Least Squares Approximate Feedback Linearization. Abstract. We study the least squares approximate feedback linearization problem:

Transcription

1 Least Squares Approximate Feedback Linearization Andrzej Banaszuk y Andrzej Swiech z John Hauser x Abstract. We study the least squares approximate eedback linearization problem: given a single input nonlinear system, nd a linearizable nonlinear system that is close to the given system in a least squares (L 2 ) sense. A linearly controllable single input ane nonlinear system is eedback linearizable i and only i its characteristic distribution is involutive (hence integrable) or, equivalently, any characteristic one-orm (a one-orm that annihilates the characteristic distribution) is integrable. We study the problem o nding (least squares approximate) integrating actors that make a xed characteristic one-orm close to being exact in an L 2 sense. One can decompose a given one-orm into exact and inexact parts using the Hodge decomposition. We derive an upper bound on the size o the inexact part o a scaled characteristic one-orm and show that a least squares integrating actor provides the minimum value or this upper bound. We also consider higher order approximate integrating actors that scale a nonintegrable one-orm in a way that the scaled orm is closer to being integrable in L 2 together with some derivatives and derive similar bounds or the inexact part. One can use least squares approximate integrating actors in approximate eedback linearization o nonlinearizable single input ane systems. Moreover, least squares approximate integrating actors allow a unied approach to both least squares approximate and exact eedback linearization. Keywords. Nonlinear systems, eedback linearization, dierential orms, calculus o variations, Sobolev spaces, elliptic PDE's. Introduction Feedback linearization o nonlinear control systems has proven to be a useul tool or the design o controllers guaranteeing good perormance and stability over a large region o operation. When the system is close to being eedback linearizable, one may still be able to guarantee satisactory perormance and stability. For this, one needs to develop some Research supported in part by NSF under grant PYI ECS , by AFOSR under grant AFOSR F , and by a grant rom Hughes Aircrat Company. y School o Mathematics, Georgia Institute o Technology, Atlanta, GA 30332, banaszuk@math.gatech.edu z School o Mathematics, Georgia Institute o Technology, Atlanta, GA 30332, swiech@math.gatech.edu x Electrical and Computer Engineering, University o Colorado, Boulder, CO , hauser@boulder.colorado.edu 1

2 measures o distance between nonlinearizable systems and linearizable ones. So ar most o the work on approximate eedback linearization has ocused on applying a change o coordinates and a preliminary eedback so that the resulting system looks like linearizable part plus nonlinear terms o highest possible order around an equilibrium point [19, 17] or an equilibrium maniold [13, 22, 23]. By neglecting these nonlinear terms one obtains a linearizable system approximating the original system. I one applies to the original system a controller designed or the approximating linearizable system, perormance and stability are guaranteed in a neighborhood o the equilibrium or equlibrium maniold. However, in many applications one requires a large region o operation or the nonlinearizable system. In such a case, demanding the nonlinear terms to be neglected to be o highest possible order may, in act, be quite undesirable. One might preer that the nonlinear terms to be neglected be small in some average or uniorm sense over the region o operation. In the present paper we derive an approach that allows one to obtain upper bounds on the norm o these nonlinear terms in Sobolev spaces and thus, via Sobolev embeddings, in all L p spaces. Consider a single-input ane nonlinear system _x = (x) + g(x)u (1) where, g are smooth vector elds dened on the closure o an open, bounded, and contractible region M o R n containing the origin and having a smooth The classical problem o eedback linearization (c. [15, 16]) can be stated as ollows: nd in a neighborhood o the origin a smooth change o coordinates z = (x) (a local dieomorphism) and a smooth eedback law u = k(x) + l(x)u new such that the closed loop system in the new coordinates with new control is linear: _z = Az + Bu new ; (2) and controllable. In the paper we will assume that the system (1) has the linear controllability property dim span g; ad g; : : :; ad n?1 gg(x) = n; x 2 M: (3) We dene the characteristic distribution or (1) D := span g; ad g; :::; ad n?2 gg (4) (it is a smooth (n? 1)-dimensional distribution since (1) is linearly controllable). We call any nowhere vanishing one-orm annihilating D a characteristic one-orm or (1). Frobenius' theorem implies that the system (1) is exactly eedback linearizable i and only i we can nd among all characteristic one-orms one which is exact, i.e., it is the exterior derivative d o some zero-orm. Let! 0 be a characteristic one-orm or (1). All the other characteristic one-orms are multiples o! 0 by a smooth nowhere vanishing unction (zero-orm). Without loss o generality we can assume that is everywhere positive. The system (1) is exactly eedback-linearizable i and only i there is a nowhere vanishing 2

3 such that! 0 is exact. Such, i it exists, is called an (exact) integrating actor or! 0. A orm! 0 that admits an integrating actor (i.e., whose multiple by some nonzero unction is exact) is called integrable. The Poincare Lemma says that, on a contractible region, a orm! 0 is exact i and only i it is closed, i.e., d! 0 = 0: (5) Thus, to nd an integrating actor or an integrable! 0, one needs to nd a nontrivial solution to rst order PDE (5). The conditions or integrability o a one-orm are nontrivial or n > 2 so that, in general, no exact integrating actor will exist. One implication o this is that a generic nonlinear system (1) will not be eedback linearizable. To construct an approximate integrating actor that is optimal in some precise sense we need a Riemannian metric a nondegenerate pointwise inner product on the tangent space to M. A Riemannian metric induces an inner product and norm on k-orms. In the present paper we use a special Riemannian metric constructed as ollows. The linear controllability property allows one to locally dene a special set o coordinates called s-coordinates (c. [18, 16, 21]). In a neighborhood o the origin one can \reach" any point x by \traveling" along vector elds ad n?1 g; : : :; ad g; gg with \times" s 1 ; s 2 ; : : :; s n. We dene a metric on vector elds and orms on M ; : : n to be a positively oriented orthonormal basis or the tangent space to M at every point. This metric will be reerred to as the s-metric. In this paper we will use a specic characteristic orm! 0 xed by the requirements j! 0 j = 1 and! 0 (ad n?1 g) > 0, where j j is the (pointwise) norm dened by the s-metric. A global L 2 norm kk o a orm on M is obtained by integrating the pointwise one: Z kk := ( M jj2 ) 1 2 ; (6) where := ds 1 ^ : : : ^ ds n is the standard volume element corresponding to the s-metric. Let be the codierential the ormal adjoint to the exterior derivative d with respect to the chosen metric. In this paper we consider nding so that d! 0 and! 0 are the smallest possible in a least squares sense. The motivation or making both d! 0 and! 0 small is as ollows. In the exactly linearizable case we may nd > 0 such that d! 0 = 0. However, i one exact integrating actor or! 0 exists, then there are innitely many. Namely, i is an integrating actor or! 0, i.e.,! 0 = d or some, then or any nowhere R vanishing unction (x) h(), h() is also an integrating actor or! 0, or h()! 0 (x) = d 0 h(y)dy. I we do not impose an assumption on the smoothness o h(), the integrating actors h() include some discontinuous ones (or integrable but discontinuous h()). We see that the discontinuity may occur in the direction transversal to the integral maniolds o! 0. The reason is that the exterior derivative in d! 0 controls only behavior o along the integral maniolds o! 0. Now requiring also! 0 = 0 has a smoothing eect on, or the codierential in! 0 controls the behavior o in the direction transversal to the integral maniolds o! 0. Thus, i we can assure both d! 0 = 0 and! 0 = 0 or some > 0, we will have a smooth 3

