The importance of “control variations” for obtaining local approximations
of the reachable set of nonlinear control systems is well known.
Heuristically, if one can construct control variations in all possible directions,
then the considered control system is small-time locally controllable
(STLC). Two concepts of control variations of higher order are introduced
for the case of smooth control systems. The relation between these variations
and the small-time local controllability is studied and a new sufficient
STLC condition is proved.* This work is partly supported by the Bulgarian Ministry of Science and Higher Education
– National Fund for Science Research under contract DO 02–359/2008

Krastanov, Mikhail

Bulgarian Digital Mathematics Library at IMI-BAS

Serdica J. Computing 4 (2010), 85–92HIGH-ORDER CONTROL VARIATIONS AND SMALL-TIMELOCAL CONTROLLABILITY*Mikhail Ivanov KrastanovAbstract. The importance of “control variations” for obtaining local ap-proximations of the reachable set of nonlinear control systems is well known.Heuristically, if one can construct control variations in all possible direc-tions, then the considered control system is small-time locally controllable(STLC). Two concepts of control variations of higher order are introducedfor the case of smooth control systems. The relation between these varia-tions and the small-time local controllability is studied and a new sufficientSTLC condition is proved.1. High-order variations and small-time local controllabiity.The traditional approach to obtaining local approximations of the reachable setof nonlinear control systems has been to construct “control variations”. If onecan construct control variations in all possible directions, then the reachableACM Computing Classification System (1998): G.1.6, G.1.7.Key words: local approximations of the reachable set, smooth control systems, small-timelocal controllability.*This work is partly supported by the Bulgarian Ministry of Science and Higher Education– National Fund for Science Research under contract DO 02–359/2008.86 Mikhail Ivanov Krastanovset should to be a full neighborhood of the starting point. Let us consider thefollowing control system:(1) x˙(t) = f0(x(t)) +m∑i=1ui(t)fi(x(t)), ui(t) ∈ [−1, 1],where fi, i = 0, . . . ,m, are smooth vector fields defined on a neigbourhood of thepoint x0 ∈ Rn with f0(x0) = 0. Let u(·) = (u1(·), . . . , um(·)) be an integrablefunction defined on the interval [0, T ] whose components take values from [−1, 1].An absolutely continuous function x(·) with x(0) = x0 and satisfying (1) foralmost every t from [0, T ] is called an admissible trajectory of (1) defined on[0, T ], starting from the point x0 and corresponding to the control u(·). ByR(x, T ) we denote all points of Rn reachable from the point x by means ofadmissible trajectories of (1) defined on [0, T ] and starting from the point x.We introduce two concepts of high-order control variations to the reach-able set of the control system (1) at the point x0: the sets of H- and S-controlvariations. First we denote by Exp(Zt)x0 the value of the solution of the equationx˙(τ) = Zt(x(τ)), x(0) = x0,at time τ = 1, where {Zt : t ∈ R+} is a given family of smooth vector fields,defined on Rn and depending continuously on t ≥ 0.Definition 1. The smooth vector field g is said to be an H-control vari-ation to the reachable set of the control system (1) at the point x0 if there exist apositive number T , a neighbourhood Ω of x0, two families of smooth vector fieldsat and bt parameterized by t > 