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Some New Results for the Semidefinite Linear Complementarity Problem
Abstract
In this paper, we present some new results for the semidefinite linear complementarity problem (SDLCP). In the first part, we introduce the concepts of (i) nondegeneracy for a linear transformation $L:{\cal S}^n \rightarrow {\cal S}^n$ and (ii) the locally-star-like property of a solution point of an SDLCP(L,Q) for $Q\in {\cal S}^n$, and we relate them to the finiteness of the solution set of SDLCP(L,Q) as Q varies in ${\cal S}^n$. In the second part, we show that for positive stable matrices A1,. . ., Ak, the linear transformation L:=LA1 \circ L_{A_2} \circ \cdots \circ L_{A_k} $ has the Q-property where LAi(X):= AiX + XAiT. A similar result is proved for the transformation $S:=S_{A_1} \circ S_{A_2} \circ \cdots \circ S_{A_k}$, where each Ai is Schur stable and $S_{A_i}(X):=X-A_iXA_i^{T}$. We relate these results to the simultaneous stability of a finite set of matrices.
... Various aspects of the above problem have been studied in recent years. Gowda making connection between control theory/dynamical systems, complementarity problems and matrix theory, see [18,19,21,22]. They introduce and study various properties of linear transformations in the semidefinite setting, similar to the matrix classes studied in LCP over R n + . ...
... Below, we state some properties of a linear transformation in the semidefinite setting from Gowda et al. [18,19,21]. ...
... Finally, we unify and prove the results of Gowda and Song [21], Malik [45] and ...
... With respect to the usual inner product in R n and the nonnegative cone R n + , a nonsingular M-matrix has the Q-property (equivalently, the P-property) [4]. It has been observed in [10] that any finite product of such matrices has the Q-property. The proof is based on the observation that the inverse of a nonsingular M-matrix is a (entry-wise) nonnegative matrix with positive diagonal. ...
... It has been shown in [9] that these properties are equivalent to the Q-property (and also the P-property) of L A . In [10], Gowda and Song prove, using degree-theoretic ideas, that any finite product of Lyapunov transformations corresponding to positive stable matrices has the Q-property with respect to S n + . (3) For a given matrix A ∈ R n×n , consider the Stein transformation S A defined on the space S n by ...
... In [8], Gowda and Parthasarathy show that these properties are further equivalent to the Q-property (and also the P-property) of S A with respect to the semidefinite cone S n + . In [10], Gowda and Song show that any finite product of Stein transformations corresponding to Schur stable matrices has the Q-property with respect to S n + . The proof is based once again on degree-theoretic ideas. ...
Motivated by the Q-property of nonsingular M-matrices, Lyapunov and Stein transformations (corre-sponding to positive stable and Schur stable matrices, respectively) and their products [A.], in the first part of the paper we present a unifying result on the product of Q-transformations defined on self-dual closed convex cones. The second part deals with the P-property of the linear transformation I − S on a Euclidean Jordan algebra where S leaves the corresponding symmetric cone invariant and ρ(S) < 1. We prove the P-property for the Lorentz cone and present some partial results in the general case.
... We introduce some concepts and results for SDLCP(L, Q). [13,16,17] Let L : S n → S n be a linear transformation. We say that (2) L has the cross commutative property if for all Q ∈ S n , and the solutions Z 1 and Z 2 of SDLCP(L, Q), it holds ...
... Theorem 2.1 [13,16,17] Let L : S n → S n be a linear transformation. The following statements are equivalent: ...
In this paper, we present some novel observations for the semidefinite linear complementarity problems, abbreviated as SDLCPs. Based on these new results, we establish the modulus-based matrix splitting iteration methods, which are obtained by reformulating equivalently SDLCP as an implicit fixed-point matrix equation. The convergence of the proposed modulus-based matrix splitting iteration methods has been analyzed. Numerical experiments have shown that the modulus-based iteration methods are effective for solving SDLCPs.
... These transformations are very useful in the study of Euclidean Jordan algebras and complementarity problems. For example, Gowda et al. gave some complementarity forms of the famous Lyapunov's theorem and described the nondegeneracy property of the Lyapunov transformation L A over the Hermitian matrix space [12,13]. Li et al. extended these results to any Jordan algebras [22]. ...
