V. RAMAKRISHNA èf; gè-invarianceèor local controlled invarianceè condition, ëf; æë ç æ + G hold. Here F is in any of the one of the vector æelds f; g

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1 Journal of Mathematical Systems, Estimation, and Control Vol. 6, No. 3, 1996, pp. 1í29 cæ 1996 Birkhíauser-Boston Disturbance Decoupling Via Diæerential Forms æ Viswanath Ramakrishna y Abstract We formulate and study the question of ænding a static feedback which will cause an aæne control system èwith some restrictions on the number of inputsè to leave a given codistribution invariant. Speciæcally, our contribution is threefold: iè we show that such a study has numerous beneæts even when the codistribution is nonsingular. For instance, one can recover both the standard results and results as manifestations of the same phenomenon; iiè for analytic data and singular codistributions we relate this question to the work of ëë18ëë; and iiiè in the process of studying the same question for partially smooth data, we develop a method for solving degenerate, overdetermined systems of ærst-order partial diæerential equations which is reminiscent of the method of successive integration. Key words: one forms, integrability conditions, blow ups AMS Subject Classiæcations: 93C10, 93C15, 93C25 1 Introduction A problem that has attracted quite some research in control theory is the controlled invariance problem. Its standard formulation runs as follows èsee ë16, 21ë and the references thereinè: given a nonlinear control system _x = fèxè+ X i=1 g i èxèu i è1.1è and an involutive distribution èor vector æeld systemè æ, ænd a feedback control law of the form u = æèxè +æèxèv, where v is the new control, so that the closed loop system will leave æ invariant. A necessary condition èif the feedback is to be regularè for its solvability is that the so-called weak æ Received January 18, 1994; received in ænal form March 28, Summary appeared in Volume 6, Number 3, y Supported in part by a grant from the ARO. 1

2 V. RAMAKRISHNA èf; gè-invarianceèor local controlled invarianceè condition, ëf; æë ç æ + G hold. Here F is in any of the one of the vector æelds f; g i ;i=1;:::;m and G is the distribution spanned by the g i ;i =1;:::;m. It is also known to be suæcient under the following regularity conditions: aè æ has constant rank in a neighborhood U of the operating point, and bè the distribution èg + æènæ has constant rank on U of the operating point. Furthermore, the feedback æèxè and æèxè may be found locally èi.e., on some open subset V ç Uè by solving the system of equations: X i æ j èxè =c i jèxè;i=1;:::;d;j =1;:::;m and èon denoting by æ l èxè the lth column of æèxèè X i æ l èxè =C i èxèæ l èxè;i=1;:::;d where X 1 ;:::;X d form a basis for the distribution æ. The c j i èxèèresp.c ièxèè are smooth functions èresp. matrices of smooth functionsè on U concocted out of the data èsee ë22ëè. A natural problem that arises is the study of the same question with æ replaced by a codistribution æ: èthroughout this paper we will use the term codistribution to mean the span of a set of one-forms, since this terminology is by now established in the control literature.è Our aim is to show that an answer to this question has umpteen practical beneæts as well. To begin with, it is known that in most applications of the controlled invariance the point of departure is actually a codistribution which is spanned by exact diæerentials of smooth functions obtained directly from the data of the design problem. The vector æeld system, æ, of the paragraph above is merely the annihilator of this codistribution. Passage to the annihilator is indeed one of the main reasons for imposing the regularity conditions mentioned in the ærst paragraph. Furthermore, passing to the annihilator imposes the additional burden of pairing down; i.e, one has to ænd basis for the annihilating distribution. This involves solving systems of analytic equations. We shall see in the balance of the paper that there are innumerable other advantages of analysing this problem via diæerential forms. One such prominent advantage is the fact that the analysis of singularities in the controlled invariance problem is rendered very elegant and rather complete, so much so that our earlier attempt at analysing this issue via distributions èsee ë23ëè seems innocuous in comparison. We will assume that æ has a basis of exact diæerentials of functions h 1 ;:::;h p which are real analytic on U a connected open set of interest in R n which contains the reference point èassumed to be the origin in R n è, and on which all the data are deæned. For details on when one may take æ to be spanned by exact diæerentials see, for instance, ë18, 19ë. By abuse of language, we will allude to the h i ;i =1;:::;p as the outputs. As we shall show below, it is more useful to consider these outputs as being more intrinsic to the 2

3 SINGULARITIES AND DECOMPOSITION problem than æ itself ècf. P2 belowè. There are two reasons for taking the h i to be analytic. First, in the application to the disturbance decoupling problem it is almost necessary to take the actual outputs of the plant tobe analytic èprimarily because one cannot guarantee the completeness of the closed loop vector æeldsè. Since, at least some of the h i ;i = 1;:::;p are the actual outputs èin the disturbance decoupling problemè this partially justiæes taking æ to be analytic. Another reason is that we will have to make certain density arguments and this necessitates analyticity. Finally, at a later stage in the paper we will need a canonical form for the h i èeither before or after blowing-upè. In this paper we will restrict attention to the case that p ç m. Since practically all the techniques in the mé1 case are exactly the same as those for m =1;we will, for the most part, deal only with the single-input case. The case for multiple inputs will be illustrated via an example which will clearly reveal the intervening calculations. The case where m é p is physically the easier situation. We speculate that this case could also be subsumed by our methods if we employ generalised inverses. The ærst question that one has to address is that of ænding the correct analog of the controlled invariance problem. Below we will list three versions. There are two motivating factors in arriving at these versions. First, any generalization must reduce to the standard formulation under the usual regularity conditions, and second, must lead to a resolution of the disturbance decoupling problem èwhich, in this author's opinion, is the archtypal application of the controlled invariance problemè. P Denoting by F, either of the closed-loop vector æelds, èi.e, either f + i g iæ i or ègæè l ;l=1;:::;mè, these versions are: ændafeedback control law uètè = æèxètèè + æèxètèèvètè, so that: P1 dl F h i èxè = P p l=1 a lièxèdh l i =1;:::;p for some smooth èresp. analyticè functions a li èxè. P2 aè dl F h i^dh 1^:::^dh p =0;i=1;:::;p;bèdL k F h i^dh 1^:::^dh p = 0;i=1;:::;p;k =0;1;:::; and cè the L k F h i are constant on the connected components of the level sets of the h i for all k =0;1;:::and all i =1;:::;p: P3 The L F h i are each some functions, A i;f èh 1 ;:::;h p è of the outputs. The regularity of the A i;f as a function of its arguments is for the moment unspeciæed. It will be clariæed below. Remark 1.1 We have been deliberately vague about the domains of validity of the statements in P1, P2 and P3. Ideally, we would like these to hold on all of U. In the discussion which will immediately follow this remark, we will clarify when this is likely to be true. For instance, if F and the h i ;i=1;:::;p are all globally real analytic èi.e., we have succeeded in 3

