Optimal control on hybrid ODE Systems with application to a tick disease model

  • Received: 01 September 2006 Accepted: 29 June 2018 Published: 01 August 2007
  • MSC : 49J15, 34K35, 92D25.

  • We are considering an optimal control problem for a type of hybrid system involving ordinary differential equations and a discrete time feature. One state variable has dynamics in only one season of the year and has a jump condition to obtain the initial condition for that corresponding season in the next year. The other state variable has continuous dynamics. Given a general objective functional, existence, necessary conditions and uniqueness for an optimal control are established. We apply our approach to a tick-transmitted disease model with age structure in which the tick dynamics changes seasonally while hosts have continuous dynamics. The goal is to maximize disease-free ticks and minimize infected ticks through an optimal control strategy of treatment with acaricide. Numerical examples are given to illustrate the results.

    Citation: Wandi Ding. Optimal control on hybrid ODE Systems with application to a tick disease model[J]. Mathematical Biosciences and Engineering, 2007, 4(4): 633-659. doi: 10.3934/mbe.2007.4.633

    Related Papers:

    [1] Valentin Duruisseaux, Melvin Leok . Time-adaptive Lagrangian variational integrators for accelerated optimization. Journal of Geometric Mechanics, 2023, 15(1): 224-255. doi: 10.3934/jgm.2023010
    [2] Jordi Gaset, Arnau Mas . A variational derivation of the field equations of an action-dependent Einstein-Hilbert Lagrangian. Journal of Geometric Mechanics, 2023, 15(1): 357-374. doi: 10.3934/jgm.2023014
    [3] Robert I McLachlan, Christian Offen . Backward error analysis for conjugate symplectic methods. Journal of Geometric Mechanics, 2023, 15(1): 98-115. doi: 10.3934/jgm.2023005
    [4] Álvaro Rodríguez Abella, Melvin Leok . Discrete Dirac reduction of implicit Lagrangian systems with abelian symmetry groups. Journal of Geometric Mechanics, 2023, 15(1): 319-356. doi: 10.3934/jgm.2023013
    [5] Maulik Bhatt, Amit K. Sanyal, Srikant Sukumar . Asymptotically stable optimal multi-rate rigid body attitude estimation based on lagrange-d'alembert principle. Journal of Geometric Mechanics, 2023, 15(1): 73-97. doi: 10.3934/jgm.2023004
    [6] Elias Maciel, Inocencio Ortiz, Christian E. Schaerer . A Herglotz-based integrator for nonholonomic mechanical systems. Journal of Geometric Mechanics, 2023, 15(1): 287-318. doi: 10.3934/jgm.2023012
    [7] William Clark, Anthony Bloch . Existence of invariant volumes in nonholonomic systems subject to nonlinear constraints. Journal of Geometric Mechanics, 2023, 15(1): 256-286. doi: 10.3934/jgm.2023011
    [8] Francesco Bonechi, Jian Qiu, Marco Tarlini . Generalised Kähler structure on CP2 and elliptic functions. Journal of Geometric Mechanics, 2023, 15(1): 188-223. doi: 10.3934/jgm.2023009
    [9] Marcin Zając . The dressing field method in gauge theories - geometric approach. Journal of Geometric Mechanics, 2023, 15(1): 128-146. doi: 10.3934/jgm.2023007
    [10] Yunhe Sheng . Categorification of VB-Lie algebroids and VB-Courant algebroids. Journal of Geometric Mechanics, 2023, 15(1): 27-58. doi: 10.3934/jgm.2023002
  • We are considering an optimal control problem for a type of hybrid system involving ordinary differential equations and a discrete time feature. One state variable has dynamics in only one season of the year and has a jump condition to obtain the initial condition for that corresponding season in the next year. The other state variable has continuous dynamics. Given a general objective functional, existence, necessary conditions and uniqueness for an optimal control are established. We apply our approach to a tick-transmitted disease model with age structure in which the tick dynamics changes seasonally while hosts have continuous dynamics. The goal is to maximize disease-free ticks and minimize infected ticks through an optimal control strategy of treatment with acaricide. Numerical examples are given to illustrate the results.


