A notion of equivalence for linear complementarity problems with application to the design of non-smooth bifurcations
AA notion of equivalence for linearcomplementarity problems with applicationto the design of non-smooth bifurcations
Fernando Casta˜nos ∗ Felix A. Miranda-Villatoro ∗∗ Alessio Franci ∗∗∗∗
Automatic Control Department, Cinvestav-IPN. Av. InstitutoPolit´ecnico Nacional 2508, 07360, CDMX, Mexico. email:[email protected] ∗∗ Department of Engineering, University of Cambridge. TrumpingtonStreet, CB2 1PZ, Cambridge, UK. email: [email protected] ∗∗∗
Department of Mathematics, Universidad Nacional Aut´onoma deM´exico, Circuito Exterior S/N, C.U., 04510, CDMX, Mexico. email:[email protected]
Abstract:
Many systems of interest to control engineering can be modeled by linear comple-mentarity problems. We introduce a new notion of equivalence between linear complementarityproblems that sets the basis to translate the powerful tools of smooth bifurcation theory tothis class of models. Leveraging this notion of equivalence, we introduce new tools to analyze,classify, and design non-smooth bifurcations in linear complementarity problems and theirinterconnection.
Keywords:
Linear complementarity problems, bifurcations, topological equivalence, piecewiselinear equations.1. INTRODUCTIONBifurcation theory is one of the most successful tools forthe analysis of nonlinear dynamical systems that dependon a control parameter. The theory is firmly groundedon the classical implicit function theorem (Dontchev andRockafellar, 2014; Golubitsky and Schaeffer, 1985), andtherefore, it requires smoothness of the maps under study.However, from a practical viewpoint, it is common to ap-proximate complicated nonlinear maps by simpler models.In such situations, the resulting approximation may benon-smooth.Linear complementarity problems are non-smooth prob-lems that arise in fields of science such as economics(Nagurney, 1999), electronics (Acary et al., 2011), mechan-ics (Brogliato, 1999), mathematical programming (Murty,1988), general systems theory (van der Schaft and Schu-macher, 1998), etc. They serve as a departing point inthe analysis of problems with unilateral constraints, andalso arise as piecewise linear approximations of nonlinearmodels (Leenaerts and Bokhoven, 1998).Recently, there have been some attempts to extend bi-furcation theory towards the non-smooth setting, see e.g.Di Bernardo et al. (2008); Leine and Nijmeijer (2004);Simpson (2010). However, the emphasis has been directedtowards analysis of discontinuous systems, and very littleis known on bifurcations in complementarity systems.The purpose of this paper is to provide a methodology forthe realization of equilibrium bifurcations in linear comple-mentarity problems. The proposed framework mimics, up to certain extent, the smooth program proposed by Arnoldet al. (1985) and relies on tools from non-smooth analysisand linear algebra. To achieve this, the concept of topolog-ical equivalence in complementarity systems is introduced.We focus on static models that arise as the steady-stateequations of piecewise linear dynamical systems. Thanksto the piecewise linear structure of the problem, the intro-duced equivalence is always global, which constitutes a ma-jor difference with respect to smooth bifurcation theories.This fundamental concept allows us to provide a completeclassification of planar complementarity problems.The paper is organized as follows. Section 2 describesthe linear complementarity problem and related concepts.Section 3 constitutes the main body of the paper andaddresses the problem of topological equivalence betweenLCP’s. Afterwards, an interconnection approach for therealization of bifurcations is presented, together with anexample applied to the non-smooth pleat and the pitchforksingularity. Finally, the paper ends with some conclusionsand future research directions in Section 4.2. PRELIMINARIES
The linear complementarity problem (LCP) is defined asfollows.
Definition 1.