4 integrating actor or! 0. We will also show that the integrating actor satisying both d! 0 = 0 and! 0 = 0 is unique up to multiplication by a nonzero constant. It is not clear at the moment that, or integrable! 0, such an integrating actor exists. We will show that it does, i the corresponding Riemannian metric is an s-metric. In the case o nonlinearizable (1) we will look or minimizing the ollowing unctional. For the xed (by the requirements j! 0 j = 1 and! 0 (ad n?1 g) > 0) characteristic orm! 0, we dene I 1 () := kd! 0k 2 + k! 0 k 2 k! 0 k 2 : (7) Note that minimizing the unctional I 1 () orces both d! 0 and! 0 to be small. Making d! 0 small makes! 0 close to being closed and hence exact (c. [7]). Control over the norm o! 0 assures smoothness o. We will show that there is a smooth positive minimizer 0 or I 1 (), to be called the least squares approximate integrating actor or! 0, uniquely dened up to multiplication by a constant. One can obtain uniqueness by a proper normalization, or instance, k 0 k 2 = 1. The Hodge decomposition o a characteristic one-orm! 0 has orm! 0 = d + ; (8) where d is a least squares approximation o! 0 by an exact orm and (called the antiexact part o! 0 ) is a one-orm whose (global) norm is a measure o how ar! 0 is rom being exact in a least squares sense (c. [1, 20]). We also show how to use the Hodge decomposition to obtain approximations o nonlinearizable systems (1) by linearizable ones (c. [4, 7]). We will show that, or the approximation o! 0 by an exact orm in the Sobolev space H 1 (the space o orms with coecients in L 2 together with all rst order partial derivatives), it makes good sense to choose = 0 where 0 is an approximate integrating actor or! 0 (o order 1). More precisely, we show that the H 1 norm o the error one-orm in the Hodge decomposition o! 0 can be bounded by q kk 1 C I 1 (): (9) when is normalized so that kk = 1. Choosing to be the (normalized) least squares approximate integrating actor provides the smallest value o this upper bound. We will also consider higher order approximate integrating actors or! 0. To be precise, any minimizer o the unctional m m z } { z } { I m () = kd! 0k 2 + k! 0 k k dd! 0 k 2 + k d! 0 k 2 (10) k! 0 k 2 is a least squares approximate integrating actor o order m or! 0. Similar to the order 1 case, we show that using an approximate integrating actor o order 2m provides or a guaranteed level o approximation o! 0 by an exact orm in the Sobolev space H 2m. To be 4

5 precise, we show that the H 2m norm o the error one-orm in the the Hodge decomposition o! 0 can be bounded by q kk M0 2m C I 2m (); (11) on any interior region M 0 o M. Finally, we show how one can use higher order approximate integrating actors in the approximate eedback linearization o the nonlinear system (1). Namely, it was shown in [7] that i (obtained via the Hodge decomposition (8) o! 0 ) and its rst n? 1 Lie derivatives along have linearly independent dierentials one can use them to dene a change o coordinates taking (1) to a normal orm _z = Az + Bru + Bp + Eu (12) where E depends linearly on and its derivatives up to order n? 1. By neglecting E one obtains a linearizable system _z = Az + Bru + Bp (13) approximating (12) with \error" Eu. The results o this paper allow one to obtain upper bounds on the H k norms o E o the orm q kek M0 k C I 2m1 (); (14) on any interior region M 0 o M, where m 1 depends on n and k. Moreover, Sobolev embedding theorems allow one to obtain upper bounds on the C k norms (uniorm norm together with k derivatives) o E o the orm kek M0 k;1 C q I 2m2 (); (15) on any interior region M 0 o M, where m 2 depends on n and k. Thereore, the use o least squares integrating actors provide a means or nding sensible approximations o nonlinear single-input systems by linearizable ones. Moreover, the minimum values o the unctionals I k () provide measures o linearizability o nonlinear single-input systems in various unction spaces. The paper is organized as ollows. In Section 1 we introduce notation and present some auxiliary results. In Section 2 we recall the construction o s-coordinates or a single-input ane nonlinear system. The construction is illustrated by a system in R 3. In Section 3 we show that the minimum value o unctionals I m () is zero i and only i the corresponding system is exactly linearizable. We also show that in this case all minimizers o I m () are constant unctions. In Section 4 we show that the minimum value o I 1 () is attained or a smooth and positive unction. All minimizers o I 1 () are unique up to a multiplication by nonzero constants. We also show that the minimizers o I 1 () satisy an elliptic PDE o second order with mixed boundary conditions. We provide an example o construction o a minimizer o I 1 () or a system in R 3. We also propose an approximation scheme or the minimizers o I 1 (). In Section 5 show that minimum value o unctionals I m (), or 5

