International audienceIn this talk we present the Generalized Lotka–Volterra dynamics associated to themodel of C-difficile infection of the intestine microbiote and aiming to transfer the systemfrom an infected state to an healthy state. The control inputs are of two types : fecalinjection or bactericides which act as Dirac pulses and prebiotics or antibiotics which act ascontinuous controls. An uniform frame can be introduced using the tools from geometriccontrol to analyze the accessibility set as the orbit of a pseudo-semi group. Optimalcontrol can be considered in the frame of permanent control or sampled-data control. Thelater being adapted to the practical constraints of a finite set of medical interventions. Inboth case the optimal control problems can be analyzed using direct and indirect schemesaiming to reach an healthy state. Such methods are tested on toys models in dimension 2and 3 related to the construction of reduced dynamics. Even those simple situations leadto interesting questions of accessibility and integrability issues in relation with the studyof dynamical systems

Bonnard, Bernard

Rouot, Jérémy

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HAL Id: hal-03727810https://hal.inria.fr/hal-03727810Submitted on 20 Jul 2022HAL is a multi-disciplinary open accessarchive for the deposit and dissemination of sci-entific research documents, whether they are pub-lished or not. The documents may come fromteaching and research institutions in France orabroad, or from public or private research centers.L’archive ouverte pluridisciplinaire HAL, estdestinée au dépôt et à la diffusion de documentsscientifiques de niveau recherche, publiés ou non,émanant des établissements d’enseignement et derecherche français ou étrangers, des laboratoirespublics ou privés.Geometric Optimal Control of the GeneralizedLotka-Volterra Model with Applications ControlledStability of MicrobiotaJérémy Rouot, Bernard BonnardTo cite this version:Jérémy Rouot, Bernard Bonnard. Geometric Optimal Control of the Generalized Lotka-VolterraModel with Applications Controlled Stability of Microbiota. French-German Conference on Opti-mization 2022, May 2022, Porto, France. ￿hal-03727810￿Geometric Optimal Control of the GeneralizedLotka-Volterra Model with Applications ControlledStability of Microbiota.Jérémy Rouot 1LMBA & Univ. Brest, France15th Viennese Conference on Optimal Control and Dynamic GamesJuly 12-15, 2022, Vienna1joint work with B. Bonnard (INRIA) & C.J. Silva (CIDMA, Univ. Aveiro).•1Controlled GLV-Model and Microbiotax = (x1, . . . ,xn) n-interacting species.Species x1: infected populationDynamics.ẋ = (diagx) (Ax + r ) ,diagx =x1x2 0. . .0 . . .xnA = (aij )ij : interacting parameters matrixr = (r1, . . . , rn)ᵀ: growth rates coefficients•2Origin of the model. Lotka–Volterra prey-predator model (Volterra, 1931).Interpretation: Describe the interaction of a population of n-species: x withlinearly computable equilibria and stability.Number of equilibria: at most 2n.Algorithm:(to be compared with Models where the computations of equilibria are intricate).•3Examples.Chemical networks2. Dynamics depends upon the temperature using Arrheniuslaw.A→ B→ C and T +M A Bk1 k2k3k4Feinberg, Horn & Jackson3: compute the equilibria and their stability with theconcept of deficiency.Biological network. Interaction with Michaelis-Menten functions generated usinghomographic functions.Conclusion.We bypass the problem of computing equilibria and their stability.2E.D. Sontag, Global Stability of McKeithan’s Kinetic Proofreading Model for T-Cell ReceptorSignal Transduction, 1999.3Feinberg and Horn-Jackson: articles in Archive Rational Mechanics, 1972.