    The Skinner–Rusk formalism was introduced by R. Skinner and R. Rusk in 1983 [1] (although a previous description in local coordinates had been developed by K. Kamimura in [2]) in order to deal with mechanical systems described by singular Lagrangian functions. This formulation combines both the Lagrangian and Hamiltonian formalism and this is why it is sometimes called unified formalism. The Skinner–Rusk formalism has been extended to time-dependent systems [3,4,5], nonholonomic and vakonomic mechanics [6], higher-order mechanical systems [7,8,9,10], control systems [11,12] and field theory [13,14,15,16,17,18]. Recently, the Skinner–Rusk unified formalism was extended to contact [19] and k-contact [20] systems. The Skinner–Rusk unified formalism has several advantages. In first place, we recover the second-order condition even if the Lagrangian of the system is singular. We also recover the definition of the Legendre map from the constraint algorithm. Also, both the Lagrangian and Hamiltonian formulations can be recovered from the Skinner–Rusk formalism by projecting onto their respective phase spaces.

    The use of contact geometry [21,22,23] to model geometrically the time-dependence in mechanical systems is very well-known [24,25,26] and it is, alongside with cosymplectic geometry [27], the natural way to do it. However, in the last decade, the application of contact geometry to the study of dynamical systems has grown significantly [28,29]. This is due to the fact that one can use contact structures to describe many different types of dynamical systems which cannot be described by means of symplectic geometry and standard Hamiltonian dynamics in a natural way. The dynamical systems which can be modelled using contact structures include mechanical systems with certain types of damping [30,31,32], some systems in quantum mechanics [33], circuit theory [34], control theory [35] and thermodynamics [36,37], among many others [23,25,38,39,40,41,42,43].

    Although contact geometry is a suitable framework when working with systems of ordinary differential equations, a generalisation is required in order to deal with systems of partial differential equations describing classical field theories. This generalisation is the so-called k-contact structure [20,44,45]. This formulation allows to describe many types of field theories both in the Lagrangian and in the Hamiltonian formalisms. The k-contact framework allows us to describe geometrically field theories with damping, some equations from circuit theory, such as the so-called telegrapher's equation, or the Burgers' equation. Recently, a geometric framework has been developed [46] in order to deal with time-dependent mechanical systems with dissipation using the so-called cocontact geometry. It is still an open problem to find a geometric setting to describe non-autonomous field theories with damping.

    The main goal of this paper is to extend the Skinner–Rusk formalism to time-dependent contact systems, studying the dynamical equations and the submanifold where they are consistent, and showing that the Lagrangian and Hamiltonian formalisms can be recovered from this mixed formalism. In first place, we introduce the phase space of this formulation: the Pontryagin bundle W=R×TQ×TQ×R endowed with natural coordinates (t,qi,vi,pi,s). This manifold has a natural precocontact structure (dt,η) inherited from the natural cocontact structure of R×TQ×R (see [46]). The Hamiltonian function associated to a Lagrangian function LC(R×TQ×R) is defined as

    H=piviL(t,qi,vi,s).

    Since the Hamiltonian system (W,dt,η,H) is singular, we need to implement a constraint algorithm in order to find a submanifold where the Hamiltonian equations are consistent. In the first iteration of the constraint algorithm we recover the second-order condition, even if the Lagrangian function is singular (in the Lagrangian formalism, we only recover the SODE condition if the Lagrangian is regular). The first constraint submanifold is the graph of the Legendre map FL. If the Lagrangian function is regular the constraint algorithm ends in one step and we obtain the usual results by projecting the dynamics onto the Lagrangian and Hamiltonian phase spaces. If the Lagrangian function is singular, the constraint algorithm is related to the usual Lagrangian and Hamiltonian constraint algorithms (imposing the SODE condition in the Lagrangian case).

    The structure of the present paper is as follows. In Section 2, we review the basics on cocontact geometry, which is an extension of both contact and cosymplectic geometry. This geometric framework allows us to develop a Hamiltonian and a Lagrangian formulation for time-dependent contact systems [46]. Section 3 is devoted to present the Skinner–Rusk unified formulation for cocontact systems. We begin by introducing the Pontryagin bundle and its natural precocontact structure and state the Lagrangian–Hamiltonian problem. In Section 4 we recover both the Lagrangian and Hamiltonian formalisms and see that they are equivalent to the Skinner–Rusk formalisms (imposing the second order-condition if the Lagrangian is singular). Finally, in Section 5 some examples are studied in full detail. These examples are the Duffing equation [47,48], an ascending particle with time-dependent mass and quadratic drag, and a charged particle in a stationary electric field with a time-dependent constraint.

    Throughout this paper, all the manifolds are real, second countable and of class C. Mappings are assumed to be smooth and the sum over crossed repeated indices is understood.