6 m > 1, is obtained or a unction which is smooth in the interior o M. Contrary to the case m = 1, positivity, uniqueness, and smoothness up to the boundary o M is not clear. However, in Section 6 we show that positivity and uniqueness up to a multiplication by nonzero constants o minimizers o I m (), or m > 1, or systems suciently close to being linearizable. In Section 7 we provide an easy to calculate lower bound on the value o unctionals I m (). In Section 8 we use the Hodge decomposition to decompose a scaled characteristic orm into exact and antiexact parts. We provide upper bounds on H m norms o the antiexact part o! 0 in terms o the value o unctionals I m (). In Section 9 we present a decomposition o a single-input ane nonlinear system into a linearizable part and an error term. The error term represents an obstruction to linearizability. We provide upper bounds on the H k and C k norms o the error term via the values o the unctionals I m (). To make the paper more accessible or readers not amiliar with the theory o dierential orms, calculus o variations, and elliptic PDE's, we include proos o some standard results rom these theories. 1 Notation and Auxiliary Results By p (M) we denote the space o smooth p-orms on M. By (M) we will mean the algebra o exterior dierential orms on M. In the sequel we assume that M is equipped with a Riemannian metric, i.e., a positive denite (pointwise) inner product h; i on the tangent space to M. Except or the Section 6, the metric considered in this paper is standard in the so-called s-coordinates s i (see Section 2 o the present paper and [21]) which are global on M (i.e., M can be covered by one patch in s-coordinates). Such a metric arises ; : : n to be a positively oriented orthonormal basis or the tangent space to M at every point. The inner product on vectors induces an inner product on p-orms [1, 9] that we will denote by the same symbol. The corresponding pointwise norm will be denoted by j j. We obtain a global inner product hh; ii o p-orms on M by integrating the pointwise one over M: Z hh; ii := h; i; where := ds 1 ^ : : : ^ ds n is the volume element corresponding to the s-metric and some xed orientation o M. We dene a (global) L 2 norm o orms on M by integrating the pointwise one: Z kk := ( M jj2 ) 1 2 ; or 2 (M): (16) We will also need the Sobolev space H m (M) o k-orms which is a Hilbert space equipped with the inner product Z X X (; ) := hh; ii + Ix Ix ; (17) M M I jjm 6

7 where = P I I ds I, = P I I ds I, = 1 ; ; j is a multiindex, jj = j, and Ix j j : The corresponding norm will be denoted by k k m. Note that a p-orm = P I Ids I is in H m (M) i and only i all o its coecients (in s-coordinates) I are in H m (M). Moreover, kk 2 m = P I k I k 2 m. The (Hodge) operator (see, e.g., [9, 1]) is dened as the unique operator : p (M)! n?p (M) such that ^ = ^ = h; i, where is the volume element corresponding to the Riemannian metric and some xed orientation o M. The operator is an isomorphism between p (M) and n?p (M). For example, let M be 3-dimensional @ s 1 ; s 2 ; s 3 and the metric chosen so that the vector elds ; 3 orm a positively oriented orthonormal basis. Then the dual one-orms ds 1 ; ds 2 ; ds 3, are also orthonormal and we may choose := ds 1 ^ ds 2 ^ ds 3. We have 1 = ds 1 ^ ds 2 ^ ds 3 ; ds 1 = ds 2 ^ ds 3 ; ds 2 = ds 3 ^ ds 1 ; ds 3 = ds 1 ^ ds 2 ; ds 1 ^ ds 2 = ds 3 ; ds 2 ^ ds 3 = ds 1 ; ds 3 ^ ds 1 = ds 2 ; ds 1 ^ ds 2 ^ ds 3 = 1: Let 2 p (M). We dene the codierential o as := (?1) n(p?1)+1 d : Note that the codierential depends on the choice o Riemannian metric. The codierential is a ormal adjoint to the exterior derivative due to the ollowing result (c. [1]). Proposition 1.1 Let 2 p (M) and 2 p+1 (M). Then Z ZM ZM hd; i = h; i + ^ : Proo: Since, or any 2 p (M), = (?1) p(n?p) (see [1]), we have d( ^ ) = d ^ + (?1) p ^ d = d ^? ^ so that d ^ = ^ + d( ^ ) Integrate this expression over M and apply Stokes Theorem. 2 7

8 Note that on maniolds without boundary the above result states that hhd; ii = hh; ii where hh; ii is the global inner product o p-orms on M obtained by integrating the pointwise one over M. We will also need the dierential operators d i := i i z } { z } { dd i := d; so that d 1 = d, 1 =, d 2 = d, 2 = d, etc. With this notation we can dene or any positive integer m the quadratic orm Q m (; ) := mx (hhd i ; d i ii + hh i ; i ii) (19) i=1 By DQ m we denote the domain o Q m (; ), i.e., the set o one-orms in L 2 such that i and d i are in L 2 or i m. In the sequel we will deal with a xed one-orm! 0 that satises j! 0 j = 1 and ad n?1 g > 0. For any zero orms, we dene q m (; ) := Q m (! 0 ;! 0 ). The domain o q m (; ) is denoted Dq m. Suppose that! 0 is the unique characteristic orm or the system (1) with j! 0 j = 1 and! 0 (ad n?1 g) > 0. Using notation introduced above we may rewrite the unctional I m dened by (10) as The Laplacian o a k-orm is dened as I m () = Q m(! 0 ;! 0 ) k! 0 k 2 = q m(; ) k! 0 k 2 : (20) := (d + d) : Any orm such that d = 0 and = 0 (which implies = 0) on M will be called harmonic. We will also need the ollowing simple result. Lemma 1.1 Let a; b be any real numbers and > 0. Then jaj jbj a2 2 + b2 2 I the closure o M 0 is included in M we write M 0 M. In the sequel we will need the ollowing result. Theorem 1.1 Let M R n be an open and bounded region with a smooth and m be a positive integer. Let M 0 M. Then, or any one-orm one has q kk M0 2m C 1 ( Q 2m (; ) + kk); (21) where C 1 depends on M 0, M, and m. 8

9 Proo: From a standard interior regularity result (see, e.g., [11, Thm 16.1]) it ollows that or any unction one has kk M0 2m c 1 (k m k + kk): (22) Note that this result extends to one-orms. For this, express in s-coordinates as = P i ds i. Then m = P m i ds i and thus k m k 2 = P k m i k 2. Obviously then kk M0 2m c 2 (k m k + kk): (23) Since dd = 0 and = 0 we have m = (d + d) m = d 2m + 2m. Thereore k m k kd 2m k + k 2m k p 2Q 2m (; ). This and (23) yield the result. 2 We will also use the spaces C k o k-times continuously dierentiable vector elds equipped with the norm kvk X X k;1 := sup i x2m jv ix j: jjk The set o smooth unctions with compact support on M will be denoted by C s-coordinates (M). The linear controllability property allows one to locally dene a special set o coordinates called s-coordinates (c. [21]). In a neighborhood o the origin one can \reach" any point x by \traveling" along vector elds ad n?1 g; : : :; ad g; gg with \times" s 1 ; s 2 ; : : :; s n, i.e., x = g s n ad g s n?1 adn?1 s 1 (0) with h s () being the ow o a vector eld h. We dene the s-metric to be the Riemannian metric in which the 1 2 ; : : n g are orthonormal. Note that the construction o the s-metric makes sense only on the subset o M on which the s-coordinates are valid. I necessary, we restrict to a subset o M which is an open and bounded subset o R n containing the origin and having a smooth boundary on which the s-coordinates are valid. The ollowing example illustrates the construction o s-coordinates or a system in R 3. Example 2.0 Consider the system _x 1 = x 2 + h 1 (x 3 ) + h 2 (x 1 ; x 2 ) _x 2 = x 3 + h 3 (x 3 ) + h 4 (x 1 ; x 2 ) _x 3 = u; where h i () are smooth unctions with h i (0) j (0) = 0 (x 2 1 ; 0) (x 2 1 ; 0) = 0. We have 3 ; ad g =?h ? (1 + h ; ad 2 g = ((1 + h0 3(x 3 ))(1 2(x 1 ;x 2 2 ) + h 0 1(x 3 2(x 1 ;x ((1 + h 0 3(x 3 4(x 1 ;x h 0 1(x 3 4(x 1 ;x 2 ) : (24) (25)

