Massachusetts Institute of Technology Department of Electrical Engineering and Computer Science 6.243j(Fall 2003): DYNAMICS OF NONLINEAR SYSTEMS by A. Megretski Lecture 12: Local controllability In this lecture, nonlinear Ode models with an input are considered. Partial answers to the general controllability question(which states can be reached in given time from a given state by selecting appropriate time-dependent control action are presented More precisely, we consider systems described by i(t=a(a(t), u(t), where a: R"XRHR is a given continuously differentiable function, and u=u(t is an m-dimensional time-varying input to be chosen to steer the solution a= r(t) in a desired direction. Let U be an open subset of R, To E R. The reachable set for a given T>0 the(U-locally) reachable set R(To, T)is defined as the set of all a(T) where 0,T+R", u: 0, T]H+Rm is a bounded solution of(12. 1)such that x(0) andx(t)∈ U for all t∈[0,m Our task is to find conditions under which R(To, T)is guaranteed to contain a neig borhood of some point in R, or, alternatively, conditions which guarante that RU(o, T) has an empty interior. In particular, when Io is a controlled equilibrium of (12.1),i.e a(io, io)=0 for some io E R", complete local controllability of(12.1)at To means that for every e>0 and T>0 there exists 8>0 such that R(, T)3 B6(Eo)for every TE B(io), where U= B(io) and B,(2)={正1∈R":|z1-≤r} denotes the ball of radius r centered at I Version of October 31. 2003
Massachusetts Institute of Technology Department of Electrical Engineering and Computer Science 6.243j (Fall 2003): DYNAMICS OF NONLINEAR SYSTEMS by A. Megretski Lecture 12: Local Controllability1 In this lecture, nonlinear ODE models with an input are considered. Partial answers to the general controllability question (which states can be reached in given time from a given state by selecting appropriate time-dependent control action) are presented. More precisely, we consider systems described by x˙ (t) = a(x(t), u(t)), (12.1) where a : Rn × Rm ⇒∀ R is a given continuously differentiable function, and u = u(t) n is an m-dimensional time-varying input to be chosen to steer the solution x = x(t) in a desired direction. Let U be an open subset of Rn, ¯x0 ∗ Rn. The reachable set for a given T > 0 the (U-locally) reachable set RU (¯x0, T) is defined as the set of all x(T) where x : [0, T] , ⇒∀ R u : [0, T] is a bounded solution of (12.1) such that x(0) = x¯0 n ⇒∀ Rm and x(t) ∗ U for all t ∗ [0, T]. Our task is to find conditions under which RU (¯x0, T) is guaranteed to contain a neigborhood of some point in Rn, or, alternatively, conditions which guarante that RU (¯x0, T) has an empty interior. In particular, when x¯0 is a controlled equilibrium of (12.1), i.e. x0, ¯ u0 ∗ Rm a(¯ u0) = 0 for some ¯ , complete local controllability of (12.1) at x¯0 means that for every φ > 0 and T > 0 there exists � > 0 such that RU (¯x, T) � B�(¯x0) for every x¯ ∗ B�(¯x0), where U = Bα(¯x0) and Br(¯x) = {x¯1 ∗ R x n : |x¯1 − ¯| √ r} denotes the ball of radius r centered at x¯. 1Version of October 31, 2003
12.1 Systems with controllable linearizations A relatively straightforward case of local controllability analysis is defined by systems with controllable linearizations 12.1.1 Controllability of linearized system Let To: 0, THR, uo: 0, T]H Rm be a bounded solution of (12. 1). The standard linearization of(12.1)around the solution(ro(), uo()) describes the dependency of small state increments 8(t)=r(t)-co(t)+o(5(t)) on small input increments ou(t)=u(t) 6n(t) 6x(t)=A(t)6(t)+B(t)6a(t) (122) where A(t) B(t)= =r0(t),u=a0(t) r=o(t) u=uo(t) are bounded measureable matrix-valued functions of time Let us call system(12.2)controllable on time interval [0, T] if for every 0,dTER there exists a bounded measureable function &u: 0, T]+Rm such that the solution of (12.2 )with 5(0)=80 satisfies S(T)=8.The following simple criterion of controllability is well known from the linear system theory Theorem 12.1 System(12.2) is controllable on interval 0, T] if and only if the matric M(t-B()(t(M(ty-dt is positive definite, where M= M(t) is the evolution matric of (12.2), defined b M(t)=A(t)M(t),M(O)=I Matrix Wc is frequently called the Grammian, or Gram matriz of (12. 