aa r X i v : . [ m a t h . D S ] O c t Distributed delays stabilize negative feedback loops
Samuel Bernard ∗ November 20, 2018
Abstract
Linear scalar differential equations with distributed delays appear in the study of the localstability of nonlinear differential equations with feedback, which are common in biology andphysics. Negative feedback loops tend to promote oscillation around steady states, and theirstability depends on the particular shape of the delay distribution. Since in applications themean delay is often the only reliable information available about the distribution, it is desirableto find conditions for stability that are independent from the shape of the distribution. Weshow here that the linear equation with distributed delays is asymptotically stable if theassociated differential equation with a discrete delay of the same mean is asymptotically stable.Therefore, distributed delays stabilize negative feedback loops.
The delayed feedback system of the form˙ x = F (cid:16) x, Z ∞ [ dη ( τ )] · g (cid:0) x ( t − τ ) , τ (cid:1)(cid:17) , (1)is a model paradigm in biology and physics [1, 3, 11, 17, 19, 20]. The first argument is theinstantaneous part and the second one, the delayed or retarded part, which forms a feedback loop.The function η is a cumulative distribution of delays and F and g are nonlinear functions satisfying F (0 ,
0) = 0 and g (0 , τ ) = 0. When F : R d × R d × d → R d and g : R d × R → R d × d are smoothfunctions, the stability of x = 0 is given by the linearized form,˙ x = − Ax − Z ∞ [ B ( τ ) · dη ( τ )] x ( t − τ ) . (2)The coefficients A and B ( τ ) ∈ R d × d are the Jacobian matrices of the instantaneous and the delayedparts, η : [0 , ∞ ) → R d × d is the distribution of delays and ( · ) is the pointwise matrix multiplication.In biological applications, discrete delays in the feedback loop are often used to account for thefinite time required to perform essential steps before x ( t ) is affected. This includes maturation andgrowth times needed to reach reproductive age in a population [13, 16], signal propagation alongneuronal axons [9], and post-translational protein modifications [7, 19]. Introduction of a discretedelay can generate complex dynamics, from limit cycles to chaos [21]. Linear stability propertiesof scalar delayed equations are fairly well characterized. However, lumping intermediate steps intoa delayed term can produce broad and atypical delay distributions, and it is not clear how thataffects the stability compared to a discrete delay [10].Here, we study the stability of the zero solution of a scalar ( d = 1) differential equation withdistributed delays, ˙ x = − ax − b Z ∞ x ( t − τ ) dη ( τ ) . (3) ∗ Universit´e de Lyon; Universit´e Lyon 1; INSA de Lyon, F-69621; Ecole Centrale de Lyon; CNRS,UMR5208, Institut Camille Jordan, 43 blvd du 11 novembre 1918, F-69622 Villeurbanne-Cedex, France, [email protected] x ( t ) ∈ R is the deviation from the zero steady state of equation (1). Coefficients a = − D F (0 , ∈ R and b = − D F (0 , = 0, and the integral is taken in the Riemann-Stieltjessense. We assume that η is a cumulative probability distribution function, i.e. η : R → [0 ,
1] isnondecreasing, piecewise continuous to the left, η ( τ ) = 0 for τ < η (+ ∞ ) = 1. Additionally,we assume that there exists ν > Z ∞ e ντ dη ( τ ) < ∞ . (4)This last condition implies that the mean delay value is finite, E = Z ∞ τ dη ( τ ) < ∞ . The corresponding probability density function is f ( τ ) given by dη ( τ ) = f ( τ ) dτ , where the deriva-tive is taken in the generalized sense. The distribution can be continuous, discrete, or a mixture ofcontinuous and discrete elements. When it is a single discrete delay (a Dirac mass), the asymptoticstability of the zero solution of equation (3) is fully determined by the following theorem, due toHayes [12], Theorem 1.
