improve the bounds on the number of smooth pieces of optimal bang-bang trajectories given in [9].
2. Problem statement
The Engel algebra g is the 4-dimensional nilpotent Lie algebra with 2 generators, of step 3. In a standard basis of the Engel algebra g = span(/i, f2,/3, f4) the product table has the form [f1, f2] = f3, [f1, /3] = /4, ad f4 = 0. The simply connected Lie group G with the Lie algebra g is called the Engel group. In some coordinates G = Rt,y,zv the Engel algebra is realized by left-invariant vector fields on G:
d yd d ^ x d ^ x2 + y2 d
dx 2 dz dy 2 dz 2 dv
d d d
dz dv dv
Define vector fields G [0,^/4])
Xi = cos / + sin /, X2 = - sin / + cos $/2, X3 = /3, X4 = /4.
Consider the following family of sub-Finsler problems on the Engel group G [0,^/4]):
q = «1X1 + U2X2, q G G, u G U, (2.1)
U = {u G R2 | \\u\\oo = max(U11, «21) < 1}, (2.2)
q(0) = qo = Id, q (T )= qb (2.3)
T min . (2.4)
The existence of optimal controls follows from the Rashevsky-Chow and Filippov theorems [10].
3. Pontryagin's maximum principle
Introduce Hamiltonians hi (A) = (A, Xi), A G T *G, i = 1,..., 4, and the corresponding Hamiltonian vector fields hi G Vec(T*M).
Theorem 1 ([10, 11]). If a control u(t) and the corresponding trajectory q(t), t G [0,T], are optimal, then there exist a curve At G T*^ G and a number v ^ 0 for which the following conditions hold:
At = ui (t)hi(At) + U2(t)h2 (At), (3.1)
Ui (t)hi(At) + U2(t)h2 (At) = H (At) = (|hi| + |h21)(At),
At = 0,
H (At) + v = 0.
The Hamiltonian system (3.1) has 3 integrals — Casimir functions on the Lie coalgebra g*: h4, E = h2/2 — (sin ph1 + cos ph2)h4, and the Hamiltonian H.
4. Abnormal trajectories
Let v = 0.
Then the optimal abnormal controls are u(t) = ±(tanp, 1).
5. Classes of normal extremal arcs
Let —v = H(At) > 0. An extremal arc At, t G I = (a,fi) C [0,T], is called:
• a bang-bang arc if card{t G I | h1h2(At) = 0} < to,
• a singular arc if one of the following conditions holds: h1(At) = 0 or h2(At) = 0,
• a mixed arc if it consists of a finite number of bang-bang and singular arcs.
Remark 1. If ^(At)^^) = 0, then u^t}^^) = si := sgnhiA)^^^. All singular arcs are optimal [9].
6. Bang-bang flow
If h1h2(At)|(a,p) = 0, then u(t)|(a ^) = (s1,s2), thus bang-bang extremals satisfy the following Hamiltonian system with the maximized Hamiltonian H = + h21:
'h 1 = —S2h3, h 2 = S1h3,
< h3 = (s1 cos p — s2 sin p)h4, (6.1)
h 4 = 0,
q = S1X1 + S2X2.
In view of the symmetry (A, q) ^ (kA, q), k > 0, we assume in the sequel that H(At) = 1. Consider the cylinder
c = g* n{H = 1}.
In [9] it was shown that bang-bang trajectories can be represented as images of an exponential mapping: {q(t)} = Exp(A,t), A G C, t > 0. The exponential mapping is single-valued for generic A G C, and is multi-valued for certain special subsets of C.
Let us parameterize the square {(hi,h2) | H(A) = 1} by an angle coordinate 6 £ R/2nZ: hi = sgn(cos 6) cos2 6, h2 = sgn(sin 6) sin2 6. Then the vertical part of system (6.1) takes the form
9 =
h3
e^f,
(6.2)
sin 26|'
h3 = (s1 cos p — s2 sin p)h4, si =sgncos 6, s2 = sgnsin 6.
System (6.2) is preserved by the group of symmetries {Id,^1} = Z2, where
ei: (hi ,h2 ,h3, hi) ^ (—hi, —h2 h, —hi), (Si,S2) ^ (—Si, —S2).
We factorize by action of this group and reduce system (6.2) to the fundamental domain of this group {(hi ,h2, h3, hi) £ R4 | hi ^ 0}.
7. Phase portrait of system (6.2)
We consider system (6.2) as an oscillator, with the full energy
h2 h2
E = -±- (sin (phi + cos iph2)h4 = y + U(9)
and the potential energy
U(6) = —(sin phi + cos ph2)h4 = —(si sin p cos2 6 + s2 cos p sin2 6)h4. The function U(9) is C1-smoot.h at 9 = ™ and analytic elsewhere.
