Local Decomposition and Accessibility of Nonlinear Infinite-Dimensional Systems

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Local Decomposition and Accessibility of Local Decomposition and Accessibility of Local Decomposition and Accessibility of Nonlinear Infinite-Dimensional Systems Nonlinear Infinite-Dimensional Systems Nonlinear Infinite-Dimensional Systems

Hubert Rams ∗∗ Markus Sch¨ oberl ∗∗ Kurt Schlacher ∗∗ Hubert Rams o berl ∗∗ Kurt Schlacher ∗∗ ∗ Markus Sch¨ ∗ Hubert Rams Markus Sch¨ o Hubert Rams Markus Sch¨ oberl berl Kurt Kurt Schlacher Schlacher ∗ ∗ Institute of Automatic Control and Control Systems Technology, of Automatic Control Control Systems Technology, ∗ ∗ Institute Institute of Control and Control Systems Technology, Johannes Kepler University Linz, and Altenberger 4040 Linz, Institute of Automatic Automatic Control and Control Strasse Systems69, Technology, Johannes Kepler University Linz, Altenberger Strasse 69, 4040 Linz, Johannes Kepler University Linz, Altenberger Strasse 69, Austria (e-mail: Johannes Kepler University Linz, Altenberger Strasse 69, 4040 4040 Linz, Linz, Austria (e-mail: Austria (e-mail: (e-mail: {hubert.rams,markus.schoeberl,kurt.schlacher}@jku.at). Austria {hubert.rams,markus.schoeberl,kurt.schlacher}@jku.at). {hubert.rams,markus.schoeberl,kurt.schlacher}@jku.at). {hubert.rams,markus.schoeberl,kurt.schlacher}@jku.at). Abstract: This article deals with the local system decomposition of infinite-dimensional Abstract: This deals the localnonlinear system decomposition of infinite-dimensional Abstract: This article deals with the system decomposition of infinite-dimensional systems, which arearticle described bywith second-order partial differential equations. We show Abstract: Thisare article dealsby with the local localnonlinear system partial decomposition of equations. infinite-dimensional systems, which described second-order differential We show systems, which are described by second-order nonlinear partial differential equations. We show that if there exists a certain codistribution which is invariant under the generalized system vector systems, which areadescribed by second-order nonlinear partial differential equations. Wevector show that if there exists certain codistribution which is invariant under the generalized system that ifathere there exists a certain certain codistribution which is invariant invariant under the the we generalized system vector vector field,if local exists triangular decomposition canwhich be obtained. Furthermore, draw connections to a that a codistribution is under generalized system field, a local triangular decomposition can be obtained. Furthermore, we draw connections to field, a local local triangular decomposition can be be obtained. obtained. Furthermore, we draw draw connections to aaa different approach which is based on transformation groups. Throughout the connections article we apply field, a triangular decomposition can Furthermore, we to different approach which is based on transformation groups. Throughout article we apply different approach which is on groups. Throughout the article we differential geometric methods, highlighting the geometric picture behind thethe system description. different approach which is based based on transformation transformation groups. Throughout the article we apply apply differential geometric methods, highlighting the geometric picture behind the system description. differential methods, highlighting the geometric picture behind the system description. The article geometric is closed with a nonlinear example. differential geometric methods, highlighting the geometric picture behind the system description. The The article article is is closed closed with with a a nonlinear nonlinear example. example. The article closed with a nonlinear © 2016, IFACis(International Federation of example. Automatic Control) Hosting by Elsevier Ltd. All rights reserved. Keywords: infinite-dimensional systems, partial differential equations, local system Keywords: infinite-dimensional systems, differential equations, local Keywords: infinite-dimensional infinite-dimensional systems, partial partial differential equations, local system system decomposition, accessibility, transformation groups, differential geometry Keywords: systems, partial differential equations, local system decomposition, accessibility, transformation groups, differential geometry decomposition, accessibility, accessibility, transformation transformation groups, groups, differential differential geometry geometry decomposition, 1. INTRODUCTION AND MOTIVATION K. Schlacher, A. Kugi, K. Zehetleitner (2002) showed that 1. INTRODUCTION AND MOTIVATION K. Kugi, (2002) showed that 1. INTRODUCTION AND MOTIVATION K. Schlacher, Schlacher, A. A.groups Kugi, K. K. Zehetleitner (2002) showed that transformation areZehetleitner an appropriate tool to tackle 1. INTRODUCTION AND MOTIVATION K. Schlacher, A. Kugi, K. Zehetleitner (2002) showed that transformation groups are an appropriate tool to tackle transformation groups are an appropriate tool to tackle the accessibility and observability problem for systems transformation groups are an appropriate tool tosystems tackle and problem It is well-known that for finite-dimensional systems de- the the accessibility accessibility andK.observability observability problem for for systems described by ode’s. Rieger, K. Schlacher (2010); K. the accessibility and observability problem for systems It is well-known that for finite-dimensional systems dedescribed by ode’s. K. Rieger, K. Schlacher (2010); It is is well-known well-known that for finite-dimensional finite-dimensional systems de- Rieger scribed by ordinary differential equations (ode’s), local dedescribed by ode’s. ode’s. K. Rieger, Rieger, K. Schlacher Schlacher (2010); K. K. It that for systems (2009) successfully extended the transformation described by K. K. (2010); K. scribed by ordinary differential equations (ode’s), local de(2009) successfully extended the transformation scribed by by ordinary ordinary differential equations (ode’s), local dede- Rieger compositions obtained from certain invariant distributions Rieger (2009) successfully extended the transformation scribed differential equations (ode’s), local group approach to systems described by pde’s. We briefly Rieger approach (2009) successfully extended by thepde’s. transformation compositions obtained from invariant distributions systems described We compositions obtained from certain invariant distributions or codistributions allow tocertain highlight system properties group group approach approach tomethod, systemswith described by pde’s. We briefly briefly compositions obtained from certain invariant distributions recapitulate thisto focus by onpde’s. the systems we group to systems described We briefly or codistributions allow to highlight system properties recapitulate this method, with focus on the systems we or codistributions allow to highlight system properties such as accessibility and observability, see e.g. H. Nijmeirecapitulate this method, with focus on the systems we or codistributions allow to highlight system properties discuss in this article. The aim is to link the results of recapitulate this method, with focus on the systems we such as accessibility and e.g. in The aim of suchA. as J. accessibility and observability, observability, seeapproach e.g. H. H. NijmeiNijmeijer, van der Schaft (1991). Thissee makes discuss discuss in this this article. article. Theapproach aim is is to to link the results of such as accessibility and observability, see e.g. H. Nijmeithe transformation group to link the the one results based on discuss in this article. The aim is to link the results of jer, A. J. van der Schaft (1991). This approach makes transformation group approach to the one based on jer, of A.special J. van vancoordinate der Schaft Schaft (1991). This This approach approach makes use transformations to visualizemakes non- the the transformation group approach to the one based on jer, A. J. der (1991). invariant codistributions. the transformation group approach to the one based on use coordinate to use of of special special coordinate transformations transformations to visualize visualize nonaccessible or non-observable subsystems. Hence, one is nonable invariant invariant codistributions. codistributions. use of special coordinate transformations to visualize noninvariant codistributions. accessible or non-observable subsystems. Hence, one is able To sum up, in this contribution we are extending an existaccessible or non-observable non-observable subsystems.orHence, Hence, one is is able able to draw conclusions about accessibilty observability by To sum up, in this contribution we accessible or subsystems. one extending existto draw conclusions about accessibilty or observability by To sum up, in this contribution we are are extending an existing system decomposition approach from first toan seconddraw conclusions conclusions about accessibilty accessibilty or or observability observability by by To atostructural analysis.about sum up, decomposition in this contribution we are extending an existto draw ing system approach from first to secondaa structural analysis. ing system system decomposition approach from from first to secondsecondorder nonlinear pde’s. Furthermore, we will make a constructural analysis. ing decomposition approach first to aK.structural nonlinear pde’s. Furthermore, we will make aabased conRieger, M.analysis. Sch¨ oberl, K. Schlacher (2010) demonstrated order order nonlinear pde’s. Furthermore, we will make connection between this approach and a different one order nonlinear pde’s. Furthermore, we will make a conK. M. ooberl, (2010) between this approach and aa different one based K. Rieger, Rieger, M. Sch¨ Sch¨ berl, K. K. Schlacher Schlacher (2010) demonstrated demonstrated that the system decomposition by exploiting the existence nection nection between this approach and different one based K. Rieger, M. Sch¨ o berl, K. Schlacher (2010) demonstrated on transformation groups. nection between this approach and a different one based that system decomposition by exploiting the that the systemcodistribution decompositioncan by be exploiting thetoexistence existence of anthe invariant extended systems on on transformation transformation groups. groups. that the system decomposition by exploiting the existence on transformation groups. of an invariant codistribution can be extended to systems of an an invariant invariant codistribution can partial be extended extended to systems systems described by first-order nonlinear differential equaof codistribution can be to described by partial differential equadescribed by first-order first-order nonlinear partial differential equa- 1.1 Mathematical Preliminaries tions (pde’s). In case a nonlinear system allows such a decomposidescribed by first-order nonlinear partial differential equa1.1 tions (pde’s). In case a system allows such a decomposi1.1 Mathematical Mathematical Preliminaries Preliminaries tions system (pde’s). properties In case case aa system system allows such such a decomposidecomposition, like accessibility and observabil- 1.1 Mathematical Preliminaries tions (pde’s). In allows a tion, system properties like accessibility and observabiltion, system properties like accessibility and observability can also be examined by a structural analysis. Our In this article we are using the notation and methods of tion, system be properties likebyaccessibility and observability analysis. Our we the and methods of ity can can also also extends be examined examined by a a structural structural analysis. Our In contribution this approach to systems described In this this article article we are are using using the notation notation andapplying methodsthe of differential geometry. In particular, we are ity can also be examined by a structural analysis. Our In this article we are using the notation and methods of contribution extends this approach to systems described differential geometry. In particular, we are applying the contribution extends this approach approach to systems systems described by second-order nonlinear pde’s. It should be noted that differential differential geometry. In particular, we are applying the usual notation of jet calculus and exterior algebra. For a contribution extends this to described geometry. In particular,exterior we arealgebra. applying the by pde’s. be notation calculus For by second-order second-order nonlinear pde’s. It It should be noted noted that that from a triangularnonlinear decomposition theshould non-accessibility of usual usual comprehensive notation of of jet jettreatise calculusofand and exterior algebra. For a more theseexterior topics, we refer to e.g.a by second-order nonlinear pde’s. It should be noted that usual notation of jet calculus and algebra. For a from aa triangular the non-accessibility of comprehensive treatise of we to fromsystem triangular decomposition the converse non-accessibility of more the follows, decomposition but in general the is not true. more comprehensive treatise of these these topics, weS.refer refer to e.g. e.g. D. J. Saunders (1989); P. Griffiths, R.topics, Bryant, Chern, R. from a triangular decomposition the non-accessibility of more comprehensive treatise of these topics, we refer to e.g. the follows, in converse not (1989); R. S. Chern, R. the system system follows, but but in general general the the converse is sufficient not true. true. D. Thus, a triangular decomposition only yields ais D. J. J. Saunders Saunders (1989); P. P. Griffiths, Griffiths, R. Bryant, Bryant, S.kept Chern, R. Gardner, H. Goldschmidt (2012). Formulas are short the system follows, but in general the converse is not true. D. J. Saunders (1989); P. Griffiths, R. Bryant, S. Chern, R. Thus, a triangular decomposition only yields a sufficient Gardner, H. Goldschmidt (2012). Formulas are kept short Thus, aa triangular triangular decomposition only yields aa sufficient sufficient Gardner, condition for non-accessibility in the pde-scenario. Gardner, H. Goldschmidt (2012). Formulas are kept short by applying Einstein’s summation convention, and not Thus, decomposition only yields H. Goldschmidt (2012). Formulas are kept short condition by summation convention, not condition for for non-accessibility non-accessibility in in the the pde-scenario. pde-scenario. by applying applying Einstein’s summation convention, and not indicating theEinstein’s index ranges if they are clear from and context. condition for non-accessibility the pde-scenario. applying Einstein’s summation convention, and not In the literature there can beinfound further approaches by indicating the index ranges if they are clear from context. indicating the index ranges if they are clear from context. Moreover, to avoid mathematical subtleties we suppose all In the literature there can be found further approaches indicating the index ranges if they are clear from context. In the the literature there can be be or found further approaches approaches for tackling the there accessibility observability problem. Moreover, to avoid mathematical subtleties we suppose all In literature can found further Moreover, to avoid mathematical subtleties we suppose suppose all manifolds to avoid be smooth, and all subtleties system functions to defor tackling the accessibility or observability problem. Moreover, mathematical we all for tackling the accessibility or observability problem. For tackling instance, the R. accessibility F. Curtain, H. J. Zwart (1995) draw manifolds to be smooth, and all system functions to defor or observability problem. manifolds to be smooth, and all system functions to depend smoothly on their arguments. Pullback bundles will For instance, R. F. Curtain, H. J. Zwart (1995) draw to be smooth, and all system functions to will deFor instance, instance,about R. F. F.accessibility Curtain, H. H.orJ. J.observability Zwart (1995) (1995)bydraw draw conclusions use manifolds pend their Pullback bundles For R. Curtain, Zwart pendbesmoothly smoothly onindicated their arguments. arguments. Pullback bundles will not explicitlyon if they are clear from context, conclusions about accessibility or observability by use pend smoothly on their arguments. Pullback bundles will conclusions about accessibility or observability by use of methods from the area of functional analysis, i.e. by not be explicitly indicated if they are clear from context, conclusions about accessibility or observability by use to notavoid be explicitly explicitly indicatednotation. if they they are are clear from context, an exaggerated Weclear use from the standard of from area be indicated if context, of methods methodscertain from the the area of of functional functional analysis, analysis, i.e. i.e. by by not examining maps. to avoid an exaggerated notation. We use the standard of methods from the area of functional analysis, i.e. by to avoid an exaggerated notation. We use the standard symbol ∧ for the wedge product (exterior product), and  examining certain maps. to avoid∧ an exaggerated notation. We use the standard examining certain maps. maps. for the wedge product (exterior product), and examining symbol ∧ for the wedge product (exterior product), andIn denotes the natural contraction between tensor fields. Moreover, acertain weaker notion of the accessibility or observ- symbol symbol ∧the fornatural the wedge product (exterior product), and denotes contraction between tensor fields. In Moreover, a weaker notion of the accessibility or observdenotes the natural natural contraction between tensortofields. fields. In Moreover, weaker notion of the thebeaccessibility accessibility or aobservobservthe following, we arecontraction using bundle structures be able ability problem can notion for instance analyzed by finite- denotes the between tensor In Moreover, aa weaker of or the following, we are using bundle structures to be able ability problem can for instance be analyzed by aa finitethe following, we are using bundle structures to be able ability problem can for instance be analyzed by finiteto distinguish between dependent and independent coordimensional approximation of the pde’s, e.g. by means of the following, we are using bundle structures to be able ability problem can for instance be analyzed by a finitebetween and coordimensional approximation of by of to distinguish distinguish between dependent and independent independent coorapproximation of the the pde’s, pde’s, e.g. by means means of to dinates. Throughout thedependent whole article we will use Greek adimensional modal approximation as proposed in E. e.g. D. Gilles (1973). to distinguish between dependent and independent coordimensional approximation of the pde’s, e.g. by means of dinates. Throughout the whole article we will use Greek aa modal approximation as proposed in E. D. Gilles (1973). dinates. Throughout the whole article we will use Greek modal approximation as proposed in E. D. Gilles (1973). a modal approximation as proposed in E. D. Gilles (1973). dinates. Throughout the whole article we will use Greek Copyright © 2016, 2016 International Federation of 170Hosting by Elsevier Ltd. All rights reserved. 2405-8963 © IFAC (International Federation of Automatic Control) Copyright © 2016 International Federation of 170 Automatic Control Copyright © 2016 International Federation of 170 Peer review under responsibility of International Federation of Automatic Copyright © 2016 International Federation of 170Control. Automatic Control 10.1016/j.ifacol.2016.07.430 Automatic Control Automatic Control