Let f ( τ ) = δ ( τ − E ) a Dirac mass at E . The trivial solution of equation (3) isasymptotically stable if and only if a > | b | , or if b > | a | and E < arccos( − a/b ) √ b − a . (5)There is a Hopf point if the characteristic equation of equation (3) has a pair of imaginaryroots and all other roots are negative. For a discrete delay, the Hopf point occurs when equalityin (5) is satisfied. Moreover, for any distribution η , there is a zero root along the line − a = b . At − a = b = 1 /E , there is a double zero root. When a > − /E , all other roots have negative realparts, but when a < − /E , there is one positive real root. Thus, the stability depends on η if andonly if b > | a | . Moreover, only a Hopf point can occur when b > | a | . Therefore, a distributionof delays can only destabilize equation (3) through a Hopf point, and only when b > | a | . This isa common situation when the feedback acts negatively on the system (D F (0 , <
0) to causeoscillations.Assuming b > t → bt , we have a → a/b , b → η ( τ ) → η ( bτ ). Equation (3) can be rewritten as˙ x = − ax − Z ∞ x ( t − τ ) dη ( τ ) . (6)The aim of this paper is to study the effect of delay distributions on the stability of the trivialsolution of equation (6), therefore, we focus on the region | a | <
1. To emphasize the relationbetween the stability and the delay distribution, we will say that η (or f ) is stable if the trivialsolution of equation (6) is stable, and that η (or f ) is unstable if the trivial solution is unstable.It has been conjectured that among distributions with a given mean E , the discrete delay isthe least stable one [4, 6]. If this were true, according to Theorem 1, all distributions would bestable provided that E < arccos( − a ) √ − a . (7)This conjecture has been proved for a = 0 using Lyapunov-Razumikhin functions [15], and fordistributions that are symmetric about their means [ f ( E − τ ) = f ( E + τ )] [4, 6, 14, 18]. It hasbeen observed that in general, a greater relative variance provides a greater stability, a propertylinked to geometrical features of the delay distribution [2]. There are, however, counter-examplesto this principle, and there is no proof that for a = 0 the least stable distribution is the singlediscrete delay. It is possible to lump the non-delayed term into the delay distribution using thecondition found in [15], but the resulting stability condition, E/ (1 + a ) < π/
2, is not optimal.2ere, we show that if inequality (7) holds, all distributions are asymptotically stable. That is,distributed delays stabilize negative feedback loops.In section 2, we set the stage for the main stability results. In section 3, we show the stabilityfor distributions of discrete delays. In section 4, we present the generalization to any distributionsand in section 5, we provide illustrative examples.
Let η be a distribution with mean 1. We consider the family of distributions η E ( τ ) = ( η ( τ /E ) , E > ,H ( τ ) , E = 0 . (8)where H ( τ ) is the step or heaviside function at 0. The distribution η E has a mean E ≥
0. Thecharacteristic equation of equation (6), obtained by making the ansatz x ( t ) = exp( − λt ), is λ + a + Z ∞ e − λτ dη E ( τ ) = 0 . (9)When condition (4) is satisfied, the distribution η E is asymptotically stable if and only if all rootsof the characteristic equation have a negative real part Re( λ ) < E = 0, i.e. when there is nodelay, there is only one root, λ <
0. When
E >
0, the characteristic equation has pure imaginaryroots λ = ± iω only if 0 < ω < ω c = √ − a . Thus, the search for the boundary of stability canbe restricted to imaginary parts ω ∈ (0 , ω c ] [6].We define C ( ω ) = Z ∞ cos( ωτ ) dη E ( τ ) , (10) S ( ω ) = Z ∞ sin( ωτ ) dη E ( τ ) . (11)We use a geometric argument to bound the roots of the characteric equation of equation (6) bythe roots of the characteristic equation with a discrete delay. More precisely, if the leading rootsassociated to the discrete delay are a pair of imaginary roots, then all the roots associated to thedistribution of delays have negative real parts. We first state a criterion for stability: if S ( ω ) < ω whenever C ( ω ) + a = 0, then f is stable. The larger the value of S ( ω ), the more “unstable” thedistribution is. We then show that a distribution of n discrete delays f n is more stable than ancertain distribution with two delays f ∗ , i.e. S n ( ω ) ≤ S ∗ ( ω ). We construct f ∗ and determine thatone of the delays of this “most unstable” distribution f ∗ is τ ∗ = 0, making it easy to determineits stability using Theorem 1. We then generalize for any distribution of delays.The next proposition provides a necessary condition for instability. It is a direct consequenceof theorem 2.19 in [22]. We give a short proof for completeness. Proposition 1.