    In this first section we will briefly review the basics on cocontact manifolds introduced in [46] and how this structure can be used to geometrically describe time-dependent contact mechanical systems.

    Definition 2.1. A cocontact structure on a (2n+2)-dimensional manifold M is a couple of 1-forms (τ,η) on M such that dτ=0 and τη(dη)n0. In this case, (M,τ,η) is said to be a cocontact manifold.

    Example 2.2. Let Q be an n-dimensional smooth manifold with local coordinates (qi). Let (qi,pi) be the induced natural coordinates on its cotangent bundle TQ. Consider the product manifolds R×TQ, TQ×R and R×TQ×R with natural coordinates (t,qi,pi), (qi,pi,s) and (t,qi,pi,s) respectively. Let us also define the following projections:

    Now consider θ0Ω1(TQ) be the canonical 1-form of the cotangent bundle with local expression θ0=pidqi and let θ1=π1θ0 and θ2=π2θ0.

    Then we have that (dt,θ2) is a cosymplectic structure in R×TQ and η1=dsθ1 is a contact form on TQ×R. Furthermore, considering the one-forms in R×TQ×R given by θ=ρ1θ2=ρ2θ1=πθ0, τ=dt and η=dsθ, we have that (τ,η) is a cocontact structure in R×TQ×R with local expression:

    τ=dt,η=dspidqi.

    In a cocontact manifold (M,τ,η) we have the so called flat isomorphism

    :TMTMv(i(v)τ)τ+i(v)dη+(i(v)η)η,

    which can be extended to a morphism of C(M)-modules:

    :XX(M)(i(X)τ)τ+i(X)dη+(i(X)η)ηΩ1(M).

    Proposition 2.3. Given a cocontact manifold (M,τ,η) there exist two vector fields Rt, Rs on M such that

    {i(Rt)τ=1,i(Rt)η=0,i(Rt)dη=0,{i(Rs)τ=0,i(Rs)η=1,i(Rs)dη=0. (2.1)

    Equivalently, they can be defined as Rt=1(τ) and Rs=1(η). The vector fields Rt and Rs are called time and contact Reeb vector fields respectively.

    Moreover, on a cocontact manifold we also have the canonical or Darboux coordinates, as the following theorem establishes:

    Theorem 2.4 (Darboux theorem for cocontact manifolds). Given a cocontact manifold (M,τ,η), for every pM exists a local chart (U;t,qi,pi,s) containing p such that

    τ|U=dt,η|U=dspidqi.

    In Darboux coordinates, the Reeb vector fields read Rt=/t, Rs=/s.

    Definition 2.5. A cocontact Hamiltonian system is family (M,τ,η,H) where (τ,η) is a cocontact structure on M and H:MR is a Hamiltonian function. The cocontact Hamilton equations for a curve ψ:IRM are

    {i(ψ)dη=dH(LRsH)η(LRtH)τ,i(ψ)η=H,i(ψ)τ=1, (2.2)

    where ψ:IRTM is the canonical lift of ψ to the tangent bundle TM. The cocontact Hamiltonian equations for a vector field XX(M) are:

    {i(X)dη=dH(LRsH)η(LRtH)τ,i(X)η=H,i(X)τ=1, (2.3)

    or equivalently, (X)=dH(LRsH+H)η+(1LRtH)τ. The unique solution to this equations is called the cocontact Hamiltonian vector field.

    Given a curve ψ with local expression ψ(r)=(f(r),qi(r),pi(r),s(r)), the third equation in (2.2) imposes that f(r)=r+cnt, thus we will denote rt, while the other equations read:

    {˙qi=Hpi,˙pi=(Hqi+piHs),˙s=piHpiH. (2.4)

    On the other hand, the local expression of the cocontact Hamiltonian vector field is

    X=t+Hpiqi(Hqi+piHs)pi+(piHpiH)s.

    Given a smooth n-dimensional manifold Q, consider the product manifold R×TQ×R equipped with canonical coordinates (t,qi,vi,s). We have the canonical projections

    τ1:R×TQ×RR ,τ1(t,vq,s)=t,τ2:R×TQ×RTQ ,τ2(t,vq,s)=vq,τ3:R×TQ×RR ,τ3(t,vq,s)=s,τ0:R×TQ×RR×Q×R ,τ0(t,vq,s)=(t,q,s),

    which are summarized in the following diagram:

    The usual geometric structures of the tangent bundle can be naturally extended to the cocontact Lagrangian phase space R×TQ×R. In particular, the vertical endomorphism of T(TQ) yields a vertical endomorphism J:T(R×TQ×R)T(R×TQ×R). In the same way, the Liouville vector field on the fibre bundle TQ gives a Liouville vector field ΔX(R×TQ×R). The local expressions of these objects in Darboux coordinates are

    J=vidqi,Δ=vivi.