4 V. RAMAKRISHNA ænding global real analytic feedbacksè we cannot still guarantee that the conditions which locally ensure the existence of C! germs A i;f to satisfy P3 extend to a yield the global real-analyticity of these A i;f. On the other hand, the reason for which we are demanding that P3 be true will not require the global real-analyticity of these A i;f. In fact, we could get by with weaker smoothness assumptions èas will be clariæed belowè, in which case the chances of the global existence of the A i;f 's increase dramatically. The point of the discussion below would be to argue that part aè of P2 yields the best possible conclusions and at the same time leads to a tractable theory of solvability. So, the discussion below will clarify, to the extent possible, the situations where the global solvability of part aè of P2 will lead to the global solvability of the other formulations. In seeking global solvability of these other formulations, we will be willing to sacriæce the regularity of the various objects involved, so long as they are adequate for the resolution of the associated control problems. 1.1 Making the case for part aè of P2 Now P1 clearly is the dual of the standard formulations of the controlled invariance ëat least when æ can be written as ker èdh i è;i =1;:::;pë. It also implies parts aè and bè of P2. That it implies part aè is trivial. For part bè we just observe that if P1 is valid, then by taking Lie derivatives with respect to the vector æeld F of the inclusion in P1, we get iteratively: dl k F h ièxè = px l=1 a k lièxèdh l i =1;:::;p;k =1;2;::: which, of course, implies part bè of P2. Now, the reverse implication èviz., P2 a implies P1è does not always hold. This is precisely the content of the so-called division theorems èsee ë7, 18, 19, 26ëè. They assert, under certain conditions, that the equality da^dh 1 ^ :::^dh p = 0, for germs of functions P èsmooth, real analytic, etc.è p A and h i ;i=1;:::;p, implies that daèxè = i=1 a ièxèdh i èxè, for germs of èsmooth, real analytic, etc.è functions, a i èxè. These ëcertain conditions" are on the ideal generated by the h i ;i =1;:::;p in the ring of ègermsè of C 1 èresp C! è functions. In ë19ë, a global division theorem is also proved for the C 1 case; i.e., if A and the h i ;i = 1;:::;p are globally deæned C 1 functions which satisfy the technical hypotheses for a local division theorem, then the functions a i èxè may be taken to be globally deæned. Since this argument makes use of partitions of unity, it cannot be expected to hold in the C! case, in general. Assuming that the division theorems hold for the h i ;i =1;:::;p, then part cè of P2 follows from part bè of P2. This in turn implies that F preserves, for small t, the level sets of the h i ;i=1;:::;p provided the Lie 4

5 SINGULARITIES AND DECOMPOSITION series for h i èç F èt; x 0 èè converges. It is this convergence of the Lie series that leads to the resolution of the disturbance decoupling problem. Now if all the h i are real analytic, then indeed F preserves the level sets of the h i for all t for which the quantity h i èç F èt; x 0 èè is well deæned. If the division theorems are not valid globally, then these assertions are only valid for those initial conditions x 0 in the open set V where these division theorems hold. This point is pertinent since, as we stated before, it is unlikely that they hold globally in the C! case. On the other hand, real analyticity is typically required for any Lie series argument to be of any validity. In this regard, we note that if both the h i ;i=1;:::;p and F are real analytic on their common domains, then so are the h i èç F èt; x 0 èè, èsee ë27ëè. This issue makes the solvability of P3 ègloballyè that much more pertinent. Indeed, assuming that P3 is globally solvable, let us show that F preserves the level sets of the h i for all t for which the quantity h i èç F èt; x 0 èè is deæned èregardless of x 0 è. To see this, consider the initial value problem for the system of ordinary diæerential equations y_ i = A i;f èy 1 ;:::;y p è;i=1;:::;p with initial conditions y i è0è = h i èx 0 è;i=1;:::;p. It is clear that h i èç F èt; x 0 èè satisæes this system. Assuming uniqueness of solutions yields the desired property. Thus, the only regularity property that we need impose on the A i;f, as functions of the h i ;i=1;:::;p is that they be regular enough to guarantee uniqueness of the associated IVP. Now, if part cè of P2 is valid, then the A i;f ;i=1;:::;p certainly exist as set theoretical objects. The question of when they are C 1 is the subject of the so-called composite diæerentiable functions theorems, ë4, 11, 18, 19ë. Let us brieæy explain the contents of ë4, 11ë These papers give suæcient conditions for the global existence of C 1 A i;f, assuming èamongst other thingsè the real analyticity oftheh i ;i=1;:::;p. Unfortunately, the suæcient conditions they require are not immediately related to an easily veriæable condition like part aè of P2. Since we will illustrate how to use a condition like part aè of P2 to arrive at the suæcient conditions of ë11ë in the penultimate section of this paper, we will not dwell on it here. Remark 1.2 æ If æ is nonsingular on U; then P1, P2 and P3 are at least locally equivalent, even if the h i are only C 1 on U. æ In our opinion, P3 èfor suæciently regular A i;f è is at the basis of the appearance of the techniques of the disturbance decoupling problem in many synthesis problems which are not really related to it. 5