If the distribution η E is asymptotically unstable, then there exists ω s ∈ [0 , ω c ] such that C ( ω s ) + a = 0 and S ( ω s ) ≥ ω s .Proof. Suppose that the distribution η E is asymptotically unstable. The roots of the characteristicequation depend continuously on the parameter E and cannot appear in the right half complexplane. Thus there is a critical value 0 < ρ < η ρE loses its stability, and this happens whenthe characteristic equation (9) has a pair of imaginary roots λ = ± iω with 0 ≤ ω < ω c = √ − a ,i.e. through a Hopf point. Splitting the characteristic equation in real and imaginary parts, wehave Z ∞ cos( ωτ ) dη ρE ( τ ) + a = 0 , Z ∞ sin( ωτ ) dη ρE ( τ ) = ω. η E , we obtain Z ∞ cos( ωρτ ) dη E ( τ ) + a = 0 , Z ∞ sin( ωρτ ) dη E ( τ ) = ω. Finally, denoting ω s = ρω ≤ ω < ω c , we have Z ∞ cos( ω s τ ) dη E ( τ ) + a = 0 , Z ∞ sin( ω s τ ) dη E ( τ ) = ω = ω s /ρ ≥ ω s . This completes the proof.Proposition 1 provides a sufficient condition for stability:
Corollary 1.
The distribution η E is asymptotically stable if (i) C ( ω ) > − a for all ω ∈ [0 , ω c ] orif (ii) C ( ω ) = − a , ω ∈ [0 , ω c ] , implies that S ( ω ) < ω . Proposition 1 suggests that the scaling η E = η ( τ /E ) is appropriate for looking at the stabilitywith respect to the mean delay. The mean delay scales linearly, and unstable distributions thereforelose their stability at a smaller values of the mean delay, under this scaling. The condition S ( ω s ) <ω s is however not necesssary for stability, as one can find cases where S ( ω s ) > ω s even though thedistribution is stable. This happens when an unstable distribution switches back to stability as E is further increased (see for instance [8] or [5] and example 5.3). We define a density of n discrete delays τ i ≥
0, and p i > i = 1 , ..., n , n ≥
1, as f n ( τ ) = n X i =1 p i δ ( τ − τ i ) (12)where δ ( t − τ i ) is a Dirac mass at τ i , and n X i =1 p i τ i = E, and n X i =1 p i = 1 . In this section, we show that f n is more stable than a single discrete delay. We do that by observingthat among all n -delay distributions, n ≥
2, that satisfy C n ( ω s ) + a = 0 for a fixed value ω s < ω c ,the distribution f ∗ that maximizes S n ( ω s ),max f n (cid:8) S n ( ω s ) | C n ( ω s ) + a = 0 (cid:9) = S ∗ ( ω s ) , (13)has 2 delays. We show that S ∗ ( ω s ) < ω s , implying that all distributions are stable. The followinglemma shows how to maximize S ( ω s ) for distributions of two delays. Lemma 1.
Let f be a delay density with mean E . Assume in addition that there exists ω s ∈ (0 , ω c ) such that C ( ω s ) = − a < cos( ω c E ) . Then there exists τ ∗ , p ∗ and p ∗ such that τ ∗ = 0 , (14) p ∗ + p ∗ = 1 , (15) p ∗ + p ∗ cos( ω s τ ∗ ) = p cos( ω s τ ) + p cos( ω s τ ) , (16) p ∗ τ ∗ = E. (17)4 oreover, there is at most two solutions for τ ∗ with τ ∗ < π/ω s . If τ ∗ is the smallest solution, wehave that τ ∗ ≤ τ and S ∗ ( ω s ) ≡ X i =1 p ∗ i sin( ω s τ ∗ i ) ≥ S ( ω s ) . Proof.