2)over [ 0, T It is easy to see why Theorem 12.1 is true: the variable change 8(t)=M(t)z(t)reduces (122)to 2(t)=M()-B(t)6n(t) Moreover. since x(T)=M(t)-B(t)ou(t)dt is a linear integral dependence, function Ou can be chosen to belong to any subclass which is dense in L(0, T). For example, du(t) can be selected from the class of polynomials class of piecewise constant functions, etc Note that controllability over an interval A implies controllability over every interval A+ containing A, but in general does not imply controllability over all intervals A contained in A. Also, system(12. 2)in which A(t)= Ao and B(t)= Bo are constant is equivalent to controllability of the pair(A, B)
2 12.1 Systems with controllable linearizations A relatively straightforward case of local controllability analysis is defined by systems with controllable linearizations. 12.1.1 Controllability of linearized system Let x0 : [0, T] , ⇒∀ R u0 : [0, T] be a bounded solution of (12.1). The standard n ⇒∀ Rm linearization of (12.1) around the solution (x0(·), u0(·)) describes the dependency of small state increments �x(t) = x(t) − x0(t) + o(�x(t)) on small input increments �u(t) = u(t) − �u(t): � ˙ x(t) = A(t)�x(t) + B(t)�u(t), (12.2) where ⎛ ⎛ da⎛ da⎛ A(t) = ⎛ , B(t) = ⎛ (12.3) ⎛ dx⎛ du x=x0(t),u=u0(t) x=x0(t),u=u0(t) are bounded measureable matrix-valued functions of time. Let us call system (12.2) controllable on time interval [0, T] if for every � ¯0 � ¯T ∗ Rn x, x there exists a bounded measureable function �u : [0, T] ⇒∀ R such that the solution of m ¯ (12.2) with �x(0) = � ¯0 satisfies �x(T) = �T x x . The following simple criterion of controllability is well known from the linear system theory. Theorem 12.1 System (12.2) is controllable on interval [0, T] if and only if the matrix � T Wc = M(t) −1 B(t)B(t) � (M(t) � ) −1 dt 0 is positive definite, where M = M(t) is the evolution matrix of (12.2), defined by M˙ (t) = A(t)M(t), M(0) = I. Matrix Wc is frequently called the Grammian, or Gram matrix of (12.2) over [0, T]. It is easy to see why Theorem 12.1 is true: the variable change �x(t) = M(t)z(t) reduces (12.2) to z˙(t) = M(t) −1 B(t)�u(t). Moreover, since � T z(T) = M(t) −1 B(t)�u(t)dt 0 is a linear integral dependence, function �u can be chosen to belong to any subclass which is dense in L1(0, T). For example, �u(t) can be selected from the class of polynomials, class of piecewise constant functions, etc. Note that controllability over an interval � implies controllability over every interval �+ containing �, but in general does not imply controllability over all intervals �− contained in �. Also, system (12.2) in which A(t) = A0 and B(t) = B0 are constant is equivalent to controllability of the pair (A, B)
12.1.2 Consequences of linearized controllability Controllability of linearization implies local controllability. The converse is not true: a nonlinear system with an uncontrollable linearization can easily be controllable Theorem12.2Leta:R"×R→ r be continuously differentiable. Let To:[0,→ R, uo: 0,THRm be a bounded solation of (12.1). Assume that system(12. 2 ), defined by(12.3), is controllable over 0, T]. Then for every e>0 there erists 8>0 such that for all To, ir satisfying o-x00)0 let B={∈R:|0 is sufficiently small. The derivative of S with respect to w at w=U=0 is identity. Hence, by the implicit mapping theorem, equation S(, v)=i has a solution w a 0 whenever l and I-To(T)I are mall enough