•4Introduction of the control actionsContinuous controls: u(·) : [0, tf ]→ [0,1] corresponds to probiotics orantibiotics.Effect: Add to the dynamics : u(t) (diagx)ε ,ε = (ε1, . . . ,εn)ᵀ : sensitivity vector.Impulsive controls: δ(t− t ′) associated to transplantations or bactericidsEffect: Jump at time t ′ in the current initial condition:x(0)→ x(0)+λvv = (v1, . . . ,vn)ᵀ constant v = 1,λ: height of the jumps.Practically they correspond to the limit as n→∞ of un = n on [t ′, t ′+1/n].Controlled GLV–model.ẋ = (diagx) (Ax + r +uε)+∑i λiδ(t− t ′i )v•5ConstraintsSince impulsive controls are associated to invasive treatments only a finitenumber of medical interventions are allowed over [0, tf ] and are defined by(t ′1,λ1, t ′2,λ2, . . . , t ′p′ ,λp′).This leads to a mixture of permanent control actions and sampled-data control.Aim: A general sampled-data control frame can be introduced since due tomedical constraints continuous controls have to be replaced by (u0, . . . ,up), uiconstant in [0,1] on 0 = t0 < t1 < · · ·< tp < tf .⇒ sampled-data control problem with sampled defined byΞ = (t1, . . . , tp,u0, . . . ,up, t ′1, . . . , tp′ ,λ1, . . . ,λp′).•6Original MotivationReduce the C. difficile infection of the intestinal microbiota. 4n = 11 species. Controlled stability of the x1 C. difficile bacteria usingantibiotic treatmentssingle fecal injectionMethod. Reduce to a 2d–modelUnstableStableStableStableStableFigure: Schematic effect of Jones et al. protocol : antibiotic prior to infection & final fecalinjection.4Stein et al., PLOS Comp. Bio., 2013 · Jones et al., PLOS Comp. Bio, 2018•7Geometric Control Theory•8Mathematical difficulty : the Maximum Principle and theclassification of geodesicsBack to controlled GLV-model written asẋ = F (x)+uG(x)u =−1: no treatment X = F −G.u = +1: maximal dosing treatment Y = F +G.Introduce the accessibility set at time tf :A(x0, tf ) = ⋃u∈Ux(tf ,x0,u)where x(tf ,x0,u) is the response to u(·).•9Mathematical difficulty : the Maximum Principle and theclassification of geodesicsIntroduce the accessibility set at time tf :A(x0, tf ) = ⋃u∈Ux(tf ,x0,u)where x(tf ,x0,u) is the response to u(·).Basic fact: Under some mild assumptions one can restrict to piecewise constantmappings u(·) and the accessibility set A(x0, tf ) is the orbits for the action of thepseudogroupS({X ,Y }) = {exp t1X ◦exp t2Y ◦ · · · ◦exp tkY , t1 + · · ·+ tk = tf , ti > 0, k ∈ N}(with X = F −G, Y = F +G) and geodesics belong to the boundary of theaccessibility set.•10GeodesicsHence it is sufficient to compute A(x0, tf ) and its boundary.Bad news : This boundary has no real nice structure. Lipschitz ? Stratifiedmanifold ?•11Problem 1. Complexity of the dynamics of dxdt = Z (x).LocalSaddleregular point regular singular point non regular singular pointNodesectorEllipticsectorSaddle sectorGlobal•12G ∼ ∂∂xn in log–coordinatesF : at most 2n computable equilibria with their linear stabilitynon regular singular point occurs at 0 related to confluence of the interactions.Problem 2. Classification of pairs of vector fields {X ,Y } in relation withaccessibility properties.Transverse at 0 Collinear at 0 (contact of order 2)fold case cusp case•13Maximum Principle and Classification of Extremals up to fold pointsTheorem (Pontryagin et al. 1958)Parameterization of the boundary of the accessibility set.Introduce H(x ,p,u) = HF +uHG with HF = p ·F (x), HG(x) = p ·G(x) theHamiltonian lifts, z = (x ,p) and p is the adjoint vector.The geodesics candidates to minimizers are projection of extremal curves: t → z(·)solutions ofẋ = ∂H∂p , ṗ =−∂H∂x ,H(x ,p,u) = max|v |≤1H(x ,p,v ).H(x ,p,u) = max|v |≤1 H(x ,p,v ) is a nonnegative constant M .Transversality condition.x(tf )∈N and p(tf )⊥ Tx(tf )N.