7.1. Case 1): hA > 0
7.1.1. Subcase 1a): ^ = 0
The phase portrait of system (6.2) is drawn as a set of curves hi = ±y/2 (E-U(9)), see Fig. 1.
h3
Fig. 1. Phase portrait of system (6.2) in case 1a).
We have a decomposition of a section of the cylinder C = g* n {H = 1} into domains with qualitatively different trajectories of system (6.2):
{X e C | h4 > 0} = uf=1Ci,
Ci = E-1(-h4), C2 = E-1(-h4,0), C3 = E-1 (0), C4 = E-1(0,h4), C5 = E-1(h4), Co = E-1(h4, +ro).
7.1.2. Subcase 1b): p = n/4
The phase portrait of system (6.2) is shown in Fig. 2.
We have a decomposition of a section of the cylinder C = g* n {H = 1}:
[X e C | h4 > 0} = U4=1CV
Ci = E-1(-
C3 = E
1
C2 = E-1(-f C4 = E-1(h4/у
h3
7.1.3. Subcase 1c): p G (0,n/4)
The phase portrait of system (6.2) is shown in Fig. 3.
h3
Fig. 3. Phase portrait of system (6.2) in case 1c). RUSSIAN JOURNAL OF NONLINEAR DYNAMICS, 2020, 16(2), 355 367_
We have a decomposition of a section of the cylinder C = g* n [H = 1}: [X e C | h4 > 0} = U8i=1C8,
Ci = E-1(-h4 cos p), C2 = E-1(-h4 cos p, -h4 sin p), C3 = E-1(-h4 sin p), C4 = E-1(-h4 sin p, h4 sin p), C5 = E-1(h4 sin p), C6 = E-1 (h4 sin p, h4 cos p), C7 = E-1(h4 cos p), C8 = E-1 (h4 cos p,
7.2. Case 2): h4 = 0
In this case the phase portrait of (6.2) is shown in Fig. 4.
hs Co
Ci
7T 7T 7T 3tt
2 2 2
Fig. 4. Phase portrait of system (6.2) in case 2).
The critical level line C1 = E-1(0) consists of fixed points, and the domain of regular values of energy is C2 = E-1(0, We have
[X e C I h4 = 0} = C1 U C2.
8. Optimality of bang-bang trajectories
8.1. Bang-bang trajectories with low energy E
In [9] the following optimality result was obtained for bang-bang trajectories with low energy E.
Theorem 2 ([9]). If a bang-bang extremal Xt,t e [0, satisfies the conditions
p e [0,^/4), -Ih4I cos p < E ^ -Ih41 sinp,
then it is optimal.
8.2. Bang-bang trajectories with high energy E
Further, in [9] the following optimality result was obtained for bang-bang trajectories with high energy E.
Theorem 3 ([9]). If (p e [0,^/4) and -\h4I sinp < E) or p = n/4, then optimal trajectories have not more than 10 switchings.
The main goal of this paper is to obtain detailed optimality results for each pattern of bang-bang trajectory and to improve Theorem 3.
9. The Agrachev — Gamkrelidze theorem
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 4 ([4, 12]). Let (q(-),u(-)) be an extremal pair for problem (2.1)-(2.4) 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 < t1 < t2 < ... < tk < Tk+1 = T and u0,...,uk G U such that u(-) is constantly equal to uj on (Tj,Tj+1) for j = 0,...,k.
Fix j = 1,.. .,k. For i = 0,...,k let Yi = u\X1 + u2X2 and define recursively the operators
Pj = Pj-i = Id ,
j J Vec(M)
Pi = Pi-1 O e(ti-ti-l)adYi-1, i = j + i,...,k, Pi = Pi+1 o e-(ti+2-ti+i)adYi+i, i = 0,...,j - 2.
Define the vector fields
Zi = Pi(Yi), i = 0,...,k.
Let Q be the quadratic form
Q(a)= aiai(Xtj, [Zi,Zi](q(tj))),
defined on the space
( k k \
W = i a = (ao,...,ak) G Rk+1 | ^a = 0, ^a.iZi(q(tj)) = 0 I.
If Q is not negative-semidefinite, then q(-) is not optimal.
We will check the sign of the quadratic form Q\W via the following test. Consider a quadratic form
n
A.(x) — ^ ^ a%jXiXj, aij — aji, Xi G i,j=l
Denote a minor
In 12 ... ip Ul i2 ... ip.
Theorem 5 ([13]). A quadratic form A(x) is negative-semidefinite iff the following inequalities hold:
(i1 i2 ... iP\
\ ^ 0, 1 ^ i1 <i2 < ... <ip ^ n, p = 1,2,...,n. i1 i2 ... ip)
ailil aili2 . . ah ip
ai2il ai2i2 . . ai2 ip
aiph aipi2 . . aipip
10. Bounds on the number of switchings on optimal bang-bang trajectories
We apply necessary optimality conditions for bang-bang trajectories of A. A. Agrachev and R. V. Gamkrelidze given by Theorem 4 and improve the bound of Theorem 3.
