A state-dependent model with unimodal feedback Qingwen Hu Center - PowerPoint PPT Presentation

A state-dependent model with unimodal feedback Qingwen Hu Center for Nonlinear Analysis Department of Mathematical Sciences The University of Texas at Dallas Richardson, Texas October 2014, UTD . . . . . . . . . . . . . . . . . . . . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . Qingwen Hu (UTD) October 2014, UTD 1 / 33

Examples Bellman’s Questions (Book: Differential -difference equations (1963), page 80): . . Under what conditions does 1 u ′ ( t ) = au ( t − u ( t )) , have a unique solution? . . Under what conditions does 2 u ′ ( t ) = 1 2 + 1 2 u ( t ) − u ( t − u ( t )) , have a unique solution? . . . . . . . . . . . . . . . . . . . . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . Qingwen Hu (UTD) October 2014, UTD 1 / 33

Driver’s Existence and Uniqueness Results: Consider the system { y ( t ) = f ( y ( t ) , y ( t − z ( t ))) , ˙ (1) z ( t ) = g ( y ( t ) , y ( t − z ( t )) , z ( t )) ˙ where ( y ( t ) , z ( t )) ∈ R n × R . Let D be an open set in R n × R n and z 0 ≤ ¯ r be positive constants. Assume that ( i ) f : D ∋ ( θ 1 , θ 2 ) → f ( θ 1 , θ 2 ) ∈ R n and g : D × R ∋ ( θ 1 , θ 2 , θ 3 ) → g ( θ 1 , θ 2 , θ 3 ) ∈ R are continuous and each is Lipschitz continuous with respect to θ 1 ; ( ii ) g ( θ 1 , θ 2 , θ 3 ) < 1 for all ( θ 1 , θ 2 , θ 3 ) ∈ D × R ; r , 0 ] → R n with ( ϕ ( 0 ) , ϕ ( − z 0 )) ∈ D . ( iii ) There is a continuous map ϕ : [ − ¯ Then system ( 1 ) with given initial conditions z ( 0 ) = z 0 and y ( t ) = ϕ ( t ) for t ∈ [ − ¯ r , 0 ] has a unique solution on [ 0 , β ) for some β > 0. . . . . . . . . . . . . . . . . . . . . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . Qingwen Hu (UTD) October 2014, UTD 2 / 33

Rekhlitskii’s problem: (Dokl, Akad. Nauk SSSR, vol 118, 1958, 447–449) Consider the scalar equation u ′ ( t )+ λ u ( t − a ( t )) = v ( t ) , t ≥ 0 u ( t ) = ϕ ( t ) , t ≤ 0 , a ( t ) ≥ 0 . If a ( t ) → a 0 > 0 as t → + ∞ , under which conditions are all solutions of the equation bounded as t → + ∞ ? . . . . . . . . . . . . . . . . . . . . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . Qingwen Hu (UTD) October 2014, UTD 3 / 33

The model We consider attractivity of the equilibrium states of the following model { x ( t ) = − µx ( t )+ b ( x ( t − τ ( t ))) ˙ (SDDE) τ ( t ) = h ( x ( t ) , τ ( t )) , ˙ where i) µ > 0, x ( t ) ∈ R ; ii) h is C 1 (continuously differentiable); iii) The time delay τ depends on the system state ( x , τ ) and hence called a state-dependent delay. iv) b is C 2 and is unimodal on [ 0 , + ∞ ) with the following properties (H1, H2, H3): . . . . . . . . . . . . . . . . . . . . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . Qingwen Hu (UTD) October 2014, UTD 4 / 33

b b b y y = b ( ξ ) y = µ ξ ξ 0 u ξ 0 (H1) b ( 0 ) = 0 and there exists M 0 > 0 or M 0 = + ∞ such that M 0 = inf { ζ : b ( ξ ) > 0 for every ξ ∈ ( 0 , ζ ) } ; (H2) There exists a unique ξ 0 ∈ ( 0 , M 0 ) such that { b ′ ( ξ ) > 0 if 0 ≤ ξ < ξ 0 , (2) b ′ ( ξ ) < 0 if ξ > ξ 0 ; (H3) b ′′ ( ξ ) < 0 if ξ ∈ ( 0 , ξ 0 ) . . . . . . . . . . . . . . . . . . . . . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . Qingwen Hu (UTD) October 2014, UTD 5 / 33