6 V. RAMAKRISHNA It seems reasonable, therefore, to consider these three versions to be equivalent èat least locallyè under a fairly wide set of circumstances. However, as we saw, it is part aè of P2 which is instrumental in leading to P3. It is also cast in the form of an equality èas opposed to P1 the dual of all the standard formulation of the controlled invariance problemè. On the basis of the above discussion, we will, therefore, concentrate our eæorts on solving on part aè of P2 in the remainder of this paper. The balance of the paper is organised as follows. In the next section we will detail certain results on diæerential forms which we will need for the analysis of the problem at hand. This section also contains a theorem of Malgrange èë18ëè which is crucial for the results in this paper. In the third section we will obtain our analog of the weak èf; gè-invariance criterion. As the reader will notice, our analog is an equality condition and not an inclusion condition. In the next section we will illustrate, via the m = 2 case, how several inputs may be handled. Furthermore, in this section we recover, as examples, the standard results and also the theorem of ë8ë when m =1. The remainder of the paper concentrates on solving the controlled invariance problem for systems with partially smooth data when p is just 1. We will take f to be C 1 and h, the sole output, to be C!. This is as weak as one can relax the smoothness assumptions when p = 1 if the results are to be of practical relevance. For this reason it is important to note that after a blow-up, any C! function is a monomial. There is, however, another reason for our considering only C 1 f. We wish to obtain a result of the ëformal + Hyperbolic Implies Smooth" genre for certain degenerate and overdetermined systems of partial diæerential equations èwhich arise in connection with the controlled invariance problemè. Monomials provide outputs which are annihilated by linear hyperbolic èwith parametersè vector æelds, thereby providing a relatively simple model for such questions and at the same time continuing to be of some practical relevance. During the course of obtaining our result we also develop a procedure for solving certain ærst order overdetermined systems of partial diæerential equations which degenerate, which is reminiscent of the method of successive integration. Mathematically speaking, this is the novel feature of this paper. We hope this method will be of independent interest. Our technique is partly inspired by Nelson's spectacular proof of the Sternberg linearization theorem èsee ë20ëè. The contents of this section should really be thought of as a method of obtaining C 1 solutions to part aè of P2, once a formal solution is known èwith the understanding that the additional hypotheses needed to make the method work will vary from case to caseè. 6

7 SINGULARITIES AND DECOMPOSITION 2 Preliminaries on Diæerential Forms We assume that the reader is conversant with the basic facts about the Lie derivative, the interior product and the exterior derivative èsee ë1ëè. Let æ be a k-form and æ be a l-form. We will allow these forms to have, in local coordinate charts, coeæcients that are meromorphic functions. More precisely, each of their coeæcients can be a quotient ofc 1 or real-analytic functions. We will say two such p-forms are equal if their coeæcients agree at every point that the two are well deæned. Under this convention all the standard results relating the exterior, interior and Lie derivatives to wedge products continue to be valid. We assume that the reader is conversant with these basic facts èsee ë1ë. We will only record the following not so well-known, though easily deduced, consequence of Cartan's magic formula èl X = d æ i X + i X æ dè: Proposition 2.1 ë1ë Let ç be a p-formèwith meromorphic coeæcientsè and X; Y two smooth vector æelds. Then we have: L X èi Y çè, L Y èi X çè, i ëx;y ë ç = dèi X èi Y çèè, i X èi Y èdçèè: Next we turn to the result of Malgrange, alluded to in the introduction. Theorem 2.1 èë18ëè Let h 1 ;:::;h p be germs of p holomorphic functions at the origin in C n. Let Sèdh 1 ;:::;dh p è be the germ of the analytic set where the rank of the codistribution spanned, over the ring of germs of holomorphic functions, by the dh 1 ;:::;dh p is less than p. Assume that the codimension of Sèdh 1 ;:::;dh p è is at least 3. Let ç be the germ of a holomorphic 1-form which satisæes the condition dç ^ dh 1 ^ :::dh p =0: Then there exists a germ of a holomorphic function æ at so that èç, dæè ^ dh 1 ^ :::^dh p =0: It is useful to make the following deænition: Deænition 2.1 The real analytic functions h i ;i = 1;:::;p, deæned on some connected open subset U containing the origin in R n,aregood outputs if their complexiæcations, h C i ;i=1;:::;p, have the property that the set Sèæè, consisting of points where the holomorphic p-form dh C 1 ^:::^dhc p vanishes, has codimension at least three inu C, the complexiæcation of the set U. Several remarks are in order regarding Theorem è2.1è: aè Malgrange's result is a local existence theorem in the holomorphic category. However, the usual practice of complexiæcation enables us to 7

8 V. RAMAKRISHNA deduce the existence of real valued solutions when the data is real analytic provided the h i ;i = 1;:::;p are good outputs. The texts ë15, 17ë contain readable expositions of complexiæcation. For explicit statements in the smooth category on a related problem èsee also ècè belowè we refer the reader to ë19ë. We will not state these results here, since it is not our purpose to obtain results of the utmost generality onp2. We only wish to illustrate, via applying Theorem è2.1è, the utility of these ideas to nonlinear control problems. We also direct the attention of the reader to the material in ë2ë which could be construed as solving the controlled invariance problem for outputs which are multivalued èin the sense of complex variablesè. These authors also relate our problem to the 16th problem of Hilbert. bè It is important to notice that the form ç is required to be holomorphic. This may, in our applications, be a little excessive. It is this author's belief that a result analogous to Theorem è2.1è would hold if only some components of this form were holomorphic. To bemore precise, we conjecture that these results would still hold even if only ç ^ dh 1 ^ :::dh p is the germ of a holomorphic form. The main step towards proving such a theorem would be to extend Saito's generalisation of the De-Rham division lemma to partially meromorphic forms, since his generalisation is a crucial ingredient in the proof of Theorem è2.1è, see ë26ë. We can, however, prove such a result if the domain over which the data is prescribed has certain desirable attributes èsee Proposition è3.1è later in the paperè. cè Malgrange and Moussu èë18, 19ëè study a similar and related problem. They consider the generalization of the Frobenius' theorem to the case when the forms in question have singularities. More precisely, consider a collection of the germs at 0 of p holomorphic 1-forms! i ;i=1;:::;p which also satisfy the integrability condition d! l ^! 1 ^ :::! p =0;l =1;:::;p. Then, under the hypothesis that the codimension of the set where the codistribution spanned by the! i ;i = 1;:::;p has rank less than p is at least 3, Malgrange shows that there exist germs P of holomorphic functions f i ;g ik ;i =1;:::;p;k =1;:::;p so that! i = k g ikdf k i =1;:::;p. Furthermore detèg ik è0èè 6= 0 and df i è0è = 0 for all i =1;:::;p. Moussu also proves similar results in the smooth category. The connection between this result and our problem is the following. Let ç be the germ of a holomorphic 1-form satisfying dç ^ dh 1 ^ :::dh p =0. Now on the space coordinatized by the dependent and independent variables èx; æè èwith the exterior derivative on this space denoted by d e è consider the collection of germs of one form fç, dæ; dh 1 ;:::;dh p g. Note that d e èç, dæè = dç. Therefore the integrability conditions d e! l ^! 1 ^ :::! p = 0 follow from the condition dç ^ dh 1 ^ :::dh p =0. Clearly the codimension of the singular set of these 1-forms is the same as the codimension of the singular set of the collection dh 1 ;:::;dh p as a set in the space coordinatized by the x's alone. Thus 8