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2. SYSTEM REPRESENTATION

letters to indicate the indices of dependent coordinates, and Latin letters for the indices of independent ones. Let us consider the bundle π : X → D, where π is a surjective submersion, calledprojection, from a manifold X with coordinates X i , xα to a manifold D with coordinates X i . In this setting xα are the dependent and X i the independent coordinates, with the corresponding index ranges α = 1, . . . , n and i = 1, . . . , q. Here D is the spatial domain of the considered pde’s, which is supposed to be a compact manifold, with a global volume form VOL = dX 1 ∧ . . . ∧ dX q and a coherently orientable boundary ∂D. Complementary,  ∂D is endowed with the adapted coordinates X∂ = X i , with i = 1, . . . , q − 1. Furthermore, a map φ : D → X , xα = φα (X i ) is called a section of the bundle π : X → D, and Γ(D, X ) indicates the set of all sections. The first and second jet manifold J 1 (X ), J 2 (X ) can be introduced endowed with the coordinates (X i , xα , xα i ), α α α resp. (X i , xα , xα i , xij ). The coordinates xi and xij are socalled derivative coordinates (jet variables) of first and second order (derivatives of the dependent coordinates with respect to the independent ones). Moreover, for a section φα (X i ) the following relations regarding the α derivative coordinates xα i and xij ∂φα ∂ 2 φα , xα , ij ◦ φ = i ∂X ∂X i ∂X j α hold. Partial derivatives with respect to X i , xα , xα i , xij will i ij be denoted as ∂i , ∂α , ∂α , ∂α for short. With the help of the jet manifolds several bundles can be formed. For our purpose the two bundle structures π02 : J 2 (X ) → X and π 1 : J 1 (X ) → D play an important role. Based on the already mentioned bundles, we are able to introduce several tangent bundle structures. We will restrict ourselves to the vertical tangent bundle νX : V(X ) → X , and the cotangent bundle τX∗ : T ∗ (X ) → X . A typical element of the latter one reads as ω = ωα dxα + ωi dX i , and is called a one-form. A vertical vector field v : X → V(X ) is given in local coordinates by v = v β (X i , xα )∂β . Additionally, with v(·) we denote the Lie derivative of a function or a one-form with respect to the vector field v. The first and second jet-prolongation of a vertical vector field read as j 1 (v) = v β ∂β + di (v β )∂βi ,   j 2 (v) = v β ∂β + di (v β )∂βi + dj (di v β )∂βij , xα i ◦φ=

