Sequential gradient dynamics in real analytic Morse systems
SSequential gradient dynamics in real analytic Morse systems
H. I¸sıl Bozma and Ferit ¨Ozt¨urkAbstract. Let Ω ⊂ R M be a compact connected M -dimensionalreal analytic domain with boundary and ϕ be a primal naviga-tion function; i.e. a real analytic Morse function on Ω with aunique minimum and with minus gradient vector field G of ϕ on ∂ Ω pointed inwards along each coordinate. Related to a roboticsproblem, we define a sequential hybrid process on Ω for G startingfrom any initial point q in the interior of Ω as follows: at eachstep we restrict ourselves to an affine subspace where a collection ofcoordinates are fixed and allow the other coordinates change alongan integral curve of the projection of G onto the subspace. Weprove that provided each coordinate appears infinitely many timesin the coordinate choices during the process, the process convergesto a critical point of ϕ . That critical point is the unique minimumfor a dense subset of primal navigation functions. We also presentan upper bound for the total length of the trajectories close to acritical point.
1. IntroductionWe consider the d -dimensional closed unit ball D ⊂ R d ( d ≥
1) and N objects (point particles or small disks with various radii) that wecall robots . Each robot has an initial position and a target positionin the interior of D . A problem stated in e.g. [4], [5], [8], [12], is tonavigate the robots from their initial to the target positions withoutoverlapping each other nor the boundary of D . The configuration spaceof nonoverlapping positions of the robots considered as point particlesis a noncompact manifold in R dN , and is connected except for d =1 , N > D N ⊂ R dN . Connectedness depends on the relativevalues of the radii; in particular if the radii of the robots are sufficiently Mathematics Subject Classification.
Primary 34C07; Secondary 58A07,37C10.
Key words and phrases.
Real analytic gradient flows, Hybrid navigation, Morsesystems, Descent methods, Lojasiewicz gradient inequality. a r X i v : . [ m a t h . D S ] O c t H. I¸sıl Bozma and Ferit ¨Ozt¨urk small then the configuration space is connected (see e.g. [2]). In thesequel we assume that Ω is connected.The initial and target positions determine two points, say q and p respectively, in the interior of Ω. Let ˜Ω be a small, open neighborhoodof Ω. Suppose there is a C function ϕ : ˜Ω → R such that(i) ϕ is Morse on Ω;(ii) −∇ ϕ is inwards on ∂ Ω;(iii) ϕ has a unique minimum at p ; there ϕ ( p ) = 0.Note that the condition that the domain of ϕ contains Ω ⊂ ˜Ω prop-erly guarantees that the derivative of ϕ on ∂ Ω is defined. Now it is wellknown through Morse theory that the integral curve γ q of the vectorfield −∇ ϕ through q ∈ Ω converges to a point q ∈ Ω and providedthat q is not chosen in some set of measure zero, we have q = p (seee.g. [13]). More precisely, the measure zero set is the complement ofthe stable manifold W sp of p in Ω. This solves the navigation problemof the N robots; they move in D simultaneously along trajectories de-termined by γ q and each converges to its target point provided that q ∈ W sp . In [8], such a Morse function has been constructed using thebasic idea that each robot will repel each other and will be attracted byits target point. Furthermore the function has been constructed realanalytic.In the present article, we ask a related question: suppose that therobots are allowed to move one at a time cyclically rather than simulta-neously. Can the same function ϕ and its minus gradient still be usedto solve this navigation problem? Let us make precise what we mean.Let ϕ be a real analytic function satisfying (i)-(iii) above on a com-pact connected real analytic domain with boundary (i.e. analyticallydiffeomorphic to a closed ball) Ω in R dN and