9 SINGULARITIES AND DECOMPOSITION if this codimension is at least 3, then we can ænd ærst integrals for this system in particular for ç, dæ. Since the ëæ" coeæcient of this form is non-zero, we can apply the implicit function theorem to its ærst integral to conclude the existence of a solution to èç, duè ^ dh 1 ^ :::^dh p =0. For a related but diæerent notion of integrability we refer the reader to the paper ë10ë. We next discuss two theorems which relate to the contents of the ænal three sections of the paper. èit is presumed that the reader is familiar with the concept of æat functions and Borel's lemma see, for instance, ë12ëè. The ærst is a result of Glaesser ë11ë on composed functions. Theorem 2.2 ë11ë Consider a p-tuple of real analytic functions h 1 ;:::;h p, deæned onanopen neighborhood, U, of the origin in R n, such that: 1. The codistribution spanned by their diæerentials has maximal rank p on an open dense subset of U: 2. The map h =èh 1 ;:::;h p è from U to R p is semi-proper; i.e., it satisæes: aè its image is closed; and bè for every compact subset K in its range there is a compact subset, L, of U so that K = hèlè. Let A 2 C 1 èuè be such that it satisæes the ëappartenance biponctuelle" condition; i.e., for every set, S, consisting of two points in U there exists a C 1 function F S on U such that the function A, F S æ h is a æat function on S. Then, there exists a C 1 function F satisfying A = F æ h on all of U. Remark 2.1 æ We can dispense with ærst hypothesis, namely the density of the set of regular points, if U is connected and if dh 1 ^:::^dh p is non-zero at at least one point of U-something which we will assume from this point onwards. This follows from the the real analyticity of the map h. æ All proper maps are semi-proper. So are monomials æ n i=1 xpi i ;p i ç0, even if some of the p i are zero. Remark 2.2 As is pointed out in the ëremarque" at the end of Section 2 of ë11ë one can replace the ëappartenance biponctuelle" condition for the set fx; yg by the veriæcation of the existence of functions F x and F y, smooth in neighborhoods of hèxè and hèyè respectively, so that A, F x æ h èresp A, F y æ hè is a æat function on fxg èresp. fygè, if hèxè 6= hèyè. One may not be able to do this if hèxè =hèyè, because one cannot guarantee that F x = F y èsee, in this regard the discussion on Page 143, and also section 4 of ë3ëè. In our situation, however, this is not a problem because we will use 9

10 V. RAMAKRISHNA a condition of the type da ^ dh = 0 to obtain these functions F x, F y. We will do this by ærst setting up an explicit formal power series solution and then invoking Borel's lemma. If h x = h y, it will be obvious that the formal powers series èthis will be a formal power series in the indeterminate hè which solves da ^ dh = 0 about x will be identical to the one about the point y. We end this section with the following version of Hironaka's resolution of singularities theorem èsee ë5ëè. Theorem 2.3 èë5ë, Corollary 4.9è Let M be areal analytic manifold. Let h be a real analytic function on M, which is not identically zero on any component of M. Then there isareal analytic manifold N andaproper, surjective real analytic map ç : N! M so that: aè h æ ç is locally normal crossings on N; i.e., about each point p 2 N, there isacoordinate system èu p ;x 1 ;:::;x n è so that on U p, h æ ç is of the form tèxèæ n i=1 xni i, where the n i ;i =1;:::;n are all non-negative integers and tèxè is real-analytic and non-vanishing on all of U p ; and bè ç isareal-analytic diæeomorphism on an open dense subset of N. In ë28ë a simultaneous desingularization theorem is proved for a ænite collection of real-analytic functions. Remark 2.3 The version of Theorem è2.3è we will use will have M replaced by an open neighborhood of the origin in R n. Furthermore we, will take the origin to be a critical point of the function h, since otherwise one does not need to blow up to get a locally normal crossings model. Also we may take ç to be a real analytic diæeomorphism on an open dense subset whose complement contains ç,1 è0è. For a further simpliæcation in the statement see Theorem è7.1è later. Remark 2.4 The factor tèxè in Th è2.3è can be taken to be 1, as was pointed out to us by Prof. H. Sussmann. As ought to be clear, we may assume that tèxè é 0 for our purposes. If it is not, we will work with,dh. Indeed, if we keep in mind that the purpose of using Th è2.3è is to pull dh back via ç æ and then work with the total diæerential of a simpler C! function, we see that: ç æ è,dhè ==,dèhæçè=dè,hæçè: So, now assuming n 1 é 0 without loss of generality, we make achange of coordinates according to: è 1 = t 1 n 1 èxèx 1 ;è k =x k ;k =2;:::;n: This is a real analytic change of coordinates, which renders tèxè = 1 in the new set of coordinates. 10