    Definition 2.6. Given a path c:RR×Q×R with c=(c1,c2,c3), the prolongation of c to R×TQ×R is the path ˜c=(c1,c2,c3):RR×TQ×R, where c2 is the velocity of c2. Every path ˜c which is the prolongation of a path c:RR×Q×R is called holonomic. A vector field ΓX(R×TQ×R) satisfies the second-order condition (it is a SODE) if all of its integral curves are holonomic.

    The vector fields satisfying the second-order condition can be characterized by means of the canonical structures Δ and J introduced above, since X is a SODE if and only if J(Γ)=Δ.

    Taking canonical coordinates, if c(r)=(t(r),ci(r),s(r)), its prolongation to R×TQ×R is

    ˜c(r)=(t(r),ci(r),dcidr(r),s(r)).

    The local expression of a SODE in natural coordinates (t,qi,vi,s) is

    Γ=ft+viqi+Givi+gs. (2.5)

    Thus, a SODE defines a system of differential equations of the form

    dtdr=f(t,q,˙q,s),d2qidr2=Gi(t,q,˙q,s),dsdr=g(t,q,˙q,s).

    Definition 2.7. A Lagrangian function is a function LC(R×TQ×R). The Lagrangian energy associated to L is the function EL=Δ(L)L. The Cartan forms associated to L are

    θL=tJdLΩ1(R×TQ×R),ωL=dθLΩ2(R×TQ×R), (2.6)

    where tJ denotes the transpose of the canonical endomorphism introduced above. The contact Lagrangian form is

    ηL=dsθLΩ1(R×TQ×R).

    Notice that dηL=ωL. The couple (R×TQ×R,L) is a cocontact Lagrangian system.

    The local expressions of these objects are

    EL=viLviL,ηL=dsLvidqi,dηL=2Ltvidtdqi2Lqjvidqjdqi2Lvjvidvjdqi2Lsvidsdqi.

    Not all cocontact Lagrangian systems (R×TQ×R,L) result in the family (R×TQ×R,τ=dt,ηL,EL) being a cocontact Hamiltonian system because the condition τη(dηL)n0 is not always fulfilled. The Legendre map characterizes which Lagrangian functions will result in cocontact Hamiltonian systems.

    Definition 2.8. Given a Lagrangian function LC(R×TQ×R), the Legendre map associated to L is its fibre derivative, considered as a function on the vector bundle τ0:R×TQ×RR×Q×R; that is, the map FL:R×TQ×RR×TQ×R with expression

    FL(t,q,v,s)=(t,FLt,s(q,v),s),

    where vTqQ and FLt,s:TQTQ is the usual Legendre map associated to the Lagrangian function Lt,s=L(t,,s):TQR with t and s freezed.

    The Cartan forms can also be defined as θL=FL(πθ0) and ωL=FL(πω0), where θ0 and ω0=dθ0 are the canonical one- and two-forms of the cotangent bundle and π is the natural projection π:R×TQ×RTQ (see Example 2.2).

    Proposition 2.9. Given a Lagrangian function L the following statements are equivalent:

    (i) The Legendre map FL is a local diffeomorphism.

    (ii) The fibre Hessian F2L:R×TQ×R(R×TQ×R)(R×TQ×R) of L is everywhere nondegenerate (the tensor product is understood to be of vector bundles over R×Q×R).

    (iii) The family (R×TQ×R,dt,ηL) is a cocontact manifold.

    This can be checked using that FL(t,qi,vi,s)=(t,qi,L/vi,s) and F2L(t,qi,vi,s)=(t,qi,Wij,s), where Wij=(2L/vivj).

    A Lagrangian function L is regular if the equivalent statements in the previous proposition hold. Otherwise L is singular. Moreover, L is hyperregular if FL is a global diffeomorphism. Thus, every regular cocontact Lagrangian system yields the cocontact Hamiltonian system (R×TQ×R,dt,ηL,EL).

    Given a regular cocontact Lagrangian system (R×TQ×R,L), the Reeb vector fields RLt,RLsX(R×TQ×R) are uniquely determined by the relations

    {i(RLt)dt=1,i(RLt)ηL=0,i(RLt)dηL=0,{i(RLs)dt=0,i(RLs)ηL=1,i(RLs)dηL=0,

    and their local expressions are

    RLt=tWij2Ltvjvi,RLs=sWij2Lsvjvi,

    where Wij is the inverse of the Hessian matrix of the Lagrangian L, namely WijWjk=δik.