Given a vector q ∈ R n and a matrix M ∈ R n × n , the LCP ( M, q ) consists in finding vectors z, w ∈ R n such that a r X i v : . [ m a t h . D S ] N ov w = M z + q R n + (cid:51) w ⊥ z ∈ R n + (1)where the second relation, called the complementary con-dition , is the short form of the following three conditions: w ∈ R n + , z ∈ R n + , and w (cid:62) z = 0.In what follows, we introduce some concepts that willbe useful for studying the geometric structure of LCPs.Given M and an index set α ⊆ { , . . . , n } , we define the complementary matrix C M ( α ) as C M ( α ) · j = (cid:26) − M · j if j ∈ αI · j if j (cid:54)∈ α , where the subscript · j denotes the j -th column. Now definethe piecewise-linear function f M ( x ) = C − M ( α ) x , x ∈ pos C I ( α ) , (2)where pos C I ( α ) is the cone generated by the columns of C I ( α ). Note that the cones pos C I ( α ) are simply the 2 n orthants in R n indexed by α ⊆ { , . . . , n } , and that f M (pos C I ( α )) = pos C M ( α ) . Proposition 2. (Cottle et al. (2009)). Let z ∈ R n be asolution of the LCP ( M, q ), then x = w − z ∈ R n is asolution of f M ( x ) = q . (3)Conversely, let x ∈ R n be a solution of (3), then z =Proj R n + ( − x ) ∈ R n + is a solution of the LCP ( M, q ).Henceforth, we treat the LCP (
M, q ) and (3) as identicalproblems, in the sense that we only need to know thesolution of one of them in order to know the solution ofthe other.The solutions of the LCP (
M, q ) depend on the geometryof the complementary cones pos C M ( α ). More precisely,there exists at least one solution x of (3) for every α such that q ∈ pos C M ( α ). If C M ( α ) is nonsingular thesolution is unique, whereas there exists a continuum ofsolutions if C M ( α ) is singular. Thus, for a given q , therecan be no solutions, there can be one solution, multipleisolated solutions, or a continuum of solutions, dependingon how many complementary cones q belongs to and theirproperties. In practical applications, the vector q depends on a control,or bifurcation parameter λ ∈ R . The bifurcation parametercan be an applied voltage or current in electronic circuits,a force or a torque in a mechanical system, or the amountof capital injection in an economic system. The goal ofbifurcation theory is to understand how the number ofsolutions changes as the bifurcation parameter is varied.In LCPs we let q = ¯ q ( λ ) , where ¯ q : R → R n is at leastcontinuous, although more regularity constraints can beimposed as needed. The mapping ¯ q defines a continuouscurve, or path in R n . As λ lets q move along this path,the number of solution to the LCPs might change. Pointswhere the number of solutions change are called bifurcationpoints . Example 3.
Let us illustrate this idea in the simple casewhere the path is a line segment joining two distinct points q i ∈ R , i ∈ { , } , that is,¯ q ( λ ) = (1 − λ ) q + λq , λ ∈ [0 , . Fig. 1. Cone configuration for matrix M in (4). The thickblack lines depict the generators of the complemen-tary cones pos C M ( α ), α ⊆ { , } , whereas the arcsdenote the complementary cones.In addition, let us set the matrix M as M = (cid:20) (cid:21) (4)and proceed to analyze the two cases shown in Fig. 1. Case a)
We take the path ¯ q a ( λ ) given by¯ q a ( λ ) = (1 − λ ) (cid:20) − (cid:21) + λ (cid:20) − (cid:21) , λ ∈ [0 , . (5)According to Proposition 2, solving the LCP ( M, ¯ q a ( λ )) isequivalent to finding x ∈ R satisfying C − M ( α ) x = ¯ q a ( λ ) , s.t. x ∈ pos C I ( α ) , (6)for α ⊆ { , } . Noting that C M ( α ) = C − M ( α ) C I ( α ) forany α ⊆ { , } , it follows that the solutions to (6) aregiven by (cid:91) α ⊆{ , } S α (7)where S α = (cid:8) ( x, λ ) ∈ R × [0 , | ∃ p λ ( α ) ∈ R : x = C I ( α ) p λ ( α ) and ¯ q a ( λ ) = C M ( α ) p λ ( α ) (cid:9) (8)Roughly speaking, in order to solve the parametrized LCP( M, ¯ q ( λ )) we need to find p λ ( α ) (the representation of¯ q ( λ ) in terms of the generators of the α -th complementarycone). Computing these explicitly and taking α = ∅ ⊂{ , } we get p λ ( ∅ ) = C M ( ∅ )¯ q a ( λ ) = (cid:20) λ − − λ (cid:21) , and it follows that p λ ( ∅ ) / ∈ R for any λ ∈ R . Therefore, S ∅ = ∅ . Now, for α = { } ⊂ { , } we get that p λ ( { } ) = C M ( { } )¯ q a ( λ ) = (cid:20) − λ − λ (cid:21) . It follows that p λ ( { } ) ∈ R for λ ∈ ( −∞ , / S { } = (cid:26) ( x, λ ) ∈ R × [0 , / | x = (cid:20) λ − − λ (cid:21)(cid:27) Similarly, for α = { } and α = { , } we have, respectively, S { } = (cid:26) ( x, λ ) ∈ R × [1 / , | x = (cid:20) − λ − λ (cid:21)(cid:27) ,S { , } = (cid:26) ( x, λ ) ∈ R × [1 / , / | x = (cid:20) − λ λ − (cid:21)(cid:27) . By putting all the pieces together, one gets the bifurcationdiagrams (in the x -coordinate) shown on the left-hand sideof Fig. 2.