To see that there is always a solution, let c > θ ) ≥ − cθ is verified for all θ . [ c = 0 . ... by solving c = sin( θ ), with 1 − θ sin( θ ) = cos( θ ).]We have that 1 − cωE ≤ C ( ω ) ≤ cos( ωE ). Thus, the line 1 − dωE that goes through C ( ω s ) satisfies d = (1 − C ( ω s )) / ( ω s E ) ≤ c , and therefore crosses the curve cos( ω s E ) at some points. The smallestsolution τ ∗ is the one such that 1 − dω s τ ∗ = cos( ω s τ ∗ ). This way, p ∗ + p ∗ cos( ω s τ ∗ ) = 1 − (1 − C ( ω s )) p ∗ τ ∗ /E, = C ( ω s ) . These new delay values maximize S ( ω s ) under the constraints that C ( ω s ) + a = 0 and that themean remains E . That is, we will prove that S ∗ ( ω s ) ≡ X i =1 p ∗ i sin( ω s τ ∗ i ) ≥ X i =1 p i sin( ω s τ i ) , for all admissible p i , τ i . Two show that, we recast the problem in a slightly different way. Writing u = ω s τ , v = ω s τ and T = ω s E , we can express parameters p i in terms of ( u, v ): p = v − Tv − u and p = T − uv − u , where u < T < v . We consider C and S as functions of ( u, v ). C ( u, v ) = v − Tv − u cos( u ) + T − uv − u cos( v ) , (18) S ( u, v ) = v − Tv − u sin( u ) + T − uv − u sin( v ) . (19)Equation (19) is to be maximized for ( u, v ) along the curve h = { u, v } implicitly defined bythe level curves C ( u, v ) = − a . There are either two solutions in v , including multiplicity, ofthe equation C (0 , v ) = − a or none, so the curve can be parametrized in a way that ( u ( ξ ) , v ( ξ ))satisfies ( u (0) = 0 , v (0) = v max ) and ( u (1) = 0 , v (1) = v min ), with v min ≤ v max . We claim that S is maximized for ξ = 1, i.e. u = 0 and v = v max . This is true only if S ( u ( ξ ) , v ( ξ )) is increasingwith ξ . That is, the curve h must cross the level curves of S upward. It is clear that S is adecreasing function of v , for u fixed and an increasing function of u , for v fixed. Thus the levelcurves S ( u, v ) = k can be expressed as an increasing function v S,k ( u ) such that S ( u, v S,k ( u )) = k, when k is in the image of S . Likewise, equation (18) can be solved locally to yield v C,a ( u ) suchthat C ( u, v C,a ( u )) = − a, whenever − a is in the image of C . The function v C,a ( u ) could take two values on the domain ofdefinition. Because S is decreasing in v , we choose the lower solution branch for v C,a ( u ). If, alongthat lower branch, the slope of v C,a ( u ) is larger than that of v S,k ( u ), then as v decreases along thecurve c , S increases. Therefore, to show that (0 , v C,a (0) = v min ) maximizes S , we need to showthat dv C,a ( u ) du > dv S,k ( u ) du > . (20)5 a u vT v (cid:1) S (cid:2) Figure 1: How delays are replaced to get an maximal value of S ∗ . In this example, a = − .
1. Parameters are u = 0 . v = 2, p = 0 . p = 0 .
63 and T = 1 .
33. The parameters maximizing S are u ∗ = 0, v ∗ = 1 . p ∗ = 0 . p ∗ = 0 . It is clear that dv S,k ( u ) /du >
0. The pointwise derivatives of the level curves at ( u, v ) are dv C ( u ) du = v − TT − u − cos( u ) + cos( v ) + ( v − u ) sin( u )cos( u ) − cos( v ) − ( v − u ) sin( v ) ,dv S ( u ) du = v − TT − u − sin( u ) + sin( v ) − ( v − u ) cos( u )sin( u ) − sin( v ) + ( v − u ) cos( v ) . Because only the lower branch of v C is considered, we restrict ( u, v ) where dv C ( u ) /du < + ∞ . Thisis done without loss of generality since S is strictly larger on the lower branch than on the upperbranch. Along the lower branch, v C ( u ) < π . Inequality (20) then holds if( v − u ) (cid:2) − (cid:0) cos( u ) cos( v ) + sin( u ) sin( v ) (cid:1) + ( v − u ) (cid:0) sin( u ) cos( v ) − cos( u ) sin( v ) (cid:1)(cid:3) > . Notice that this inequality does not depend on T , which cancels out, nor on a , since comparisonis made pointwise, for any level curves. The inequality can be simplified and rewritten in terms of z = v − u > z (cid:2) − z ) − z sin( z ) (cid:3) > . It can be verified that this inequality is satisfied for z ∈ (0 , π ]. Therefore, S is maximized when u = 0 and v = v C,a (0) = v min . Theorem 2.
