An algorithm for computing a neighborhood included in the attraction domain of an asymptotically stable point
Nicolas Delanoue
a,⇑, Luc Jaulin
b, Bertrand Cottenceau
aaLARIS, Université d’Angers, 62 avenue Notre Dame du Lac, 49000 Angers, France
bENSTA-Bretagne, 2 rue Franois Verny, 29806 Brest Cedex 09, France
a r t i c l e i n f o
Article history:
Available online 16 September 2014
Keywords:
Interval computations Reliable algorithm Nonlinear stability theory
a b s t r a c t
Many methods exist to detect stable equilibrium pointsxof nonlinear dynamical systems x_¼fðxÞ. Most of them also prove the existence of a neighborhoodN ofxsuch that all trajectories initialized inN converge to x. This paper provides a numerical method combining Lyapunov theory with interval analysis which makes to find a setN which is included in the attraction domain ofx.
Ó2014 Elsevier B.V. All rights reserved.
1. Introduction
Consider a nonlinear dynamical system described by a differential equationx_¼fðxÞ, wheref:Rn!Rnis a smooth vector field. The pointxis an equilibrium point iffðxÞ ¼0. To find the equilibrium points it suffices to solvennonlinear equations withnunknowns. This can be solved using elimination theory-based methods[18], or any local numerical algorithm[20]. A pointxis asymptotically stable if for all neighborhoodMofx, there exists a neighborhoodN ofxsuch that all trajectories initialized inN converge toxand remain insideM.
From the theoretical point of view, the Hartman–Grobman theorem states that iffis sufficiently regular around a hyper- bolic equilibrium statexthen there exists a local homeomorphism between the solutions of thex_¼fðxÞand its linearization x_¼DfðxÞðxxÞ. In other words, the qualitative behavior of the dynamical systemfaroundxis the same that ofDfðxÞ.
Therefore, the existence ofNis usually provided by studying the eigenvalues of the Jacobian matrix offatx. Interval based methods have already been used to study the stability of dynamical systems. In the case of linear system, a classical result from control theory states that the origin (which is always an equilibrium state) is stable if and only if all roots of the char- acteristic polynomial offhave a negative real part. Such a polynomial is said to be Hurwitz stable. In[16], Khraritonov gives a necessary and sufficient effective condition to the Hurwitz stability of a polynomial with interval coefficients. Whenfis lin- ear with unknown bounded coefficients (i.e.fcan be represented by a matrix whose entries are intervals), the Khraritonov’s condition only offers a sufficient condition to check that the origin is stable. More recently, Wang and al[17]determine a necessary and sufficient effective condition to the Hurwitz stability of an interval matrix.
The present paper deals with nonlinear dynamical system. Contrary to the linear case, the stability of an equilibrium state is, most of the time, only local: the trajectories must be initialized sufficiently close to the equilibrium statexto converge to x. The set of initial states for which the trajectory converges toxis theattraction domainofx. The main contribution of this paper is an algorithm which provides a neighborhoodN ofxincluded in the attraction domain ofx.
http://dx.doi.org/10.1016/j.cnsns.2014.08.034 1007-5704/Ó2014 Elsevier B.V. All rights reserved.
⇑Corresponding author.
E-mail addresses: nicolas.delanoue@univ-angers.fr (N. Delanoue), luc.jaulin@ensta-bretagne.fr (L. Jaulin), bertrand.cottenceau@univ-angers.fr (B. Cottenceau).
Contents lists available atScienceDirect
Commun Nonlinear Sci Numer Simulat
j o u r n a l h o m e p a g e : w w w . e l s e v i e r . c o m / l o c a t e / c n s n s
This section introduces notations and definitions related to interval analysis. An interval½x;xofRnis a set which can be written asfx2Rn; x6x6xgwithxandxinRn. Here the relation6has to be understood component-wise. Note that this definition implies that intervals are bounded. The set intervals is usually denoted byIRn.
Definition 1.A map ½f:IRn!IRm is said to be an inclusion map of f:Rn!Rm if 8½x 2IRn;fð½xÞ ½fð½xÞ (where fð½xÞ ¼ ffðxÞjx2 ½xg).