13 23 − λ x
13 23 − λ x
14 34 λ x
14 34 λ x Fig. 2. Solutions to problem (6) for M given by (4) andpaths ¯ q a as in (5) (left), and ¯ q b as in (9) (right), bothwith λ ∈ [0 , Case b)
We take the path¯ q b ( λ ) = (1 − λ ) (cid:20) − (cid:21) + λ (cid:20) − (cid:21) . (9)As in the previous case, we need to solve a family ofconstrained linear problems. Simple computations lead usto S ∅ = (cid:26) ( x, λ ) ∈ R × [1 / , / | x = (cid:20) λ − − λ (cid:21)(cid:27) ,S { } = (cid:26) ( x, λ ) ∈ R × [0 , / | x = (cid:20) − λ − λ (cid:21)(cid:27) ,S { } = (cid:26) ( x, λ ) ∈ R × [3 / , | x = (cid:20) λ − − λ (cid:21)(cid:27) ,S { , } = ∅ . The right-hand side of Fig. 2 depicts the solution set ofLCP ( M, ¯ q b ( λ )) (in the x -variable).It is clear that, as long as the path ¯ q ( λ ) lies in theinterior of the same cone, or set of cones, the numberof solutions cannot change. Exiting and/or entering acone, that is, crossing a cone face is thus a necessarycondition for a bifurcation to occur. It is not sufficientthough. For instance, in Example 3 Case b) above, thepath ¯ q b ( λ ) crosses through different cones at the points λ ∈ { , } . However, there is no change in the numberof solutions, see Fig. 2, right. This last observation posesthe following question: How can we characterize the faceat which bifurcations occur?The non-smooth Implicit Function Theorem (see Corollaryat page 256 of Clarke (1990)) provides an answer tothis question. Let Ω f be the set of measure zero wherethe Jacobian Df ( x ) of a Lipschitz continuous function f : R n → R n does not exist. Definition 4. (Clarke generalized Jacobian). The general-ized Jacobian of f at x is the set ∂f ( x ) = co (cid:110) lim i →∞ Df ( x i ) | x i → x, x i (cid:54)∈ S, x i (cid:54)∈ Ω f (cid:111) , where S is any set of measure zero and co denotes convexclosure. Definition 5. ∂f ( x ) is said to be maximal rank if every M in ∂f ( x ) is non-singular.For a function F : R n × R m → R n , F : ( x, y ) → F ( x, y ), thegeneralized Jacobian with respect to the first argument,denoted by ∂ x F ( x, y ), is the set of all n × n matrices M such that [ M N ] belongs to ∂F ( x, y ) for some n × m matrix N . Theorem 6.
Suppose that F ( x , y ) = 0 and its general-ized Jacobian ∂ x F ( x , y ) is maximal rank. Then thereexist a neighborhood U of y and a Lipschitz function¯ x : U → R n such that F (¯ x ( y ) , y ) ≡ y ∈ U .By specializing this theorem to (3) with F ( x, q ) = f M ( x ) − q , it follows that a solution ( x , q ) to an LCP can be abifurcation point only if ∂f M ( x ∗ ) is not maximal rank,that is, if there exists a singular matrix M belonging tothe set ∂f M ( x ∗ ). This motivates the following definition. Definition 7.
A solution point ( x , q ) of (3) such that ∂f M ( x ) is not maximal rank is called a non-smoothsingularity.Observe that ∂f M ( x ) = co { C − M ( α ) | x ∈ pos C I ( α ) } .Thus, ∂f M ( x ) is a singleton if x belongs to the interior ofan orthant or the convex closure of a (finite) set of matricesif x belongs to the face between two or more orthants.The following proposition helps in finding non-smoothsingular points. Proposition 8.
Let x be a solution of the LCP ( M, q ). Ifthere exists M + ∈ ∂f M ( x ) such that det( M + ) > M − ∈ ∂f ( x ) such that det( M − ) <
0, then ∂f M ( x ) isnot maximal rank. Proof.