Quadratic representations are very useful in the study of Euclidean Jordan algebras and complementarity problems. In this paper, we provide some characterizations of the complementarity properties for the quadratic representation P
a
. For example, P
a
has the E0-property; P
a
is monotone iff \({\pm a \in {\mathcal K}}\). In addition, the algebra and cone automorphism invariance of some E-properties are studied. By use of the quadratic representations, the Jordan quad E-property is proved to keep cone automorphism invariant in simple Jordan algebras. The pseudomonotone property is shown to be cone automorphism invariant.
... This problem can be also viewed as a generalization of the standard linear complementarity problem (LCP) and also included the geometric monotone semi-definite linear complementarity introduced by Kojima et al., [9] and which contains the pair of primal-dual semidefinite programming problems (SDP). For more details on (SDLCP) we refer the reader to the references [6][7][8]11] and the thesis of Phd of Song [20]. Moreover it turns out that primal-dual path-following interior point algorithms can solve efficiently many problems such as linear, semidefinite programming, convex quadratic programming, conic and complementarity problems. ...
In this paper a primal-dual path-following interior-point algorithm for the monotone semidefinite linear complementarity problem is presented.
The algorithm is based on Nesterov-Todd search directions and on a suitable proximity for tracing approximately the central-path.
We provide an unified analysis for both long and small-update primal-dual algorithms.
Finally, the iteration bounds for these algorithms are obtained.
... However, the characterization of the GUS-property of the Stein transformation is still open. The known results so far are in 2003, Gowda, Song, and Ravindran (Thm 3 [11]) showed that if S A is strictly monotone (that is, ⟨X, S A (X)⟩ > 0 for all 0 ̸ = X ∈ S n ), then S A has the GUS-property; If A is normal, the converse also holds. Moreover, in 2013, J. Tao [26] examined conditions which gives the so-called Cone-GUS-property (that is, SDLCP(S A , Q) has a unique solution for all Q ∈ S n + . ...
In the setting of semidenite linear complementarity problems on , we focus on the Stein Transformation , and show that is (strictly) monotone if and only if (
... If such an X exists, then we call X to be the solution of SDLCP(L, Q ). In the last decade significant work done in the area of semidefinite linear complementarity problem starting from Kojima et al. [16] in 1997 and Gowda with others, see the articles [9,8,12,13,7]. SDLCP is also a special case of semidefinite programming, see Shida et al. [19]. ...
Motivated by the so-called P2P2-property in the semidefinite linear complementarity problems, in this article, we introduce the concept of P2′-property for a linear transformation on the space of real n×nn×n symmetric matrices. While these two properties turn out to be different, we show that they are equivalent for the Lyapunov transformation LALA, double-sided multiplicative transformation MAMA and a particular class of Stein transformations. We also show that P2′ implies the SSMSSM and QQ-properties.
... Motivated by these considerations and the observation that the Lorentz cone is a symmetric cone in the Euclidean Jordan algebra L n , in this article we consider some special types of transformations on Euclidean Jordan algebras and describe their cone spectra. Our focus here is on Lyapunov transformations and quadratic representations which have been extensively studied in the literature on complementarity problems5678910. Interestingly enough, for these special transformations, the cone spectrum is contained in the spectrum, thus proving the finiteness. ...
Let L be a linear transformation on a finite dimensional real Hilbert space H and K be a closed convex cone with dual K* in H. The cone spectrum of L relative to K is the set of all real lambda for which the linear complementarity problem x is an element of K, y = L(x) - lambda x is an element of K*, and < x, y > = 0 admits a nonzero solution x. In the setting of a Euclidean Jordan algebra H and the corresponding symmetric cone K, we discuss the finiteness of the cone spectrum for Z-transformations and quadratic representations on H. (c) 2009 Elsevier Inc. All rights reserved.
... In particular, when H = S n and K = S n + and L i = L A T i , we can relate the existence of a common quadratic Lyapunov function to a solution of a complementarity problem. We refer the reader to [9] and [12] for results highlighting this connection. See also, Section 25.5. ...
Using duality, complementarity ideas, and Z-transformations, in this chapter we discuss equivalent ways of describing the existence of common linear/quadratic Lyapunov
functions for switched linear systems. In particular, we extend a recent result of Mason–Shorten on positive switched system
with two constituent linear time-invariant systems to an arbitrary finite system.