•14Classification of extremalsFirst step. Classification of singular extremals.u(t) = sign(HG(z(t))) a.e. ⇒ Regular extremalsHG(z(t)) = 0 identically ⇒ Singular extremalsExceptional M = 0 ⇒ Regular or SingularComputing Singular ExtremalsPoisson bracket: {HF ,HG}= p · [F ,G].HG = {HG,HF}= 0{{HG,HF},HF}+us {{HG,HF},HG}= 0 .Once us is computed, define the singular extremal flowż = ~Hs(z), Hs = HF +usHG.One shall study this dynamics restricted to HG = {HG,HF}= 0 (codim. 2).•15Second step.High Order Maximum Principle (Krener, SICON, 1977).Discrimination of the extremals (assuming p unique up to a scalar):Small time minimizers Hyperbolic case∂∂ud2dt2 ∂H∂u (z) = {{HG,HF},HG}(z)> 0Small time maximizers Elliptic case∂∂ud2dt2 ∂H∂u (z) = {{HG,HF},HG}(z)< 0Exceptional M = 0 (then can be both hyperbolic and elliptic).Assuming us strictly admissible i.e. us ∈]−1,1[.Saturating case: |us|= 1 at t = tf .•16Third step.Classification of regular extremals near Σ : HG = 0. (Ekeland – IHES – 1977,Kupka – TAMS – 1987)Denote:σ+: bang arc with u = +1σ−: bang arc with u =−1σs: singular arc u = us ∈]−1,1[σ1σ2 is the arc σ1 followed by σ2.Switching surface:Σ : {(x ,p) | HG(x) = 0}Σ′ : {(x ,p) | HG(x) = H[G,F ](x) = 0} ⊂ ΣSwitching function: Φ(t) = HG(x(t),p(t))•17Ordinary Switching time: t ∈]0, tf [ such that Φ(t) = 0 and Φ̇(t) 6= 0.LemmaNear z(t) every extremal solution projects onto σ+σ− if Φ̇(t)< 0 and σ−σ+ ifΦ̇(t)> 0Fold case: t ∈]0, tf [ such that Φ(t) = Φ̇(t) = 0 (then z(t)∈ Σ′).Φ̈ε = H [[G,F ],F ]+ε [[G,F ],G], ε =±1Assumption: Σ′ : surface of codimension two, Φ̈ε(z(t)) 6= 0 for ε =±1.parabolic case: Φ̈+(t)Φ̈−(t)> 0hyperbolic case: Φ̈+(t)> 0 and Φ̈−(t)< 0elliptic case: Φ̈+(t)< 0 and Φ̈−(t)> 0•18Fold caseParabolic, Hyperbolic, EllipticIn the parabolic case |u0|> 1 and the singular arc is not admissible.Theorem ( I. Kupka, 1987 (TAMS) )In the neighborhood of z(t)∈ Σ′ every extremals projects onto:Parabolic case: σ+σ−σ+ or σ−σ+σ−Hyperbolic case: σ±σsσ±Elliptic case: every extremal is of the form σ+σ−σ+σ− . . . (Bang-Bang) butthe number of switches is not uniformly bounded.•19u = +1u =−1singular Σ′p ·G(x)< 0p ·G(x)> 0Figure: Fold case in the hyperbolic case and the turnpike phenomenonConclusion: Existence of the set Σ′ : HG = H[F ,G] = 0 leads to complicatedextremal policies. This is coming from a pathology of the accessibility set in thesingular directions.•20Time minimal synthesis of  min tf |u| ≤ 1ẋ = F (x)+uG(x)x1(tf )∈N = {x1 = d}Methods: Two steps:1 Calculation of the time minimal syntheses near the terminal manifold2 Bounds on the number of switchesStep 1: Take x0 ∈N , z0 = (x0,n(x0)) where n(x0) is the normal vector of N at x0.Find, in a small neighborhood U of x0, the time minimal closed loop control u∗(x)to reach N starting from x in minimal time.NUx0n•21Whitney Charts and Unfolding.5Construction. (U,φ,x0) with x0 ∈ U an open set,φ coordinate system (x ,y,y1, . . . ,yn−3,z)such thatG = ∂∂z ,N : (1/2k s2,w ,w1, . . . ,wn−3,s) (k is the curvature),The (local) time minimal synthesis