� 3 12.1.2 Consequences of linearized controllability Controllability of linearization implies local controllability. The converse is not true: a nonlinear system with an uncontrollable linearization can easily be controllable. Theorem 12.2 Let a : Rn × Rm be continuously differentiable. Let x0 : [0, T] Rn, u0 : ⇒∀ Rn ⇒∀ [0, T] be a bounded solution of (12.1). Assume that system (12.2), defined by (12.3), is controllable over [0, T]. Then for every φ > 0 there exists � > 0 such that for ⇒∀ Rm all x¯0, x¯T satisfying |x¯0 − x0(0)| | 0 let x ∗ Rn Bα = {¯ ¯ : | | x 0 is sufficiently small. The derivative of S with respect to w at w = v = 0 is identity. Hence, by the implicit mapping theorem, equation S(w, v) = x¯ has a solution w � 0 whenever | | | | v and x¯ − x0(T) are small enough
12.2 Controllability of driftless models In this section we consider OdE models in which the right side is linear with respect to the control variable, i.e. when(12.1) has the special form i(t)=g(x()()=∑9(x()(),(0)= 12. k=1 functions deli."are given Coo(i.e. having continuous derivatives of arbitrary order) where gk functions defined on an open subset Xo of R, and u(t)=u(t);.; um(t)) is the vector control input. Note that linearization(12.2)of(12.4)around every equilibrium solution co(t)=io= const, uo(t)=0 yields A=0 and B= g(io), which means that the linearization is never controllable unless m n. Nevertheless. it turns out that. for a generic"function g, system(12. 4)is expected to be completely controllable, as long as m>1 12.2.1 Local controllability and Lie brackets Let us say that system(12. 4)is locally controllable at a point o E Xo if for every e>0 T>0, and I E Xo such that Ii-iol e there exists a bounded measureable function u:0, T]H+Rm defining a solution of(12.4)with (0)=Io such that a(r)=T and r()- q we have hk=[h,h。」for some i, s< h, and the vectors hi(a)with i=1,., N span the whole R system(12.4) is locally controllable at io.k R"form a complete set at Lo E Xo then Theorem 12. 3 If Co functions gk
� 4 12.2 Controllability of driftless models In this section we consider ODE models in which the right side is linear with respect to the control variable, i.e. when (12.1) has the special form m x˙ (t) = g(x(t))u(t) = gk(x(t))u(t), x(0) = x¯0, (12.4) k=1 where gk : are given C� X0 ⇒∀ R (i.e. having continuous derivatives of arbitrary order) n functions defined on an open subset X0 of Rn, and u(t) = [u1(t); . . . ; um(t)] is the vector control input. Note that linearization (12.2) of (12.4) around every equilibrium solution x0(t) ≥ x¯0 = const, u0(t) = 0 yields A = 0 and B = g(¯x0), which means that the linearization is never controllable unless m = n. Nevertheless, it turns out that, for a “generic” function g, system (12.4) is expected to be completely controllable, as long as m > 1. 12.2.1 Local controllability and Lie brackets Let us say that system (12.4) is locally controllable at a point x¯ if 0 ∗ X0 for every φ > 0, T > 0, and x¯ ∗ X0 such that x¯ − x¯0| q we have hk = [hi, hs] for some i, s < k, and the vectors hi(¯x) with i = 1, . . . , N span the whole Rn. Theorem 12.3 If C� functions gk : X0 ⇒∀ Rn form a complete set at x¯0 ∗ X0 then system (12.4) is locally controllable at x¯0
Theorem 12.3 provides a sufficient criterion of local controllability in terms of the span of all vector fields which can be generated by applying repeatedly the Lie bracket operation to gk. This condition is not necessary, as can be seen from the following example: the order system o(x1)u2, where function o: R H R is infinitely many times continuously differentiable and such that 0(0)=0,d(y)>0fory≠0,()(0)=0Vk is locally controllable at every point io E r despite the fact that the corresponding set of vector fields 9()=/1 0 o(x1) is not complete at i=0. On the other hand, the example of the system L hich is not locally controlable at I=0, but is defined by a(single element) set of vector fields which is complete at every point except i=0, shows that there is