11 SINGULARITIES AND DECOMPOSITION We mention, in passing, that resolution of singularities was suggested as a tool for problems in geometric control in ë6ë. The approach taken there, however, diæers from that in this paper signiæcantly. 3 The Analog of the Weak èf; gè-invariance Condition In this section, we will obtain a necessary condition for the solvability of P2. We will assume that èafter a preliminary feedback has been appliedè dl g h i^ dh 1 ^ :::^dh p =0;i =1;:::;p. We will not carry out the construction of such afeedback here, in order to keep the length of this paper within reasonable limits. We just remark that construction of æ is usually easier and typically goes through under weaker regularity hypotheses than those for æ èsee ë25ëè. Consider, therefore, the problem of ænding æèxè so that dl F h i ^ dh 1 ^ :::^dh p =0;i=1;:::;p, where F èxè =fèxè+gèxèæèxè. The last equation reads as: èdl f h i + ædl g h i + L g h i dæè ^ dh 1 ^ :::^dh p =0;i=1;:::;p; which yields: èdl f h i + L g h i dæè ^ dh 1 ^ :::^dh p =0i=1;:::;p: è3.1è If we formally divide by L g h i wehave the following problem to solve: P: Find a smooth function æèxè in a neighborhood of the operating point èassumed to be the originè so that : è, dl f h i L g h i, dæè ^ dh 1 ^ dh 2 :::^dh p : =0i=1;:::;p è3.2è For P to make sense we havetomake some smoothness assumptions on the forms ç i =, dl f h i L gh i ;i =1;:::;p. There are either of two conditions which we will impose from this point onwards: S1 At least one of the forms ç i ;i =1;:::;p are smooth in a neighborhood U of the operating point. S2 The forms ç i ^ dh i ^ :::^dh p ;i =1;:::;p are smooth on a neighborhood U of the operating point. Remark 3.1 There is a disparity of sorts between S1 and S2 in that we require only one of the forms in the former to satisfy the prescribed condition whereas in the latter all the forms have tosatisfy the imposed condition. As the reader will notice, later we only require S2 for the results to go through. S1 will serve only the purposes of a comparison, to be made in the sequel, with the standard formulation. On the other hand, in the light of è3.4è below, we could even aæord to demand that S2 be only 11

12 V. RAMAKRISHNA satisæed a priori by one of the forms, ç i, since it will then be satisæed by the rest as well. We seek æ on a neighborhood V ç U. Note that as long as æ exists on this subset V, P2 will have asolution on V. Indeed, the only points where the ç i ;i=1;:::;pcould conceivably fail to be smooth are the points where L g h i = 0;i = 1;:::;p. But if æ exists at these points, then the equation in P2 holds there. Indeed, under either S1 or S2 we must have dl f h i^dh 1 :::^dh p =0;i=1;:::;pat points where L g h i =0;i=1;:::;p. If we take the exterior derivative of both sides of è3.2è, we arrive at the following necessary condition for the solvability of the above problem: dç i ^ dh 1 :::^dh p =0;i=1;:::;p: è3.3è However a more careful analysis shows that there is a stronger condition which implies è3.3è, and which is also necessary for the solvability of the problem P. Indeed, if there exists an æèxè which solves P, then we necessarily have èç 1,ç i è^dh 1 :::^dh p = 0 for i =1;:::;p. If now in addition è3.3è is satisæed a priori only for i = 1, then exterior diæerentiation of the preceding equality shows that è3.3è holds for the remaining indices as well. Furthermore, it is easy to see that if this condition holds and P is solvable for just i =1,then its solution æ solves the remaining equations in P as well. We, therefore, take these two conditions as our analogue of the weak èf; gè-invariance condition, viz: dç 1 ^ dh 1 :::^dh p = 0 èç 1, ç i è ^ dh 1 :::^dh p = 0;i=1;:::;p: è3.4è Remark 3.2 aè If one writes out every component of the equality expressed by P, then an overdetermined system of partial diæerential equations for the unknown æ is obtained. The important thing to observe about this system is that it is obtained without having to pair down from æ ècf. section 1è. Once æ is given, these equations are extremely simple to describe. bè The necessary conditions è3.4è are equalities and not inclusion conditions. cè When p = 1 è3.4è always holds. The ærst equality holds due to the assumption dl g h ^ dh = 0, and the second for trivial reasons. Next we will relate P to an equivalent problem which uses the set of vector æelds which annihilate each of the dh i ;i = 1;:::;p. We assume such a distribution exists for the moment. If such a distribution exists only locally, then the results below are valid only on its domain of existence. If it does not exist at all, then this circle of ideas is irrelevant. So, let X i ;i2i èwhere I is some indexing setè be the collection of all vector æelds 12

13 SINGULARITIES AND DECOMPOSITION which satisfy L Xi h j = i Xi dh j = 0;i 2 I;j = 1;:::;p. If æ denotes this collection, then it is certainly involutive. We further assume that æ has maximal rank p on a dense subset of U. Denote by c i the smooth functions deæned as i Xi ç 1 èthe smoothness follows from S2 and the formula for the interior product of a vector æeld and a wedge productè. We claim that the solution, if one exists, to the overdetermined system of partial diæerential equations: X i æ = c i ;i2i è3.5è is the desired feedback. Indeed, a solution to è3.5è satisæes i Xi èç 1, dèæèè = 0;i2I: This follows from Cartan's magic formula. Now the h l ;l =1;:::;p being analytic, æ has maximal rank equal to p on a dense set in U. Therefore, the distribution spanned by the X i has maximal rank equal to n, p on a dense subset. This implies that ç, dæ is in the span of the dh i ;i=1;:::;p èpossibly with meromorphic coeæcientsè on this dense subset. Hence its wedge product with dh 1 ^ :::^dh p equals 0 on the same dense set. But the wedge product being 0 on a dense subset implies its being 0 on all of U since it is, after all, a system of equalities amongst everywhere deæned continuous functions. Remark 3.3 Observe that the second condition in è3.4è also yields the equality: i Xi ç j = i Xi ç k ;i2i; j; k =1;:::;p: This follows from taking the interior product of both sides of the second equation in è3.4è with respect to the X i ;i 2 I: In the context of è3.5è this means that there is no ambiguity inchoosing h 1 in writing down the system of partial diæerential equations è3.5è. Any other choice of output would yield exactly the same set of equations. Next, we will show that the ærst equation of è3.4è, provides the integrability conditions for the overdetermined system of partial diæerential equations è3.5è. For purposes of brevity let us denote, by ç, the p-form dh 1 ^ :::^dh p. Let us apply Proposition è2.1è to the p + 1-form ç 1 ^ ç. Doing so yields, as a ærst step: èl Xk c l, L Xl c k, i ëxk ;X l ëç 1 èç = dëèi Xk i Xl ç 1 è ^ çè, i Xk i Xl èdç 1 ^ çè: In arriving at the left hand side we have made use of Leibnitz formulae for the interior product and the fact that all the X i ;i 2 I annihilate the dh j ;j =1;:::;p: The second term on the right hand side of the equation 13