with di as the total derivative vector field with respect to X i . Since there exists a symmetry in the second-order derivative coordinates we have to use a slightly modified version of the summation convention. Thus, we agree on the index range 1 ≤ i ≤ j ≤ q for all occurring double indices to avoid multiple appearing equal terms. To handle second-order pde’s we introduce so-called generalized vertical vector fields. These vector fields v : α J 2 (X ) → π02,∗ (V (X )) read as v = v β (X i , xα , xα i , xij )∂β and can be introduced with the help of a pullback bundle structure 1 . Finally, systems described by pde’s will be denoted as pde-systems, and their accompanying boundary conditions as bc’s for short. 1

the total space manifold is defined as  More precisely, 2 2 (a, b) ∈ J (X ) × V(X ) : π0 (a) = τX (b)

with

the

projection

π02,∗ (τX )(a, b) = a.

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169

In this contribution we will focus on pde-systems with a boundary control input, which are represented by system equations of the type   α x˙ β = f β X i , xα , xα i , xij , β = 1, . . . , n, (1) κ 0 = g ν X∂i , xα , xα , ν = 1, . . . , b ≤ n. i ,u The system equations (1) correspond to a set of secondorder pde’s with appropiate bc’s. It is worth stressing that the time t does not correspond to a coordinate in this setting, it remains in the role of an evolution parameter for the pde’s. The underlying geometric structure of the system equations is given by the bundles π : X → D and ρ : U → ∂D, where the latter one is equipped with the coordinates uκ , κ = 1, . . . , m and X∂ . To include the bc’s in the geometric picture we introduce the pullback bundle ι∗ (J 1 (X )) → ∂D 2 with the inclusion map ι : ∂D → D. Thereby, we build the fibred product bundle ι∗ (J 1 (X )) × U → ∂D as an appropriate underlying geometric structure of the bc’s. The geometric picture of the system (1) is given by a generalized vector field f = f β ∂β : J 2 (X ) → π02,∗ (V (X )), with f β ∈ C ∞ J 2 (X ) explicitly depending on first and secondorder derivative coordinates. This generalized vector field defines a submanifold S ⊂ π02,∗ (V (X )). Moreover, the generalizedsystem vector field  f is subject to the bc’s with g ν ∈ C ∞ ι∗ (J 1 (X )) × U , which define a submanifold S∂ ⊂ ι∗ (J 1 (X )) × U . For the bc’s, we assume that the condition rank [∂κ g ν ] = m 3 holds. Thus, by the implicit function theorem we can locally rewrite the bc’s in the form   uκ = g1κ X∂i , xα , xα i  , κ = 1, . . . , m,  (2) 0 = g2ξ X∂i , xα , xα i , ξ = 1, . . . b − m. Remark 1. We suppose that the system equations (1) are well-posed in the sense of Hadamard, i.e. there are suitable Banach spaces for the solutions and the inputs, and the solutions depend continuously on their initial data and the inputs. Additionally, we suppose the well-posedness of all pde-systems which occur in the following.

A solution of (1) for an input η(t) : R+ → Γ(∂D, U) is given by a map Φ(t) : R+ × Γ(D, X ) → Γ(D, X ) with the properties γ(t) = γt = Φt (γ0 ), γt1 +t2 = Φt2 ◦ Φt1 (γ0 ), and the initial condition γ0 = γ(0). Moreover, Φt satisfies the system equations ∂t Φβt (γ0 )(X) = f β ◦ j 2 (Φt (γ0 ))(X),   0 = g ν ◦ ηt , j 1 (Φt (γ0 ))(X∂ ) . After this brief introduction, we start with the system decomposition. 3. LOCAL DECOMPOSITION This section is devoted to the decomposition of the pdesystems (1). More precisely, we are trying to find a 2

 Here,

the total space manifold is defined as  (a, b) ∈ ∂D × J 1 (X ) : ι(a) = π 1 (b) with the projection ι∗ (π 1 )(a, b) = a. 3 [·] denotes the associated matrix representation, and ∂ κ is a shortcut for ∂/∂uκ as already mentioned in the introduction.

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coordinate chart in which the pde-systems allow for a triangular decomposition on the domain and boundary as follows   α1 α2 , (3) , z α2 , ziα2 , zij z˙ β1 = f¯β1 X i , z α1 , ziα1 , zij  i α2 α2 α2  β2 β 2 , (4) z˙ = f¯ X , z , z , z i

ij

with α1 , β1 = 1, . . . , n − d, α2 , β2 = n − d + 1, . . . , n, and   uκ = g¯1κ z α1 , ziα1 , z α2 , ziα2 , X∂i , (5) 0 = g¯2ν1 z α1 , ziα1 , z α2, ziα2 , X∂i , 0 = g¯3ν2 z α2 , ziα2 , X∂i , the bc’s with κ = 1, . . . , m, ν1 = 1, . . . , b − m − d, and ν2 = 1, . . . , d. We confine ourselves to bundle preserving coordinate transformations   (X i , z α1 , z α2 ) = idD (X i ), Ψα1 (X, x) , Ψα2 (X, x) , (6) where idD (·) is the identity map on D. Coordinate transformations of the type (6) will be denoted with Ψ for short. As shown in K. Rieger, M. Sch¨ oberl, K. Schlacher (2010), suitable transformations can be constructed from certain codistributions which are invariant under the generalized system vector field f β ∂β . In the following, we extend this approach to second-order pde’s.   Definition 2. A codistribution ∆∗ = span dX j , ω k ⊂ T ∗ (X ), j = 1, . . . , q, k = 1, . . . , d, on X is called (locally) invariant under a generalized vector field f : J 2 (X ) → π02,∗ (V (X )) if for any ω ∈ ∆∗       f (ω) ∈ span dX j , ω k , di ω k , di dj ω k   is satisfied, whereby f (ω) ⊂ T ∗ J 2 (X ) holds.