The determinant function det : R n × n → R iscontinuous and the set ∂f M ( x ) is connected (since itis convex). It follows that, because det( M ) takes bothpositive and negative values in ∂f M ( x ), it must alsovanish in some subset of ∂f M ( x ). (cid:50) As an application of Proposition 8, let us consider Ex-ample 3 above. Note that ∂f M can be set-valued onlyfor { x ∈ R n | f M ( x ) ∈ bdr pos C M ( α ), α ⊆ { , . . . , n }} .Therefore, with M as in (4), the generalized Jacobian (atthe coordinate axes) is ∂f M ( x ) = (cid:26)(cid:20) − µ (cid:21) , µ ∈ [0 , (cid:27) ,x ∈ pos C I ( ∅ ) ∩ pos C I ( { } ) (cid:26)(cid:20) − µ (cid:21) , µ ∈ [0 , (cid:27) ,x ∈ pos C I ( ∅ ) ∩ pos C I ( { } ) (cid:26)(cid:20) µ (cid:21) , µ ∈ [0 , (cid:27) ,x ∈ pos C I ( { , } ) ∩ pos C I ( { } ) (cid:26)(cid:20) µ (cid:21) , µ ∈ [0 , (cid:27) ,x ∈ pos C I ( { , } ) ∩ pos C I ( { } )(10)whereas ∂f M is single-valued and nonsingular for all of theothers points x ∈ R . It follows from Proposition 8 and(10) that solutions of f M ( x ) − q = 0 satisfying x = 0 or x = 0 are non-smooth singular points, see the expressionfor S { , } and the left-hand side of Fig. 2. In contrast,it follows directly from Definition 7 that all solutions ofCase b ) in Example 3 are regular, see the right-hand sideof Fig. 2. It is worth to remark that to have a singularityit is not necessary that det( C − M ( α )) = 0 for some α .hen det( C − M ( α )) = 0 for some α such that q ∈ pos C − M ( α ), another source of singularities appears. Inthis case, the cone pos C − M ( α ) is degenerate , in the sensethat its n -dimensional interior is empty (Danao, 1994).We expect the crossing of degenerate cones to induce non-smooth bifurcations because at the crossing of degener-ate cones there is necessarily a continuum of solutions.Indeed, if det( C − M ( α )) = 0, the full orthant pos C I ( α ) ismapped by f M onto the (lower-dimensional) degeneratecone pos C − M ( α ). Thus, given q ∈ pos C − M ( α ), theremust exist a (locally linear) subset of pos C I ( α ) that ismapped by f M to q (Danao, 1994; Murty, 1972). Example 9.
Let us consider the degenerate matrix M = (cid:20) (cid:21) and the path ¯ q a as in (5). For α = { , } , solutions of (6)are characterized by the expression (cid:20) λ − − λ (cid:21) = (cid:20) (cid:21) (cid:20) x x (cid:21) , x ∈ pos C I ( { , } )Note that the above equation has a nonempty solution set S { , } if and only if 4 λ − − λ , that is, if and only if λ = . Hence, for λ = the solution set is given by S { , } = (cid:110) ( x, λ ) ∈ R × (cid:110) (cid:111) | x = (cid:20) µ − − µ (cid:21) , µ ∈ [ − , (cid:111) , whereas for the other subsets α ⊂ { , } , the solutions are S ∅ = ∅ S { } = (cid:26) ( x, λ ) ∈ R × [0 , / | x = (cid:20) λ − − λ (cid:21)(cid:27) S { } = (cid:26) ( x, λ ) ∈ R × (1 / , | x = (cid:20) λ − − λ (cid:21)(cid:27) Therefore, for α = { , } the solution set S α has aninfinite number of solutions for a single value of λ , whichcorresponds to the situation in which the path ¯ q a intersectsthe degenerate cone C M ( α ).We summarize the results of this section as follows. • Non-smooth bifurcations can happen when the pathdefined by q = ¯ q ( λ ) crosses a face of non-degeneratecones, or at the crossing of degenerate cones. • Crossing a degenerate cone always leads to bifurca-tions. • The presence and nature of a bifurcation when cross-ing a face of a non-degenerate cone depends on thenature and disposition of the other cones that sharethat face.It follows that non-smooth bifurcations in LCPs are essen-tially determined by: i) the complementary cone configu-ration; ii) how the path moves across them.3. MAIN RESULTSSimilarly to smooth bifurcation theory, it is possible touse equivalence relations to provide an exhaustive list ofthe possible bifurcation phenomena. We start here thisprogram by deriving a notion of equivalence between LCPs , which will provide equivalence classes of cone configu-rations. The relevance of this notion in classifying non-smooth bifurcation problems will be then illustrated.
Our notion of equivalence between LCPs (
M, q ) and (
N, r )has a topological and an algebraic component. The alge-braic component captures the relations among the com-plementary cones that M and N generate. The relevantalgebraic structure is that of a Boolean algebra, a subjectthat we now briefly recall (see Givant and Halmos (2009);Sikorski (1969) for more details).Let X be a set and P ( X ) the power set on X . A fieldof sets is a pair ( X, F ) where F ⊂ P ( X ) is closed underintersections of pairs of sets and complements of individualsets (this implies closure under union of pairs of sets).Let G be a subset of P ( X ). The field of sets generated by G is the intersection of all the fields of sets that contain G .A field of sets is a concrete example of a Boolean algebra,and as such, the usual algebraic concepts apply to them. Definition 10. A Boolean homomorphism from the field( X, F ) onto the field ( X (cid:48) , F (cid:48) ) is a mapping h : F → F (cid:48) such that h ( P ∩ P ) = h ( P ) ∩ h ( P ) and h ( − P ) = − h ( P )(11)for all P , P ∈ F . Here, − P denotes the complementof P . A one-to-one Boolean homomorphism h is called a Boolean isomorphism . An isomorphism of a field onto itselfis called a
Boolean automorphism . Definition 11.