little room for relaxing the sufficient conditions of Theorem 12.3 12.2.2 Proof of heorem 12.3 Let s denote the set of all continuous functions s: Qs H+ Xo, where Qs is an open subset of R x Xo containing 0x Xo(Qs is allowed to depend on s). Let Sk E S be the elements of s defined b Sk(r,)=x(7):i(t)=9k(x(t),x(0)= Let Sg be subset of s which consists of all functions which can be obtained by recursion sk+1(,T)=Sa()(sk(,T),k(7),0(z,7)= where a(k)E (1, 2,..., m) and Ok: R-Rare continuous functions such that Ok(0)=0 One can view elements of Sg as admissible state transitions in system(12. 2)with piecewise constant control depending on parameter T in such a way that T=0 corresponds to the identity transition. Note that for every s∈ S, there exists an“ Inverse”s'∈ S such that s(S(,T),7)=V(z,)∈9, defined by applying inverses Sa()(, -ok(T)of the basic transformations Sa()(, %(T) n the reverse order
� � � � 5 Theorem 12.3 provides a sufficient criterion of local controllability in terms of the span of all vector fields which can be generated by applying repeatedly the Lie bracket operation to gk. This condition is not necessary, as can be seen from the following example: the second order system x˙ 1 = u1, x˙ 2 = α(x1)u2, where function α : R ⇒∀ R is infinitely many times continuously differentiable and such that α(0) = 0, α(y) > 0 for y = 0, α(k) ∈ (0) = 0 � k, is locally controllable at every point x¯0 ∗ Rn despite the fact that the corresponding set of vector fields 1 0 g1(x) = , g2(x) = 0 α(x1) is not complete at x¯ = 0. On the other hand, the example of the system x˙ = xu, which is not locally controlable at x¯ = 0, but is defined by a (single element) set of vector fields which is complete at every point except x¯ = 0, shows that there is little room for relaxing the sufficient conditions of Theorem 12.3. 12.2.2 Proof of Theorem 12.3 Let S denote the set of all continuous functions s : �s ⇒∀ X0, where �s is an open subset of R×X0 containing {0}×X0 (�s is allowed to depend on s). Let Sk ∗ S be the elements of S defined by Sk(δ, x¯) = x(δ ) : x˙ (t) = gk(x(t)), x(0) = x. ¯ Let Sg be subset of S which consists of all functions which can be obtained by recursion sk+1(¯x, δ ) = S�(k)(sk(¯x, δ ), αk(δ )), β0(¯x, δ ) = ¯x, where �(k) ∗ {1, 2, . . . , m} and αk : R ⇒∀ R are continuous functions such that αk(0) = 0. One can view elements of Sg as admissible state transitions in system (12.2) with piecewise constant control depending on parameter δ in such a way that δ = 0 corresponds to the identity transition. Note that for every s ∗ Sg there exists an “inverse” s� ∗ Sg such that s(s� (¯ x x, δ ) ∗ �s x, δ ), δ ) = ¯ � (¯ �, defined by applying inverses S�(k)(·, −αk(δ )) of the basic transformations S�(k)(·, αk(δ )) in the reverse order
Let us call a Coo function h: Xo HR" implementable in control system(12.4)if for every integer k>0 there exists a function s E So which is k times continuously differentiable in the region T>0 and in the region T 0 for all I E Xo. One can say that the value h(i) of an implementable function h( at a given point I E Xo describes a direction in which solutions of(12.4) can be steered fron元 We will prove Theorem 12. 3 by showing that Lie bracket of two implementable vec- tor fields is also an implementable vector field. After this is done, an implicit function argument similar to one used in the proof of Theorem 12. 2 shows local controllability of (124) Now we need to prove two intermediate statements concerning the set of implementable vector fields. Remember that for the differential How(t, t)HS(, t) defined by a smooth vector field h we have S(S(, t1, t2)=S(i, ti+t2) which, in particular, implies that sh(,t)=2+th()+h()h(z)+O(t3) as t-0. This is not necessarily true for a general transition s from the definition of an implementable vector field h. However, the next Lemma shows that s can always be chosen to match the first k Taylor coefficients of Sh Lemma 12.1 If h is implementable then for every integer k>0 there erists a k times continuously differentiable function s E Sa such that (,7)=Sh(,r)+O(r) (12.6) Proof By assumption, (12.6)holds for k=2 and T>0, where s is N times continuously differentiable in the region T >0, and N can be chosen arbitrarily large. Assume that for s(E,T)=sh(o, T)+rw(a)+O(rR ( which is implied by(12.6)), where 20 the function sa.b(E,T)=s(s(s(i, ar), br, aT