    If the Lagrangian L is singular, the Reeb vector fields are not uniquely determined, actually, they may not even exist [46].

    Definition 2.10. Given a regular cocontact Lagrangian system (R×TQ×R,L) the Herglotz–Euler–Lagrange equations for a holonomic curve ˜c:IRR×TQ×R are

    {i(˜c)dηL=(dEL(LRLtEL)dt(LRLsEL)ηL)˜c,i(˜c)ηL=EL˜c,i(˜c)dt=1, (2.7)

    where ˜c:IRT(R×TQ×R) is the canonical lift of ˜c to T(R×TQ×R). The cocontact Lagrangian equations for a vector field XLX(R×TQ×R) are

    {i(XL)dηL=dEL(LRLtEL)dt(LRLsEL)ηL,i(XL)ηL=EL,i(XL)dt=1. (2.8)

    The only vector field solution to these equations is the cocontact Lagrangian vector field.

    Remark 2.11. The cocontact Lagrangian vector field of a regular cocontact Lagrangian system (R×TQ×R,L) is the cocontact Hamiltonian vector field of the cocontact Hamiltonian system (R×TQ×R,dt,ηL,EL).

    Given a holonomic curve ˜c(r)=(t(r),qi(r),˙qi(r),s(r)), equations (2.7) read

    ˙t=1, (2.9)
    ˙s=L, (2.10)
    ˙t2Ltvi+˙qj2Lqjvi+¨qj2Lvjvi+˙s2LsviLqi=ddr(Lvi)Lqi=LsLvi. (2.11)

    The fact that ˙t=1 justifies the usual identification tr. For a vector field XL with local expression XL=ft+Fiqi+Givi+gs, equations (2.8) are

    (Fjvj)2Ltvj=0, (2.12)
    f2Ltvi+Fj2Lqjvi+Gj2Lvjvi+g2LsviLqi(Fjvj)2Lqivj=LsLvi, (2.13)
    (Fjvj)2Lvivj=0, (2.14)
    (Fjvj)2Lsvj=0, (2.15)
    L+Lvj(Fjvj)g=0, (2.16)
    f=1. (2.17)

    Theorem 2.12. If L is a regular Lagrangian, XL is a SODE and equations (2.12)–(2.17) become

    f=1,g=L,2Ltvi+vj2Lqjvi+Gj2Lvjvi+L2LsviLqi=LsLvi,

    which, for the integral curves of XL, are the Herglotz–Euler–Lagrange equations (2.9), (2.10) and (2.11). This SODE XLΓL is the Herglotz–Euler–Lagrange vector field for the Lagrangian L.

    The coordinate expression of the Herglotz–Euler–Lagrange vector field is

    ΓL=t+viqi+Wji(Lqj2Ltvjvk2LqkvjL2Lsvj+LsLvj)vi+Ls.

    An integral curve of ΓL fulfills the Herglotz–Euler–Lagrange equation for dissipative systems:

    ddt(Lvi)Lqi=LsLvi,˙s=L.

    Consider a cocontact Lagrangian system with configuration space R×Q×R, where Q is an n-dimensional manifold, equipped with coordinates (t,qi,s). Consider the product bundles R×TQ×R with natural coordinates (t,qi,vi,s) and R×TQ×R with natural coordinates (t,qi,pi,s), and the natural projections

    τ2:R×TQ×RTQ ,τ0:R×TQ×RR×Q×R,π2:R×TQ×RTQ ,π0:R×TQ×RR×Q×R.

    Let θ0Ω1(TQ) denote the Liouville 1-form of the cotangent bundle and let ω0=dθ0Ω2(TQ) be the canonical symplectic form of TQ. The local expressions of θ0 and ω0 are

    θ0=pidqi ,ω0=dqidpi.

    We will denote by θ=π2θ0Ω1(R×TQ×R) and ω=π2ω0Ω2(R×TQ×R) the pull-backs of θ0 and ω0 to R×TQ×R.

    Definition 3.1. The extended Pontryagin bundle is the Whitney sum

    W=R×TQ×QTQ×R

    equipped with the natural submersions

    ρ1:WR×TQ×R,ρ2:WR×TQ×R,ρ0:WR×Q×R.