0, and a continuous function p : R+ → R+ suchthat for each x ∈ Ω and each t ∈ [0, T ]Exp (tg + at + bt) (x) ∈ R(x, p(t)),where‖at(x)‖ ≤ M tθ ‖x− x0‖, ‖bt(x)‖ ≤ N tσ, p(t) < ν tλ,for some positive constants M , N , ν, θ, σ > 1 and λ. We denote by H+(x0) theset of all H-control variations to the reachable set of the control system (1) at x0.Remark 1. To our knowledge, Hermes was the first to realize (cf. [5])the importance of the control variations for obtaining local approximations ofthe reachable set of control systems. Similar definitions of high-order controlvariations can be found in [6], [8], [12], [13] etc. All these definitions use theHigh-Order Control Variations and Small-Time Local Controllability 87notion of a Lie bracket. Let f and g be smooth vector fields defined on Rn. Thenthe Lie bracket [f, g] is defined as[f, g](x) :=∂g∂x(x)f(x) −∂f∂x(x)g(x).It should be mentioned that the notion of Lie brackets is extended to the non-smooth case in [3].The next proposition provides constructions of elements of the setH+(x0):Proposition 1. The following assertions hold true:a) The set H+(x0) is a convex cone;b) The vector fields fi, ± [fi, fj ], [fi, [fi, f0]], i, j = 1, . . . ,m, are elements ofthe set H+(x0);c) Let ±g ∈ H+(x0). Then ±[g, f0] ∈ H+(x0).Slightly different versions of this proposition can be found in [5], [6],[8], [12]. The corresponding proofs are based on the so called Campbell-Baker-Hausdorff formula.Definition 2. Let H ={h1, . . . , hk}be a finite set of smooth vector fields.H is said to be a set of S-control variations of the control system (1) at the pointx0 if there exist positive reals ν, λ, γ0 and γ1 with γ0 < γ1, a neighbourhood Ωof x0, two families of smooth vector fields ct and dt parameterized by t > 0 suchthat for each γ from the open interval (γ0, γ1) and for each vector s = (s1, . . . , sk)whose components belong to the interval [0, 1] there exists a continuous functionps,γ : R+ → R+ with ps,γ(η) < ν η−λ such that for each x ∈ Ω and for eachsufficiently large positive integer η(2) Expη−γk∑j=1sjhj + cη + dη (x) ∈ R(x, ps,γ(η)),where‖cη(x)‖ ≤ Q1ηθ‖x− x0‖, ‖dη(x)‖ ≤ R1ησ,for some positive constants Q, R, θ and σ > γ1. We denote by S+(x0) the setof all sets of S-control variations to the reachable set of the control system (1) atthe point x0.88 Mikhail Ivanov KrastanovRemark 2. It is shown in [9] that the general sufficient STLC conditionobtained by Sussmann in [11] (cf. also [1]) is closely related to the above definedset of high-order control variations. More precisely, the proofs of the sufficientSTLC conditions in [1] and [11] can be used to construct elements of the setS+(x0). For different applications of the control theory, it is very important toimplement them explicitly by using admissible controls.The following proposition shows how these high-order control variationsare related to the small-time local controllability (STLC):Proposition 2. Let Hi ={h1i , . . . , hkii}, i = 1, . . . , α, be sets of S-controlvariations at x0, let g1, . . . , gk ∈ H+(x0) and let the origin belong to the interiorof the convex hull of the set(3){h11(x0), . . . , hk11 (x0), . . . , h1α(x0), . . . , hkαα (x0), g1(x0), . . . , gk(x0)}.Then the control system (1) is STLC.P r o o f. According to Definition 2, for each index i ∈ {1, . . . , α} thereexist positive reals νi, λi, γi0 and γi1 with γi0 < γi1, a neighbourhood Ωi of x0,families of smooth vector fields cit and dit parameterized by t > 0 such that foreach γi from the open interval(γi0, γi1)and for each vector si = (si1, . . . , skii )whose components belong to the interval [0, 1] there exists a continuous functionpsi,γi : R+ → R+ with psi,γi(η) < νi η−λi such that for each x ∈ Ωi and for eachsufficiently large positive integer η(4) Expη−γiki∑j=1sjihji + ciη + diη (x) ∈ R(x, psi,γi(η)),where‖ciη(x)‖ ≤ Qi η−θi ‖x− x0‖, ‖diη(x)‖ ≤ Ri η−σi ,for some positive constants Qi, Ri, θi and σi > γi1, i = 1, . . . , α.Similarly, according to Definition 1, for each index j ∈ {1, . . . , k}, thereexist a positive real T , a neighbourhood Ωj of x0, two families of smooth vectorfields ajt and bjt parameterized by t > 0, and a continuous function pj : R+ → R+such that for each x ∈ Ωj and each t ∈ [0, T ](5) Exp(tgj + ajt + bjt)(x) ∈ R(x, pj(t)),where‖ajt (x)‖ ≤ Mj tθj ‖x− x0‖, ‖bjt (x)‖ ≤ Nj tσj , pj(t) < ν˜j tλ˜j ,High-Order Control Variations and Small-Time Local Controllability 89for some positive constants Mj , Nj , ν˜j , θj, σj > 1 and λ˜j .Without loss of generality, we may think thatγ10 = max{γi0 : i = 1, . . . , α}.Let us fix an arbitrary γ from (γ10 , γ11) and an arbitrary vectors˜ := (s˜1, . . . , s˜k, s1, . . . , sk) with s˜i := (s˜1i , . . . , s˜kii ),where each s˜ji ∈ [0, 1], j = 1, . . . , ki, i = 1, . . . , α and each sj ∈ [0, 1], j = 1, . . . , k.For each index i ∈ {1, . . . , α} two cases are possible: a) γ < γi1; b) γ ≥ γi1.In the case a) we have that γi0 ≤ γ10 < γ < γi1. By setting γi := γ and sji := s˜ji ,j = 1, . . . , ki, we obtain that (4) is fulfilled for each sufficiently large positiveinteger η > 0. Let us assume that the case b) holds true, i.e., γ ≥ γi1. We fixa positive number γi from (γi0, γi1). Then γi < γ and for each positive integer ηthere exists a positive integer µi = µi(η) ≥ η such that the following inequalitieshold true: (1µi(η))γi≥(1η)γ>(1µi(η) + 1)γi.We set s¯i := (s1i , . . . , skii ), wheresji :=s˜ji (µi(η))γiηγ, j = 1, . . . , ki.Clearly, 0 ≤ sji ≤ 1. For this choice of s¯i and γi and replacing η by µi(η), theinclusion (4) can be written as follows(6) Expη−γki∑j=1s˜jihji + ciµi(η)+ diµi(η) (x) ∈ R(x, pis¯i,γi(µi(η)),where‖ciµi(η)(x)‖ ≤ Q µi(η)−θi ‖x− x0‖, ‖diµi(η)(x)‖ ≤ Ri µi(η)σi ,for the positive constants Qi, Ri, θi and σi > γi1, i = 1, . . . , α. Our choice of µi(η)and the inequalities σi > γi1 > γi imply thatηγµi(η)σi<(µi(η) + 1)γiµi(η)σi=(1 +1µi(η))γiµi(η)σi−γi−→η→∞0,90 Mikhail Ivanov Krastanovand (1η)γ>(1µi(η))σifor all sufficiently large positive integers η. Thus we have shown that‖diµ(η)(x)‖ = o (1/ηγ) ,where by definitiono(t)t→ 0 as t ↓ 0.Hence, for each index i the inclusion (4) or the inclusion (6) is fulfilled(depending on the cases a) and b)). Then there exists a neighborhood Ω of x0such that for each point x ∈ Ω and for all sufficiently large positive integers ηE(s˜, η, γ)(x) := Expη−γk1∑j=1s˜j1hj1 + c1η + d˜1η · · ·Expη−γkα∑j=1s˜jαhjα + cαη + d˜αη Exp(s1η−γg1 + a1s1η−γ + b1s1η−γ)· · ·Exp(skη−γgk + akskη−γ + bkskη−γ)(x) ∈ RN~X (x, p˜s˜,γ˜(η)) ,where by d˜iη, ,i = 1, . . . , α, we have denoted the corresponding diη (in the case a))or diµi(η)(in the case b)). Also, here we have setp˜s˜,γ˜(η) = p˜1sˆ1,γ1(η) + · · ·+ p˜αsˆα,γα(η) + p1(s1η−γ)+ · · · + pk(skη−γ),where p˜isˆi,γi(η) is equal to pis˜i,γi(η) (in the case a)) or to pis¯i,γi(µi(η)) (in the caseb)). Taking into account the estimations for pisi,γi(·), i = 1, . . . , α, for pj(·),j = 1, . . . , k, and the inequalities µi(η) ≥ η for all indices i for which the case b)holds true, we obtain that(7) p˜s˜,γ˜(η) <α∑i=1νi η−λi +k∑j=1ν˜j(sjη−γ)λ˜j < ν η−λ,where ν := α∑i=1νi +k∑j=1ν˜jsλ˜jj and λ := min(λ1, . . . , λα, γλ˜1, . . . , γλ˜k). Ap-plying the C-B-H formula, we obtain that E(s˜, η, γ)(x) =(8)= Expη−γ α∑i=1ki∑j=1s˜jihji +k∑j=1sjgj+Θη,γ +∆η,γ,s˜ (x) ∈ R (x, p˜s˜,γ˜(η)) ,High-Order Control Variations and Small-Time Local Controllability 91where p˜s˜,γ(η) < ν η−λ,(9) ‖Θη,γ‖ ≤ Q(1η)θ‖x− x0‖ and ‖∆η,γ,s˜‖ = o (1/ηγ) .If we expand the right-hand side of (8), it turns out thatE(s˜, η, γ)(x) = 1+η−γ α∑i=1ki∑α=1s˜jihji +k∑j=1sjgj (x)+Y2(η, γ)(x)+Y3(η, s˜, γ)(x),where Y2(η, γ) is a sum of powers of Θη,γ and Y3(η, s˜, γ) is a sum of productsof the factors η−γ α∑i=1ki∑α=1s˜jihji +k∑j=1sjgj, Θη,γ and ∆η,γ,s˜ and at least onefactor is ∆η,γ,s˜ or η−γ α∑i=1ki∑α=1s˜jihji +k∑j=1sjgj. The inequalities (9) imply thatΘη,γ(x0) = 0, and hence every power Θη,γ(x0) = 0 vanishes as well. Also, everyterm of the sum Y3(η, s˜, γ) is a product containing at last one factor o(1/η−γ).So, we have that E(s˜, η, γ)(x0) == 1 + η−γ α∑i=1ki∑α=1s˜jihji +k∑j=1sjgj (x) + Y3(η, s˜, γ)(x) ∈ R (x, p˜s˜,γ˜(η)) ,where p˜s˜,γ˜(η) < ν η−λ and Y3(η, s˜, γ˜)/ηγ tends to zero as η → ∞ ( the conver-gence being uniform with respect to the vector s˜ whose components vary in theinterval [0, 1]). From here (cf. for example, [10]), it follows that R(x0, t) containsa neighbourhood of the point x0 for each t > 0. This completes the proof. REFERENCES[1] Agrachev A., R. Gamkrelidze. Local controllability and semigroups ofdiffeomorphisms. Acta Applicandae Mathematicae, 32 (1993), 1–57.[2] Clarke F. H., Yu. S. Ledyaev, R. J. Stern, P. R. Wolensky. Non-smooth analysis and control theory. Graduate Texts in Mathematics 178,Springer, New York, Berlin, Heidelberg, 1998.92 Mikhail Ivanov Krastanov[3] Frankowska H. Local controllability of control systems with feedback. J.Optimization Theory Appl., 60 (1989), 277–296.[4] Hermes H. On the synthesis of a stabilizing feedback control via Lie alge-braic methods. SIAM J. Control Optim., 16 (1978), 715–727.[5] Hermes H. Lie algebras of vector fields and local approximation of attain-able sets. SIAM J. Control Optim., 18 (1980), 352–361.[6] Krastanov M. I., M. Quincampoix. Local small-time controllability andattainability of a set for nonlinear control systems. ESAIM: Control. Optim.Calc. Var., 6 (2001), 499–516.[7] Krastanov M. I. A sufficient condition for small-time local attainabilityof a set. Control and Cybernetics, 31 (2002), No 3, 739-750.[8] Krastanov M. I., V. M. Veliov. On the controllability of switching linearsystems. Automatica, 41 (2005), 663–668.[9] Krastanov M. I. A Sufficient Condition for Small-Time Local Controlla-bility. SIAM J. Control Optim., 48 (2009), No 4, 2296–2322.[10] H. Sussmann. Lie brackets and local controllability: a sufficient conditionfor scalar-input systems. SIAM J. Control Optim., 21 (1983), 686–713.[11] Sussmann H. J. A general theorem on local controllability. SIAM J. ControlOptim., 25 (1987), 158–194.[12] Veliov V., M. I. Krastanov. Controllability of piecewise linear systems.Systems & Control Letters, 7 (1986), 335–341.[13] Veliov V. M. On the controllability of control constrained linear systems.Math. Balk., New Ser., 2 (1988), 147–155.Institute of Mathematics and InformaticsBulgarian Academy of SciencesAcad. G. Bonchev Str., Bl. 81113 Sofia, Bulgaria,e-mail: krast@math.bas.bgReceived October 10, 2009Final Accepted February 4, 2010

High-Order Control Variations and Small-Time Local Controllability

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High-Order Control Variations and Small-Time Local Controllability

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