Analytic Left Inversion of Multivariable Lotka Volterra Models α

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1 Analytic Left Inversion of Multivariable Lotka Volterra Models α W. Steven Gray β Mathematical Perspectives in Biology Madrid, Spain February 5, 6 β Joint work with Luis A. Duffaut Espinosa (GMU) and Kurusch Ebrahimi-Fard (ICMAT) α Research supported by the BBVA Foundation Grant to Researchers, Innovators and Cultural Creators (Spain). W. S. Gray February 5, 6 ICMAT

2 Problem Trajectory Generation Problem: explicitly compute the input to drive a nonlinear system to produce some desired output. Fliess operators: F c : u y are analytic multivariable input-output maps, which are described by coefficients (c, η) and corresponding iterated integrals (M. Fliess, 983). Left inversion problem: given a multivariable Fliess operator F c and a function y in its range, determine an input u such that y = F c [u]. Hopf algebra antipode: group (G, ) of unital Fliess operators and its corresponding Hopf algebra H of coordinate functions; G F c = F c S, S : H H S id = id S = ɛ Lotka Volterra Model: ż i = β i z i + n j= α ijz i z j, i =,,..., n Input-Output systems are obtained by introducing time dependence on the parameters β i (t) and α ij (t) (inputs u k ), and assuming that y = h(z) (outputs y = F c [u]). W. S. Gray February 5, 6 ICMAT

3 Setting Fliess operator y = F c [u](t, t ) = alphabet: X = {x, x,..., x m } system: c := (c, η) η X }{{} R l η R l X controls: u : [t, t ] R m, u := η X (c, η) E η [u](t, t ) x i u i E xi η[u](t, t ) = t t u i (s)e η [u](s, t ) ds E [u] := W. S. Gray February 5, 6 ICMAT

4 System interconnections I F c u y F d product connection: F c F d = F c d F c u + y F d parallel connection: F c + F d = F c+d W. S. Gray February 5, 6 ICMAT 3

5 System interconnections II Cascade connection u F d v F c y d := (d, η)η, (d, η) R m, d := η X (F c F d )[u](t, t ) = [ (c, η) E η Fd [u] ] (t, t ) η X E xi η[f d [u]](t, t ) = t t F di [u](s, t )E η [F d [u]](s, t ) ds (F c F d )[u] = F c d [u] x i η d := x ( di (η d) ) W. S. Gray February 5, 6 ICMAT 4

6 System interconnections III Feedback loop u v F c y v = u + Fd c [v] F d F c d [u] = F c (ɛ d c) [u] Involves an extension of Fliess operators: unital Fliess operators F c ɛ[u] := u + F c [u] = (I + F c )[u] c ɛ := ɛ + c F c ɛ F d ɛ[u] = F c ɛ dɛ[u] This composition defines a group (G, ) with unit ɛ on R X ɛ [G-DE]. W. S. Gray February 5, 6 ICMAT 5

7 Faà di Bruno type Hopf algebra Coordinate functions I Coordinate functions: a i η : G R, ai η (c ɛ) := c ɛ, a i η = (c ɛ, η) i R c ɛ d ɛ, a i η = c ɛ d ɛ, (a i η ) = c ɛ d ɛ, (η) a i η a j η Theorem: Coordinate functions form a connected graded commutative non-cocommutative Hopf algebra (H,, ɛ, S, m, ι). Antipode: S : H H c ɛ, a i η = c ɛ, S(a i η ) S(a i η ) = ai η (η) S(a i η )a j η = a i η (η) a i η S(a j η ) W. S. Gray February 5, 6 ICMAT 6

8 Coordinate functions II Coproduct and antipode calculations : H H H S : H H (a i ) = ai + ai S(a i ) = ai (a i x j ) = a i x j + a i x j S(a i x j ) = a i x j (a i x ) = a i x + a i x + a i x l a l S(a i x ) = a i x + a i x l a l (a i x j x k ) = a i x j x k + a i x j x k S(a i x j x k ) = a i x j x k c ɛ d ɛ, a i x j x k = (c ɛ d ɛ, x j x k ) i = a i x (c ɛ ) + a i x (d ɛ ) + a i x l (c ɛ )a l (d ɛ) = (c ɛ, x ) i + (d ɛ, x ) i + (c ɛ, x l ) i (d ɛ, ) l c ɛ, a i x j x = (c k ɛ, x j x k ) i = a i x (c ɛ ) + a i x l (c ɛ )a l (c ɛ) = (c ɛ, x ) i + (c ɛ, x l ) i (c ɛ, ) l W. S. Gray February 5, 6 ICMAT 7

9 Left Inversion of MIMO Fliess operators I Observe: c R X can be written as c = c N + c F, where c N := k (c, xk )xk and c F := c c N. Definition: Given c R k X, let r i be the largest integer such that supp(c F,i ) x r i X, i =,,..., m. Then c i has relative degree r i if x r i x j supp(c i ), for j {,..., m}, otherwise it is not well defined. In addition, c has vector relative degree r = [r r r m ] if each c i has relative degree r i and the m m matrix A = (c, x r x ) (c, x r x ) (c, x r x m ).... (c m, x r m x ) (c m, x r m x ) (c m, x r m x m ) has full rank. Otherwise, c does not have vector relative degree. This definition coincides with the usual definition of relative degree given in a state space setting. But this definition is independent of the state space setting. W. S. Gray February 5, 6 ICMAT 8

10 Left Inversion of MIMO Fliess operators II Lemma: The set of series R m m X having invertible constant terms is a group under the shuffle product. In particular, the shuffle inverse of any such series C is C = ((C, )(I C )) = (C, ) (C ), where C = I (C, ) C is proper, i.e., (C, ) =, and (C ) := k (C ) k. Lemma: For any C R m m X with an invertible constant term, F C, which is defined componentwise by [F C ] i,j = F Ci,j, has a well defined multiplicative inverse given by (F C ) = F C. Notation: Let R[[X ]] be all commutative series over X := {x }. When c R[[X ]], F c [u](t) = k (c, xk )E x k [u](t) = k (c, xk )tk /k!. W. S. Gray February 5, 6 ICMAT 9

11 Left Inversion of MIMO Fliess operators III y (r) = F (x r ) (c) [u] + F C[u]u R m u = (F C [u]) F (x r ) (c cy) [u], y(t) = F cy[u](t) = k (c y, x k )tk /k! u = F d [u], d = C (x r ) (c c y ) (x r ) (c c y ) i = (x r i ) (c i c yi ) and C i,j = (x r i x j ) (c i ) Theorem: Suppose c R m X has vector relative degree r. Let y be analytic at t = with generating series c y R m LC X satisfying (c y, x (r) ) = (c, x (r) ). Then the input u(t) = (c u, x k )tk k! k= with c u = ((C (x r ) (c c y )) ) N, is the unique solution to F c [u] = y on [, T ] for some T >. Note: the condition on c y ensures that y is in the range of F c. W. S. Gray February 5, 6 ICMAT

12 Multivariable I/O Lotka Volterra Models I ż i = β i z i + n j= α ijz i z j, i =,,..., n ż ż ż 3 = β z α z z β z + α z z α 3 z z 3 β 3 z 3 + α 3 z 3 z Predators - Prey The systems within the first octant have: periodic orbits around (β /α, β /α, ) if β α 3 = β 3 α extinction of one population if β α 3 < β 3 α unbounded growing if β α 3 > β 3 α. ANSATZ: Input-output models are obtained by introducing time dependence on the parameters β i (t) s or α ij (t) s (inputs), and assuming y = h(z) (outputs). W. S. Gray February 5, 6 ICMAT

13 Multivariable I/O Lotka Volterra Models II Vector relative degree r for three LV systems with y = z and y = z 3 : F c : F c : F c : I/O map r range restrictions ] [ ] y not defined [ β β [ β β 3 [ β β 3 ] ] y [ y y [ y y ] [ ] ] [ ] (full) (c y, ) = (c, ) (c y, ) = (c, ) (c y, ) = (c, ) (c y, x ) = (c, x ) (c y, ) = (c, ) Consider case : r = [ ], u := β, u := β 3 ż ż = ż 3 β z α z z α z z α 3 z z 3 α 3 z 3 z z u u, z 3 ( y y ) = ( z z 3 ) with z i () = z i, >, i =,, 3. Normalizing all parameters to. W. S. Gray February 5, 6 ICMAT

14 Multivariable I/O Lotka Volterra Models III Inputs: u = β, u = β 3 Vector relative degree r = [ ]. c = z, + (α z, z, α 3 z, z 3, )x (z, )x + x +, c = z 3, + (α 3 z, z 3, )x + x (z 3, )x +. Extinction vs. periodic orbit Top-level predator population Equilibrium (,, ) Initial orbit Top-level predator population Equilibrium (,, ) Final orbit Prey population Mid-level predator population Prey population Mid-level predator population.5 3 u =, u =.. u = u =. W. S. Gray February 5, 6 ICMAT 3

15 Output function Multivariable I/O Lotka Volterra Models IV One must select an output function y(t) = k= (c y, x k )tk k!, where c y = [c y, c y ] T is the generating series of y. Consider a polynomial of degree 4: (c yj, x i ) = v ij, i =,, 3, 4, j =,. [ ] (c A = (C, ) =, x r x ) (c, x r x ) (c, x r x ) (c, x r = diag{ z,, z 3, } has full rank x ) d = C (x r ) (c c y ) = [d d ] T. c u = (d ) N W. S. Gray February 5, 6 ICMAT 4

16 Numerical Simulation Multivariable I/O Lotka Volterra Models V Note that u (t ) = u (t ) = and y (t ) = z (t ) = and y (t ) =, Top-level predator population Initial orbit Transition path Final orbit Prey population Mid-level predator population Fig. 5.: Orbit transfer. W. S. Gray February 5, 6 ICMAT 5

17 Conclusions The general multivariable left inverse problem for input-output systems represented as Fliess operators was solved explicitly via methods from combinatorial Hopf algebras: cancellation-free antipode formula. The technique was then illustrated for an orbit transfer problem in a three species Lotka Volterra system: orbit transfer in order to avoid the extinction of the top-predator: System parameters β, α become controls. Efficiency of the software used for calculations/simulations is currently being improved. Thank you for your attention! W. S. Gray February 5, 6 ICMAT 6