The following lemma gives necessary and sufficient conditions to test whether a codistribution is invariant under a generalized vector field in the sense of Definition 2.   Lemma 3. A codistribution ∆∗ = span dX j , ω k ⊂ ∗ T (X ), j = 1, . . . , q, k = 1, . . . , d, on X is called (locally) invariant under a generalized vector field f : J 2 (X ) → π02,∗ (V (X )) iff for any ω k , k = 1, . . . , d the conditions f (ω k ) ∧ Ω ∧ Ω1 ∧ Ω2 ∧ VOL = 0 (7)  1 1 d . . ∧ω , Ω1 = d1 ω ∧ . . . ∧ are satisfied with Ω = ω ∧ .  1 Ω d ω ∧ . . . ∧ d i dj ω l ∧ . . . ∧ = d dq ω d  and 2 1 1  d dq dq ω .

Proof. This statement directly results from the definition of invariance of a codistribution under a generalized vector field, and standard results of exterior algebra.  In case of a nonsingular and involutive codistribution, one can always find functions αk such that the codistribution is spanned by exact one-forms dαk = ω k . This is a result of the Frobenius Integrability Theorem. As a consequence, the conditions (7) can be rewritten as follows. Lemma involutive codistribution ∆∗ =   4. A nonsingular span dX j , ω k ⊂ T ∗ (X ), j = 1, . . . , q, k = 1, . . . , d, of constant dimension q + d, spanned by exact one-forms ω k = dΨk with Ψ ∈ C ∞ (X ), is (locally) invariant under a generalized vector field f : J 2 (X ) → π02,∗ (V (X )) iff for any ω k , k = 1, . . . , d the conditions   ∂αij f ω k dxα ∧ Ω ∧ VOL = 0 (8)  i     α ∂α f ω k − 2dj ∂αij f ω k dx ∧ Ω ∧ VOL = 0 (9)   α k δα f ω dx ∧ Ω ∧ VOL = 0 (10)

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are satisfied, with Ω = ω 1 ∧.. . ∧ω dand the  Euler-Lagrange   operator δα (·) : ∂α (·) − di ∂αi (·) + di dj ∂αij (·) .

Proof. Lemma 4 we use Cartan’s magic formula     To prove k k + f dω , which simplifies in our case f ω k = d f ω   k k to f ω = d f ω since the one-forms ω k are exact. Calculating the exterior derivative yields the expression       d f ω k = ∂j f ω k dX j + ∂α f ω k dxα + . . .     α ij k dxij , + ∂αi f ω k dxα i + ∂α f ω which can be simplified to       d f ω k = ∂j f ω k dX j + δα f ω k dxα + . . .       α  + di ∂αi f ω k − 2dj ∂αij f ω k dx + . . .   ij   α  k + di dj ∂α f ω dx by use of di (dxα ) = dxα i and the product rule. 

In the following, we will connect the invariance criterion to the system decomposition. First, we start with the decomposition within the spatial domain of the pde’s. Second, we continue with the decomposition of the bc’s. Finally, these single results are combined to one theorem. 3.1 Domain We begin the system decomposition on the domain of the pde’s by stating the following theorem.   Theorem 5. Let ∆∗ = span dX j , ω α2 ⊂ T ∗ (X ), j = 1, . . . , q, α2 = 1, . . . , d be a nonsingular involutive codistribution of constant dimension q + d spanned by exact one-forms ω α2 = dΨα2 with Ψ ∈ C ∞ (X ). If ∆∗ is (locally) invariant under a generalized vector field f α ∂α , then there exists a local coordinate chart (X, z α1 , z α2 ) together with a diffeomorphism of the form (6) in which the vector field f α ∂α exhibits the triangular decompostion (3) - (4). β2 Proof. Since the one-forms ω β2 = dΨ  are  exact,we can β1 β2 = Ψβ1 , Ψβ2 (X, x) introduce new coordinates z , z such that ω β2 = dz β2 . In the following, we show that in these coordinates f¯β2 does not depend on z α1 , ziα1 and α1 α1 zij . We will do this one after the other for zij , ziα1 and z α1 . The idea is to translate the conditions for the independence of f¯β2 of the resp. variables back into the original coordinates, and show that these conditions are equivalent to the ones of Lemma 4. α1 (1) Clearly, f¯β2 is independent of zij iff   −1 = 0. (11) ∂αij1 f¯β2 = ∂αij1 f (Ψβ2 ) ◦ j 2 (Ψ)

To translate this condition in the (original) coordinates (X, x), we transform the holonomic frames ∂αij1 and the functions f¯β2 back separately. The transformation of the holonomic frames ∂αij1 can be found in the appendix. The functions f¯β2 in original coordinates take the form f (Ψβ2 ) = f α ∂α Ψβ2 , which is just the contraction f ω β2 . With this, the above condition is equivalent to the condition that   (12) v α ∂αij f ω β2 = 0 ⊥

∗ holds for all v α which meet v α∂α ∈ (∆  ) . With the help of the one form σ = ∂αij f ω β2 dxα , (12) can then be written as vσ=0, which implies that σ ∈ ∆∗ . However, this is equivalent to   ∂αij f ω β2 dxα ∧ Ω ∧ VOL = 0,

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which is exactly the condition (8) of Lemma 4. (2) Independence of ziα1 is equivalent to (13) ∂αi 1 f¯β2 = 0, in the (X, z)-chart. If we translate this in the original coordinates, we get the condition that    α i (14) v ∂α + 2dk (v α ) ∂αik f ω β2 = 0 ⊥

must hold for all v ∈ (∆∗ ) . By use of the product rule and the condition (12) this can be rewritten in the form    α   i dx ∧ . . . (15) ∂α f ω β2 − 2dk ∂αik f ω β2 ∧Ω ∧ VOL = 0, which matches (9) of Lemma 4. (3) Finally, f¯β2 is independent of z α1 iff (16) ∂α f¯β2 = 0. 1

In original coordinates we get the condition   ⊥ j 2 (v) f ω β2 = 0, ∀v ∈ (∆∗ ) . (17) By means of the product rule and the results (12) and (14), this can be reformulated as   (18) v α δα f ω β2 = 0, ⊥

for all v ∈ (∆∗ ) (resp. their coefficients v α , to be precise). Emphasizing the exact accordance between (10) and (18) we can write (18) as   δα f ω β2 dxα ∧ Ω ∧ VOL = 0, which completes the proof.