� 6 Let us call a C� function h : X0 ⇒∀ Rn implementable in control system (12.4) if for every integer k > 0 there exists a function s ∗ Sg which is k times continuously differentiable in the region δ → 0 and in the region δ √ 0, such that s(¯x, δ ) = ¯x + δh(¯x) + o(δ ) (12.5) as δ ∀ 0, δ → 0 for all x¯ ∗ X0. One can say that the value h(¯x) of an implementable function h(·) at a given point x¯ describes ∗ X0 a direction in which solutions of (12.4) can be steered from x¯. We will prove Theorem 12.3 by showing that Lie bracket of two implementable vector fields is also an implementable vector field. After this is done, an implicit function argument similar to one used in the proof of Theorem 12.2 shows local controllability of (12.4). Now we need to prove two intermediate statements concerning the set of implementable vector fields. Remember that for the differential flow (t, x¯ (¯ ) ⇒∀ S x, t) defined by a smooth h vector field h we have x, t1, t2) = Sh S (¯ h(Sh(¯ x, t1 + t2), which, in particular, implies that t 2 Sh(¯x, t) = ¯x + th(¯x) + h x)h(¯ ˙(¯ x) + O(t 3 ) 2 as t ∀ 0. This is not necessarily true for a general transition s from the definition of an implementable vector field h. However, the next Lemma shows that s can always be chosen to match the first k Taylor coefficients of Sh. Lemma 12.1 If h is implementable then for every integer k > 0 there exists a k times continuously differentiable function s ∗ Sg such that x, δ ) = Sh s(¯ (¯x, δ ) + O(δ k). (12.6) Proof By assumption, (12.6) holds for k = 2 and δ → 0, where s is N times continuously differentiable in the region δ → 0, and N can be chosen arbitrarily large. Assume that for δ → 0 s(¯ x, δ ) + δ k x, δ ) = S w(¯ h(¯ x) + O(δ k+1, (which is implied by (12.6)), where 2 √ k 0 the function x, δ ) = s(s� sa,b(¯ (s(¯x, aδ ), bδ ), aδ )
satisfies Sab(,r)=S(2,(2*a-b)r)+(2a4-b)ru(2)+Oxk+1 Since k>2, one can choose a, b in such way that b=1. 20k=bk which yields(12.6)with h increased by 1 After(12.6)is established for T >0, s can be defined for negative arguments by (Z,-7)=8(z,7),T≥0, which makes it k-l times continuously differentiable ae ext lemma is a key result explaining the importance of Lie brackets in controllability Lemma 12.2 If vector fields h1, h2 are implementable then so is their Lie bracket h Proof By Lemma 12. 1, there exist 2*k+2 times continuously differentiable(for+0) functions s1. s? E Sa such that (,r)=2+Th1(2)+r2h2(2)h(2)+o(r2) Hence(check this!), S3 E Sg defined by s3(,7)=S2(81(s2(81(z,r),T),-7),-7) satisfies S3(x,)=2+72h(2)+0(r2) Now for i=3.4..,2*k+2 let s1+1(,7)=s1(s(,r/√2),-7/ By induction s+(,)=2+∑2A2(2)+0(72) the transformation from si to si+1 removes the smallest odd power of T in the Taylor expansion for si. Hence >0 defines a k times continuously differentiable function for sufficiently small T >0, and s(,7)=2+Th(z)+o(r) forr≥0,7→