G = −∇ ϕ . By defini-tion the vector field G is real analytic. We will denote the points of R dN by x = ( x , . . . , x N ) = (( y , . . . , y d ) , . . . , ( y dN − d +1 , . . . , y dN )) with x m = ( y dm − d +1 , . . . , y dm ) ∈ R d . (Now and below, x m is a d -tuple and y j ∈ R ; m always denotes an index in 1 , . . . , N and j an index in J = { , . . . , dN } .)Let n be a subset of J . For a point a ∈ Ω, we let R n a . = { ( y , . . . , y dN ) | y t = a t for all t ∈ J − n } ∩ Ωand call it the slice of type n through a . To keep the notation simplewe will denote the particular slice through a of type n = { dm − d +1 , . . . , dm } by R ma and we will call it a slice of type m . equential gradient dynamics in real analytic Morse systems 3 Let us consider an initial point q ∈ int(Ω) and the vector field G q over R q defined for each x ∈ R q by G q ( x ) = pr ( G ( x )) = −∇ ( ϕ | R q )Here pr i is the projection map onto the i -th d -tuple of coordinates.Observe that G q is a real analytic gradient vector field. In the sequel wewill drop the subscript and write G in case no confusion arises. Now,(cid:32)Lojasiewicz’s theorem states that any integral curve of the gradientflow of a real analytic function that has a nonempty ω -limit set has aunique limit [7]. Thus in our case if the orbit of the vector field G q through q stays in R q , it converges to a unique point, say q ∈ R q ,which is a singular point of the vector field G q . Let us denote theclosure of that orbit by γ , which starts from q and ends at q .At this point, we will have an extra condition to guarantee that theorbits stay in Ω. For each a ∈ ∂ Ω let f a ∈ C ω ( R dN , R ) be a defininganalytic function for ∂ Ω in an open subset U a containing a . Supposethat ∇ f a is outward. Then we impose the following condition which isstronger than condition (ii) above:(ii) (cid:48) For every a ∈ ∂ Ω and x ∈ U a ∩ ∂ Ω, the angle between ∇ ϕ ( x ) and ∇ f a ( x ) is sufficiently small so that for every j ∈ J , the j -th componentof ∇ ϕ ( x ) and ∇ f a ( x ) have the same sign (or are zero).This condition is equivalent to having the projected vector field −∇ ( ϕ | R n a ) inwards (or zero) at the boundary of every slice R n a , for each a ∈ Ω and n ⊂ J . Observe that (ii) (cid:48) is satisfied if, for example, theangle between ∇ ϕ and ∇ f a is identically zero on ∂ Ω. If, furthermore, ∂ Ω is given as a level set of a unique real analytic function, this situa-tion is nothing but the admissibility of a navigation function referredin e.g. [9], [10], [8].Now recursively, at the k -th step, let m k = ( k mod N ) + 1. Weconsider the orbit through the point q k − of the vector field G m k q k − overthe slice R m k q k − . Again by Lojasiewicz’s theorem and thanks to thecondition (ii) (cid:48) , the orbit converges to a unique singular point, say, q k .We denote the closure of that orbit by γ k . Note that R m k q k = R m k q k − . Wecall q k ’s the stationary points of the process. We will say that the point q k is of type m k ; i.e. type- m points are the limits of integral curves ontype- m slices. This process producing the sequence ( q k ) ∞ k =0 of pointswill be called a sequential gradient dynamical process associated to theslices R mx . Note that there are infinitely many points of the sequence( q k ) ∞ k =0 of type m for each m ∈ { , , . . . , N } .We deal with the following question: For what values of q does thesequence ( q k ) converge to the unique minimum p ? For other values H. I¸sıl Bozma and Ferit ¨Ozt¨urk of q does ( q k ) converge? Our purpose in this article is to prove thefollowing theorem that answers these two questions. To state that, wewill need a definition: we call a real analytic function satisfying theconditions (i), (ii) (cid:48) , (iii) above a primal navigation function . Theorem 1.