14 V. RAMAKRISHNA above vanishes on account of è3.4è. The ærst term also equals 0, since i Xl i Xk èçè = 0 for any 1-form, ç. Thus we get: L Xk c l, L Xl c k, i ëxk ;X l ëç 1 =0; k; l 2 I on the dense subset where ç 6= 0, and hence everywhere. These are indeed the integrability conditions for the system è3.5è. We nowhave all the tools we need to prove a slightly improved version of Theorem è2.1è. Proposition 3.1 Let h 1 ;:::;h p be p real analytic functions deæned on a connected open subset U of R n whose complexiæcation, U C, is a Stein manifold in C n. Let Sèdh 1 ;:::;dh p è be the analytic set where the rank of the codistribution spanned, over the ring of germs of holomorphic functions, by the complexiæcations of the dh 1 ;:::;dh p is less than p. Assume that the codimension of Sèdh 1 ;:::;dh p è in U C is at least 3. Let ç be areal analytic 1-form which satisæes the condition dç ^ dh 1 ^ :::dh p =0: Then there exists a real analytic function æ deæned on all of U so that èç, dæè ^ dh 1 ^ :::^dh p =0: Proof: The linearity of the question allows us to complexify. Let us denote by O the set U C nsèæè and cover it by open subsets O i ;i2i,over which the systems è3.5è corresponding to the data at hand are solvable èwith solutions denoted by æ i è. Now since the h l ;l =1;:::;p and their complexiæcations h C l ;l = 1;:::;p are globally deæned, we have the following equality on O ij = O i ë O j : dèæ i, æ j è ^ dh C 1 ^ :::^dh C p =0 for each pair of indices i; j 2 I for which O ij is non-empty. By the nonsingularity of the complexiæcation of æ on O; we conclude that for such indices we have æ i,æ j =t ij èh C 1 ;:::;hc pè for some function t ij which is holomorphic on O ij. Since on triple overlaps æ i, æ j + æ j, æ k + æ k, æ i =0; the identity t ij + t jk +t ki = 0 follows. Now on the complements of analytic subsets of Stein spaces which have codimension at least 3, Cousin's Problem A is solvable èë14ë, p. 138è. We deduce that there exist functions t i which are holomorphic on the O i for i 2 I such that t ij = t i, t j. 14

15 SINGULARITIES AND DECOMPOSITION Furthermore, an examination of the proof on page 132 of ë14ë reveals that the t i are functions of h C 1 ;:::;hc p. Indeed, the functions t i are constructed in page 132 of ë14ë out of the Taylor series expansions of the functions t ij. Since the t ij are dependent only on h i ;i=1;:::;p, and on O the map h C =èh C 1 ;:::;hc pè has full rank, it is easy to see that the functions t i are also functions only of h. Now, deæning ^æ i as æ i, t i, we see that the ^æ i patch up to yield a function which is holomorphic on all of O and also satisæes èç C, dæè ^ dh C 1 ^ :::^dhc p = 0 on the subset O. Since the complement of O is an at least codimension 3 subset in U C ; the holomorphic function constructed above can be extended holomorphically to all of U C to yield a function, again denoted by æ, which by density satisæes the equation èç C, dæè ^ dh C 1 ^ :::^dhc p =0on U C. This last extension result may befound in standard several complex variables texts èe.g., ë14ëè. This completes the proof of the proposition. We make the following remarks regarding Proposition è3.1è: æ R n itself satisæes the requirement onu. æ The proof above contains a global result which depends on two crucial assumptions : aè the analyticity of the data at hand; bè the fact that the h i ;i=1;:::;pare globally deæned. Whilst aè could perhaps be circumvented under special circumstances for C 1 data è e.g., the extension past the singularities could fail without additional hypothesesè, bè cannot. It is here that the global results of ë9ë are beyond the scope of our techniques. On the other hand, it is reasonable to assume the global existence of the h i ;i=1;:::;p in applications. æ We still believe that this version is valid for germs of the h i ;i = 1;:::;p; i.e., we need not assume that U C is Stein the method of the proof used here will, of course, not work in this case. æ In ë18ë Malgrange proves the solvability èin the holomorphic categoryè even if CodSèæè = 2 provided ç satisæes S1, and P is known to be formally solvable. We cannot improve this result to the case that only ç ^ dh C is holomorphic èat least via the techniques of Proposition è3.1è è, since we do not know if Cousin's Problem A is solvable on domains which are the complements of codimension 2 subsets of Stein manifolds. The global problem considered in Proposition è3.1è is called a Cousin A problem in the several complex variables literature èsee, for instance, ë14ëè. The paper ëë2ëë also proves a ëlocal implies global" result by solving a Cousin A problem. However, the hypotheses and the conclusions are diæerent from ours. 15

16 V. RAMAKRISHNA Example A - The Nonsingular Case: We ærst treat the results from the standard formulation of the controlled invariance problem. Recall the regularity assumptions from the introductory section regarding the distributions æ and G. Since m = 1, the second regularity condition translates to G ë æ = 0; i.e, L g h i 6=0;i=1;:::;p. Let æ be the annihilator of æ. By the Frobenius theorem we may assume that there exists a local coordinate system in which h i èxè =x i ;i =1;:::;p. Since L g h i 6=0;i =1;:::;p the corresponding ç i ;i=1;:::;p satisfy the assumption S1. The assumptions G ë æ=0is thus the strongest requirement in order that S1 be valid. We assume that è3.4è holds. Clearly, then P is solvable. If the reader wishes he may appeal to è3.5è to see this. Example B: In this example we will show that one can recover the results of èë8ëè on the solvability of the controlled invariance problem in the presence of singularities in the inputs. Of course, ëë8ëë obtains results even when m 6= 1. The most general result of this type can be found in ë13ë. More precisely, the situation considered is that of a smooth nonlinear control system, and an involutive, nonsingular distribution æ which the authors seek to render controlled invariant locally via feedback. They make the following assumption: ëf; X i ë=v i +c i èxègèxè;i=1;:::;n,p: è3.6è Here p is the codimension of æ and the V i ;i=1;:::;n,pare vector æelds which take values in æ, and most importantly the c i èxè;i =1;:::;n,p are C 1 functions on U. They do not make the standard assumption that G ë æ = 0. Under the assumption è3.6è, they demonstrate the existence of a smooth feedback æèxè which causes the closed loop vector æeld fèxè+ æèxègèxè to leave æinvariant. We will now recover their result using the formalism of this section. To that end, let æ be the annihilator of æ. Since æ is nonsingular and involutive, we may assume that there exists a system of coordinates under which the h i = x i ;i = 1;:::;p. In the same coordinates the vector j ;j =p+1;:::;n constitute, visibly, a basis for æ. Denote by f i ;g i ;i = 1;:::;n the components of f and g respectively in these coordinates. We then have: ç i = nx j g i dx j i =1;:::;p: Using the formula ëx; Y ë=èdxè:y, èdy è:x for the Lie bracket of two vector æelds X and Y to express è3.6è in coordinates shows that i Xk ç l = c k èxè; k = p +1;:::;n; l = 1;:::;p. Thus we see that è3.6è yields, in addition to the integrability conditions for è3.5è, the fact that the ç i ;i = 1;:::;p satisfy S2. Thus the system of partial diæerential equations, è3.5è, 16