... It was shown in [13] that the analog of (4) ⇒ (2) holds in this setting. The above property in S n and its nonsymmetric version were studied extensively in [12,[14][15][16]27]. A question arose as to whether the above property could be introduced in other Euclidean Jordan algebras, in particular in R n where the cone of squares is the Lorentz cone (also called the second-order cone or ice-cream cone). ...
A real square matrix is said to be a P-matrix if all its principal minors are positive. It is well known that this property is equivalent to: the nonsign-reversal property based on the componentwise product of vectors, the order P-property based on the minimum and maximum of vectors, uniqueness property in the standard linear complementarity problem, (Lipschitzian) homeomorphism property of the normal map corresponding to the nonnegative orthant. In this article, we extend these notions to a linear transformation defined on a Euclidean Jordan algebra. We study some interconnections between these extended concepts and specialize them to the space of all n×n real symmetric matrices with the semidefinite cone and to the space Rn with the Lorentz cone.
We extend the power penalty method for linear complementarity problem(LCP) (Wang and Yang, 2008) to the semidefinite linear complementarity problem(SDLCP). We establish a family of low-order penalty equations for SDLCPs. Under the assumption that the involved linear transformation possesses the Cartesian P-property, we show that when the penalty parameter tends to infinity, the solution to any equation of this family converges to the solution of the SDLCP exponentially. Numerical experiments verify this convergence result.
Let K⊆Rn be the n-dimensional Lorentz cone. Given an n×n matrix M and q∈Rn, the Lorentz-cone linear complementarity problem LCLCP(M,q) is to find an x∈Rn that satisfiesx∈K,y:=Mx+q∈KandyTx=0.
We show that if M is a Z-matrix with respect to K, then M is positive stable if and only if LCLCP(M,q) has a non-empty finite solution set for all q∈Rn.
In this paper, we introduce the concepts of w-P and w-uniqueness properties for a linear transformation defined on a Euclidean Jordan algebra V and study some interconnections between these concepts. We also specialize them to the space Sn of all n × n real symmetric matrices with the semidefinite cone Sn+ and to the space Rn with the Lorentz cone Ln+.
Generalizing the w-P property of a matrix, Tao [Some w-P properties for linear transformations on Euclidean Jordan algebras, Pacific J. Optimization, 5 (2009), 525-547] recently introduced and studied the w-P and the w-uniqueness properties for linear transformations defined on Euclidean Jordan algebras V. In this paper, we further study these properties. In particular, we study them for Lyapunov-like and Z transformations on V. We also present a sufficient condition for the w-uniqueness property on V and S(n) respectively.
The well-known Lyapunov’s theorem in matrix theory/continuous dynamical systems asserts that a square matrix A is positive stable if and only if there exists a positive definite matrix X such that AX+XA* is positive definite. In this paper, we extend this theorem to the setting of any Euclidean Jordan algebra V. Given any element a ∈ V, we consider the corresponding Lyapunov transformation L
a
and show that the P and S-properties are both equivalent to a being positive. Then we characterize the R0-property for L
a
and show that L
a
has the R0-property if and only if a is invertible. Finally, we provide L
a
with some characterizations of the E0-property and the nondegeneracy property.
In the setting of semidefinite linear complementarity problems on Sn, the implications strictmonotonicity⇒P2⇒GUS⇒P are known. Here, P and P2 properties for a linear transformation L:Sn→Sn are respectively defined by: X∈Sn,XL(X)=L(X)X⪯0⇒X=0 and X⪰0,Y⪰0, (X−Y)[L(X)−L(Y)](X+Y)⪯0⇒X=Y; GUS refers to the global unique solvability in semidefinite linear complementarity problems correspond- ing to L. In this article, we show that the reverse implications hold for any self-adjoint linear transformation, and for normal Lyapunov and Stein transformations. By introducing the concept of a principal subtransformation of a linear transformation, we show that L:Sn→Sn has the P2-property if and only if for every n×n real invertible matrix Q, every principal subtransformation of L has the P-property where L(X):=QTL(QXQT)Q. Based on this, we show that P2,GUS, and P properties coincide for the two-sided multiplication transfor- mation.