10.1. Case 1a): h4 > 0, p = 0
Theorem 6. Let h4 > 0, p = 0, and X G uf=1Ci. Then the bang-bang trajectory Exp(A, t) is optimal.
Proof. Apply Theorem 2. □
Theorem 7. Let h4 > 0, p = 0, and X G Ue=4Ci. Then the bang-bang trajectory Exp(X, t) with k switchings is not optimal, where k is given by the following tables:
• X G C4 ^ Table 1,
• X G C5 ^ Table 2,
• X G Ce ^ Table 3.
Table 1. X G C4
Start (+,+)+ (-,+)-
k 8 9 7 7 9 7
Table 2. X G C5
Start (-,+)+ (+,+)+ (+,-)+ (-,-)+ (-,+)- (+,+)- (+,-)-
-- 7 8 7 7 6 5 8 7
- + 8 8 7 7 6 5 8 8
+ - 8 5 6 8 7 8 8 8
+ + 8 5 6 7 7 8 8 7
Table 3. X G C6
Start
k 6 5 6 5
Remark 2. We explain now how Tables 1-3 should be read. Consider Table 1. The first line — Start — gives the values of (w1(0),w2(0)) = (sgnhi(0),sgnh2(0)). For example, the first column of Table 1 corresponds to
(ui(0),«2(0)) = (sgn hi(0), sgn h2(0)) = (+1, +1).
The second column of Table 1 corresponds to the initial values (w1(0),w2(0)) = (sgnh1(0),sgnh2(0)) = = ( — 1, +1). The lower index ± near (±, ±) indicates the value of sgn h3(0).
The same agreement on reading similar tables is used in subsequent subsections.
We prove Theorem 7.
Proof. Let A G C4, the cases A G C5 and A G C6 are considered similarly. Then system (6.2) has the phase portrait shown in Fig. 1.
Consider the first column of Table 1 — a control starting from (1,1)+ and having k = 8 switchings (controls starting from other values are considered similarly). We apply Theorem 4 and show that such control is not optimal. We have 0 = t0 <ti < ... <tg = T, where
ti G (0,rij, t2 — ti = t4 — t3 = t5 — t4 = tj — t6 = tg — tj = Ti, ta — t2 = t6 — t5 = T2, tg — tg G (0,T2],
and
y/2(E + h4) - V2E _ 2
t\ — - = -, t~2 = -
hi sJ'2(E + h4) + \/2E' h4
Further, we have
U\(to ,ti) = u\(t4 ,t5) = u\(t6 ,t7) = (!) 1)> U\(tl ,t2) = U\(t3 M) = U\(t7 ,ts) = (_1>
U\(t2 ,t3) = U\(t8 ,t9) = (-1) -1)) U\(t5,t6) = (1) -1))
see Fig. 1. We apply Theorem 4 in the case k = 8, j = 4. We use the basis (X+,X-,X3,X4) in the Lie algebra g, where X+ = Xi + X2, X- = Xi — X2. Then
Fq = —12 = ^4 = F = —= X+, Yi = 13 = —15 = 17 = —X-.
Further,
Thus,
P4 = P3 = Id, P5 = eT1 ad ,
P6 = P5 o eT2 ad , P7 = P6 o eT1 ad x+,
p2 = eTi ad x-, pi = p2 o eT2 ad X+,
PQ = P1 o eT1 ad x-, P8 = P7 o e-T1 ad x-
zq = X+ + 4r 1X3 + (4t2 + 2t 1 T2)X4, Zi = —X- + 2X2X3 + (t22 + 2T1T2)X4, Z2 = —X+ — 2x2X3 — T22 X4, Z3 = —X-,
Z4 = X+,
Z5 = X- — 2t 1X3 — t2x4 , Z6 = X+ + 2T2 X3 + (T22 + 2T1T2)X4, Z7 = —X- + 4T1X3 + (4t2 + 2T1T2 )X4, Z8 = —X+ + (2t 1 — 2T2 )X3 + (3t2 — T22 )X4.