Let Ω ⊂ R dN be a compact connected real analytic domainwith boundary; ϕ : Ω → R be a primal navigation function with theunique minimum at p ∈ int (Ω) and q ∈ int (Ω) be an arbitrary initialpoint. Then the sequence ( q k ) ∞ k =0 of the sequential gradient dynamicalprocess converges to a critical point of ϕ .Moreover, there is a primal navigation function ψ on Ω with thesame minimum p such that ψ is arbitrarily C ω -close to ϕ , and that theprocess for ψ converges to p for all initial points q except when q is asaddle point.In any case of convergence above, it is possible to have ( q k ) ultimatelyconstant or there may be infinitely many nontrivial steps of the process. The proofs of the first two claims are given in Section 2 as Lemma 3,Corollary 4 and Lemma 5. They depend heavily on ϕ ’s being realanalytic and Morse. The main tool in the proofs is the (cid:32)Lojasiewiczinequality [6]: For q a critical point of the real analytic function ϕ and ϕ ( q ) = 0, there are a neighborhood U of q , c > µ ∈ [0 , x ∈ U ,(1) |∇ ϕ ( x ) | > c | ϕ ( x ) | µ . We will call the triple (
U, c, µ ) a (cid:32)Lojasiewicz neighborhood. BeingMorse will be used only in Lemma 5. The examples for the last claimof the theorem are presented in Section 3.2.The proof of Theorem 1 is valid for a more general claim regardinga more general set-up for the process. Consider a sequence ( n i ) ∞ i =1 ofsubsets of { , . . . , M } ( M ∈ Z + ) and the slices R n i a ⊂ R M through a ∈ Ω of type n i We define the process associated to the sequence ( n i )as follows: at the i -th step the dynamics occurs in the slice R n i q i − andconverges to the point q i . Then we have the following Theorem 2. ϕ : Ω → R , p , q and ( n i ) ∞ i =1 be as above. Suppose eachindex j ( ≤ j ≤ M ) appears infinitely many times as elements of thesets of the sequence ( n i ) . Then the sequence ( q k ) ∞ k =0 of the sequentialgradient dynamics associated to the sequence ( n i ) converges to a criticalpoint of ϕ .Moreover, there is a primal navigation function ψ on Ω with thesame minimum p such that ψ is arbitrarily C ω -close to ϕ , and that theprocess for ψ converges to p for all initial points q except when q is asaddle point. equential gradient dynamics in real analytic Morse systems 5 The proof of Theorem 2 follows closely that of Theorem 1, both ofwhich we will present below.2. Proof of the theoremsBefore stating and proving the claims, we will need several definitionsand observations.First, for the sake of simplicity of the notation, let us restrict our-selves (except in Lemma 5) to the case d = 2. ( d = 1 oversimplifies theproblem [3].) Whatever we present and claim below applies similarlyfor other dimensions as well. Now, let Γ j ⊂ Ω denote the set of pointsat which the vector field G has the j -th component zero; i.e.Γ j = { x ∈ Ω | ϕ j ( x ) = 0 } where ϕ j ( x ) = ∂ϕ∂y j ( x ). Observe that every critical point of ϕ lies inΓ j for each j = 1 , . . . , N . Let z be a nondegenerate critical pointof ϕ . Since the Hessian of ϕ at z is nondegenerate by assumption, ∇ ϕ j ( z ) is nonzero and ∇ ϕ j ( z ) and ∇ ϕ i ( z ) are linearly independentfor every pair of distinct j and i . Hence we conclude that around z each Γ j and Γ j,i := Γ j ∩ Γ i are smooth real analytic submanifolds ofΩ of codim 1 and 2 respectively. Furthermore, again thanks to thenondegeneracy assumption, the tangent spaces of Γ j ’s at z constitutea central hyperplane arrangement of coordinate planes. Therefore, theintersection of any collection consisting of n of Γ j ’s is a smooth realanalytic submanifold around z as well, of codim n .Now let us investigate the process more closely in case ϕ is a realanalytic Morse function. The point q k of type m k which the processconverges to at k -th step lies in the codim-2 (in general codim- d ) sub-manifold Γ m k − , m k ; in other words, k -th step converges to a point inΓ m k − , m k ∩ R m k q k . At that point G is perpendicular to the slice. Wemay view the process as a chase within the subspaces Γ m ’s, which aresubmanifolds of codim 2 near the critical points. So, for example, in aneighborhood of some q ∈ Ω if Γ m ∩ ϕ − ( R ≥ ϕ ( q ) ) is empty for some m then the process cannot converge to q .Now we are ready for the claims and proofs. In the first two state-ments, we assume that ϕ is a real analytic function, not necessarilyMorse. Observe that the process is still well-defined in the absence ofbeing Morse. Lemma 3.