A more comprehensive computation of the transformed holonomic frames can be found in the appendix.  Remark 6. To motivate the given definition   of invariance, note that if we calculate f¯(dΨβ2 ) = d f¯dΨβ2 for an invariant, nonsingular and involutive codistribution, we get f¯(dΨβ2 ) = ∂i f¯β2 dX i + ∂α2 f¯β2 dz α2 + . . . + ∂ i f¯β2 di (dz α2 ) + ∂ ij f¯β2 dj (di (dz α2 )) . α2

α2

This shows that       f¯(dΨβ2 ) ∈ span dX i , dΨβ2 , di dΨβ2 , dj di dΨβ2 holds, which justifies the given definition of invariance. Next, we move on to the decomposition on the boundary ∂D of D, resp. to the decomposition of the bc’s. 3.2 Boundary Starting point for our considerations are the bc’s in the form of (2). In the following, we only consider the part g2ξ of the bc’s, which is independent of uκ . First, we rearrange   the bc’s in the form Θνξ 2 g2ξ = 0 with Θνξ 2 ∈ C ∞ J 1 (X ) . Secondly, we transform them into the (X, z)-chart, and ¯ ν2 g ξ ◦ j 1 (Ψ)−1 = 0. To split off an autonomous obtain Θ ξ 2 subsystem we must claim that   ¯ ν2 g ξ ◦ j 1 (Ψ)−1 = 0, ∂ α1 Θ (19) 2 ξ   ¯ ν2 g ξ ◦ j 1 (Ψ)−1 = 0, ∂αi 1 Θ (20) ξ 2

are satisfied in the new coordinates, with ν2 = 1 . . . d. If we translate these conditions into the original coordinates, we get the conditions that 173

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  v α ∂αi Θνξ 2 g2ξ = 0,   v α δα Θνξ 2 g2ξ = 0,

(21) (22)

⊥

must hold for all v ∈ (∆∗ ) . Remark 7. In original coordinates (19) would correspond to j 1 (v) (Θνξ 2 g2ξ ) = 0. By use of the product rule and (21) we obtain (22). Fact 8. One possible solution of the conditions (21) α2 2 ξ 2 for some suitable functions Θα (22) is Θα ξ g2 = Ψ ξ . This represents the special case where the bc’s of the autonomous subsystem do not depend on jet variables. An equivalent formulation of the conditions (21) - (22) is   ∂D ∂αi Θνξ 2 g2ξ dxα ∧ ω k ∧ dX∂ = 0, k = 1 . . . d, (23)   ∂D δα Θνξ 2 g2ξ dxα ∧ ω k ∧ dX∂ = 0, k = 1 . . . d. (24)

As a consequence it is possible to reformulate the bc’s to (5), which exhibits a triangular decomposition on the boundary ∂D. In the next section we connect the so far obtained results to the accessibility problem by a structural analysis. 3.3 Local Decomposition and Accessibility

The results of the Sections 3.1 and 3.2 can be summarized to the following theorem on the decomposition of a pdesystem by means of an invariant codistribution.   Theorem 9. Let ∆∗ = span dX j , ω α2 ⊂ T ∗ (X ), j = 1, . . . , q, α2 = 1, . . . , d, be a nonsingular involutive codistribution of constant dimension q + d, spanned by the exact one-forms ω α2 = dΨα2 with Ψ ∈ C ∞ (X ). If ∆∗ is (locally) invariant under a generalized vector field f α ∂α , then there exists a local coordinate chart (X, z α1 , z α2 ) together with a diffeomorphism (6) in which the vector field f α ∂α exhibits a triangular decompostion in the form of (3) - (4). Furthermore, if the conditions (23) and (24) are satisfied on the boundary ∂D, a triangular decomposition of the bc’s in the form of (5) is possible. Proof. This directly follows from the deductions above.  Remark 10. Theorem 9 states sufficient conditions for the non-accessibility of a nonlinear second-order pde-system. The pde’s z˙ β2 = f¯β2 together with the bc’s g¯3ν2 = 0 are representing a non-accessible (autonomous) subsystem. It should be noted that Theorem 9 is based on a structural analysis - however, non-accessibility does not necessarily allow for such a triangular decomposition. It is worth noting that a similar analysis can be carried out regarding the non-observability analysis. 4. CONNECTION TO A GROUP THEORETICAL APPROACH In this section we use the concept of transformation groups, see e.g. P. J. Olver (1993), as well as so-called invariants in order to analyze the accessibility problem similary as in K. Rieger (2009); K. Rieger, K. Schlacher (2010). It should be noted that the approach based on

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transformation groups and invariants is suitable for tackling several different problems, e.g. (1) accessibility along a single fixed trajectory, and (2) accessibility along any possible system trajectory (equivalent to an arbitrary trajectory). In the following, we will show that the latter problem is able to highlight a connection between the group and the invariant codistribution approach, i.e. we will link the existence of an invariant to the triangular decomposition (3) - (5). Inspecting the decomposition (3) (5) it can be deduced that a variation of the input by fixing the initial conditions leads to different solutions for the z α1 components, whereas z α2 remains unchanged. Thus, we conclude that z α2 serves as an invariant regarding to such a variation. To connect the two examined methods we are considering special functions of the type I (γ) = I(X i , xα ) ◦ γ, (25) with I(X, x) = I ∈ C ∞ (X ), which are left invariant under such a variation (a transformation group action). Therefore I(¯ γ ) = I(γ) must hold for any trajectory. Remark 11. To be more precise, we are trying to find a sum of invariants I α2 (X i , xα ). In K. Rieger, K. Schlacher (2010); K. Rieger (2009) a slightly different ansatz for the invariant function I (·) was made. With the help of transformation groups we are able to map solutions (γ(t), η(t)) on new distorted solutions (γ(t), η(t)) of (1). The distorted solutions are given as follows γ¯ α (t) = Φα X , (X, γ(t)) , κ η¯ (t) = ΦκU , (X, η(t)) , with the transformation group Φ = (ΦX , , ΦU , ). Remark 12. Here, X consists of the time t and the spatial coordinates X i . Since we want to link the group approach to the non-accessibility problem, we require the group action not to modify the initial data, resp. γ(0) = γ¯ (0). By differentiating the transformation group with respect to the group parameter , we obtain the infinitesimal generators of the group  α  v χ = ∂  Φα X , =0 ∂α = vχ ∂α ,  vU = ∂ ΦκU ,  ∂κ = vUκ ∂κ . =0