This article studies some geometrical aspects of the semidefinite linear complementarity problem (SDLCP), which can be viewed as a generalization of the well-known linear complementarity problem (LCP). SDLCP is a special case of a complementarity problem over a closed convex cone, where the cone considered is the closed convex cone of positive semidefinite matrices. It arises naturally in the unified formulation of a pair of primal-dual semidefinite programming problems. In this article, we introduce the notion of complementary cones in the semidefinite setting using the faces of the cone of positive semidefinite matrices and show that unlike complementary cones induced by an LCP, semidefinite complementary cones need not be closed. However, under R 0-property of the linear transformation, closedness of all the semidefinite complementary cones induced by L is ensured. We also introduce the notion of a principal subtransformation with respect to a face of the cone of positive semidefinite matrices and show that for a self-adjoint linear transformation, strict copositivity is equivalent to strict semimonotonicity of each principal subtransformation. Besides the above, various other solution properties of SDLCP will be interpreted and studied geometrically.
Motivated by the equivalence of the strict semimonotonicity property of the matrix A and the uniqueness of the solution to the linear complementarity problem LCP(A,q) for q∈R
+
n
, we study the strict semimonotonicity (SSM) property of linear transformations on Euclidean Jordan algebras. Specifically,
we show that, under the copositive condition, the SSM property is equivalent to the uniqueness of the solution to LCP(L,q) for all q in the symmetric cone K. We give a characterization of the uniqueness of the solution to LCP(L,q) for a Z transformation on the Lorentz coneℒ+
n
. We study also a matrix-induced transformation on the Lorentz space ℒ
n
.
In this article, we study the positive principal minor (PPM) property of linear transformations on Euclidean Jordan algebras.
Specifically, we give a characterization of the PPM property on the Lorentz space ℒ
n
and show that the PPM property implies the Q property. We also study a matrix-induced transformation onℒ
n
.
We introduce a Cartesian P-property for linear transformations between the space of symmetric matrices and present its applications to the semidefinite linear complementarity problem (SDLCP). With this Cartesian P-property, we show that the SDLCP has GUS-property (i.e., globally unique solvability), and the solution map of the SDLCP
is locally Lipschitzian with respect to input data. Our Cartesian P-property strengthens the corresponding P-properties of Gowda and Song [15] and allows us to extend several numerical approaches for monotone SDLCPs to solve more
general SDLCPs, namely SDLCPs with the Cartesian P-property. In particular, we address important theoretical issues encountered in those numerical approaches, such as issues
related to the stationary points in the merit function approach, and the existence of Newton directions and boundedness of
iterates in the non-interior continuation method of Chen and Tseng [6].
Motivated by the similarities between the properties of Z-matrices on $$R^{n}_+$$ and Lyapunov and Stein transformations on the semidefinite cone $$\mathcal {S}^n_+$$ , we introduce and study Z-transformations on proper cones. We show that many properties of Z-matrices extend to Z-transformations. We describe the diagonal stability of such a transformation on a symmetric cone by means of quadratic representations.
Finally, we study the equivalence of Q and P properties of Z-transformations on symmetric cones. In particular, we prove such an equivalence on the Lorentz cone.
We introduce the notion of a complementary cone and a nondegenerate linear transformation and characterize the finiteness of the solution set of a linear complementarity problem over a closed convex cone in a finite dimensional real inner product space. In addition to the above, other geometrical properties of complementary cones have been explored.
Dedicated to Hoang Tuy on the occasion of his seventieth birthday Abstract. This short note presents a constructive way of reducing monotone LCPs (linear complementarity problems) over cones to LPs (li- near programs) over cones. In particular, the monotone semideflnite lin- ear complementarity problem (SDLCP) in symmetric matrices, which was recently proposed by Kojima, Shindoh and Hara, is reducible to an SDP (semideflnite program). This gives a negative answer to their question whether the monotone SDLCP in symmetric matrices is an essential gen- eralization of the SDP.
Kojima, Shindoh and Hara proposed a family of search directions for the semidefinite linear complementarity problem (SDLCP) and established polynomial convergence of a feasible shortstep path-following algorithm based on a particular direction of their family. The question of whether polynomiality could be established for any direction of their family thus remained an open problem. This paper answers this question in the affirmative by establishing the polynomiality of primal-dual interior-point algorithms for SDLCP based on any direction of the Kojima, Shindoh and Hara family of search directions. We show that the polynomial iterationcomplexity bounds of two well-known algorithms for linear programming, namely the short-step path-following algorithm of Kojima et al. and Monteiro and Adler, and the predictor-corrector algorithm of Mizuno et al., carry over to the context of SDLCP. keywords: Semidefinite programming, interior-point methods, polynomial complexity, pathfollowing methods,...