A Boolean mapping h : F → F (cid:48) is said tobe induced by a mapping ϕ : X (cid:48) → X if h ( P ) = ϕ − ( P ) (12)for every set P ∈ F .Allow us to present a simple corollary to a theorem bySikorski. Corollary 12.
Let F be a field generated by G . If a bijec-tion g : G → G (cid:48) is induced by a bijection ϕ : X (cid:48) → X , then g can be extended to a Boolean isomorphism h : F → F (cid:48) . Proof.
Define h as in (12). Since ϕ is bijection, h sat-isfies (11), that is, g can been uniquely extended to aBoolean homomorphism from F into F (cid:48) . Likewise, g − canbe extended to a Boolean homomorphism from F (cid:48) into F .It follows from (Sikorski, 1969, Thm. 12.1) that h is indeeda Boolean isomorphism. (cid:50) Now, consider the collection G M = { pos C M ( α ) } α , and let( R n , F M ) be the field of sets generated by G M . We are nowready to state our main definition. Definition 13.
Two matrices
M, N ∈ R n × n are said tobe LCP equivalent , M ∼ N , if there exists topologicalisomorphisms (i.e., homeomorphisms) φ, ψ : R n → R n such that f M = ϕ ◦ f N ◦ ψ , (13)where ψ induces a Boolean automorphism on F I .Condition (13) is the commutative diagram n R n R n R nψf M f N ϕ . It is standard in the literature of singularity theory (Arnoldet al., 1985), and ensures that we can continuously mapsolutions of the problem f M ( x ) = q into solutions of theproblem f N ( x (cid:48) ) = ϕ − ( q ). The requirement on ψ being aBoolean automorphism implies that ψ maps orthants intoorthants, intersections of orthants into intersections of or-thants, and so forth; and this ensures that the complemen-tarity condition is not destroyed by the homeomorphisms. Theorem 14.
The matrices
M, N ∈ R n are LCP equivalentif, and only if, there exists a bijection g : G M → G N induced by a homeomorphism ϕ : R n → R n . Proof.
Suppose that M is LCP equivalent to N . Define g as g (pos C M ( α )) = ϕ − (pos C M ( α )) , where ϕ satisfies (13). By (13), ϕ − ◦ f M (pos C I ( α )) = f N ◦ ψ (pos C I ( α )) . Since ψ induces a Boolean automorphism on F I , ψ (pos C I ( α )) = pos C I ( β )for some β . Thus, ϕ − (pos C M ( α )) = f N (pos C I ( β )) , so that g (pos C M ( α )) = pos C N ( β ) . (14)This shows that ϕ necessarily induces a bijection g from G M onto G N .For sufficiency, suppose that there is a bijection g : G M →G N induced by a homeomorphism ϕ : R n → R n . We willconstruct ψ explicitly. Use the equation g (pos C M ( α )) = pos C N ( ˆ β ( α ))to define the bijection ˆ β on the power set of { , . . . , n } anddenote its inverse by ˆ α . Now, define ψ as ψ ( x ) = C − − N ( ˆ β ( α )) · ϕ − ( C − M ( α ) · x ) ,x ∈ int pos C I ( α ) . Note that ψ ( x ) ∈ pos C I ( ˆ β ( α )), so the application of f N on both sides of the equation shows f N ◦ ψ ( x ) = ϕ − ◦ f M ( x )for x in the interior of any orthant.Clearly, ψ is piecewise continuous in the interior of theorthants. Its continuity at the boundaries follows from thecontinuity of f N . More precisely, x (cid:48) = C − − N ( β i ) · C − N ( β j ) · x (cid:48) for any indexes β i , β j such that x (cid:48) ∈ pos C I ( β i ) ∩ pos C I ( β j ) . Thus, for x in the boundary pos C I ( α i ) ∩ pos C I ( α j ), wehave C − − N ( ˆ β ( α i )) · ϕ − ( f M ( x )) = C − − N ( ˆ β ( α j )) · ϕ − ( f M ( x ))so that the image of x is the same, regardless of whether α i or α j is used in the definition of ψ .It is not difficult to verify that ψ ( x ) is invertible withinverse ψ − ( x (cid:48) ) = C − − M (ˆ α ( β )) · ϕ ( C − N ( β ) · x (cid:48) ) ,x (cid:48) ∈ int pos C I ( β ) . Indeed, the composition ψ − ◦ ψ ( x ) gives C − − M (ˆ α ( β )) ϕ (cid:16) C − N ( β ) C − − N ( ˆ β ( α )) ϕ − ( C − M ( α ) x ) (cid:17) ,x ∈ int pos C I ( α ) . This expression reduces to the identity by the fact that ˆ α is the inverse of ˆ β .By similar arguments, we can show that ψ ◦ ψ − is theidentity, and that ψ − is continuous at the boundaries ofthe orthants. (cid:50) Remark 15.
It follows from Corollary 12 that a necessarycondition for M ∼ N is the existence of a bijection g : G M → G N that extends to an isomorphism h : F M → F N . Example 16.