Background of the model • Fish dynamics (Arino, Lhassan, and Bravo); • Internet conjestion model (Briat, Hjalmarsson, et al); • Counterpart model with constant delay and with different assumptions on the feedback is extensively investigated in literature; • Similar model with increasing feedback b was investigated by [Bartha 2001, Chen 2003]. . . . . . . . . . . . . . . . . . . . . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . Qingwen Hu (UTD) October 2014, UTD 6 / 33

Outline " Purposes of the work; " Existence and positivity of solutions; " Invariant intervals and attractively of solutions; " Examples; " Summary, Remarks and References. . . . . . . . . . . . . . . . . . . . . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . Qingwen Hu (UTD) October 2014, UTD 7 / 33

Purposes of the work • Invariant order intervals; • Positivity of the solutions; • Attractivity of the nonnegative stationary states; • Hopf bifurcation from nonnegative stationary states (for two examples). . . . . . . . . . . . . . . . . . . . . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . Qingwen Hu (UTD) October 2014, UTD 8 / 33

Existence and positivity of solutions We further assume that (H4) Let u be the positive stationary state of equation (SDDE) if b ′ ( 0 ) > µ . There exists a constant A with A > max { u , ξ 0 , b ( ξ 0 ) / µ } such that b ( x ) < µx for all x > A ; L (H5) There exists a constant L > 0 such that h ( x , τ ) < L + 1 for all ( x , τ ) ∈ R n + 1 ; (H6) h ( x , 0 ) > 0 for all x ∈ R . . . . . . . . . . . . . . . . . . . . . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . Qingwen Hu (UTD) October 2014, UTD 9 / 33

We choose r > 0 and let X 0 be the set of continuous real functions φ : [ − r , 0 ] → [ 0 , M 0 ] endowed with the supremum norm ∥ φ ∥ = max s ∈ [ − r , 0 ] | φ ( s ) | , where M 0 is defined in (H1). Using Driver’s work on state-dependent DEs, we obtain . Theorem . Assume that ( H1–H6 ) hold. Then for every φ ∈ X 0 and τ 0 ∈ ( 0 , r ] , there exists a unique solution ( x , τ ) : [ 0 , + ∞ ) ∋ t → ( x ( t ) , τ ( t )) ∈ [ 0 , M 0 ] × ( 0 , + ∞ ) , for system ( SDDE ) through ( φ , τ 0 ) . . . . . . . . . . . . . . . . . . . . . . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . Qingwen Hu (UTD) October 2014, UTD 10 / 33

Invariant sets and attactivity of stationary states (H7) For every x ∈ R + , there exists a unique positive y : = l ( x ) such that h ( x , y ) = 0. Let the set of stationary states of x -component in (SDDE) be E : = { x ∈ R : b ( x ) = µx } . Then by (H7), every stationary state of (SDDE) can be denoted by ( x , l ( x )) for some x ∈ E . . . . . . . . . . . . . . . . . . . . . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . Qingwen Hu (UTD) October 2014, UTD 11 / 33

We have the following elementary results . Lemma . Assume ( H1–H6 ) hold. Let ¯ u be the maximal nonnegative stationary state of u < ξ 0 , then there exists T > 0 large enough so that x in (SDDE). If ¯ x ( t ) ∈ [ 0 , ξ 0 ] for all t > T . Moreover, for every solution ( x , τ ) of system ( SDDE ) with initial value ( φ , τ 0 ) ∈ X 0 × ( 0 , r ] , the ω -limit set ω ( φ ) satisfies ω ( φ ) ∩ [ 0 ∗ , ξ 0 ∗ ] ̸ = / 0 , where we use ∗ to indicate constant functions. . . . . . . . . . . . . . . . . . . . . . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . Qingwen Hu (UTD) October 2014, UTD 12 / 33

The Fluctuation Method One important use of invariant intervals, as well as the boundedness of solutions in the current work is for the fluctuation method (Thieme, Zhao, Hsu) which has been repeatedly employed to prove existence of certain limits: Let x : [ a , + ∞ ) → R be a continuously differentiable function. If x ( t ) = x ∞ < ∞ , liminf t → + ∞ x ( t ) = x ∞ < ∞ are finite with x ∞ < x ∞ , limsup t → + ∞ Then there exist sequences { t n } and { s n } in R with lim n → + ∞ t n = lim n → + ∞ s n = + ∞ such that n → + ∞ x ( t n ) = x ∞ ,  x ( t n ) = 0 , lim ˙  x ( s n ) = 0 , lim n → + ∞ x ( s n ) = x ∞ . ˙  . . . . . . . . . . . . . . . . . . . . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . Qingwen Hu (UTD) October 2014, UTD 13 / 33