Let ϕ : R N → R be a real analytic function. Considerthe sequence ( q k ) ∞ k =0 of stationary points of the process described above.If q ∈ R N is an accumulation point of the sequence ( q k ) then it is acritical point of ϕ . H. I¸sıl Bozma and Ferit ¨Ozt¨urk
Proof.
Suppose that ( q k ) is not ultimately constant so that every neigh-borhood of q contains infinitely many distinct q k ’s. Either every neigh-borhood of q contains infinitely many q k ’s of type m for every m =1 , . . . , N or not. In the former case by the continuity of G , we musthave G ( q ) = 0 which proves the lemma. Now we prove by contradictionthat the latter case is not possible. Indeed, suppose there is an openball U centered at q which does not contain any point of the sequenceof, say, type 1 and contains infinitely many points q k i − of type N .We assumed that each curve γ k i , starting from the point q k i − , leaves U and converges to the point q k i (cid:54)∈ ¯ U . For any given (cid:15) >
0, there is asufficiently large K such that if i > K we have(2) (cid:15) > ϕ ( q k i − ) − ϕ ( q k i ) > (cid:90) γ ki ∩ U | G k i | >
12 diam( U ) · avg( | G k i | )where avg( | G k i | ) is the average of | G k i | over γ k i ∩ U . Let A = (cid:83) i γ k i and γ = (closure( A ) − A ) ∩ U . The set γ can be considered as the setof limit points in U of all point sequences ( p l ) with p l ∈ γ k l , ( l ) beingany increasing sequence of positive integers. Observe that γ lies in theintersection of the type-1 slice R q and the ϕ ( q )-level set. Moreover itfollows from (2) that G q is zero on γ , i.e. all points of γ are criticalpoints of G q . (Loosely speaking γ is the limit of the arcs γ k i ∩ ¯ U howeverwe do not claim and do not need that γ is a real analytic arc.)Now since ¯ γ ⊂ ¯ U is compact and all its points are critical points of G q and ϕ (¯ γ ) = ϕ ( q ), then one can choose a (cid:32)Lojasiewicz neighborhood W ⊂ R q ∩ U of γ such that, assuming ϕ ( q ) = 0, there are c > µ ∈ [0 ,
1) satisfying | G q ( x ) | > c | ϕ ( x ) | µ , for all x ∈ W . (The triple ( W, c, µ ) is found as follows: first choose a(cid:32)Lojasiewicz neighborhood ( W x , c x , µ x ) of every point x ∈ γ , then takea finite subcollection ( W l , c l , µ l ) covering ¯ γ and set W = ∪ W l ∩ U , c = min l c l and µ = max l µ l .) By the continuity of G u with respect topoint u , one can extend W to an open V ⊂ U such that(3) | G u ( x ) | ≥ c | ϕ ( x ) | µ , for all u ∈ V and x ∈ R u . The set V can be chosen smaller (as aneighborhood of γ ) such that sup V ϕ < δ where δ > U ) ≥ diam( V ) ≥ diam( γ ) ≥ diam( U ) / γ k i ∩ V ) ≥ diam( V ) / . equential gradient dynamics in real analytic Morse systems 7 for large i ’s. On the other hand using (3) we observe, as in the proofof (cid:32)Lojasiewicz inequality (see e.g. [11]), that there is a finite upperbound, for sufficiently large i but independent from i , for the length of γ k i in V : (cid:90) γ ki ( t ) ∩ V (cid:54) = ∅ | ˙ γ k i | dt = (cid:90) t | ˙ γ k i ( t ) | dt ≤ c ( ϕ ( γ k i ( t ))) − µ < cδ − µ . where c = ( c (1 − µ )) − . Here t is the moment when γ k i leaves U .This upper bound can be made arbitrarily small (that holds for larger i ), in particular, much less than diam( V ). This contradicts with (4).Therefore there must be infinitely many points of type 1 in U and theproof by contradiction follows recursively. (cid:3) From the above proof, it follows that sufficiently small neighborhoodsof an accumulation point do not let trajectories out. This proves thefollowing statement.