in U is C0 anddescribed with foliations by 2d or 3d synthesis.Stratification of S.Switching locus. W = W +∪W− of ordinary switching point.Switching locus. W s = W s+∪W s− associated to hyperbolic singular arcs.Cut locus. C where at least two minimizers with equal length are reachingthe terminal manifold N .5(Bonnard–Kupka, Forum Math. 1991), (B.Bonnard–J.R., Annual Review Control, 2019)•22Whitney Charts and Unfolding.6Construction. (U,φ,x0) with x0 ∈ U an open set,φ coordinate system (x ,y,y1, . . . ,yn−3,z)such thatG = ∂∂z ,N : (1/2k s2,w ,w1, . . . ,wn−3,s) (k is the curvature),The (local) time minimal synthesis in U is C0 anddescribed with foliations by 2d or 3d synthesis.Stratification of S.Switching locus. W = W +∪W− of ordinary switching point.Switching locus. W s = W s+∪W s− associated to hyperbolic singular arcs.Cut locus. C where at least two minimizers with equal length are reachingthe terminal manifold N .6(Bonnard–Kupka, Forum Math. 1991), (B.Bonnard–J.R., Annual Review Control, 2019)•23Classification and the concept of unfolding.Example: 2 dimensional hyperbolic case.Hyperbolic singular arc terminating at 0.n ·G = n · [G,F ] = 0.Normalize the singular extremal to: σs : t 7→ (t ,0).Two generic cases :σs optimal σs not optimal(existence of a cut locus)•24From 3d-case to 2d-case: Concept of Unfolding.One can unfold the 2d-case where σs is optimal asN = ⋃λN(λ)where N(λ) is one–dimensional so that the 3d–synthesis are foliated:Whitney chart Whitney chart Whitney chartλ < 0 λ= 0 λ > 0•25This leads to compute Whitney neighborhoodn–dimensional case−→ 2d–casewhere the switching locus is stratified intoW +∪W−∪W s+∪W s−W +: σ−σ+W−: σ+σ−W s+: singular locus terminating at N+W s−: singular locus terminating at N−Other classifications are associated to elliptic, parabolic fold points.Existence of a cut locus.•26The limit of unfolding 3d to 2d caseNon existence of 2d–foliationContact theoryExample: Regular Exceptional codimension 2 case.N = (0,w ,s), G = ∂∂z .Non-plane foliation with parameter : w2−8bs/3.•27Every point of the target is accessible from the domain x > 0 by an arc σ−.For x < 0 the optimal policy is σ+.In particular the dashed arc σ− decomposes into two optimal subarcs in x > 0 andone subarc in x < 0 which is not optimal.Such a case is not coming from an unfolding of 2d–case.TheoremEvery case up to codimension 2 Lie brackets relation can be unfolded as either a2d or 3d cases.•28Figure: Strata of the surface S in the exceptional case.•29Gluing chartsConstruction of time minimal syntheses in the n–dimensional case by gluingtogether Whitney charts....Intermediate ManifoldsWhitney charts•30ConclusionThis allows to construct patchy time minimal regular synthesis based on thedictionary from B. Bonnard–J.R., Annual Review of Control, 2019.The methods relies only on the Lie brackets computations and uses singular arcs.Formal computations are used to construct seminormal forms and compute thetime minimal synthesis.Compare previous results with:Ancona–Bressan, “Patchy subotpimal synthesis, Annales IHP”.Earliest works in 2d–case : H. Sussmann–H. Schättler, Boscain–Piccoli.•31

Geometric Optimal Control of the Generalized Lotka-Volterra Model with Applications Controlled Stability of Microbiota

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Geometric Optimal Control of the Generalized Lotka-Volterra Model with Applications Controlled Stability of Microbiota

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