Let f n be a density with n ≥ discrete delays and mean E satisfying inequality (7).The density f n is asymptotically stable.Proof. Single delay distributions ( n = 1) are asymptotically stable by Theorem 1. We first showthe case n = 2.Consider a density f , with τ < τ . Suppose C ( ω s ) + a = 0 for a value of ω s < ω c (ifnot, Corollary 1 states that f is stable). Remark that − a = C ( ω s ) < cos( ω s E ). Indeed, frominequality (7) and ω s ≤ ω c = √ − a , we have cos( ω s E ) ≥ cos( ω c E ) > − a . Replace the two6elays by two new delays with new weights: τ ∗ = 0 and τ ∗ ≥ p ∗ τ ∗ = p τ + p τ , (21) p ∗ + p ∗ cos( ω s τ ∗ ) = p cos( ω s τ ) + p cos( ω s τ ) , (22) p ∗ + p ∗ = p + p (= 1) . (23)Lemma 1 ensures that there always exists a solution when C ( ω s ) ≤ cos( ω s E ). Additionally, τ ∗ ≤ τ and S ∗ ( ω s ) ≡ X i =1 p ∗ i sin( ω s τ ∗ i ) ≥ X i =1 p i sin( ω s τ i ) . That is, the new distribution ∗ maximizes the value of S . Therefore, if we are able to showthat distributions with a zero and a nonzero delay satisfy S ( ω s ) < ω s , then by Corollary 1, alldistributions with two delays are stable. Consider f ( τ ) = (1 − p ) δ ( τ ) + pδ ( τ − r ). Suppose thatthere is ω s ≤ ω c such that C ( ω s ) = 1 − p + p cos( ω s r ) = − a. We must show that S ( ω s ) = p sin( ω s ) < ω s . Summing up the squares of the cosine and the sine,we obtain p = ( − a + p − + S ( ω s ) , so S ( ω s ) = p p − ( − a + p − . By assumption, the mean delay statisfies inequality (7), pr < arccos( − a ) √ − a . Thus, ω s = arccos (cid:0) − ( a + 1 − p ) p − (cid:1) r > p p − a arccos (cid:0) − ( a + 1 − p ) p − (cid:1) arccos( − a ) . Because ( a + 1 − p ) /p ≥ a for p ∈ (0 ,
1] and a ∈ ( − , − a ) √ − a ≤ arccos (cid:0) − ( a + 1 − p ) p − (cid:1)q − (cid:0) ( a + 1 − p ) p − (cid:1) . Thus, S ( ω s ) = p p − ( − a + p − ≤ p p − a arccos (cid:0) − ( a + 1 − p ) p − (cid:1) arccos( − a ) < ω s . This completes the proof for the case n = 2.For distributions f n with n > C ( ω s ) constant and increases S ( ω s ), assuming that C ( ω s ) + a = 0. This requires two steps.In the first one, all pairs of delays τ i < τ j for which the inequality X k ∈{ i,j } p k cos( ω s τ k ) ≤ cos (cid:18) ω s X k ∈{ i,j } p k τ k (cid:19) , (24)holds are iteratively replaced by new delays τ ∗ i = 0 and τ ∗ j < τ j , as done in Lemma 1. Thistransformation preserves E , C ( ω s ) and increases S ( ω s ). This is repeated until there remains m < n delays with τ i > i = 2 , ..., m such that X k ∈{ i,j } p k cos( ω s τ k ) > cos (cid:18) ω s X k ∈{ i,j } p k τ k (cid:19) , rccos( (cid:3) a/b) (cid:4) b (cid:5) a stableconditionally stable stableunstable( (cid:6) >0) ( (cid:7) (cid:8) =0 (cid:9) =0 (cid:10) >0 (cid:11) =0 (cid:12) <0 (cid:13) E exp0.6 (cid:14) +0.4 (cid:15) r -1.25 -1 -0.75 -0.5 -0.25 0.25 0.5 0.75 1 1.252.557.510 a/bbE Figure 2: Stability chart of distributions of delay in the ( a/b, bE ) plane. The distribution-independent stabilityregion is to the right of the blue curve. The distribution-dependent stability region is the shaded area. All stabilitycurves leave from the point ( a = − b, E = 1 /b ). The signs of the real roots of the characteristic equation λ , λ along a = − b are distribution-independent. for i = j ∈ { , ..., m } , and τ = 0. (The τ i are not the same as in the original distribution, the ∗ have been dropped for ease of reading.) The positive delays τ i > m X i =2 p i cos( ω s τ i ) > cos (cid:16) ω s m X i =2 p i τ i (cid:17) . while, by assumption, m X i =1 p i cos( ω s τ i ) = − a < cos( ω s E ) . The second step is to replace all delays τ i , i = 2 , ..., m with the single delay ¯ τ = P mi =2 p i τ i .We now have a 2-delay distribution with ¯ τ = 0 and ¯ τ >