Corollary 4.
Let ϕ : R N → R be a real analytic function and q an accumulation point of the sequence ( q k ) of stationary points of theprocess. Then lim k → + ∞ q k = q . To finish the proof of the main theorem, we need the following
Lemma 5.
Let ϕ : ˜Ω → R be a primal navigation function on Ω ⊂ ˜Ω .Then except when the initial point is a saddle point, convergence to asaddle point is impossible for a C ω -dense subset of primal navigationfunctions.Proof. Let q = 0, ϕ ( q ) = 0, s be the index of ϕ at q and ( U, f ) bea real analytic Morse neighborhood of q , i.e. f : ( R dN , → ( R dN , g . = ϕ ◦ f − : ( U, → ( R , , ( y , . . . , y dN ) (cid:55)→ − y − . . . − y s + y s +1 + . . . + y dN .Now in this new setting, −∇ g has the first s components nonnega-tive. Therefore the first s coordinates of the position cannot convergeto 0 unless they are already 0. A necessary condition for that is when q k ∈ U = U ∩ ( { } × R dN − s ) for some k >
0. Furthermore in sucha case, convergence is possible only when there are slices which con-tain integral curves in their intersection with U ; more precisely, forthe slice S with the projected vector field G S , U ∩ S contains someintegral curves of G S . We will argue that such a case does not occurfor a C ω -dense subset of primal navigation functions.First we perturb ϕ to a primal navigation function ψ with the sameminimum point p such that the local stable manifold U q of each criticalpoint q of ψ and the slices R mx (for every m and x ∈ U q sufficiently H. I¸sıl Bozma and Ferit ¨Ozt¨urk close to q ) intersect transversely. It is straightforward to see that asufficiently small generic real analytic coordinate change in a smallneighborhood of q achieves this locally. For example, a small genericrotation centered at the critical point would work. However we needfurther that this real analytic perturbation can be done globally, fix-ing p . We propose the following construction in case Ω is the closedunit ball. The construction in the general case will follow since Ω isanalytically diffeomorphic to the closed unit ball. Let o be a point inthe interior of Ω such that for each critical point q of ϕ the line oq istransverse and non-orthogonal to (cid:83) ≤ m ≤ N R mq ∩ U q , i.e. each intersec-tion R mq ∩ U q is non-tangent to oq and has a radial (i.e. oq ) component.Since these conditions are open, the set of such o ’s is open and densein Ω. Below we take o as the origin in the unit closed ball Ω. Theconstruction below can be made for the general case similarly.Now we consider the real analytic diffeomorphism h : R dN → R dN , h ( x ) = x + bx sin ar r where r is the distance | ox | and a, b ∈ R . We require a satisfy | op | = 2 πk/a for some k ∈ Z + . The radial perturbance h fixes p and moves the other critical points slightly, the extent of which canbe controlled by b . One can choose a such that the radial perturbanceof each slice at each critical point is nontrivial, which is required fortransversality. We also choose the domain ˜Ω of ϕ so that the condition(ii) (cid:48) is satisfied in ˜Ω − V where V is a compact subset of Ω. By making b small, one can have h arbitrarily C ω -close to id and h ( ˜Ω) ⊃ Ω. Thesechoices guarantee that the function ψ = ϕ ◦ h : h − (Ω) → R is definedand is a real analytic Morse function with the same minimum as ϕ ; itscritical points are in one to one correspondence with, arbitrarily closeto and of the same index as those of ϕ . Moreover the condition (ii) (cid:48) is satisfied by ψ on ∂ Ω provided that h is sufficiently C ω -close to id.Thus we obtain a primal navigation function satisfying the requiredtransversality conditions.Finally let us note that since the condition that R mq ∩ U q containsintegral curves of G mq is a closed condition, the construction abovegives the perturbation required for almost all o , infinitely many a andsufficiently small b . (cid:3) Proof of Theorem 1.