17 SINGULARITIES AND DECOMPOSITION corresponding to this situation can indeed be solved. This yields the desired conclusion. Before proceeding to the case of a singular æ, we wish to illustrate by means of two examples that it is possible that both S1 holds and fails, and yet in both situations P is solvable. Example C: Let hèxè =x 1 x 2 x 3,x2R 3. Let the control vector æeld be gèxè = 1 3 èx 1;x 2 ;x 3 è t and the drift vector æeld, fèxè equal èx 2 1 x 2x 2 3 ; 0; 0èt. Then L g h = h, so that dl g h ^ dh = 0. However, dl f h ^ dh 6= 0. In fact, L f h = h 2 x 3, so that dl f h = 2hx 3 dh + h 2 dx 3. Therefore, ç 1 =,èx 3 dh + hdx 3 è, which equals,dèhx 3 è. Hence, S1 is valid. Trivially, è3.4è holds, and a solution to P is æèxè =,hx 3. Now consider, the same data except for the drift, fèxè, which is now given by the vector æeld èx 2 1 x 2;x 2 2 ;x 3è t. Now, the diæerential form ç 1 equals,ëè4x 2 + x,1 1 èdx 1 +è4x 1 +x,1 2 èdx 2 +è 2èx1x2+1 x 3 èdx 3 ë. Clearly, S1 does not hold. However, ç 1 ^ dh is smooth as can be easily seen. Hence, S2 is valid. In addition, è3.4è holds and a solution, æèxè, to P is,2x 1 x 2. We now analyse the singular case under the assumption of real analyticity of all data. The Analytic Case: We now allow æ to become singular. Let us make the following assumptions about the data of the problem: 1. The 1-forms ç i ^ dh 1 ^ :::^dh p ;i =1;:::;p are real-analytic on a connected open subset U of R n. 2. The h i ; 1=1;:::;p are all good outputs on a connected open subset U whose complexiæcation is a Stein manifold. 3. Equation è3.4è is satisæed by the data of the problem. We then have: Proposition 3.2 Under the above assumptions there exists a real analytic feedback æèxè on U under which the closed loop system corresponding to the control system è1.1è leaves æ controlled invariant. Proof: The result is a consequence of Proposition è3.1è and complexiæcation. Of course, if U's complexiæcation is not a Stein manifold, then we can prove the local existence of such a feedback iftheç i ;i =1;:::;p are themselves real analytic. 4 The Case of Multiple Inputs We will brieæy illustrate how all of the foregoing analysis can be extended to the case of multiple inputs if p ç m. We will display the intervening calculations for the case p =3; m=2. 17

18 V. RAMAKRISHNA We form m æ C p m one forms in the following fashion. We ærst take h 1 and h 2 together and go through the steps in the single-input case. More precisely, we deæne a pair of one forms: èç 12 1 ;ç12 2 è tr = A,1 èxèè,dl f h 1 ;,dl f h 2 è tr è4.1è where, the matrix Aèxè has for its ijth element the quantity L gj h i ;i;j = 1;2. By considering h 2 ;h 3 together we form, likewise, a second pair of one forms denoted èç1 23 ;ç23 2 è tr. Similarly we form a pair of one forms èç1 31 ;ç31 2 è tr. Of course, all these forms will typically be meromorphic. We can demand that they either satisfy S1 or S2. The æ i can then be found by solving, say, èçi 12, dæ i è ^ dh 1 ^ :::^dh p =0;i=1;2: è4.2è The version of the equations è3.4è now become: èç 12 l dç 12 l ^ dh 1 :::^dh p = 0, çl rs è ^ dh 1 :::^dh p = 0;l=1;2 è4.3è for each index rs from the set fè23è; è31èg. We maynow apply the reasoning of the previous sections to the above system of equations - see ë25ë for the exact technical hypotheses needed. 5 Outline for Partially Smooth Data In this section we will outline the structure of the proof of the solvability of P èwhen p = 1è when f and g are only C 1, so that ç = ç 1 ^ dh 1 is just C 1. We take h = h 1 to be a good output deæned on an open subset U of R n with a Stein complexiæcation U C. The solution to P will then be carried out in the following steps: Step 1: Since ç satisæes S2 and h is a good output there exists a formal solution to P about each point ofu. After choosing a operating point, call this formal solution æ. By Remark è2.4è h is locally normal crossings. Consider the eæect of the map ç æ on the equation èç, dæè ^ dh = 0. Denoting, by abuse of notation, ç æ èçè, also by ç we see that we have a formal solution to the equation èç, dæè ^ dèæ n i=1 xpi i è = 0 about each point of N. For use in the last section, we denote by ç, ç the object ç,dèçæè, where çæ is the smooth function whose Taylor jet is the series æ. Thus, ç ^ dh has all its components æat at the origin. Step 2: In the next section we will construct a C 1 solution to the equation èç, dæè ^ dèæ n i=1 xpi i è=0. We need only consider the case where the functions c i in the system è3.5è corresponding to this data all vanish to inænite order at the origin. However, we will also brieæy examine the 18