10 Note that ad 2 g(x 1; 0; 1 and ad 2 g(x 1; x 2 ; 2. We see that the s-coordinates s 1 := x 1 s 2 :=?x 2 s 3 := x 3 : are valid on all o R 3. 2 Remark 2.1 It is possible to consider a more general construction o s-coordinates than the one using the ow o ad n?1 g; : : :; ad g; gg. Actually, all results o this paper, except those o Section 6, remain true, i to construct the s-coordinates one uses the ow o any set o vector elds g 1 ; g 2 ; g 3 ; : : :; g n g with the property that g 1 is transversal to the characteristic distribution D in a neighborhood o 0 and g 2 ; g 3 ; : : :; g n g is any basis or D. In [8] we construct another set o s-coordinates (in this more general sense) or the system (24). 2 3 Least Squares Exact Integrating Factors Clearly, the value o the unctional I 1 () given by (7) is always nonnegative. We will show that it attains zero i and only i the system (1) is exactly eedback linearizable. Thus the minimum value o the unctional provides a measure o linearizability o (1) in a least squares sense. Theorem 3.1 Let M R n be an open, bounded, and contractible region with a smooth Then the system (1) is exactly eedback linearizable i and only i there exists a smooth positive zero-orm such that I 1 () = 0. Moreover, in this case any minimizer has the orm = c = const or some c 6= 0. Proo: ()) Note that in the case o exactly eedback linearizable system (1) the vector elds ad n?2 g; : : :; ad g; gg span the tangent spaces to the n? 1 dimensional maniolds s 1 = const. Thus, we @ ; ; : : n g = D. Thereore, in s-coordinates any characteristic orm! 0 has the orm! 0 =! 1 0 ds 1; where! 1 0 is a smooth nonvanishing unction o s 1 ; : : :; s n. Moreover, the normalization j! 0 j = 1 and! 0 (ad n?1 g) > 0 implies! 1 = 1.Thus 0 d! 0 = d ^ ds 1 = nx i=2! i ds i ^ ds 1 ; (26) 10

11 Note that in the s-metric (or, more precisely in the metric on orms induced by s-metric on vector elds) ds i ^ ds 1 are orthonormal. Thereore jd! 0 j 2 + j! 0 j 2 = jdj 2 : (27) Choosing 0 = c, where c is a nonzero constant, we obtain I 1 ( 0 ) = 0. Changing sign o c, i necessary, we may assume that 0 is strictly positive. (() I 1 () = 0 implies in particular d! 0 = 0 on M. Thus! 0 is exact in a neighborhood o the origin. 2 One can also show that Theorem 3.2 The system (1) is exactly eedback linearizable i and only i there exists a smooth positive zero-orm such that I m () = 0. Moreover, in this case any minimizer has the orm = c = const or some c 6= 0. Proo: It is similar to that o Theorem Corollary 3.1 Let M R n be an open, bounnded, and contractible region with a smooth A linearly controllable system (1) is exactly eedback linearizable i and only i there exists a characteristic orm! or (1) which is harmonic in the s-metric. A nice eature o an s-metric is that in the case o linearizable systems the value o s 1 parametrizes the integral maniolds o D. The calculation o terms d! 0 and! 0 in the proo o Theorem 3.1 shows precisely that the exterior derivative in d! 0 controls the behavior o along the integral maniolds o! 0, while the codierential in! 0 controls the behavior o in the direction transversal to the integral maniolds o! 0. A natural question arises o what happens i we skip the assumption that the Riemannian metric is an s-metric. As we shall see in the next section, the minimum o the integral is still obtained or some smooth and positive. However, we can no longer guarantee that the minimum value o the unctional will be zero or a linearizable system. (Actually, or a generic choice o metric this will not be the case.) Hence, we cannot claim that the minimum value o the unctional is a measure o linearizability i we do not dene it with an s-metric. 4 Least Squares Approximate Integrating Factors o Order One It can be shown that the minimum o I 1 () exists or any positive denite Riemannian metric. To simpliy calculations we will use the standard metric associated with some coordinate system x i. In applications we would seek or the minimum o the unctional in some s-metric, since only or those metrics we know that we obtain an exact integrating actor or a characteristic orm! 0 i one exists (i.e. when the corresponding system (1) is exactly linearizable). 11

12 Theorem 4.1 Let M R n be an open, bounded, and contractible region with a smooth with the standard metric associated with some coordinate system x i. Then, among all zero-orms in H 1 (M) there exists a smooth positive zero-orm that minimizes the unctional I 1 () dened by (7), determined uniquely up to a multiplication by a positive constant. Dene 0 := in I 1 () : (28) 2H 1 (M) Any minimizing zero-orm is a solution to the boundary value problem with the boundary conditions h! 0 ; (! 0 )i? 0 = 0; on M; (29)! 0 ^ d(! 0 )? (! 0 ) ^! 0 = 0; (30) Moreover, any 6 0 in H 1 (M) satisying (29) and (30) is a minimizer o I 1 (). Step 1. Boundedness. Note that I 1 () 0; 8 2 H 1 (M), so that 0 0. Let Proo: i 2 H 1 (M) be any minimizing sequence, i.e. lim I 1 ( i ) = 0. Without loss o generality we can assume that the minimizing sequence is normalized in L 2 (M), so that Note also that we can assume that or some real C 0 we have k i! 0 k = k i k = 1: (31) kd i! 0 k 2 + k i! 0 k 2 C 0 : (32) Let l() := jd! 0 j 2 + j! 0 j 2. Note that l() is a quadratic orm in and its rst partial derivatives. We will show that there are positive constants C 1 ; C 2 such that For this, by direct calculation we veriy that l() C 1 jdj 2? C 2 2 8x 2 M: (33) l() = jdj 2 + l(1) 2 + nx i=1 c i ; (34) where c i (x) are smooth unctions (depending on! 0 and the partial derivatives o its components). Thus, X n l() jdj 2 + l(1) 2? C 4 j; i=1 i where C 4 jc i (x)j 8i 8x 2 M Now, using Lemma 1.1 we i j i ) 2 8i; 8 > 0; (36) 12