Since Ω is compact, the sequence ( q k ) hasan accumulation point, which is a critical point of ϕ by Corollary 4. Theproof follows from Lemma 5 except the last sentence of the theorem.We assert the last sentence by giving explicit examples for each case inSection 3.2 below. (cid:3) equential gradient dynamics in real analytic Morse systems 9 Proof of Theorem 2.
The proof will be immediate after modi-fying the definition of the type of a point : we say that q k has type j if G ( q k ) has the j -th component 0. Note that q k has type j only if theprevious slice of the process has type n k such that j ∈ n k .Now we start as in Lemma 3. Suppose that every neighborhoodof q contains infinitely many distinct sequence points. Either everyneighborhood of q contains infinitely many q k ’s of type j for every j = 1 , . . . , M or not. In the former case by the continuity of G , wemust have G ( q ) = 0 so that we reach the conclusion of Lemma 3. Inthe latter case suppose there is an open ball U centered at q whichcontains a subsequence ( q k i − ) of points of type, say, 1. Suppose infin-itely many n k i ’s contain the index, say, 2. Denote the correspondingpoint subsequence by ( q l i − ). Then the main body of Lemma 3 provesthat all type 2 sequence points q l i are contained in U . The sets n k i and n l i +1 corresponding to the type 1 points q k i − and type 2 points q l i respectively, either contain a different index than 1 and 2 infinitelymany times, or ultimately the indices that appear are nothing but 1 or2. In the latter case the subsequence ( q l i − , q l i ) constitute a tail of ( q k )so that this latter case presents a contradiction to the assumption ofthe theorem that all indices appear infinitely many times. Hence we areleft with the former case and the claim of Lemma 3 follows recursivelyas before.Proof of Lemma 5 works exactly the same. (cid:3)
3. Further discussion and examples3.1.
Convergence analysis.
Considering the proof of Lemma 3 above,more can be said about convergence to a critical point. Through amore explicit convergence analysis one can control the lengths of thetrajectories. This analysis shall reflect the nature of the process neara critical point.Let q be a critical point, ϕ ( q ) = 0. We will bound from above thetotal length of the trajectories once the sequence gets sufficiently closeto q in U . More precisely here is our claim. Proposition 6.