Then Q(a) = ^aua*ai, where
Then
001 = h3 + (2t 1 + T2)h4, 024 = T1h4,
002 = T1h4, 025 = h3 + 2t h,
003 = h3 + 2t 1h4, 026 = (T1 - T2)h4,
004 = -2T1h4, 027 = -h3 - 3t1h4,
005 = -h3 - 3t 1 h4, 028 = (T2 - 2t 1)h4,
006 = (T2 - 2t 1 )h4, 034 = -h3,
007 = h3 + 4t 1h4, 035 = T1h4,
008 = (3t 1 - T2)h4, 036 = -h3 - T2h4,
012 = h3 + (T1 + T2)h4, 037 = -2t 1h4,
013 = T2h4, 038 = h3 + (T2 - T1)h4,
014 = -h3 - T2h4, 045 = -h3 - T1h4,
015 = (T1 - T2)h4, 046 = T2h4,
016 = -h3 - 2T2h4, 047 = h3 + 2t 1h4,
017 = (-2t 1 + T2)h4, 048 = (T1 - T2)h4,
018 = h3 + (2T2 - T1)h4, 056 = h3 + (T1 + T2)h4,
023 = -h3 - T1h4, 057 = T1h4,
058 = -h3 - T2h4, 067 = h3 + (2t 1 + T2)h4
068 = T1h4, 078 = h3 + (T1 + T2)h4.
Further,
W = j (ao, ...,a8) e R9 I ^ a* = 0, ^ alZl(q(t1)) = 0 j
I l=0 l=0 J
= {(ao, ...,a8) e R8 I a1 = 27ao - ao - 2^aj + (1 - 2^)a8, a3 = 2jao - a2 + a6 + (27 - 1)a7 + (27 - 2)a8, a4 = -ao + a2 - a6 + a8, a5 = -a2 - a8}, Y = T1 /T2,
Q\w = -4\/2/a(/i + a/2 - \/a(a + 1 )/3), a = E/h4 e (0,1), /1 = (ao + a7 + a8)2,
f2 = ao + a2 - 2a2a6 + 2a6 + a7(2a2 - 4a6 + 3a7)
+ (4a2 - 7a6 + 9a7)a8 + 8a8 + 2ao(a7 + a8), /3 = ao + ao(a2 - 2a6 + 3a7 + 5a8 + (a7 + a8)(a2 - 2a6 + 3a7 + 5a8).
A
27 =
8Vl + a/y/a (lly/a(l+a) - 4 - 12a) < 0.
a32 a25 a52 a55
By Theorem 5, the quadratic form qIw is not negative semidefinite. By Theorem 4, the control u is not optimal. □
10.2. Case 1b): > 0, p = n/4
Theorem 8. Let h4 > 0, p = n/4, and X G C1. Then the bang-bang trajectory Exp(A,t) is optimal.
Proof. Apply Theorem 2. □
Theorem 9. Let h4 > 0, p = 0, and X G U4=2Ci. Then the bang-bang trajectory Exp(A, t) with k switchings is not optimal, where k is given by the following tables:
• X G C2 ^ Table 4,
• X G C3 ^ Table 5,
• X G C4 ^ Table 6.
Proof. Similarly to the proof of Theorem 7. □
Table 4. X G C2
Start (+,+)-
k 7 6 5 4
Table 5. A G C3
Start
k 4 5
Table 6. A G C4
Start
k 6 6 5 7
10.3. Case 1c): h4 > 0, p G (0,n/4)
Theorem 10. Let h4 > 0, p G (0,n/4), and X G u3=1Ci. Then the bang-bang trajectory Exp(X, t) is optimal.
Proof. Apply Theorem 2. □
Theorem 11. Let h4 > 0, p G (0,n/4), and X G uS=4Ci. Then the bang-bang trajectory Exp(X, t) with k switchings is not optimal, where k is given by the following tables:
• X G C4 ^ Table 7,
• X G C5 ^ Table 8,
• X G Co ^ Table 9,
• X G C7 ^ Table 10,
• X G Cs ^ Table 11.
Proof. Similarly to the proof of Theorem 7. □
Table 7. X e C4
Start (+,+)-
k 7 8 8 7
Table 8. X e C5
Start (+,+)-
k 7 8 8 7
Table 9. X e C6
Start (+,+)+ (-,+)-
k 8 9 8 8 9 8
Table 10. X e C7
Start (+,+)+ (-,+)+ (-,-) (-,+)- (+,+)- (+,-)
-- 10 9 8 7 8 7
- + 8 9 7 10 9 8
+ - 10 9 8 10 9 8
+ + 8 8 7 10 9 8
Table 11. X e C8
Start
k 8 8 7 7
10.4. Case 2): h4 = 0
Theorem 12. Let h4 = 0, and X e C2. Then the bang-bang trajectory Exp(X,t) with 7 switchings is not optimal.
Proof. Similarly to the proof of Theorem 7. □
Now Theorems 7-12 imply the following statement.
Corollary 1. If (p e [0,n/4) and -Ih41 sin p < E) or p = n/4, then optimal bang-bang trajectories have not more than 9 switchings.
11. Conclusion
Many interesting questions on the sub-Finsler problem on the Engel group considered in this paper remain open:
• precise description of the cut time along extremal trajectories,
• optimal synthesis,
• sub-Finsler sphere and distance.
We hope to address these questions in the forthcoming works.
Acknowledgments
The author is grateful to the reviewer whose suggestion improved the exposition of the paper.
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