Keeping the notation short and simple, we use from now on γ (t) = ΦX , ◦ γ(t) and η (t) = ΦU , ◦ η(t) for the distorted solution. Because (γ , η ) must again fulfill the system equations (1), we are able to obtain the local criteria for the infinitesimal generators of the transformation group as     dt vχβ = j 2 (vχ ) f β , (26)

0 = j 1 (vχ ) (g ν ) + vUκ ∂κ g ν , (27) where dt (·) is the total time derivative. These criteria can be derived by plugging in the transformation group into the system equations and differentiation with respect to the group parameter. (26) and (27) represent a set of pde’s which must hold along any system trajectory. By differentiation of the relation I(γ )= I(γ) with respect to  we obtain ∂ I (γ ) |=0 = γ(t)∗ {vχ (I)} = γ(t)∗ {vχ dI} = 0. This demand can be rewritten to α γ(t)∗ {vX ωα } = 0, (28) with use of ω = dI = ∂i IdX i + ∂α Idxα = ∂i IdX i + ωα dxα . (29) 174

α To incorporate the restrictions (26) on vX we calculate the total time derivative of (28)   γ(t)∗ {dt vχα ωα + vχα dt (ωα )} = 0.

After including the local conditions (26) we obtain      γ(t)∗ {[vχα ∂α f β + di vχα ∂αi f β + di dj vχα . . .

(30) ∂αij f β ]ωβ + vχα dt (ωα )} = 0. Due to the ansatz of the function I we conclude that besides ∂t ωα = 0 also the relations ∂βi ωα = ∂βij ωα = 0 are valid. Hence, dt (ωα ) simplifies to f β ∂β ωα , and substituting this relation in (30) yields to      γ(t)∗ {[vχα ∂α f β + di vχα ∂αi f β + di dj vχα . . . (31) ∂αij f β ]ωβ + vχα f β ∂β ωα } = 0.

Equation (31)  further simplified because of the  can be restriction d ωβ dxβ ∧ dX=0, therefore we consider the following expression   f β ∂β ωα + ωβ ∂α f β = ∂α ωβ f β + f β (∂β ωα − ∂α ωβ ) .    =0   For that reason (31) simplifies to γ(t)∗ {j 2 (vχ ) f β ωβ } = γ(t)∗ {j 2 (vX ) (f dI)} = 0. We require this to hold independently of the choice of (γ, η), therefore j 2 (vX ) (f dI) = 0 (32) must vanish on its own. To prove that the group approach also allows for a local decomposition as in the case of an invariant codistribution, we use a special coordinate transformation. Therefore we assume the existence of a invariant function I, resp. functions I α2 , thus (26) and (27) allow for non-trivial solutions. Exploiting the fact that the functions I α2 stay invariant under any system trajectory we use z α2 = Ψα2 = I α2 (X, x). Since Ψ must be a diffeomorphism the functions Ψα1 have to be chosen appropriately. In new coordinates (X, z) the invariant functions and their exterior derivative are given by I α2 = z α2 , resp. dI α2 = dz α2 . Now we make the general ansatz for the symmetry α1 α2 vX = v X ∂ α1 + v X ∂ α2 , α1 α2 ∞ with vX , vX ∈ C (X ). Based on vX dz α2 = 0, one α2 simply concludes vX = 0. Remark 13. In case the system allows for a triangular decompostion it is sufficient to consider variational fields vX that are independent of the time coordinate. Moreover, transforming the expression f dI β2 results in f¯β2 , thus (32) read as  α1       vχ ∂α1 + di vχα1 ∂αi 1 + di dj vχα1 ∂αij1 f¯β2 = 0, (33)

in the coordinate chart (X, z). Using the special ansatz and the existence of the invariants I α2 together with the fact α1 that all restrictions on the symmetry vX ∂α1 are already included in (33), it follows that (33) equates to zero, if the relations ∂α1 f¯β2 = ∂αi 1 f¯β2 = ∂αij1 f¯β2 = 0 are satisfied. Fact 14. In consequence of (33) we conclude that the conditions (16), (13) and (11) are satisfied. Hence, the system cannot be accessible along any system trajectory if non-trivial invariant functions (25) exist. Fact 14 ensures that a local triangular system decompostion within the domain can be achieved if non-trivial invariant functions I α2 exist. Furthermore, the structural