Let ( U = B r ( q ) , c, µ ) be a (cid:32)Lojasiewicz neighborhood of q . Suppose that for some c (cid:48) > / ( c (1 − µ )) there is a q l ∈ U satisfying (5) c (cid:48) ϕ − µ ( q l ) < r and that q i ∈ U for l ≤ i ≤ n for some n > l . Then the total length ofthe trajectories ( γ i ) l (cid:15) i -neighborhood B i = B (cid:15) i ( q i ) ∩ R iq i onthe slice R iq i (that contains the points q i − and q i and the curve γ i ), thecurve γ i satisfies the angle condition :(6) ddt ϕ ( γ i ( t )) = (cid:104)∇ ϕ ( γ i ( t )) , ˙ γ i ( t ) (cid:105) ≤ − δ |∇ ϕ ( γ i ( t )) | · | ˙ γ i ( t ) | . Such (cid:15) i ’s exist since at point q i , ∇ ϕ ( q i ) is perpendicular to the slice R iq i .Now if necessary, we make each (cid:15) i smaller so that the corresponding B i becomes a (cid:32)Lojasiewicz neighborhood for the vector field G m i q i too (here q i is of type m i ), i.e. there are c i > µ i ∈ [0 ,
1) such that for all x ∈ B i , we have(7) | G m i q i ( x ) | ≥ c i | ϕ ( x ) − ϕ ( q i ) | µ i . Set c (cid:48) i = 1 / ( c i δ (1 − µ i )).Finally let ∆ > (cid:15) i ’s further smaller, if necessary,so that for every a ∈ { µ, µ l +1 , . . . , µ n } and for every x ∈ ¯ B i :(8) | ϕ ( x ) − a − ϕ ( q i ) − a | < ∆ C ( n − l ) , where C = max { c (cid:48) i | l < i ≤ n } . This can be fullfilled since each ϕ − a is continuous. Then we have the total length of the trajectories γ i , equential gradient dynamics in real analytic Morse systems 11 ( l < i ≤ n ): L = n (cid:88) i = l +1 (cid:90) γ i | ˙ x | = n (cid:88) i = l +1 (cid:18)(cid:90) t i | ˙ x | + (cid:90) ∞ t i | ˙ x | (cid:19) , t i is when γ i enters B i the last time ≤ n (cid:88) i = l +1 c (cid:48) (cid:0) ϕ ( q i − ) − µ − ϕ ( γ i ( t i )) − µ (cid:1) using (1), (6), (7)+ n (cid:88) i = l +1 c (cid:48) i (cid:0) ϕ ( γ i ( t i )) − µ i − ϕ ( q i ) − µ i (cid:1) , and [1, Equation 2.6] ≤ c (cid:48) ϕ ( q l ) − µ − c (cid:48) ϕ ( γ n ( t n )) − µ + n − (cid:88) i = l +1 c (cid:48) (cid:0) ϕ ( q i ) − µ − ϕ ( γ i ( t i )) − µ (cid:1) (these terms are negative)+ n (cid:88) i = l +1 c (cid:48) i (cid:0) ϕ ( γ i ( t i )) − µ i − ϕ ( q i ) − µ i (cid:1) < c (cid:48) ϕ ( q l ) − µ other terms are negative+ n (cid:88) i = l +1 c (cid:48) i ∆ C ( n − l ) , (8) < r + ∆ , (5) and c (cid:48) i ≤ C .Since ∆ was arbitrary, we deduce L ≤ r . (cid:3) We close this section with a consequence of the proposition.
Corollary 7.
With the hypotheses of Proposition 6 suppose q i ∈ U forall i ≥ l (so that lim q i = q ). Then the total length of the trajectories ( γ i ) i>l is less than r . Examples.