19 SINGULARITIES AND DECOMPOSITION formal solvability of these equations because we need a slightly stronger property than the vanishing to inænite order at the origin of the c i. Remark 5.1 We leave it to the reader to verify the fact that we may take the p i all to be relatively prime. Clearly if Step 2 can be executed in this case, then so can it in the general case. As a matter of fact, we will need the p i to be relatively prime only for the next step. More precisely, we will need it to be able to conclude the ëappartenance biponctuelle" condition of Theorem è2.2è from a condition like da^dh = 0, where A is a C 1 function obtained from the C 1 functions of Step 2. Step 3: The smooth solution obtained in the step above will have to be blown down. We will carry this out in the last section. 6 Successive Integration and the Monomial Case Let h =æ n i=1 xpi i. Assume that the ærst r of the p i's are not zero and the remaining are. One has to then solve the following system of equations: X l æ = p l 1, p 1 l = c l = i Xl ç; l =2;:::;r è6.1è l = c l = i Xl ç; l = r +1;:::;n: è6.2è This follows either from writing down directly the equations è3.2è, or by using the equations è3.5è and noticing that the X l form a basis for the vector æelds which annihilate dh. We ærst claim that we may suppose that r = n. This is a special case of the following theoremèsee ë24ëè: Theorem 6.1 Suppose that è6.1è has a C 1 solution when one sets èx r+1 ;:::;x n è=è0;:::;0è: Then the entire system of equations è6.1è and è6.2è also has a C 1 solution. Furthermore, if the c i vanish to inænite order at the origin, and the solution to the corresponding è6.1è with èx r+1 ;:::;x n è =è0;:::;0è also is æat at the origin, then so does the solution to the complete system è6.1è and è6.2è. Therefore, from this point onwards, we shall concentrate our energies on è6.1è with r = n. In the subsections which follow, we will construct the solution to the system è3.5è, corresponding to H, as the sum of three smooth functions Uèxè, V èxè and W èxè. 19

20 6.1 Construction of Uèxè V. RAMAKRISHNA We deæne Uèxè to be the C 1 function whose Taylor jet equals the formal power series solution to è6.1è èwith r = nè. The formal solvability, of course, follows from the hypotheses that ç satisæes the condition S1 and Theorem è2.1è. The existence of U èxè then follows from Borel's lemma. Let m =èm 1 ;:::;m n è beamulti-index of non-negative integers. Associate to the vector æeld X i ;i = 1;:::;n and every multi-index m the quantity P i èmè;i = 1;:::;n deæned P to be the sum p 1 m 1, p i m i. It is easily seen that if Aèx 1 ;:::;x n è= m A mx m is any formal power series then X X i Aèxè = A m P i èmèx m : è6.3è m The reader should use this fact and the formal solvability of è6.1è to set up a tangible formal power series solution. We will not carry out the details here for reasons of brevity. Doing so, however, is very useful to understand the motivation behind the steps of the remaining two subsections. Notice that the formal power series solution obtained this way will diæer from any other formal solution only by a formal ærst integral of all the X i ;i = 2;:::;n. One can use è6.3è and the last observation to ascertain: Remark 6.1 Let us denote by d i the functions c i, X i U; i = 2;:::;n. Then it is easy to see from the structure of the formal power series solution that the d i vanish to inænite order with respect to èx 1 ;x i è;i =2;:::;n at èx 1 ;x i è=è0;0è. This will be of importance in the subsequent subsection. 6.2 Construction of the function V èxè For later use we observe that the æows of the vector æelds X i ;i=2;:::;n denoted ç i èt; xè;i =1;:::;n, consist of diæeomorphisms èdenoted ç i tè deæned for all t: We ærst form the auxiliary system of equations X i v = d i ;i=2;:::;n è6.4è for the unknown function v. Let us ærst observe that the integrability conditions for the system è6.4è, namely X k d l = X l d k ;k;l=2;:::;n, follow from those for the system è6.1è. The function V èxè will now be concocted out of the sum of n, 1integrals I i ;i=2;:::;n. R 1 Let I 2 =, 0 d 2 èç 2 t èxèèdt. I 2 is well deæned if x 1 =0. Indeed, since d 2 vanishes at the origin, we have the estimate: j d 2 èxè jç K j x j è6.5è 20

21 SINGULARITIES AND DECOMPOSITION and since x 1 = 0 this, in turn, implies that for x 1 =0 ji 2 jç KL Z 1 e,p1t j x j dt 0 è6.6è where L is another constant. The latter estimate shows that I 2 is well deæned if x 1 = 0. Furthermore, since d 2 vanishes to inænite order with respect to èx 1 ;x 2 è whenever èx 1 ;x 2 è=è0;0è we have estimates similar to è6.5è for derivatives of all possible orders of d 2. These yield estimates similar to è6.6è and justify diæerentiation under the integral sign, and thereby, shows that I 2 is actually a C 1 function. We will next show that I 2 actually R satisæes X 2 I 2 = d 2 on the hyperplane 1 fx 1 = 0g. We have X 2 I 2 =, X 0 2 d 2 èç 2 t èxèdt: ènotice that we have already justiæed diæerentiation under the integral sign.è We will now use the following facts èsee ë1ëè: Proposition For any diæeomorphism ç, any smooth vector æeld Y and any smooth function y we have: ç æ L X yèxè =L èç æ èxèç æ yèxèè: è6.7è 2. Let X; Y be two smooth vector æelds with respectiveèlocalè æows ç t and è t. Then ëx; Y ë=0iæ aè ç æ t Y = Y and bè èæ t X = X. 3. Let X be avector æeld and ç t be its èlocalè æow. Let y be a smooth function. Then: d dt èçæ t yè=çæ t L Xyèxè: è6.8è Applying è6.7è to each of the diæeomorphisms ç 2 t, along with Proposition è6.1è, we get: X 2 I 2 =, Z 1 èç 2 t è æ èl X2 d 2 èdt: 0 Upon using the formula è6.8è, we now get X 2 I 2 =, Z 1 0 d dt èç2 t è æ d 2 èxèdt: This ænally yields X 2 I 2 = d 2 èxè, Lim t!1 èç 2 t è æ d 2 èxè. Since x 1 = 0, this limit tends to d 2 è0; 0;x 3 ;:::;x n è, which equals 0, since d 2 vanishes whenever èx 1 ;x 2 è=è0;0è. Thus I 2 indeed satisæes the equation X 2 v = d 2 on fx 1 =0g. 21