13 so that 8 > 0 we have l() (1? C 4 2 )jdj2 + (l(1)? nc 4 2 )2 : (37) Put C 1 := 1? C 4 2 ; C 2 := nc 4? l(1): (38) 2 Note that the constants are positive or small enough. (Actually, we only need positivity o C 1.) The normalization (31) and the inequalities (32) and (33) imply that kd i k C 3 (39) or some real C 3, so that the mimimizing sequence i is bounded in H 1 (M). In particular, we proved that Dq 1 = H 1 (M): (40) Step 2. Compactness. By Rellich-Kondrasov Theorem (a ball in H 1 (M) is precompact in L 2 (M)), i has a subsequence, to be also denoted by i, converging to some 0 2 H 1 (M) weakly in H 1 (M) and strongly in L 2 (M). Note that since k i k = 1 we also have k 0 k = 1. Since the norm k k is lower semicontinuous with respect to the weak convergence, I 1 ( 0 ) lim in I 1 ( i ) = 0 ; so that 0 is a minimizer. Step 3. Regularity. Since we know that a minimizer 0 2 H 1 (M) exists, we can write the corresponding Euler-Lagrange equations in a weak orm. We know that I 1 ( 0 ) = 0 so that the associated unctional Z J() := q 1 (; )? 0 kk 2 = M (jd! 0j 2 + j! 0 j 2? 0 2 ) (41) has value zero or = 0 and J( 0 + t) 0 or any xed 2 H 1 (M) and any real t (c. [12, Section 8.12]). Thus, or xed 2 H 1 (M) the rst 0+t) must vanish t = 0. A straightorward calculation gives the ollowing Euler-Lagrange equation in a weak 0 + t) = 2(q 1 ( 0 ; )? 0 hh 0 ; ii) = 0; (42) or all 2 H 1 (M). The estimate (33) shows that the orm q 1 (; )? 0 hh; ii dened is coercive in H 1 (M) and hence a standard elliptic regularity argument (c. [10]) shows that 0 2 C 1 (M). Thereore, integration by parts is legal. We use Proposition 1.1 to obtain the strong version o the Euler-Lagrange PDE (29) with the boundary conditions (30). Step 4. Positivity. One can check that i 0 is a minimizer o I 1 (), so is j 0 j. From Step 3 we see that 0 2 H 1 (M) is a minimizer o I 1 () i and only i it is an eigenunction (the ground state) o the strongly elliptic operator L := h! 0 ; (! 0 )i corresponding to the minimal eigenvalue 0 o L. Thereore, i 0 is a minimizer o I 1 (), then j 0 j is an 13

14 eigenunction o L. The Harnack inequality (c. [12, Thm 8.21]) shows that j 0 j > 0 so that 0 must have been either strictly positive or negative. One can assume that 0 is strictly positive. Step 5. Uniqueness. I there were two linearly independent minimizers o I 1 (), they would be both eigenunctions o L corresponding to the eigenvalue 0. But then one could choose them to be orthogonal. Yet, Step 4 shows that each o them is either strictly positive or negative, so that they cannot be orthogonal. Thus, the eigenspace o L corresponding to the minimal eigenvalue 0 is one-dimensional, so that the minimizer is determined uniquely up to a multiplication by a nonzero constant. (c. [12, Thm 8.38]) 2 Remark 4.1 It can be shown that in the case when! 0 is not normalized, a zero-orm minimizing I 1 () is a solution to the boundary value problem h! 0 ; (! 0 )i? 0 j! 0 j 2 = 0 (43) with the boundary conditions (30). 2 Below we show an example o construction o a least squares approximate integrating actor or a system in R 3. Example 4.0 Consider the system _x 1 = x 2 + log(cos(ax 3 ) 1 a ) _x 2 = x 3 _x 3 = u; where a > 0. Let M be any open, bounded, and contractible region on which jx 3 j < 2a. Note that this system is a particular case o (24), so that the s-coordinates are s 1 := x 1, s 2 :=?x 2, and s 3 := x 3. We have 3 ; ad g = tan(ax @x 2 = tan(as 2 ; ad : The normalized characteristic one-orm is! 0 := cos(as 3 )ds 1? sin(as 3 )ds 2 : One can show that minimizing I 1 () depends only on s 3. Ater straightorward calculations we obtain jd! 0 j 2 = 02 + a 2 2 (46) j! 0 j 2 = 0: Thereore, I 1 () = R (44) (45) M (02 + a 2 2 ) R : (47) M 2 Note that or every 2 H 1 (M) we have I 1 () a 2. Moreover, or any nonzero constant c, or = c, we have I 1 () = a 2. Thus = c are the minimizers o I 1 (). We also see that a 2 is a measure o linearizability o the system (44). 2 14

15 One can rarely expect to be able to nd least squares integrating actors directly by investigating the unctional I 1 () or solving the corresponding eigenvalue problem. An important eature o our variational problem is that one can approximate least squares integrating actors by a sequence o minimizers m o the corresponding unctionals I m() 1 where I m 1 () is simply I 1 () with is restricted to a xed m-dimensional subspace E m o H 1 (M). Theorem 4.2 Let M R n be an open and bounded region with a smooth Let e j g be a basis or the space H 1 (M), which is orthonormal in L 2 (M). Let m be the unique positive minimizer o I 1 () or 2 E m := span e 1 ; : : :; e m g with k m! 0 k = 1. Then the sequence m g converges strongly in L 2 (M) to the unique positive minimizer 0 o I 1 () with k! 0 k = 1. Proo: Let b j be the coecients o 0 with respect to e j g. Then the sequence o partial sums m := P m j=1 b j e j converges to 0 strongly in H 1 (M). Thus I 1 ( m ) converges to I 1 ( 0 ). Obviously, I 1 ( m ) I 1 ( m ). As in Step 1 o the proo o Theorem 4.1 one shows that kd m k are bounded. It ollows rom the uniqueness o positive normalized minimizer 0 o I 1 () that m converges strongly in L 2 (M) to 0. 2 Note that minimizing the unctional I 1 () over a nite dimensional space E m is a standard quadratic minimization problem. The solution or the coecients o m in basis e 1 ; : : :; e m g is an eigenvector o a nonnegative denite m m matrix q 1 (e i ; e j )g corresponding to its smallest eigenvalue. By solving the problem or each m we obtain a sequence o approximate minimizers that converges in L 2 (M) to the unique positive minimizer 0 o I 1 () with k! 0 k = 1. 5 Higher Order Approximate Integrating Factors In this section we show that the minimum o I m () is attained. We begin with the ollowing important result. Proposition 5.1 For any m 1, one has Dq m H 1 (M) and kk 2 1 C(q m (; ) + kk 2 ); 8 2 Dq m : (48) Proo: In Step 1 o the proo o Theorem 4.1 we showed that Dq 1 = H 1 (M). It is obvious that Dq m Dq 1. 2 Theorem 5.1 Let M R n be an open, bounded, and contractible region with a smooth and m be a positive integer. Then, among all zero-orms in Dq m, there exists a smooth zero-orm that minimizes the unctional I m () dened by (20). 15