Consider the matrices M = (cid:20) − . − (cid:21) , N = (cid:20) − . − (cid:21) and O = (cid:20) . . (cid:21) . Their cone configurations are shown in Fig. 3.Note that (cid:92) α pos C M ( α ) = pos C M ( ∅ ) . Suppose, for the sake of argument, that there exists abijection g : G M → G N that extends to an isomorphism h : F M → F N . Then, h (cid:16)(cid:92) α pos C M ( α ) (cid:17) = h (pos C M ( ∅ )) (cid:92) α pos C N ( ˆ β ( α )) = pos C N ( ˆ β ( ∅ ))for some bijection ˆ β . However, note that the intersection ofall the complementary cones generated by N is no longera cone. This is a contradiction, from which we concludethat such a g cannot exist and that, by Remark 15, M and N are not equivalent. This is intuitively clear since,depending on the location of q , there can be none, two, orfour solutions to the LCP ( M, q ); whereas, depending onthe location of r there can be either one or three solutionsto the LCP ( N, r ).Although N and O are fairly ‘distant’ from each other,they are LCP equivalent. To see this, consider the matrices¯ N = [ N · N · ] and ¯ O = [ O · O · ] . It is lengthy but straight forward to verify that themapping ϕ : R → R , given by ϕ ( y (cid:48) ) = C ¯ N (ˆ γ ( α )) · C − O ( α ) · y (cid:48) , y (cid:48) ∈ pos C ¯ O ( α )with ˆ γ ( ∅ ) = { , } , ˆ γ ( { } ) = { } , ˆ γ ( { } ) = { } andˆ γ ( { , } ) = ∅ , maps the cones pos C ¯ O ( α ) to the conespos C ¯ N (ˆ γ ( α )) (see Fig. 3). Clearly, ϕ is a homeomorphism.Also, it induces the mapping g : G N → G O given by g (pos C N ( α )) = pos C O ( − α ) . We have verified the conditions of Theorem 14.In the example, M and N are not equivalent, even thoughthey are ‘close’ to each other. This issue takes us to thefollowing concept. Definition 17.
A matrix M ∈ R n × n is said to be LCPstable if it is LCP equivalent to every matrix that issufficiently close to it.ig. 3. Complementary cones of the matrices M , N and O in Example 16, depicted by black arcs. The cones generatedby C ¯ N ( α ) and C ¯ O ( α ) are depicted by red arcs. The matrices M and N are not equivalent, but N and O are, astheir complementary cones have the same Boolean structure. The following results will provide a characterization ofequivalence classes of stable matrices in R × . Lemma 18.
Let M ∈ R × . If M , M (cid:54) = 0 anddet( M αα ) (cid:54) = 0, for all α ⊆ { , } , then M is stable. Proof.
Let E M = [ I − M ], let A α with α ⊆ { , } be theconnected components of R − (cid:83) i =1 pos E M · i , and notethat A α are (not necessarily convex) cones that partition R , i.e., (cid:83) α ¯ A α = R and A α ∩ A β = ∅ for α (cid:54) = β . Forevery index α , define G M ( α ) ∈ R × as the submatrix of E M such thatbdr A α = pos G M ( α ) · ∪ pos G M ( α ) · and | G M ( α ) | > . Let M (cid:48) = M + ε ˜ M be a perturbationof M . The matrix E M (cid:48) varies smoothly as a function of ε .In particular, M (cid:48) · i → M · i as ε → R n ).Let ϕ : R → R be defined by ϕ ( y (cid:48) ) = G M ( α ) · G − M (cid:48) ( α ) · y (cid:48) , y (cid:48) ∈ A (cid:48) α with G M (cid:48) ( α ) the ε -perturbation of G M ( α ) such that | G M (cid:48) ( α ) | >
0. Thus, ϕ | A (cid:48) α , α ⊆ { , } , is continuous andbijective. We now show that ϕ is well-defined, continuous,and bijective at the cone boundaries, too. Let A (cid:48) α and A (cid:48) β be two contiguous cones with common boundarypos E M (cid:48) · i . Note that, if y (cid:48) ∈ A (cid:48) α ∩ A (cid:48) β , then y (cid:48) = κE M (cid:48) · i for some κ >
0, so that ϕ | A (cid:48) α ( y (cid:48) ) = κE M (cid:48) · i = ϕ | A β ( y (cid:48) ) . Thus ϕ : R → R is a homeomorphism. Moreover, since ϕ ( I · i ) = I · i , i = 1 ,
2, and ϕ ( M (cid:48) · i ) = M · i , i = 1 ,
2, it followsthat ϕ − (pos C M ( α )) = pos C M (cid:48) ( α ), that is, ϕ − inducesa bijection g : G M → G M (cid:48) . Then Theorem 14 implies M and M (cid:48) are equivalent. (cid:50) Theorem 19.