    The extended Pontryagin bundle is endowed with natural coordinates (t,qi,vi,pi,s).

    Definition 3.2. A path γ:RW is holonomic if the path ρ1γ:RR×TQ×R is holonomic, i.e., it is the prolongation to R×TQ×R of a path RR×Q×R.

    A vector field XX(W) satisfies the second-order condition (or it is a SODE) if its integral curves are holonomic in W.

    A holonomic path in W has local expression

    γ(τ)=(t(τ),qi(τ),˙qi(τ),pi(τ),s(τ)).

    The local expression of a SODE in W is

    X=ft+viqi+Fivi+Gipi+gs.

    In the extended Pontryagin bundle W we have the following canonical structures:

    Definition 3.3.

    1. The coupling function in W is the map C:WR given by

    C(w)=i(vq)pq,

    where w=(t,vq,pq,s)W, qQ, vqTQ and pqTQ.

    2. The canonical 1-form is the ρ0-semibasic form Θ=ρ2θΩ1(W). The canonical 2-form is Ω=dΘ=ρ2ωΩ2(W).

    3. The canonical precontact 1-form is the ρ0-semibasic form η=dsΘΩ1(W).

    In natural coordinates,

    Θ=pidqi ,η=dspidqi ,dη=dqidpi=Ω.

    Definition 3.4. Let LC(R×TQ×R) be a Lagrangian function and consider L=ρ1LC(W). The Hamiltonian function associated to L is the function

    H=CL=piviL(t,qj,vj,s)C(W)

    Remark 3.5. In [46] the authors introduce the notion of characteristic distribution of a couple of one-forms (τ,η)Ω1(M) as C=kerτkerηkerdη. If C has constant rank, we say that the couple (τ,η) is of class cl(τ,η)=dimMC.

    The couple (τ,η) defines a precocontact structure on M if dτ=0, the characteristic distribution of the couple (τ,η) has constant rank and cl(τ,η)=2r+22. The triple (M,τ,η) is called a precocontact manifold. It is important to point out that if C=0, we have a cocontact manifold.

    Notice that the couple (dt,η) is a precocontact structure in the Pontryagin bundle W. Hence, (W,dt,η) is a precocontact manifold and (W,dt,η,H) is a precocontact Hamiltonian system. Thus, we do not have a unique couple (Rt,Rs) of Reeb vector fields. In fact, in natural coordinates, the general solution to (2.1) is

    Rt=t+Fivi,Rs=s+Givi,

    where Fi,GiC(W) are arbitrary functions. Despite this fact, the formalism is independent on the choice of the Reeb vector fields, as proved in Theorem 5.9 and Corollary 5.10 in [46]. Since the extended Pontryagin bundle W is trivial over R×R, the vector fields /t,/s can be canonically lifted to W and used as Reeb vector fields.

    Definition 3.6. The Lagrangian–Hamiltonian problem associated to the precocontact Hamiltonian system (W,dt,η,H) consists in finding the integral curves of a vector field ZX(W) such that

    (Z)=dH(LRsH+H)η+(1LRtH)dt.

    Equivalently,

    {i(Z)dη=dH(LRsH)η(LRtH)dt,i(Z)η=H,i(Z)dt=1. (3.1)

    Thus, the integral curves γ:IRW of Z are solutions to the system of equations

    {i(γ)dη=(dH(LRsH)η(LRtH)dt)γ,i(γ)η=Hγ,i(γ)dt=1. (3.2)

    Since (W,dt,η,H) is a precocontact system, equations (3.1) may not be consistent everywhere in W. In order to find a submanifold WfW (if possible) where equations (3.1) have consistent solutions, a constraint algorithm is needed. The implementation of this algorithm is described below.

    Consider the natural coordinates (t,qi,vi,pi,s) in W and the vector field ZX(W) with local expression

    Z=At+Biqi+Civi+Dipi+Es.

    The left-hand side of equations (3.1) read

    i(Z)dη=BidpiDidqi,i(Z)η=EpiBi,i(Z)dt=A.

    On the other hand, we have

    dH=vidpi+(piLvi)dviLtdtLqidqiLsds,(LRsH)η=Ls(dspidqi),(LRtH)dt=Ltdt.