16 Let := in I m (). Note that I m () 0; 8 2 Dq m, so that 0. Let i 2 Dq m Proo: be any minimizing sequence, i.e., lim I m ( i ) =. Without loss o generality we can assume that the minimizing sequence is normalized in L 2 (M) by (31). Thus q m ( i ; i ) = I m ( i ) and or some real C 0 we have q m ( i ; i ) C 0 : (49) Thereore, by Proposition 5.1, we have k i k 1 C 1 ; (50) or some C 1. Thus, by Rellich-Kondrasov Theorem (c. [20]), there is 0 2 Dq m such that there is a subsequence o i (to be also denoted by i ) which converges to 0 strongly in L 2 and all the terms d j i! 0 and j i! 0 converge weakly in L 2 to d j 0! 0 and j 0! 0. To explain the latter, we notice that d j i! 0 and j i! 0 converge weakly in L 2 to something and in the distributional sense to d j 0! 0 and j 0! 0, respectively, so that the limits coincide. Moreover, since the norm is lower semicontinuous with respect to weak convergence, we obtain q m ( 0 ; 0 ) = I m ( 0 ) =. Note that or every 2 Dq m one has q m ( 0 ; )? hh 0 ; ii) = 0: (51) Since the smooth unctions with compact support on M are in Dq m, it ollows rom Proposition 1.1 that 0 satises the ollowing Euler-Lagrange equation in a weak orm Z M (h( m? )! 0 ;! 0 i) = 0; (52) or all 2 C 1 0 (M). Standard interior elliptic regularity arguments (see, e.g., [11, Part 1, 15-16]) show that is smooth in M. 2 Note that Theorem 5.1 is a much weaker result than Theorem 4.1. First o all, contrary to the case m = 1, we have not proved positivity and uniqueness (up to multiplication by a constant) o minimizers o I m or m > 1 in a general situation. Since we do not have positivity o minimizers o I m or m > 1, we cannot claim that! 0 is a characteristic oneorm or (1) on M. However, by a perturbation argument, we will show in the next section that we have positivity and uniqueness o higher order approximate integrating actors or system suciently close to being linearizable. In general, positivity o approximate integrating actors seems to be very dicult to prove, or there is no general theory that would guarantee positivity o the ground state o higher order elliptic operators. Another weakness o Theorem 5.1 is that it only guarantees smoothness o minimizers o I m or m > 1 in the interior o M and we do not have smoothness on the boundary. The reason is that, except when m = 1, the orm q m (; ) is not coercive in H m (M) and thus q m (; ) does not control the behavior o on the boundary o M. This, in turn, does not allow one to claim that the minimizers satisy certain boundary conditions in the classical sense. One can treat boundary conditions by abstract trace techniques (see [3]), but it does 16

17 not improve the regularity o the minimizer on the boundary. It seems to be possible to obtain smoothness up to the boundary and classical boundary conditions or the minimizer (as in the case m = 1), by modiying the unctional I m so that it involves some more higher order partial derivatives. One can derive an approximation scheme or the minimizer o I m or m > 1 similar to that given by Theorem Higher Order Approximate Integrating Factors or Systems Close to Being Linearizable In this section we show that the higher order approximate integrating actors do not vanish in M i the system is suciently close to being linearizable. Since we are going to work with several systems dened by several pairs ; g, each dening its own s ;g -coordinates, and thus a metric and a unctional, to compare them we need to introduce some common xed x-coordinates and Sobolev spaces. Denote by k k x, k k x m, k k x k;1 respectively the L 2, H m, and C k norms in some xed x-coordinates. We will denote the corresponding spaces by L x 2, H x;m, and C x;k, respectively. For every pair ; g the corresponding L 2 norms, Laplacians, unctionals, and quadratic orms dened using s ;g -coordinates will be denoted by k k ;g, ;g, Im ;g (), and qm ;g (; ), respectively. We will assume that all the s ;g -coordinates are global on M and all Jacobian determinants o changes o coordinates rom x to s ;g and rom s ;g to x ;g are uniormly bounded away rom zero. We also assume that all s ;g - coordinates have common origin. (These assumptions are not essential, they are made mainly or simplicity o presentation.) In the sequel we will need the ollowing basic result. Proposition 6.1 For any k the mapping (; g) 7! s ;g () assigning s-coordinates to ; g is a continuous mapping rom C x;k+n?1 C x;k+n?1 to C x;k. The main result o this section is as ollows. Theorem 6.1 Let 0 ; g 0 dene a linearizable system on M. Then or a suciently large integer k and a suciently small real number i k? 0 k x k;1 + kg? g 0k x k;1 < then the zero-orms minimizing the unctional Im ;g () do not vanish in M and are unique up to multiplication by a nonzero constant. Proo: Let k; k 0 be positive integers to be determined later. It ollows rom straightorward calculations that, ater change o coordinates rom s to x, the operator L ;g := ;g + 2 ;g + + m ;g? ;g is a strongly elliptic dierential operator o order 2m in coordinates x, whose coecients are smooth unctions and change continuously in C x;k0 i ; g change continuously in C x;k C x;k, or suciently large k. To prove the theorem it is enough to show that, or any M 0 M and any sequence o normalized minimizers i o the unctionals I i;g i m () corresponding to i ; g i converging in C x;k to 0 ; g 0, one has lim in M 0 j i j > 0. Suppose to the contrary that there is M 0 and 17