0, ¯ p ¯ τ + ¯ p ¯ τ = E , ¯ C ( ω s ) ≤ C ( ω s ) and¯ S ( ω s ) ≥ S ( ω s ). Replace ¯ τ by the delay τ ∗ < ¯ τ so that C ∗ ( ω s ) = − a , while keeping E constant.Existence of τ ∗ is shown using the notation from the proof of Lemma 1, and noting that C (0 , v )and S (0 , v ) are both decreasing in v . This change of delay has the effect of increasing S : S ∗ ( ω s ) ≥ ¯ S ( ω s ). Therefore, we have found a pair of discrete delays (0 , τ ∗ ) such that C ∗ ( ω s ) = C ( ω s ) and ω s > S ∗ ( ω s ) ≥ S ( ω s ). By Corollary 1, f n is asymptotically stable. From the stability of distributions of discrete delays to the stability of general distributions ofdelays, there is a small step. First we need to bound the roots of the characteristic equation forgeneral distributed delays.
Lemma 2.
Let η E be a delay distribution with mean E satisfying inequality (7). There existsa sequence { η n,E } n ≥ with distribution η n,E having n delays, such that η n,E converges weakly to η E . Then λ is a root of the characteristic equation if and only if there exists a sequence of roots λ n for η n,E such that lim n →∞ λ n = λ . Let { µ n } n ≥ be a sequence of real parts of roots of thecharacteristic equations. Additionally, lim sup n →∞ µ n = µ < . roof. Consider λ n = µ n + iω n a root the characterisitic equation for η n,E . E satisfies inequality(7), so µ n <
0. So (cid:12)(cid:12)(cid:12) λ n + a + Z ∞ e − λ n τ dη E ( τ ) (cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12) λ n + a + Z ∞ e − λ n τ d [ η E ( τ ) − η n,E ( τ )] + Z ∞ e − λ n τ dη n,E ( τ ) (cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)Z ∞ e − λ n τ d [ η E ( τ ) − η n,E ( τ )] (cid:12)(cid:12)(cid:12) → , as n → ∞ by weak convergence. Thus any converging sub-sequence of roots converges to a rootfor η E . The same way, if λ is a root for η E , (cid:12)(cid:12)(cid:12) λ + a + Z ∞ e − λτ dη n,E ( τ ) (cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12) λ + a + Z ∞ e − λτ d [ η n,E ( τ ) − η E ( τ )] + Z ∞ e − λτ dη E ( τ ) (cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)Z ∞ e − λτ d [ η n,E ( τ ) − η E ( τ )] (cid:12)(cid:12)(cid:12) → , as n → ∞ . Convergence is guarantedd by inequality (4). Therefore, each root λ lies close to acorresponding root λ n .Denote µ = lim sup n →∞ µ n . Then µ is the real part of a root of the characteristic equationassociated to η E . µ n < n , so µ ≤
0. Suppose µ = 0. Without loss of generality, we canassume that all other roots have negative real parts. Then η E is at a Hopf point, i.e. the leadingroots of the charateristic equation are pure imaginary. Consider the distribution η ¯ a,ρ ( τ ) = η ( τ /ρ )and the associated real parts µ ¯ a,ρ , where the subscript a is there to emphasize the dependence ofthe stability on a . Then, by continuity, there exists (¯ a, ρ ) in the neighborhood ε > a, E ) forwhich η ¯ a,ρ is unstable, i.e. µ ¯ a,ρ >
0. For sufficiently small ε >
0, inequality (7) is still satisfied: ρ < arccos( − ¯ a ) √ − ¯ a . Additionally, η n,ρ converges weakly to η ¯ a,ρ . However, because η ¯ a,ρ is unstable, there exists N > η n, ¯ a,ρ is unstable for all n > N , a contradiction to Theorem 2. Therefore µ < Theorem 3.