Interval arithmetic[1,13,19]provides an effective method to build inclusion maps. In [5], Neumaier proves that it is always possible to find an inclusion map½fwhenfis defined by an arithmetical expression. This possibility to enclose the image of an interval ½x under f is powerful. Indeed, let us suppose that 0R½fð½xÞ, one can conclude that 8x2 ½x; fðxÞ–0. On the other hand, if 02 ½fð½xÞ, this does not imply that9x2 ½xjfðxÞ ¼0.Fig. 1
Since Moores works[1,2]that introduced interval arithmetic, many algorithms have been developed in different areas, for example in global optimization[7], non-linear dynamical systems, etc. As interval analysis provides rigorous methods, these algorithms can prove mathematical assertion. For instance, in 2003, Hales launched the ‘‘Flyspeck project’’ (‘‘Formal Proof of Kepler’’) in an attempt to use computers to automatically verify every step of the proof (partially based on interval analysis) of the Kepler’s conjecture. Another important example is a generalization of the Newton method called Interval Newton method. This method can be applied to find all zeros of a given differentiable mapf:Rn!Rn. The interval Newton method creates a sequence of intervals containing zeros offand has very interesting properties: combined with Brouwer fixed point theorem, it can prove existence and uniqueness of a zero off[6,14].
Note that the set of inclusion maps of a given f:Rm!Rn can be partially ordered by the relation:
½f160½f2()8½x 2IRn; ½f1ð½xÞ ½f2ð½xÞ. Due to the fact that the available inclusion map is rarely minimal (related to 60), interval analysis cannot basically be used to prove the assertion8x2 ½x; fðxÞP0 in the case of existence ofx02 ½xsuch thatfðx0Þ ¼0. The next section shows how such a proof can be done by combining interval computation with algebra calculus.
3. Sufficient condition to checkf >0.
This section proposes a theorem which provides a sufficient condition to check the following assertion for a given differ- entiable real valued functionf:8x2 ½x; fðxÞP0. The main idea is close to the second derivative criterion classically used in optimization. Then, an algorithm based on the proposed theorem and interval analysis is presented. Let us recall that a sym- metric real matrixAis positive definite if8x2Rn f0g; xTAx>0. In this paper, the set of positive definite symmetricnn matrices is denoted bySnþ.
Theorem 1. Let f2C1ð½x Rn; RÞ. If there exists x2 ½xsuch that fðxÞ ¼0and DfðxÞ ¼0, and8x2 ½x; D2fðxÞ 2Snþ, then 8x2 ½x; fðxÞP0and fðxÞ ¼0)x¼x.
Proof. The assertion8x2 ½x; D2fðxÞ 2Snþ implies that fis a strictly convex function defined on a convex set½x. Since DfðxÞ ¼0, one can conclude that inf
x2½xfðxÞPfðxÞ ¼0. The proof of uniqueness is by reduction to a contradiction. Suppose that there existsx2 ½xsuch thatfðxÞ ¼0 andx–x. Asfis strictly convex, one has
Fig. 1.Illustration of inclusion function.
f xþx 2
<1 2fðxÞ þ1
2fðxÞ ¼0
Therefore, since½xis convex, we havem¼xþx22 ½xsuch thatfðmÞ<0. h
This theorem induces an effective method to prove that8x2 ½x; fðxÞP0. Indeed, iffðxÞ ¼0 andDfðxÞ ¼0 for some x2 ½xcan be proved by calculus algebras[8], one only has to check thatD2fð½xÞis included inSnþto conclude.
In practice, this inclusion is performed using results based on interval symmetric matrices. WithAandAtwo symmetric matrices such that A6A, an interval symmetric matrix is a set ½A of symmetric matrices of the form:
½A ¼ fA2Rnn; A6A6A; AT¼Ag[15]. Here the partial order relation6between matrices is understood component-wise.
(SeeFig. 2)
Definition 2. A symmetric interval matrix½Ais positive definite if½A Snþ.