Two matrices
M, N ∈ R × are equivalent if M · N > , M · N > M αα ) · det( N αα ) > , α ⊆ { , } . Proof.
Under the hypothesis of the theorem, M t := (1 − t ) M + tN is stable for all t ∈ [0 , M t satisfiesthe conditions of Lemma 18 for all t ∈ [0 , t ∈ [0 ,
1] there exists a neighborhood U t of t in [0 , By the assumptions of the lemma, every 2 × E M isnonsingular, so positivity of the determinant is ensured by suitablyarranging the columns of G M ( α ). such that M t ∼ M t (cid:48) for all t (cid:48) ∈ U t . Because [0 ,
1] iscompact, we can cover it with a finite number of U t ’s,say, for t < t < · · · < t n , with 0 ∈ U t and 1 ∈ U t n . Let τ i ∈ U t i − ∩ U t i , i = 1 , . . . , n . Then M = M ∼ M τ ∼ M τ ∼ · · · ∼ M τ n ∼ M = N . (cid:50) Corollary 20.
Let M ∈ R × with det( M αα ) = 0 for some α ⊆ { , } , then M is not stable. Proof.
Recalling that non-singular matrices are dense,there exists M (cid:48) arbitrarily close to M such that det( M (cid:48) αα ) (cid:54) =0 for all α ⊆ { , } . Then, any bijection g : G M → G M (cid:48) can-not be induced by a homeomorphism ϕ − . If it was, ϕ − would map an empty-interior complementarity cone to anon-empty interior complementarity cone, a contradiction.It follows by Theorem 14 that M cannot be stable. (cid:50) The results of this section provides a list of “normal forms”to explore equivalence classes of stable matrices in R × .The matrices M δ = (cid:20) δ δ − δ (2 δ − δ δ ) δ (cid:21) and N δ = (cid:20) δ δ − δ (0 . δ − δ δ ) δ , (cid:21) with δ i ∈ {− , } , i = 0 , . . . , M δ , for some combination of δ i ∈ {− , } , i = 0 , . . . ,
4. By varying the parameters of M δ , we can thusconstruct an explicit list of equivalence classes of stablebidimensional matrices. The constructed list might not beexhaustive, but by Corollary 20 what is left out from thisclassification is the zero-measure set of matrices satisfyingdet( M αα ) (cid:54) = 0, for all α ⊆ { , } , but M M = 0.Stability and equivalence class of matrices in this zero-measure set are assessed a posteriori on a case-by-casebasis in the normal form matrix O δ = (cid:20) δ δ δ δ (cid:21) with δ , δ ∈ {− , } and δ , δ ∈ {− , , } , δ δ = 0.After studying the cone structure of each of these matrices,we conclude that there are only four classes of LCP stablematrices in R × . Representative members of two differentclasses are the matrices M and N , defined in Example 16.Two more representative matrices are K = (cid:20) − (cid:21) and L = (cid:20) − . − − . (cid:21) . (15)ig. 4. Complementary cones of the matrices K (left) and L (right) defined in (15), depicted by black arcs.Since G K partitions R (see Fig. 4), the LCP ( K, q ) has aunique solution for every q . A matrix with this property iscalled a P -matrix (Cottle et al., 2009). The complementarycones of L are also shown in Fig. 4. Depending on q , theLCP ( L, q ) may either have two or no solutions.
The strong link between piecewise linear functions andLCP’s, pointed out in Proposition 2, motivates us torestrict ourselves to piecewise linear paths through coneconfigurations. In this setting, the path itself can begenerated from the solution set of another LCP (Eavesand Lemke, 1981; Garcia et al., 1983). This approachnaturally leads us towards an interconnection frameworkreminiscent of circuit theory, in the sense that an intricatehigh-dimensional LCP is treated as the result of the interconnection of simpler LCP’s. Proceeding in this waywe prove that, by selecting appropriate inputs and outputs ,the feedback interconnection of LCPs is again an LCP.Afterwards, we use this decomposition approach to obtainthe unfoldings of the pitchfork singularity.We start by considering two linear complementarity prob-lems in their z -coordinates, that is,LCP( M k , ¯ q k ) : (cid:26) w k = M k z k + ¯ q k R n k + (cid:51) w k ⊥ z k ∈ R n k + , where M k ∈ R n k × n k and ¯ q k ∈ R n k , for k ∈ { a, b } .Let z k ∈ R n k be the output of the k -th LCP and let¯ q k ∈ R n k take the role of input . Additionally, considerthe interconnection rule¯ q a = H a z b + ¯ θ a , ¯ q b = H b z a + ¯ θ b , (16)where H a ∈ R n a × n b , H b ∈ R n b × n a and ¯ θ k ∈ R n k areadditional inputs available for further interconnection.With this convention we have the following result. Proposition 21.