Given a linear transformation L:?
n
→?
n
and a matrix Q∈?
n
, where ?
n
is the space of all symmetric real n×n matrices, we consider the semidefinite linear complementarity problem SDLCP(L,?
n
+,Q) over the cone ?
n
+ of symmetric n×n positive semidefinite matrices. For such problems, we introduce the P-property and its variants, Q- and GUS-properties. For a matrix A∈R
n×n
, we consider the linear transformation L
A
:?
n
→?
n
defined by L
A
(X):=AX+XA
T
and show that the P- and Q-properties for L
A
are equivalent to A being positive stable, i.e., real parts of eigenvalues of A are positive. As a special case of this equivalence, we deduce a theorem of Lyapunov.
Semidefinite programming is an extension of linear programming where (some of) the vector variables are replaced by matrix variables and (some of) the nonnegativity elementwise constraints are replaced by positive semidefiniteness constraints. These are convex problems which can be solved efficiently by interior-point methods. They arise in many applications, e.g. in combinatorial optimization, matrix completion problems, stability of differential systems, and, more generally, as the dual of Lagrangian relaxations of quadratic models of numerically hard problems. 1 Introduction 1.1 Background Semidefinite programming (denoted SDP) is an extension of linear programming (LP), where (some of) the vector variables are replaced by matrix variables and (some of) the nonnegativity elementwise constraints are replaced by positive semidefiniteness constraints. (The semidefiniteness constraints are also referred to as linear matrix inequalities.) We can express a linear (primal) SDP as (PSDP )...
We discuss an approach for solving the Bilinear Matrix Inequality (BMI) based on its con- nections with certain problems defined over matrix cones. These problems are, among others, the cone generalization of the linear programming (LP) and the linear complementarity problem (LCP) (referred to as the Cone-LP and the Cone-LCP, respectively). Specifically, we show that solving a given BMI is equivalent to examining the solution set of a suitably constructed Cone-LP or Cone-LCP. This approach facilitates our understanding of the geometry of the BMI and opens up new avenues for the development of the computational procedures for its solution. © 1997 The Mathematical Programming Society, Inc. Published by Elsevier Science B.V.
Over the past decade, the field of finite-dimensional variational inequality and complementarity problems has seen a rapid development in its theory of existence, uniqueness and sensitivity of solution(s), in the theory of algorithms, and in the application of these techniques to transportation planning, regional science, socio-economic analysis, energy modeling, and game theory. This paper provides a state-of-the-art review of these developments as well as a summary of some open research topics in this growing field.
Preliminaries and statement of the problem.- P-matrices and N-matrices.- Fundamental global univalence results of Gale-Nikaido-Inada.- Global homeomorphisms between finite dimensional spaces.- Scarf's conjecture and its validity.- Global univalent results in R2 and R3.- On the global stability of an autonomous system on the plane.- Univalence for mappings with Leontief type Jacobians.- Assorted applications of univalence mapping results.- Further generalizations and remarks.
The SDLCP (semidefinite linear complementarity problem) in symmetric matrices introduced in this paper provides a unified mathematical model for various problems arising from systems and control theory and combinatorial optimization. It is defined as the problem of finding a pair $(\X,\Y)$ of $n \times n$ symmetric positive semidefinite matrices which lies in a given $n(n+1)/2$ dimensional affine subspace $\FC$ of $\SC^2$ and satisfies the complementarity condition $\X \bullet \Y = 0$, where $\SC$ denotes the $n(n+1)/2$-dimensional linear space of symmetric matrices and $\X \bullet \Y$ the inner product of $\X$ and $\Y$. The problem enjoys a close analogy with the LCP in the Euclidean space. In particular, the central trajectory leading to a solution of the problem exists under the nonemptiness of the interior of the feasible region and a monotonicity assumption on the affine subspace $\FC$. The aim of this paper is to establish a theoretical basis of interior-point methods with the use of Newton directions toward the central trajectory for the monotone SDLCP.