18 a sequence i ; g i ; i violating this statement. Observe that L i ;g i have ellipticity constant in x-coordinates uniorm in i and common bounds on C x;k0 norms o coecients. Recall that i are smooth on M and satisy L i ;g i i = 0 on M. Moreover, we can also assume k i k i;g i = k i! i ;g i k i;g i = 1 (note that we have a standing assumption j! i ;g i j i;g i = 1), and hence q i;g i m ( i ; i ) = I i;g i m ( i ). One can easily show that or suciently large i one has c 1 k i k x c 2 (53) or some positive c 1 ; c 2. Note that or xed 1 ; 2, q i;g i m ( 1 ; 2 ) change continuously i i ; g i change continuously in C x;k C x;k, or suciently large k. Thereore, since = c 6= 0 (see Theorem 3.2) is a minimizer o the unctional I 0;g 0 m have q i;g i m ( i ; i )! 0. Thus, one can prove that () and q i;g i m ( i ; i ) q i;g i m (c; c), we k i k x 1 c 3 ; (54) or some positive c 3. Choose a subsequence o i, also denoted by i, that converges strongly in L x. Expressing all terms 2 d j i! i and j i! i as explicit dierential operators in x-coordinates and arguing as in the proo o Theorem 5.1 it can be shown (passing to a subsequence, i necessary) that d j i! i and j i! i converge weakly in L x to 2 d j 0! 0 and j 0! 0, or some 0 2 Dq 0;g 0 m. Thus q 0;g 0 m ( 0 ; 0 ) = 0. Thereore, 0 is a constant that we can assume to be positive. Since the operators L i ;g i have ellipticity constant in x- coordinates uniorm in i and common bounds on C x;k0 norms o coecients, by standard interior regularity results (see, e.g., [11, Part 1, 15-16]), we get k i k x;m0 m 0 C m 0, where m 0 can be arbitrarily large by choice o k 0. From Sobolev embeddings ([2]) it ollows that i are uniormly continuous on M 0, uniorm in i. Hence, i converge uniormly to 0, which is a nonzero constant. We have reached a contradiction. Assume now that the k and are such that the corresponding minimizers o I m () are positive. To prove uniqueness, observe that one can easily show (using (51)) that the space o minimizers o I m () is linear. Thus, i the space o minimizers o I m () were o dimension two or more, there would be two linearly independent minimizers 1 and 2 o I m (). Moreover, one could choose them to be orthogonal in L 2 (M). But then at least one o them would have to change sign, contradicting the positivity. Thus, the space o minimizers o I m () has dimension one. 2 Note that the optimal value o integer k can be explicitly computed. 7 A Lower Bound In this section we establish a lower bound or the value o the unctionals I m (). For this we need a decomposition o the exterior derivative o a characteristic orm! 0 (c. [7, 6]). Proposition 7.1 Let! 0 two-orm such that be a given one-orm on M. Then there is a one-orm and a 18

19 1. d! 0 = ^! jj = jd! 0? ^! 0 j is pointwise minimal possible 3. or any one-orm the two-orms ^! 0 and are pointwise orthogonal. The two-orm is determined uniquely, while the one-orm is determined uniquely up to an addition o a multiple o the one-orm! 0 by any zero-orm. The orms and can be explicitly expressed as = (?1) n+1 (! 0^d! 0 ) j! 0 j 2 = (! 0^(d! 0^! 0 )) (55) j! 0 j 2 ; where is the Hodge star operator. Proo: To prove that and given by (55) satisy d! 0 = ^! 0 + is a tedious, but straightorward calculation. To prove the third statement, let be any one-orm on M. Note that h ^! 0 ; i = ^! 0 ^ = ^! 0 ^ (! 0 ^ (d! 0 ^! 0 )) = 0: j! 0 j 2 Thus, ^! 0 and are pointwise orthogonal. It was proved in [7] that the second statement ollows rom the third one. 2 An alternative expression or and in terms o interior products instead o -operator can be ound in [7, 6]. It can be shown that = 0 i and only i! 0 is integrable and = 0 and = 0 i and only i! 0 is closed. With those orms we can express d! 0 as Now, we can obtain the ollowing lower bound or I m () Proposition 7.2 Let m 1. For every 2 Dq m we have where the inmum is taken over M. Proo: every, we have d! 0 = (d + ) ^! 0 + : (56) I m () (in jj) 2 ; (57) Since j! 0 j = 1 and the two-orms (d + ) ^! 0 and are pointwise orthogonal or I m () I 1 () = = R M j(d+)^! 0j 2 R M j! 0j 2 R M (jd! 0j 2 +j! 0 j 2 ) R R R M j! 0j 2 M j! 0j 2 + R M jj2 R R RM jd! 0j 2 M j! 0j = 2 R M jj2 jj 2 M jj2 j! 0 j (in 2 jj)2 : R M j! 0j 2 M j(d+)^! 0+j

20 Note that the lower bound or I m () derived above is positive i and only i is everywhere nonzero. (It can be shown that the latter is the case whenever d! 0 ^! 0 6= 0 on M.) It ollows rom Frobenius' Theorem that an exact integrating actor or! 0 6= 0 exists i and only i = 0 on M. Thus, jj provides a pointwise measure o integrability o! 0. From this point o view we may say that Proposition 7.2 gives a positive lower bound or the average measure o integrability in terms o a pointwise one. This lower bound is easy to nd, or it requires only pointwise calculations. I one uses an arbitrary (i.e., not normalized) characteristic orm! 0, one should replace (in jj) 2 in j (x)j with supj! 0 (x)j in (57). It in j (x)j can be shown that the ratio supj! 0 (x)j is independent o the choice o! 0. 8 Hodge Decomposition In this section we show how to approximate a characteristic one-orm! 0 by an exact orm d in a least-squares sense. The most important result is the Hodge Decomposition Theorem (see, e.g., [1, 20] and the application in [4]) which provides a least squares approximation o a one-orm by an exact orm. Theorem 8.1 Let! 0 be a xed smooth one-orm on M. For any there is a unique (up to a constant) smooth zero-orm on M and a smooth one-orm such that! 0 = d + ; (58) and such that kk is minimized over all possible smooth zero-orms. satises = 0 The one-orm in M and is tangent to the o M, i.e., (v) = 0, where v is the unit outward normal vector eld The smooth unction is given uniquely up to an addition o a constant by the solution o Poisson's equation with Neumann boundary conditions =! 0 (59) d(v) =! 0 (v): (60) The one-orms d and appearing in the Hodge decomposition (58) o! 0 will be called, respectively, the exact part and antiexact part o! 0. In the sequel we assume that! 0 is a characteristic one-orm or (1) such that j! 0 j = 1 and! 0 (ad n?1 g) > 0, is a smooth nonvanishing unction normalized in L 2 so that kk = 1, and is the antiexact part o! 0. We can obtain the ollowing upper bound on the H 1 norm o. 20