Let η E be a delay distribution with mean E satisfying inequality (7). The distribution η E is asymptotically stable.Proof. Consider the sequence of distributions with n delays { η n,E } n ≥ where η n,E converges weaklyto η E . By Lemma 2, the leading roots of the characteristic equation of η E have negative real parts.Therefore η E is asymptotically stable.Is there a result similar to Theorem 3 for the most stable distribution? That is, is there amean delay value such that all distributions having a larger mean are unstable? When a ≥ a ≥
0. Anderson has shown that all distributions with smooth enoughconvex density functions are unconditionally stable [2], but densities do not need to be convex tobe unconditionally stable. For example, the non-convex density f ( τ ) = 0 . δ ( τ ) + δ ( τ − E )] hasmean E but is unconditionally stable. However, no distribution is unconditionally stable for allvalues of a ∈ [ − , a ≥ a ∗ with a ∗ > − orollary 2. The zero solution of equation (3) is asymptotically stable if a > − b and a ≥ | b | orif b > | a | and E < arccos( − a/b ) √ b − a . The zero solution of equation 3 may be asymptotically stable (depending on the particular distribu-tion) if b > | a | and E ≥ arccos( − a/b ) √ b − a . The zero solution of equation (3) is unstable if a ≤ − b . The exact boundary of the stability region in the ( a, E ) plane can be calculated by parametrizing (cid:0) a ( u ) , E ( u ) (cid:1) . Consider the distribution η . Then, at the boundary of stability,0 = iω + a + Z ∞ e − iωτ dη ( τ /E ) , = iω + a + Z ∞ e − iωEτ dη ( τ ) , setting u = Eτ , = i uE + a + Z ∞ e − iuτ dη ( τ ) . Separating the imaginary and the real part, we obtain a ( u ) = − C ( u ) and E ( u ) = uS ( u ) , (25)for u ≥
0. The fact that u depends on E is not a problem: u → ∞ if and only if E → ∞ , and u → E →
0. Equations (25) allows systematic exploration of the boundary ofstability in the ( a, E ) plane.
The exponential distribution f ( τ ) = e − τ has normalized mean 1, and C ( u ) = 11 + u and S ( u ) = u u . The stability boundary is given by E = − /a , for − ≤ a <
0. Therefore the exponentialdistribution is not unconditionally stable for a < The exponential distribution is also not the most stable distribution. The density with a zero anda positive delay is f ( τ ) = (1 − p ) δ ( τ ) + pδ ( τ − r ), p ∈ (0 , E = pr = arccos (cid:0) − ( a + 1 − p ) p − (cid:1)q − (cid:0) ( a + 1 − p ) p − (cid:1) This has an asymptote at a = 2 p −
1, which can be located anywhere in ( − , a E −1 −0.8 −0.6 −0.4 −0.2 0 0.210 a E unstable stable Figure 3: (
Left ) Stability chart of the three-delay distribution with τ = 16 τ , τ = 96 τ , p = 0 . p = 0 . p = 0 .
1. (
Right ) Stability chart of the second order gamma distribution, equation (26).
In general, for a distribution with n delays, a ( u ) = − n X i =1 p i cos( uτ i ) and E ( u ) = u P ni =1 p i sin( uτ i ) . The boundary of the stability region can be formed of many branches, as with a distribution withthree delays in figure 3.
As the mean E is increased, distributions can revert to stability. This is the case with the secondorder gamma distribution (also called strong kernel) with normalized mean 1, f ( τ ) = 2 τ e − τ . (26)We have C ( u ) = 1 − u (cid:0) u (cid:1) , and S ( u ) = 2 u (cid:0) u (cid:1) , The boundary of stability is given by (cid:0) a ( u ) , E ( u ) (cid:1) = (cid:18) u − u ) , (1 + u ) (cid:19) , There is a largest value ˆ a = 0 . E , a → + . Therefore the boundary ofthe stability region is not monotonous; for a ∈ (0 , ˆ a ), f first becomes unstable and then reverts tostability as the mean is increased (figure 3). Acknowledgements
The author thanks Fabien Crauste for helpful discussion.
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