Remark 1. LetVð½AÞdenote the finite set of corners of½A. Since the coneSnþand½Aare convex subsets of the set of sym- metric matrices, one has the following equivalence:
½A Snþ()Vð½AÞ Snþ
The set of symmetricnn-matrices is a vector space of dimensionnðnþ1Þ2 . ThereforeVð½AÞhas a cardinality of 2nðnþ1Þ2 . In[9], Rohn proposes a method to check½A Snþ by testing positive definiteness of only 2n1 matrices. The procedure is the following: with ½A a interval symmetric matrix, one can create two symmetric matrices Ac and D such that
½A ¼ fA; AcD6A6AcþDg where Ac¼12ðAþAÞ and D¼12ðAAÞ. Let us denote by C the finite set:
C¼ fx2Rn; jxij ¼1; 8i2 f1;. . .;ngg. One has#C¼2n. For eachzinC, let us denote byTzthe diagonal matrix defined by z,i.e. Tz¼diagðzÞand by Az the matrix AcTzDTz. Each Az, withz2C, is obviously in ½A, and since Az¼Az, the set fAz; z2Cgis finite and of cardinal 2n1. In[9], Rohn proves that
½A Snþ() fAz;z2Cg Snþ:
Fig. 2.Withn¼2, an interval symmetric matrix½A;A].
0 2 4 6 8 10 12 14 16 18 20
Z
−2 −3 0 −1
2 1
4 3 X
−3−2
−1 0 1 2
3
Y 4
Fig. 3.Graph off.
Example 1. Let f:R2!Rbe the function defined by fðx;yÞ ¼ cosðx2þ ffiffiffi p2
sin2yÞ þx2þy2þ1. This function satisfies fð0;0Þ ¼0 andDfð0;0Þ ¼0 since (SeeFig. 3)
Dfðx;yÞ ¼ 2xðsinðx2þ ffiffiffi p2
sin2yÞ þ1Þ 2 ffiffiffi
p2
cosy siny sin ffiffiffi p2
sin2yþx2
þ2y 0
@
1 A
T
: ð1Þ
The Hessian matrix is given byD2f¼ a1;1 a1;2
a2;1 a2;2
whereai;jare given by the following formulas:
a1;1¼2 sin ffiffiffi p2
sin2yþx2
þ4x2cos ffiffiffi p2
sin2yþx2
þ2;
a2;2¼ 2 ffiffiffi p2
sin2y sin ffiffiffi p2
sin2yþx2
þ2 ffiffiffi p2
cos2y sin ffiffiffi p2
sin2yþx2
þ8 cos2y sin2ycos ffiffiffi p2
sin2yþx2
þ2;
a1;2¼a2;1¼4 ffiffiffi p2
x cosy siny cos ffiffiffi p2
sin2yþx2
:
Thanks to interval analysis, it is possible to guarantee that8x2 ½0:5;0:52; D2fðxÞ ½Awhere½Ais the following inter- val symmetric matrix:
½A ¼ ½1:9;4:1 ½1:3;1:4
½1:3;1:4 ½1:9;5:4
According toRemark 1, to prove thatfðxÞP0 for allxin21;122
, one only has to check that the 2 matrices:
A1¼ 1:9 1:3 1:3 1:9
andA2¼ 1:9 1:4 1:4 1:9
are definite positive[6](this can be shown using rigorous computations). Interval symmetric matrix½Ais represented in Fig. 4with a blue box. The two matricesA1andA2used in the Rohn[9]sufficient condition are red corners.
4. Algorithm for proving stability
In this section, an efficient method able to prove asymptotic stability is given. The theorem presented in4.1combines results of the previous section with Lyapunov theory and induces an algorithm given in4.2. This algorithm also generates a subsetN of the attraction domain of the asymptotically stable point.
4.1. Theorem
To prove stability, most of the methods are based on Lyapunov theory. It consists in creating a real valuedLfunction which is energy-like. Before introducing our algorithm, let us present some definitions and theorems related to stability.
Let fgt:Rn!Rngt2R denotes the flow associated to the vector field x#fðxÞ, i.e. the 1-parameter family of functions fgt:Rn!Rngt2Rsatisfying:
Fig. 4.The interval symmetric matrix½Ais represented in blue. The two matrices used in the Rohn sufficient condition are red corners. This interval symmetric is positive definite since it is included in the coneSnþ. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
d
dtgtðxÞ ¼fðgtðxÞÞfor allt2R g0ðxÞ ¼x
Definition 3. A subsetN ofRnis stable (according tof) if8tP0; gtðN Þ N,
Definition 4. LetMandN be two subsets ofRnsuch thatN M. An equilibrium pointxis asymptoticallyðN;MÞ-stable (according tof) if
8tP0; gtðN Þ M 8x2 N; lim
t!1gtðxÞ ¼x (
This notion is illustrated byFig. 5.