21 Theorem 8.2 There is a positive constant C (depending only on M) such that q I 1 (): (61) kk 1 C Proo: It ollows rom Theorem o [20]. 2 We see that a \small" value o I 1 () guarantees a \small" value o kk 1. In particular, I 1 () = 0 implies kk 1 = 0. Since the bound (61) in monotone in I 1 (), we obtain the smallest value o this upper bound by choosing to be a least squares integrating actor. We can obtain a similar upper bound on the H 2m (M 0 ) norm o, where M 0 M. Theorem 8.3 Let M 0 M. There is a positive constant C (depending only on M 0, M, and m) such that q kk M0 2m C I 2m (): (62) Proo: It ollows rom Theorem 1.1 that kk M0 2m p C 1 ( Q 2m (; ) + kk). Observe that d = d! 0 and = 0 imply that Q 2m (; ) Q 2m (! 0 ;! 0 ). pon the other phand we have kk kk 1, andp it ollows rom Theorem 8.2 that kk 1 C 2 I1 () = C 2 Q1 (! 0 ;! 0 ). Thus kk 1 C 2 Q2m (! 0 ;! 0 ). This proves the result. 2 We see that a \small" value o I 2m () guarantees a \small" value o kk 2m. In particular, I 2m () = 0 implies kk 2m = 0. Again, since the bound (62) in monotone in I m (), we obtain the smallest value o this upper bound by choosing to be a higher order least squares integrating actor. 9 Application o Approximate Integrating Factors to Approximate Feedback Linearization In this section we show how higher order approximate integrating actors can be applied to obtain eedback linearizable systems approximating a given nonlinearizable system. Namely, in the previous section we have shown that construction o least-squares integrating actors provides good least-squares approximation o characteristic one-orms by exact orms. In this section we show that this, in turn, leads to good better approximation o nonlinearizable systems (1) by linearizable systems. To see that, recall that it was shown in [7] that i and its rst n?1 Lie derivatives along have linearly independent dierentials one can use them to dene a change o coordinates taking (1) to a normal orm _z = Az + Bru + Bp + Eu (63) 21

22 where A; B are in Brunovsky orm and r; p, and E := [e 1 ; e 2 ; ; e n ] T are given by e 1 =?(g); e 2 =?(L )(g);. e n =?(L n?1 )(g); p = (L n?1!)()? (L n?1 )(); r = (?1) n?1!(ad n?1 g): Note that by linear controllability assumption r 6= 0 on M. Observe also that E vanishes whenever does. Neglecting E one obtains a linearizable system (64) _z = Az + Bru + Bp (65) approximating (63) with \error" Eu. The results o this paper allow one to obtain upper bounds on the H k and C k norms o E providing a measure o how well (65) approximates (63) on the given region. Note that the expression or E involves derivatives o up to order n? 1. Thereore, it is possible to nd an upper bound on the H k norm o E depending on the H k+n?1 norm o. Namely, we have the ollowing simple result. Proposition 9.1 Let M 0 M. There is a positive constant C (depending only on M 0, k,, and g) such that kek M0 k Ckk M0 k+n?1 : (66) Using Sobolev embeddings it is also possible to obtain bounds on the C k norm o E depending on the H m+n?1 norm o where m > k + n=2. The next result ollows rom the Sobolev embedding H m,! C k or m > k + n=2 (see, e.g., [2]). Proposition 9.2 Let M 0 M and m > k+n=2. There is a positive constant C (depending only on M 0,, and g) such that kek M0 k;1 Ckk M0 m+n?1 : (67) Theorem 8.3, combined with Propositions 9.1 and 9.2, allows one to obtain upper bounds on the H m and C k norms o E on M 0 M depending on the value o the unctional I 2m (). Namely, we have the ollowing simple corollary. Corollary 9.1 Let M 0 M. Assume that! 0 is a characteristic orm or (1) such that j! 0 j = 1 in the s-metric. Let be any smooth nonzero unction normalized in L 2 so that kk = 1 and let be obtained rom the Hodge decomposition (58) o! 0. Then, i ; L ; ; L n?1 are valid coordinates in M 0, one can put (1) in orm (63). Let k be any be any integer such that k + n? 1 2m 1. Then, there is a positive integer and let m 1 positive constant C 1 (depending only on M 0, M,, g, and k) such that kek M0 k C 1 qi 2m1 (): (68) 22

23 Also, let m 2 be any integer such that k + 3n=2? 1 < 2m 2. Then there is a positive constant C 2 (depending only on M 0, M,, g, and k) such that kek M0 k;1 C 2 qi 2m2 (): (69) The above results allow one to calculate a minimum order o approximate integrating actor that guarantees the required bound on nonlinear perturbation term E in (63). The table below gives, or several values o the dimension n o M, a minimum value o 2m 1 and 2m 2 so that I 2m1 () and I 2m2 () provide upper bounds on, respectively, kek M0 and kek M0 0;1 (i.e. on L 2 and C 0 norms o E) in (68) and (69). Minimum order o approximate integrating actor dimension o M m 1 that provides a bound on kek M0 in (68) m 2 that provides a bound on kek M0 0;1 in (69) Conclusion In this paper we have developed a unied approach to the exact and least squares approximate eedback linearization problems. This approach uses the Hodge decomposition to obtain the closest (in a least squares sense) exact orm to a given scaled characteristic one-orm. We have studied the problem o nding approximate integrating actors that scale a nonintegrable one-orm in a way that the scaled orm is close to being integrable in L 2 together with some derivatives. Our solution(s), the (order one and higher order) least squares approximate integrating actor, possesses a number o useul properties. In particular, a least squares integrating actor will be an exact integrating actor i the one-orm is integrable and will provide a guaranteed level o approximation when the one-orm is not integrable. Our work suggests that practical approximation schemes can be developed and applied to the approximate eedback linearization o engineering systems. Acknowledgement. We acknowledge the use o the Mathematica package \Dierential Forms" created by Frank Zizza o Willamette University. Reerences [1] R. Abraham, J. E. Marsden, and T. Ratiu. Maniolds, Tensor Analysis, and Applications, Addison-Wesley, [2] R. A. Adams. Sobolev Spaces, Academic Press, [3] J. P. Aubin. Approximation o Elliptic Boundary-Value Problems, Wiley,