The interconnection of linear complemen-tarity problems under the pattern (16) is again a linearcomplementarity problem.
Proof.
The interconnection of two LCPs under the pat-tern (16) yields, (cid:20) w a w b (cid:21) = (cid:20) M a H a H b M b (cid:21) (cid:20) z a z b (cid:21) + (cid:20) ¯ θ a ¯ θ b (cid:21) R n a + n b + (cid:51) (cid:20) w a w b (cid:21) ⊥ (cid:20) z a z b (cid:21) ∈ R n a + n b + . (17)The conclusion follows directly from (17), which is an LCPof dimension n a + n b with extended input (cid:2) ¯ θ a ¯ θ b (cid:3) (cid:62) andextended output [ z a z b ] (cid:62) . (cid:50) Fig. 5. Non-smooth pleat, pitchfork path, and their pro-jection to the plane ( y , y ). The black lines on theplane are generators of complementary cones.Note that, in contrast to the framework of dynamicalsystems, we are studying static relations that may be set-valued. Thus, the conditions for well-posedness of (17) aremore relaxed in comparison with their smooth counter-part. Let us consider the class of LCPs represented by the matrix O in Fig. 3. This class gives rise to the non-smooth pleatshown in Fig. 5. The pleat is given by (cid:110) [ y y x ] (cid:62) ∈ R | ∃ x ∈ R such that f O ( x ) = y (cid:111) , where f O : R → R is the piecewise linear map defined inProposition 2. It is worth to remark that the non-smoothpleat is stable in the sense that the matrix O is LCP-stable.In complete analogy with the smooth case, see e.g. (Gol-ubitsky and Schaeffer, 1985, Chapter III.12), one canrecover a large family of bifurcations from the pleat byselecting appropriate paths through it. We illustrate thiswith the pitchfork singularity and its unfoldings, but itis also possible to obtain the hysteresis and the cuspsingularities and their unfoldings by changing the path ina suitable way.Concretely, let us consider the LCP ( M b , ¯ q b ) associatedto the non-smooth pleat shown in Fig. 5 with matrix M b = 2 O and O as in Example 16. In order to realize thepath ¯ q b , we follow the interconnection approach describedin the previous subsection. We consider the second LCP( M a , ¯ q a ) with M a = 1 and path ¯ q a ( λ ) = 2 λ −
1. The LCP( M a , ¯ q a ) has a unique solution for every λ ∈ R which iscomputed easily as z a ( λ ) = (cid:26) , λ < λ − , ≤ λ . (18)We thus set the path ¯ q b ( λ ) as¯ q b ( λ ) = R s (cid:20) z a ( λ ) λ (cid:21) + (cid:20) µ µ (cid:21) , where R s is a rotation matrix, s is the angle of rotationand the parameters µ , µ are extra degrees of freedomthat will allow us to change the path ¯ q b ( λ ) on the pleat.Equivalently, the resulting LCP can be seen as the inter-connection between LCP ( M a , ¯ q a ) and LCP ( M b , ¯ q b ) underthe interconnection rule (16) withig. 6. A sample of paths to recover the pitchfork singu-larity and its unfoldings. . . . λ z . . . λ z . . . λ z . . . λ z . . . λ z . . . λ z Fig. 7. The pitchfork singularity and its unfoldings, ob-tained from the paths depicted in Fig. 6. H a = 0 , H b = (cid:20) cos s sin s (cid:21) , ¯ θ a = µ − λ sin s , ¯ θ b = µ + λ cos s . Let us fix s = π . By varying the parameters µ and µ we are able to displace the path ¯ q b ( λ ) on the pleat whoseprojection onto the plane is depicted in Fig. 6 for differentvalues of the vector µ .The associated bifurcation diagrams to the paths on Fig. 6are shown in Fig. 7. Note that the central path in Fig. 6produces the pitchfork organizing center, whereas pertur-bations of this path lead to any of the left or right-handside diagrams.4. DISCUSSION AND FUTURE DIRECTIONSWe have presented a notion of global equivalence betweenLCPs that allows us to make a classification of this prob-lems in the planar case. In addition, an interconnectionapproach for the realization of non-smooth bifurcationswas presented. These tools are thought to be handful formany applications, as for instance, the analysis and designof neuromorphic circuits (Casta˜nos and Franci, 2017),the study of economic equilibria in competitive markets(Nagurney, 1999), and the analysis of elastic-plastic struc-tures in engineering (Pang et al., 1979), just to name afew. This work also opens the path towards the analysisof behaviors in dynamical linear complementarity systems(van der Schaft and Schumacher, 1998).REFERENCESAcary, V., Bonnefon, O., and Brogliato, B. (2011). Non-smooth modeling and simulation for switched circuits .Lecture Notes in Electrical Engineering. Springer.Arnold, V.I., Varchenko, A.N., and Gusein-Zade, S.M.(1985).
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