� 7 satisfies sa,b(¯ x,(2 ⊃ a − b)δ ) + (2ak − bk)δ k x, δ ) = S w(¯ h(¯ x) + O(δ k+1. Since k → 2, one can choose a, b in such way that 2a − b = 1, 2ak = bk, which yields (12.6) with k increased by 1. After (12.6) is established for δ → 0, s can be defined for negative arguments by x, −δ ) = s� s(¯ (¯x, δ ), δ → 0, which makes it k − 1 times continuously differentiable. Next lemma is a key result explaining the importance of Lie brackets in controllability analysis. Lemma 12.2 If vector fields h1, h2 are implementable then so is their Lie bracket h = [h2, h1]. Proof By Lemma 12.1, there exist 2 ⊃ k + 2 times continuously differentiable (for δ ∈= 0) functions s1, s2 ∗ Sg such that si(¯x, δ ) = ¯x + δhi(¯x) + δ x)hi(¯ 2 h˙ i(¯ x) + o(δ 2 ). Hence (check this!), s3 ∗ Sg defined by s3(¯x, δ ) = s2(s1(s2(s1(¯x, δ ), δ ), −δ ), −δ ), satisfies s3(¯ x + δ 2 x, δ ) = ¯ h(¯x) + o(δ 2 ). Now for i = 3, 4, . . . , 2 ⊃ k + 2 let si+1(¯ x, δ/≈ 2), −δ/≈ x, δ ) = s 2). i(si(¯ By induction, i si+2(¯ x + δ 2i x, δ ) = ¯ �iq(¯x) + o(δ 2i ), q=1 i.e. the transformation from si to si+1 removes the smallest odd power of δ in the Taylor expansion for si. Hence s(¯x, δ ) = s2k+2(¯x, ≈δ ), δ → 0 defines a k times continuously differentiable function for sufficiently small δ → 0, and s(¯x, δ ) = ¯x + δh(¯x) + o(δ ) for δ → 0, δ ∀ 0
12.2. 3 Frobenius Theorem Let gk: R"HR", k=1,., m, be k times(k>1)continuously differentiable functions We will say that gk define a regular Ck distribution D(gk )at a point to E R if vectors gk Io) are linearly independent. Let Xo be an open subset of R". The distribution D(gk))is called involutive on Xo if the value gi; (r)of every Lie bracket gi;=[i,g;] belongs to the linear span of gk (i)for every I E Xo. Finally, distribution D(gk))is called completely Ck integrable over Xo if there exists a set a set of k times continuously differentiable functions hk: Xo R,k=1,., n-m, such that the gradients Vhk() are linearly independent for all i E Xo, and Vh()9/(z)=0V∈X0 The following classical result gives a partial answer to the question of what happens o controllability when the Lie brackets of vector fields gk do not span R Theorem 12.4 Let D(gk))define a cr distribution (r>1) which is regular atToER Then the following conditions are equivalent (a)there erists an open set Xo containing io such that d(gk)is completely Cr inte grable over xo, (b)there erists an open set Xo containing To such that D(gu ) ) is involative on Xo Essentially, the Frobenius theorem states that in the neigborhood of a point where the dimension of the vector fields generated by Lie brackets of a given driftless control system is maximal but still less than n, there exist non-constant functions of the state vector which remain constant along all solutions of the system equations The condition of regularity in Theorem 12.4 is essential. For example, when n=2,m=1,n=0∈R2,g(21 the distribution defined by g is smooth and involutive(because [g, gl=0 for every vector field g), but not regular at Io. Consequently, the conclusion of Theorem 12.4 does not hold at To =0, but is nevertheless valid in a neigborhood of all other points The "locality"of complete integrability is also essential for the theorem. For exampl the vector field r2+(1-n2-x2)2 9 defines a smooth regular involutive distribution on the whole R. However, the distri- bution is not completely integrable over R, while it is still completely integrable in a neigborhood of every point
8 12.2.3 Frobenius Theorem Let gk : Rn ⇒∀ R , k = 1, . . . , m, be k times (k → 1) continuously differentiable functions. n We will say that gk define a regular Ck distribution D({gk}) at a point x¯0 ∗ Rn if vectors gk(¯x0) are linearly independent. Let X0 be an open subset of Rn. The distribution D({gk}) is called involutive on X0 if the value gij (¯x) of every Lie bracket gij = [gi, gj ] belongs to the linear span of gk(¯x) for every x¯ ∗ X0. Finally, distribution D({gk}) is called completely Ck integrable over X0 if there exists a set a set of k times continuously differentiable functions hk : X0 ⇒∀ R, k = 1, . . . , n − m, such that the gradients ≡hk(¯x) are linearly independent for all x¯ ∗ X0, and ≡hi(¯x)gj (¯x) = 0 � x¯ ∗ X0. The following classical result gives a partial answer to the question of what happens to controllability when the Lie brackets of vector fields gk do not span Rn. Theorem 12.4 Let D({gk}) define a Cr distribution (r 