A case where the process hits a critical point q aftera finite number of steps is easy to cook up locally (see Figure 1(a)). Wehave seen that this case is not generic. In fact Figure 1(b) illustrates asmall generic real analytic local perturbation of (a) where there is noinitial q for which the process hits 0.Similarly, Figure 2 depicts a situation where convergence to a saddlepoint is provided by a set of initial points of positive area. Note thatthis case occurs only if a slice lies in an stable manifold and the slicecontains an integral curve of the projected gradient system.One might ask to herself if infinitely many steps is ever requiredfor convergence to a critical point. The answer is affirmative and anexample for a minimum point is depicted in Figure 3. q q Figure 1. Integral curves of a function. (a)The processhits q after finitely many steps; (b) A small perturbationspoils the convergence; by definition the process startswith a type-1 step.Figure 2. Level curves of a function. The initial pointsin the shaded region start a process converging to thesaddle point.It is not difficult either to imagine a case where infinitely many stepsis needed for convergence to a saddle point. Let d = 1, N = 3, s = 1, b = 0 and c = 2, and the type-1, -2, -3 slices be lines parallel to x -, y -and z -axes respectively. We consider the Morse function f ( x, y, z ) =( x − z ) + 2( x + z ) − y (3 x + z ) which has a unique critical point at0. Observe that the curve β = Γ ∩ Γ lies in 0 × R . We start from aninitial point q ∈ (0 × R ) − α , where α = Λ ∩ (0 × R ). The type-1steps terminate at points on the curve α , which are critical points forthe type-2 steps as well. Thus the process makes zigzags between α and β via type-1 and type-3 steps. The behaviour on 0 × R is exactlythe same as in Figure 3. equential gradient dynamics in real analytic Morse systems 13 q q q q Γ Γ Figure 3. a ( x + y ) + b ( x − y ) = 1 , a (cid:54) = b . Infinite stepsneeded for convergence to 0.References [1] Absil P. A.; Mahony R.; Andrews B. Convergence of iterates of descent meth-ods for analytic cost functions. SIAM J. Optim. Vol. 16, No. 2, pp. 531-547,2005.[2] Baryshnikov, Y.; Bubenik, P.; Kahle, M. Min-type Morse theory for configura-tion spaces of hard spheres. Int. Math. Res. Not. IMRN 2014, no. 9, 25772592.[3] Bozma, H.I., D.E.Koditschek. Assembly as a Noncooperative Game of itsPieces: Analysis of 1D Sphere Assemblies, Robotica, pp: 93-108, Jan 2001.[4] Farber, M. Topological complexity of motion planning. Discrete and Compu-tational Geometry,Vol. 29, No. 2, 211-221, 2003[5] Cohen, D. C.; Farber, M. Topological complexity of collision-free motion plan-ning on surfaces. Compos. Math. 147 (2011), no. 2, 649-660.[6] (cid:32)Lojasiewicz, S. Une propri´et´e topologique des sous-ensembles analytiques r´eels.Colloques Internationaux du C.N.R.S. no 117, les equations aux d´eriv´ees par-tielles, Paris 25 – 30 juin (1962), p. 87-89[7] (cid:32)Lojasiewicz, S. Sur les trajectoires du gradient d’une fonction analytique.Geometry seminars, 1982-1983 (Bologna, 1982/1983), 115–117, Univ. Stud.Bologna, Bologna, 1984.[8] Karag¨oz C. S.; Bozma H. I.; Koditschek Daniel E. Coordinated Navigationof Multiple Independent Disk-Shaped Robots. IEEE Trans. on Robotics 30(2014), no. 6, 1289–1304.[9] Koditschek, D. E.; Rimon, E. Robot navigation functions on manifolds withboundary. Adv. in Appl. Math. 11 (1990), no. 4, 412–442.[10] Rimon, E.; Koditschek, D. E. The construction of analytic diffeomorphisms forexact robot navigation on star worlds. Trans. Amer. Math. Soc. 327 (1991),no. 1, 71–116.[11] Kurdyka, K.; Mostowski, T.; Parusi´nski, A. Proof of the gradient conjectureof R. Thom. Ann. of Math. (2) 152 (2000), no. 3, 763–792.[12] Latombe, J.-C. Robot motion planning (Kluwer, Dordrecht, 1991). [13] Milnor, J. Morse theory. Based on lecture notes by M. Spivak and R. Wells.Annals of Mathematics Studies, No. 51 Princeton University Press, Princeton,N.J. 1963.Bo˘gazi¸ci ¨Universitesi, Department of Electrical and Electronics Engineering,˙Istanbul, Turkey E-mail address : [email protected] Bo˘gazi¸ci ¨Universitesi, Department of Mathematics, ˙Istanbul, Turkey
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