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accordance of two apparently different approaches is emphasized. Fact 15. By virtue of vX (I α2 ) = 0 and (27), based on a suitable rearrangement of the bc’s, one is able to prove that the conditions (19) - (20) are satisfied. These conditions guarantee a triangular decomposition of the bc’s. Sketching the idea of the proof, we only consider the part of the bc’s which is independent of uκ , namely ¯ ν2 g ξ ◦ j 1 (Ψ)−1 in the (X, z)-chart. Equation (27) g¯3ν2 = Θ ξ 2   therefore simplifies to vχα1 ∂α1 g¯3ν2 + di vχα1 ∂αi 1 g¯3ν2 = 0. As there are no further restrictions on vχα1 we conclude ∂α1 g¯3ν2 = ∂αi 1 g¯3ν2 = 0 in accordance to (19) - (20). 5. EXAMPLE We want to study the nonlinear example x˙ 1 = f 1 = x1 x111 x2 + x1 , (34) 2 2 1 2 2 2 2 1 1 2 (35) x˙ = f = −(x x x1 + (x ) (x11 − x1 ) + 2x ), with D = [0, 1], and the bc’s u1 = g 1 = x1 x21 + x11 (x2 + 1), (36) 2 1 2 −1 (37) 0 = g = x − (x ) , on ∂D = {0}. As a possible candidate for an invariant codistribution we want to consider the one-form ω = x2 dx1 + x1 dx2 together with ω = dΨ2 = d(x1 x2 ). The expression f ω results in f ω = −(x1 )2 x2 x21 − x1 x11 (x2 )2 − x1 x2 , therefore ∂αij (f ω) = 0 holds, and the conditions (8) are satisfied. Considering the conditions (9) 1 1 2 2 1 2 we obtain ∂11 (f ω)dx , and ∂21 (f  ω)dx =  1 = −x (x1 ) dx 2 1 2 2 1 2 −x (x ) dx , thus ∂1 (f ω)dx + ∂2 (f ω)dx ∧ω∧dX = 0 is satisfied. Moreover, the conditions (10) are fulfilled because of δ1 (f ω) = δ2 (f ω) = 0. To achieve a system decomposition  we apply the  coordinate transformation  Ψ = Ψ1 , Ψ2 = x1 , x1 x2 according to Theorem 9. In new coordinates the system equations read as 1 2 z˙ 1 = f¯1 = z11 z + z1, (38) 2 2 2 2 ¯ (39) z˙ = f = −z (z1 + 1), emphasizing the decomposition within the spatial domain. Furthermore, to achieve a decompostion of the bc’s we rearrange (37) to g˜2 = x1 x2 − 1 = 0. The condtions (23) are satisfied because g˜2 does not depend on derivative coordinates. According to Fact 8 this yields a special case. It remains to verify the conditions (24), this is done in the following.  2 1    δ1 g˜ dx + δ2 g˜2 dx2 ∧ ω = x2 dx1 + x1 dx2 ∧ ω = 0 Thus, the bc’s can be decomposed into u1 = g¯1 = z12 + z11 , and 0 = g¯2 = z 2 − 1. Referring to Remark 10, the system is not accessible. 6. CONCLUSION The article dealt with the local decomposition of secondorder nonlinear pde-systems. Moreover, a link between the transformation group and the invariant codistribution approach was emphasized. Further works could take indomain controls into account, or tackle for instance the observability or parameter identification problem in this setting. Regarding the observability problem it should be noted that in the transformation group setting the invariants are known, as they are corresponding to the output-functions. 175

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7. APPENDIX In the following, we briefly sketch the transformation of ∂α1 , ∂αi 1 , ∂αij1 with respect to the coordinate transformation ˆ α (X i , z α1 , z α2 ), where Ψ ˆ denotes Ψ−1 . First, we xα = Ψ start with the transformation laws for the first and secondorder derivative coordinates ˆα ˆ α α1 xα (40) i = ∂ i Ψ + ∂ α 1 Ψ zi + . . . , α1 α α α α1 α ˆ ˆ ˆ xij = ∂j ∂i Ψ + ∂i ∂α Ψ z + z ∂j ∂α Ψ + . . . 1

j

i

1

ˆ α z β1 + ∂ α Ψ ˆ α z α1 + . . . , + ziα1 ∂β1 ∂α1 Ψ 1 j ij

(41)

where we only wrote out the relevant terms. Finally, we consider the transformation of the vertical vector fields in α1 z α1 , ziα1 and zij direction, reading as follows ˆ α ∂αij , ∂αij1 → ∂α1 Ψ ˆ α ∂αi + 2dj (∂α Ψ ˆ α )∂αij , ∂αi 1 → ∂α1 Ψ 1 ˆ α ∂α + di (∂α Ψ ˆ α )∂ i + dj (di (∂α Ψ ˆ α ))∂ ij . ∂α1 → ∂ α1 Ψ α α 1 1 ⊥

In the (X, z)-chart we have (∆∗ ) = span{∂α1 }, and ⊥ ˆ α ∂α }. therefore in the (X, x)-chart (∆∗ ) = span{∂α1 Ψ Consequently, with regard to the proof of Theorem 5 it should be stressed that a vector field v α ∂α is contained in ⊥ (∆∗ ) iff its coefficients stem from linear comibinations of ˆ α ]. the columns of the Jacobian matrix [∂α1 Ψ ACKNOWLEDGEMENTS This work has been partially supported by the Linz Center of Mechatronics (LCM) in the framework of the Austrian COMET-K2 programme. REFERENCES D. J. Saunders (1989). The Geometry of Jet Bundles. Cambridge University Press. E. D. Gilles (1973). Systeme mit verteilten Parametern. R. Oldenbourg Verlag M¨ unchen Wien. H. Nijmeijer, A. J. van der Schaft (1991). Nonlinear Dynamical Control Systems. Springer. K. Rieger (2009). Analysis and Control of InfiniteDimensional Systems - A Geometric and FunctionalAnalytic Approach. Phd-Thesis, JKU-Linz. K. Rieger, K. Schlacher (2010). Accessibility Conditions for PDE Systems and the Adjoint System. In Proceedings of 8th IFAC Symposium on Nonlinear Control Systems. K. Rieger, M. Sch¨oberl, K. Schlacher (2010). Local Decomposition and Accessibility of PDE Systems. In Proceedings 49th IEEE Conference on Decision and Control (CDC) , 6271–6276. K. Schlacher, A. Kugi, K. Zehetleitner (2002). A LieGroup Approach for Nonlinear Dynamic Systems Described by Implicit Ordinary Differential Equations. In 15th International Symposium on Mathematical Theory of Networks and Systems. P. Griffiths, R. Bryant, S. Chern, R. Gardner, H. Goldschmidt (2012). Exterior Differential Systems. Springer. P. J. Olver (1993). Applications of Lie Groups to Differential Equations. Springer, second edition. R. F. Curtain, H. J. Zwart (1995). An Introduction to Infinite-Dimensional Linear System Theory. Springer.