Definition 5. LetMbe a subset ofRnandxbe in the interior ofM. A differentiable real valued functionLis a Lyapunov function for the dynamical systemx_¼fðxÞif:
1. LðxÞ ¼0()x¼x, 2. 8x2 M fxg; LðxÞ>0, 3. 8x2 M fxg; hDLðxÞ;fðxÞi<0.
This theory is motivated by the following theorem which gives a sufficient condition to asymptotic stability.
Theorem 2. If L:M !Ris a Lyapunov function related to the dynamical systemx_¼fðxÞthen there exists a subsetNofMsuch that the point x2 M(the unique one satisfying LðxÞ ¼0) is asymptoticallyðN;MÞ-stable.
The proof can be found in[12]. To check stability, one merely has to:
1. find a candidate for the Lyapunov function, 2. check that this candidate is of Lyapunov.
For the first step, since the set of differentiable functions is infinite dimensional, one prefers to limit the search for the candidate to a finite dimensional subspace. For instance, we may suppose thatLis a quadratic formLðxÞ ¼xTWxwhereW is a symmetric square matrix. It is well known[12]that, in the linear case (x_¼AxwhereAis a square matrix), the origin is asymptotically stable if and only if there exists a matrixWinSnþsuch that
ATWþWA¼ I: ð2Þ
Solving this equation whose unknown isWamounts to solving linear equations. If W is positive definite, then all condi- tions of Theorem 2are fulfilled, thus the origin is asymptotically stable. In other words, in the linear case, an effective method to prove stability exists. Our idea is partially based on this effective method. The main idea is to construct a quadratic
Fig. 5.The pointxis asymptoticallyðN;MÞ-stable.
Theorem 3. Consider the dynamical systemx_¼fðxÞand a matrix W2Snþwhose maximum and minimum eigenvalues arekmax
andkminrespectively. Let Lnbe a quadratic form defined by
Ln:M ! R
x # ðxnÞTWðxnÞ; ð3Þ
withn2 M. Let½xandN be boxes such that ½x N M,
the center ofN is in½x,
the radius ofN is smaller than ffiffiffiffiffiffiffiffiffiffi nkkmin
max
q lð½x;MÞ.
We have the following implication:
If there exists a single x2 Mstrictly inside½x, such that fðxÞ ¼0, and 8x2 M; 8n2 ½x;D2hDLnðxÞ; fðxÞi 2Snþ;
then xis asymptoticallyðN;MÞ-stable.
Proof. SinceW2Snþ, one has:
1. LxðxÞ ¼0()x¼x 2. 8x2 M fxg; LxðxÞ>0
Lethbe the real valued function defined byhðxÞ ¼ hDLxðxÞ;fðxÞi. By construction, we havehðxÞ ¼0 andDhðxÞ ¼0.
Moreover, since 8x2 M; 8n2 ½x; D2hDLnðxÞ;fðxÞi 2Snþ, supposing n¼x, one can conclude that 8x2 M; D2hðxÞ 2Snþ.
ApplyingTheorem 1toh, one has8x2 M; hðxÞP0. Therefore,Lx is a Lyapunov function for the dynamical system x_¼fðxÞ. In other words, there exists a subsetN ofMandx2 N such that:
8t2Rþ; gtðN Þ M;
8x2 N; lim
t!þ1gtðxÞ ¼x: (
LetEbe the ellipsoid oriented byW, with centerx, and long axe ffiffiffiffiffiffiffiffiffi kmin
p lð½x;MÞ. Obviously, the setEis included inMand is stable. Thus, any boxesN whose center is in½xand whose radius is smaller than ffiffiffiffiffiffiffiffiffiffi
nkkmin
max
q lð½x;MÞis, by construction, included in the ellipsoidE. Therefore,xis asymptoticallyðN;MÞ-stable. h
From a dynamical systemx_¼fðxÞand a setM, the following algorithm computes a setNand proves that there exists a unique equilibrium pointxwhich is asymptoticallyðN;MÞ-stable. The computed setNis therefore included in the attrac- tion domain ofx.