1) which is regular at x¯0 ∗ Rn → . Then the following conditions are equivalent: (a) there exists an open set X0 containing x¯0 such that D({gk}) is completely Cr integrable over X0; (b) there exists an open set X0 containing x¯0 such that D({gk}) is involutive on X0. Essentially, the Frobenius theorem states that in the neigborhood of a point where the dimension of the vector fields generated by Lie brackets of a given driftless control system is maximal but still less than n, there exist non-constant functions of the state vector which remain constant along all solutions of the system equations. The condition of regularity in Theorem 12.4 is essential. For example, when ⎝� �⎡ � � x2 n = 2, m = 1, x¯0 = 0 ∗ R2 , g x1 = , x2 −x1 the distribution defined by g is smooth and involutive (because [g, g] = 0 for every vector field g), but not regular at x¯0. Consequently, the conclusion of Theorem 12.4 does not hold at x¯0 = 0, but is nevertheless valid in a neigborhood of all other points. The “locality” of complete integrability is also essential for the theorem. For example, the vector field ⎠⎦ ⎞� ⎦ ⎞ 2 2 x1 x1 + (1 − x2 2 − x2)2 g ⎤� x ⎣� = � ⎣ 2 x3 x3 −x2 defines a smooth regular involutive distribution on the whole R3 . However, the distribution is not completely integrable over R3 , while it is still completely integrable in a neigborhood of every point
12.2. 4 Proof of theorem 12.4 The implication(a)=>(b) follows straight forwardly from the reachability properties of Lie brackets. Let us prove the implication(b)=(a) Let sp denote the differential How map associated with gk, i. e. Sk(2)=a(r), where (t)is the solution of i(t)=9k(x(t),x(0)=元 Let A(i) denote the span of g1(E),., gm(i). The following stetement, which relies on both regularity and involutivity of the family igkIm_l, states that the Jacobian DK(a)of Sk at i maps A(i)onto A(St(i)). This a generalization of the(obvious) fact that, for a single vector field g: RT+ R", moving the initial condition a(0) by g(ar(0))8 of a solution z=r(t) of d c/dt=g(a)results in c(t) shifted by g(a(t))8+o(8) Lemma 12.3 Under the assumptions of Theorem 12.4 D()△(z)=△(S() Proof According to the rules for differentiation with respect to initial conditions, for a fixed I, D(t)=Di(E) satisfies the ODE dt Dk(1)=9k(x(t)Dk(t),Dk(0)=1, where r(t)=St(i), and gk()denotes the Jacobian of g atI. Hence Dk(t)=Dk(1)9(2), where g(2)=[91(z)92(2)….9m(2) satisfies ,D()=9k(x(t)D(t),D2(0)=g(z) 12 Note that the(12.7) is an OdE with a unique solution. Hence, it is sufficient to show that(12.7)has a solution of the form D()=9x()0)=∑9((t)5(t), 128) where 8=8(t) is a continuously differentiable m-by-m matrix valued function of time, and &(t)is the i-th row of &(t). Indeed, substituting into(12.7)yields ∑(x(1)g(x(1)()+9(x(1)()=()∑9(r()5() and d(0)=I. Equivalently 9(x(t)6(t)=A(t)6(1)
� � � 9 12.2.4 Proof of Theorem 12.4 The implication (a)≤(b) follows straightforwardly from the reachability properties of Lie brackets. Let us prove the implication (b)≤(a). Let S� denote the differential flow map associated with gk, i.e. Sk � (¯x) = x(δ ), where k x = x(t) is the solution of x˙ (t) = gk(x(t)), x(0) = x. ¯ Let �(¯x) denote the span of g1(¯x), . . . , gm(¯x). The following stetement, which relies on m k both regularity and involutivity of the family {gk} (¯ k=1, states that the Jacobian Dt x) of St at x¯ maps �(¯x) onto �(S x)). This a generalization of the (obvious) fact that, for k t(¯ k a single vector field g : Rn ⇒∀ R , moving the initial condition x(0) by g(x(0))� of a n solution x = x(t) of dx/dt = g(x) results in x(t) shifted by g(x(t))� + o(�). Lemma 12.3 Under the assumptions of Theorem 12.4, k(¯ x) = �(Sk t D (¯ t x)�(¯ x)). Proof According to the rules for differentiation with respect to initial conditions, for a fixed ¯ k x, Dk(t) = D (¯ t x) satisfies the ODE d Dk(t) = g˙k(x(t))Dk(t), Dk(0) = I, dt where