LetC be a pointed, solid, closed and convex cone in then-dimensional Euclidean spaceE n ,C* its polar cone,M:C→E n a map, andq a vector inE n . The complementarity problem (q|M) overC is that of finding a solution to the system$$(q|M) x \varepsilon C, M(x) + q \varepsilon C{^*} , \left\langle {x, M(x) + q} \right\rangle = 0.$$ It is shown that, ifM is continuous and positively homogeneous of some degree onC, and if (q|M) has a unique solution (namely,x=0) forq=0 and for someq=q 0 ∈ intC*, then it has a solution for allq ∈E n .
Merit functions such as the gap function, the regularized gap function, the implicit Lagrangian, and the norm squared of the
Fischer-Burmeister function have played an important role in the solution of complementarity problems defined over the cone
of nonnegative real vectors. We study the extension of these merit functions to complementarity problems defined over the
cone of block-diagonal symmetric positive semi-definite real matrices. The extension suggests new solution methods for the
latter problems.
Matrix stability has been intensively investigated in the past two centuries. We review work that has been done in this topic, focusing on the great progress that has been achieved in the last decade or two. We start with classical stability criteria of Lyapunov, Routh and Hurwitz, and Liénard and Chipart. We then study recently proven sufficient conditions for stability, with particular emphasis on P-matrices. We investigate conditions for the existence of a stable scaling for a given matrix. We review results on other types of matrix stability, such as D-stability, additive D-stability, and Lyapunov diagonal stability. We discuss the weak principal submatrix rank property, shared by Lyapunov diagonally semistable matrices. We also discuss the uniqueness of Lyapunov scaling factors, maximal Lyapunov scaling factors, cones of real positive semidefinite matrices and their applications to matrix stability, and inertia preserving matrices.
This paper addresses the common Lyapunov function problem for
discrete-time linear systems. Namely, the authors raise the question
that in order for a given set of Schur stable system matrices to share a
quadratic Lyapunov function what kind of properties are required for the
set. The authors provide a partial answer to it by showing that if the
matrices form a commuting family then they have the desired property
The paper demonstrates that a common quadratic Lyapunov function
exists for all linear systems of the form x˙=A<sub>i</sub>x,
i=1,2,···,N, where the matrices A<sub>i</sub> are
asymptotically stable and commute pairwise. This in turn assures the
exponential stability of a switching system x˙(t)=A(t)x(t) where
A(t) switches between the above constant matrices A<sub>i</sub>
In semidefinite programming one minimizes a linear function subject to the constraint that an affine combination of symmetric matrices is positive semidefinite. Such a constraint is nonlinear and nonsmooth, but convex, so semidefinite programs are convex optimization problems. Semidefinite programming unifies several standard problems (e.g., linear and quadratic programming) and finds many applications in engineering and combinatorial optimization. Although semidefinite programs are much more general than linear programs, they are not much harder to solve. Most interior-point methods for linear programming have been generalized to semidefinite programs. As in linear programming, these methods have polynomial worst-case complexity, and perform very well in practice.
The well known Lyapunov's theorem in matrix theory/ continuous dynamical systems asserts that a (complex) square matrix A is positive stable (that is, all eigenvalues lie in the open right-half plane) if and only if there exists a positive definite matrix X such that AX+XA is positive definite. In this paper, we prove a complementarity form of this theorem: A is positive stable if and only if for any Hermitian matrix Q, there exists a positive semidefinite matrix X such that AX+XA +Q is positive semidefinite and X[AX+XA +Q] = 0. By considering cone complementarity problems corresponding to linear transformations of the form I Gamma S, we show that a (complex) matrix A has all eigenvalues in the open unit disk of the complex plane if and only if for every Hermitian matrix Q, there exists a positive semidefinite matrix X such that X Gamma AXA + Q is positive semidefinite and X[X Gamma AXA +Q] = 0. By specializing Q (to GammaI ), we deduce the well known Stein's theore...
A new merit function and a descent method for semidefi-nite complementarity problems, in Reformulation: Nonsmooth, Piecewise Smooth, Semi-smooth and Smoothing Methods
- N Yamashita
- M Fukushima
N. Yamashita and M. Fukushima, A new merit function and a descent method for semidefi-nite complementarity problems, in Reformulation: Nonsmooth, Piecewise Smooth, Semi-smooth and Smoothing Methods, Kluwer Academic, Boston, 1999, pp. 405–420.
Problé géné de la stabilité des mouvement
- A M Lyapunov
A.M. Lyapunov, Problé géné de la stabilité des mouvement, Ann. Fac. Sci. Toulouse, 9 (1907), pp. 203–474;