4.2. Algorithm
The main idea, of this algorithm, is first to linearize the given system using a point close to the equilibrium state. In a second step, one checks that a Lyapunov function for the linearized one is also a Lyapunov function for the nonlinear one according to results obtained in Section3. This can be summarized in Algorithm 1.
Algorithm 1.Algorithm Require: a boxMofRn,
Require: a dynamical systemx_¼fðxÞwheref2C1ðM;RnÞ,
Ensure: a boxN, and a proof of existence and uniqueness of an equilibrium statexwhich is asymptoticallyðN;MÞ stable,
1:½x:¼Newton Interval Algorithm forfðxÞ ¼0; x2 M.
2:~x:¼an element of½x. 3:
A:¼ df dxjx¼~x
ð4Þ
4: SolveATWþWA¼ I.
5:ifW2SnþandD2hDL½xðMÞ;fðMÞi Snþ,then
6: Returnthe boxN whose center is~xand radius is ffiffiffiffiffiffiffiffiffiffiffi nkkmin
max
q
lð½x;MÞ.
7:else
8: Return‘‘Failure’’.
9:end if
Step 1 can be performed using the interval Newton method previously cited. Note that at step 3 the matrixAchosen is not the exact linearization offwithxbut is a linearization with an approximation ofxdenoted~x. This is important because, in practice, it is often impossible to compute the exact zero of a given function. The exact position of the equilibrium state is not needed for the rest of the procedure. An step 4, linear algebra is used to solve linear equations. This linear system does not need to be solved in a rigorous way to ensure the correctness of the general method. At step 5, interval analysis is used to prove that:D2hDL½xðMÞ;fðMÞi Snþ.
5. Illustrative example
In this section, the proposed method is discussed via the example:
x_¼ x_1
x_2
¼ x2
x1 ð1x21Þx2
¼fðxÞ ð5Þ
whereM ¼ ½0:6;0:62. The vector field associated to this dynamical system is represented onFig. 6.
Fig. 6.Normalized vector field and its linearization aroundx. The linearized one is represented by red dotted lines. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
First, interval Newton method is used to prove that the boxMcontains a uniquexequilibrium state. Moreover, this fixed point of the flow is proven to lie in½x ¼ ½0:02;0:022. Then, the dynamical system is linearized with~x¼ ð0:01;0:01Þ.Fig. 6 also shows the linearized one with~xin red dotted lines. In this case, the Lyapunov function created is:
LnðxÞ ¼ ðxnÞT 1;51 0;49 0;49 1;01
ðxnÞ ð6Þ
Some level curves ofLnfor some valuesn2 ½xare represented onFig. 7. In a neighborhood of½x, the functionLnseems to be a Lyapunov function since vectorsfðxÞcross the level curves form outside to inside. AsLnis of Lyapunov for the lin- earized system, the last geometrical interpretation is equivalent to8x2 M fxg; hDLðxÞ; fðxÞi<0. This last assertion is true sincehðxÞ ¼0; DhðxÞ ¼0, andD2h½xðMÞ SnþasD2h½xðMÞ ½Awith½A ¼ ½1:78;5:78 ½4:14;4:15
½4:14;4:15 ½0:56;3:45
positive definite.
6. Conclusion
This paper provides an effective rigorous method able, from a given dynamical system described byx_¼fðxÞand a given boxMto compute a boxNsuch thatNcontains a unique asymptoticallyðN;MÞ-stable equilibrium statex. These ideas has already been employed by Rauh[22]to verify stability analysis of continuous-time control systems with bounded parameter uncertainties.
Our point of view is that the marriage of Lyapunov theory and interval analysis works because of genericity. Indeed, inter- val based method succeeds in generic cases and Lyapunov functions are stable in the sense that any positive definite function in a sufficiently small neighborhood containing a Lyapunov function for a dynamical system is also of Lyapunov.
To fill out this work, different perspectives appear. It could be interesting to prove that the proposed algorithm terminates in the generic case. This method could also be combined with graph theory and guaranteed numerical integration of O.D.E.
[10,11,21]to compute a rigorous approximation of the attraction domain ofx. The computed boxN(with non empty inte- rior) could be a good first approximation of the attraction domain for an iterative scheme.
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