    Thus, the second equation in (3.1) gives

    E=(Bivi)pi+L, (3.3)

    the third equation in (3.1) reads

    A=1, (3.4)

    and the first equation in (3.1) gives the conditions

    Bi=vi(coefficients in dpi), (3.5)
    pi=Lvi(coefficients in dvi), (3.6)
    Di=Lqi+piLs(coefficients in dqi). (3.7)

    Notice that

    ● Equations (3.5) are the SODE conditions. Hence, the vector field Z is a SODE. Then, it is clear that the holonomy condition arises straightforwardly from the Skinner–Rusk formalism.

    ● Conditions (3.6) are constraint functions defining the first constraint submanifold W1W. It is important to notice that the submanifold W1 is the graph of the Legendre map FL defined previously 2.8:

    W1={(vq,FL(t,vq,s))W(t,vq,s)R×TQ×R}.

    This implies that the Skinner–Rusk formalism implies the definition of the Legendre map.

    In virtue of conditions (3.5), (3.6), (3.7), the vector fields Z solution to equations (3.1) have the local expression

    Z=t+viqi+Civi+(Lqi+piLs)pi+Ls

    on the submanifold W1, where Ci are arbitrary function.

    The constraint algorithm continues by demanding the tangency of Z to the first constraint submanifold W1, in order to ensure that the solutions to the Lagrangian–Hamiltonian problem (the integral curves of Z) remain in the submanifold W1. The constraint functions defining W1 are

    ξ1j=pjLvjC(W).

    Imposing the tangency condition LZξ1j=0 on W1, we get

    Ci2Lvivj=2Ltvjvi2LqivjL2Lsvj+Lqj+pjLs (3.8)

    on W1. At this point, we have to consider two different cases:

    ● When the Lagrangian function L is regular, from (3.8) we can determine all the coefficients Ci. In this case, we have a unique solution and the algorithm finishes in just one step.

    ● In case the Lagrangian L is singular, equations (3.8) establish some relations among the functions Ci. In this some of them may remain undetermined and the solutions may not be unique. In addition, new constraint functions ξ2jC(W1) may arise. These new constraint function define a submanifold W2W1W. The constraint algorithm continues by imposing that Z is tangent to W2 and so on until we get a final constraint submanifold Wf (if possible) where we can find solutions to (3.1) tangent to Wf.

    Consider an integral curve γ(r)=(t(r),qi(r),vi(r),pi(r),s(r)) of the vector field ZX(W). We have that A=˙t, Bi=˙qi, Ci=˙vi, Di=˙pi and E=˙s. Then, equations (3.3), (3.4), (3.5), (3.6) and (3.7) lead to the local expression of (3.2). In particular,

    ● Equation (3.5) gives vi=˙qi, namely the holonomy condition.

    ● Combining equations (3.3) and (3.5), we see that

    ˙s=L, (3.9)

    which corresponds to equation (2.10).

    ● Equations (3.7) give

    ˙pi=Lqi+piLs=(Hqi+piHs),

    which are the second set of Hamilton's equations (2.4). These equations, on the first constraint submanifold W1, read

    ddt(Lvi)Lqi=LviLs,

    which are the Herglotz–Euler–Lagrange equations (2.11). Also, the first set of Hamilton's equations (2.4) comes from the definition of the Hamiltonian function 3.4 taking into account the holonomy condition.

    ● Combining equations (3.6) and (3.9), the tangency condition (3.8) yields the Herglotz–Euler–Lagrange equations (2.11). It is important to point out that, if the Lagrangian function L is singular, the Herglotz–Euler–Lagrange equations may be incompatible.

    The aim of this section is to show the equivalence between the Skinner–Rusk formalism, presented above, and the Lagrangian and Hamiltonian formalisms.

    Let us denote ȷ1:W1W as the natural embedding. Then

    ρ1ȷ1:W1R×TQ×R and ρ2ȷ1:W1R×TQ×R

    where

    (ρ1ȷ1)(W1)=R×TQ×R and (ρ2ȷ1)(W1)=P1R×TQ×R.

    Also P1 is a submanifold of R×TQ×R whenever L is an almost regular Lagrangian (see [46] for a precise definition of this concept in the cocontact setting) and we have the equality P1=R×TQ×R when L is regular. Since W1=graph(FL), it is diffeomorphic to R×TQ×R and (ρ1ȷ1) is a diffeomorphism. Similarly, under the assumption L is almost-regular, we have

    (ρ1ȷα)(Wα)=SαR×TQ×R and (ρ2ȷα)(Wα)=PαP1R×TQ×R.

    for every Wα submanifold obtained from the constraint algorithm. Let us denote by Wf the final constraint submanifold

    WfWαW1W.

    Figure 1 summarizes all the applications we have just introduced.