x(t) = Sk t(¯x), and g˙k(¯x) denotes the Jacobian of g at x¯. Hence D¯ k(t) = Dk(t)g(¯x), where g(¯x) = [g1(¯x) g2(¯x) . . . gm(¯x)], satisfies d D¯ k(t) = g˙k(x(t))D¯ k(t), D¯ k(0) = g(¯x). (12.7) dt Note that the (12.7) is an ODE with a unique solution. Hence, it is sufficient to show that (12.7) has a solution of the form m D¯ k(t) = g(x(t))�(t) = gi(x(t))�k(t), (12.8) i=1 where � = �(t) is a continuously differentiable m-by-m matrix valued function of time, and �i(t) is the i-th row of �(t). Indeed, substituting into (12.7) yields [g˙i(x(t))gk(x(t))�k(t) + gi(x(t))� ˙ k(t)] = g˙k(x(t)) gi(x(t))�k(t) i i and �(0) = I. Equivalently, g(x(t))� ˙(t) = A(t)�(t)
where A(t)is the n-by-m matrix with columns gki((t)), gki=gk, gil. By involutivity and regularity, A(t)=g(a(t))a(t) for some continuous m-by-m matrix valued function a=a(t). Thus, the equation for 8(t) becomes 6(t)=a(t)6(1),6(0)=I, hence existence of 8(t) such that Dk(t)=g(a(t))8(t)is guaranteed Let 9m+1, .. gn be Coo smooth functions gi: R"+R" such that vectors g1(io),., gn(io) form a basis in R".(For example, the functions gi with i>m can be chosen constant Consider the map F(a)=Si(S2(.(San(io)).) defined and k times continuously differentiable for 2=[21, .. ,n in a neigborhood zero in R. F is a k times differentiable map defined in a neigborhood of z=0,i=i Since the Jacobian F(O) of F at zero, given by ()=[1(50)g2(x0)….9n(xo) not singular, by the implicit mapping theorem there exists a k times continuously differentiable function z=H(x)=hn(x);hn-1(x);……;h1(x) defined in a neigborhood of To, such that F(H()=a Let us show that functions hi= hi() satisfy the requirements of Theorem 12.4 deed, differentiating the identity F(H()=r yield F(H()H()=L, 12. where H(x)=[Vhn(x)Vhn-1(x);……;Vh1(x) is the jacobian of h at x and F(z)=[f1(x)f2(2)….fn(x is the jacobian of F at 2. Hence vectors form a basis as well as the co-vectors Vhi(a)h. By Lemma 12.3, vectors fi(a) ≤ m belong to△(F())=△(x)(and hence, by linear independence, form a basis in A(r)). On the other hand,(12.9) implies Vhr(x)f(H(x)=0forr≤n- m and i≤m. Hence Vh(x)forr≤m- m are linearly independent and orthogonal to all gi (a) for i m
10 where A(t) is the n-by-m matrix with columns gki(x(t)), gki = [gk, gi]. By involutivity and regularity, A(t) = g(x(t))a(t) for some continuous m-by-m matrix valued function a = a(t). Thus, the equation for �(t) becomes � ˙(t) = a(t)�(t), �(0) = I, hence ¯ existence of �(t) such that Dk(t) = g(x(t))�(t) is guaranteed. Let gm+1, . . . , gn be C x0), . . . , gn(¯ � smooth functions gi : Rn ⇒∀ Rn such that vectors g1(¯ x0) form a basis in Rn. (For example, the functions gi with i > m can be chosen constant). Consider the map F(z) = Sz1 (Sz2 (. . .(Szn (¯x0)). . .)), 1 2 n defined and k times continuously differentiable for z = [z1, . . . , zn] in a neigborhood of zero in Rn. F is a k times differentiable map defined in a neigborhood of z = 0, x¯ ¯ = x0. Since the Jacobian F˙(0) of F at zero, given by F x0) g2(¯ x0)] ˙(0) = [g1(¯ x0) . . . gn(¯ is not singular, by the implicit mapping theorem there exists a k times continuously differentiable function z = H(x) = [hn(x); hn−1(x); . . . ; h1(x)] defined in a neigborhood of x¯0, such that F(H(x)) ≥ x. Let us show that functions hi = hi(x) satisfy the requirements of Theorem 12.4. Indeed, differentiating the identity F(H(x)) ≥ x yields F˙(H(x))H˙ (x) = I, (12.9) where H˙ (x) = [≡hn(x); ≡hn−1(x); . . . ; ≡h1(x)] is the Jacobian of H at x, and F˙(z) = [f1(z) f2(z) . . . fn(x)] is the Jacobian of F at z. Hence vectors fi(z) form a basis, as well as the co-vectors ≡hi(x)h. By Lemma 12.3, vectors fi(z) with i √ m belong to �(F(z)) = �(x) (and hence, by linear independence, form a basis in �(x)). On the other hand, (12.9) implies ≡hr(x)fi(H(x)) = 0 for r √ n − m and i √ m. Hence ≡hr(x) for r √ n − m are linearly independent and orthogonal to all gi(x) for i √ m