    Figure 1.  Recovering the Lagrangian and Hamiltonian formalisms.

    Theorem 4.1. Consider a path σ:IRW1. Therefore σ=(σL,σH) where σL=ρ1σ and σH=FLσL. Denote also σ0=ρ0σ:IRR×Q×R. Then:

    If σ:IRW1 fulfills equations (3.2) on Wf then σL is the prolongation of the curve σ0=ρ0σ, which is a solution of (2.7) on Sf. Also the path σH=FL˜σ0 is a solution to (2.2) on Pf.

    Conversely, let the path σ0:IRR×Q×R be a solution of (2.7) on Sf. Then, the curve σ=(σL,σH) is a solution of (3.2), where σL:=˜σ0 and σH:=FL˜σ0. Also path σH is a solution to (2.2) on Pf.

    Proof. The proof of this theorem can be easily worked out by using natural coordinates and taking into account equations (3.3)–(3.8).

    As a consequence, we obtain the following recovering theorems:

    Theorem 4.2. Let ZX(W) be a solution of (3.1) at least on Wf and tangent to Wf. Then:

    The vector field XX(R×TQ×R) defined by Xρ1=Tρ1Z is a holonomic vector field (tangent to Sf) which is a solution to the equations (2.8) with H=ρ1EL.

    Conversely, every holonomic vector field solution of the equations (2.8) can be recovered in this way from a vector field ZX(W) solution of (3.1) on Wf and tangent to Wf.

    Proof. Let ZX(W) be a solution to equations (3.1) with local expression

    Z=t+viqi+Civi+(Lqi+piLs)pi+Ls,

    where the functions Ci satisfy the relations (3.8). Thus, the vector field XX(R×TQ×R) has local expression

    X=t+viqi+Civi+Ls,

    where the functions Ci satisfy the relations (3.8). It is clear that X is a holonomic vector field and that it satisfies equations (2.8) with H=viLviL, where ρ1L=L.

    Moreover, every holonomic vector field XX(R×TQ×R) solution to (2.8) comes from a vector field ZX(W) by considering that pi=Lvi.

    Similarly, we also recover the Hamiltonian formalism:

    Theorem 4.3. Let ZX(W) be a solution of (3.1) at least on Wf and tangent to Wf. Then, the vector field YX(R×TQ×R) defined by Yρ2=Tρ2Z is a solution to the equations (2.3) on Pf and tangent to Pf, assuming that ρ2H=H.

    Conversely, every vector field solution of the equations (2.3) can be recovered in this way from a vector field ZX(W) solution of (3.1) on Wf and tangent to Wf, assuming that ρ2H=H.

    The proof of this theorem is straightforward by taking natural coordinates in W and in R×TQ×R.

    Remark 4.4. The results presented in this section are analogous to those obtained for the Skinner–Rusk formalism for time-dependent dynamical systems. Intrinsic proves of the corresponding theorems can be found in [3] (see also [4]).

    Remark 4.5. It is important to remark that in the case of singular Lagrangians, the two constraint algorithms are equivalent only when the second-order condition is imposed as an additional condition in the Lagrangian side [1,49].

    The Duffing equation [47,48], named after G. Duffing, is a non-linear second-order differential equation which can be used to model certain damped and forced oscillators. The Duffing equation is

    ¨x+δ˙x+αx+βx3=γcosωt, (5.1)

    where α,β,γ,δ,ω are constant parameters. Notice that if γ=0, which means that the system does not depend on time, we are in the case of contact mechanics. On the other hand if δ=0, namely there is no damping, we have a cosymplectic system. Finally, if β=δ=γ=0, we obtain the equation of a simple harmonic oscillator. In physical terms, equation (5.1) models a damped forced oscillator with a stiffness different from the one obtained by Hooke's law.

    It is clear that this system is not Hamiltonian nor Lagrangian in a standard sense. However, we will see that we can provide a geometric description of it as a time-dependent contact Hamiltonian system.

    Consider the configuration space Q=R with canonical coordinate (x). Consider now the product bundle R×TQ×R with natural coordinates (t,x,v,s) and the Lagrangian function

    L:R×TQ×RR

    given by

    L(t,x,v,s)=12v212αx214βx4δs+γxcosωt. (5.2)

    Let W be the extended Pontryagin bundle

    W=R×TQ×QTQ×R

    equipped with natural coordinates (t,x,v,p,s). The coupling function is C(t,x,v,p,s)=pv. The couple of one-forms (dt,η=dspdx) define a precocontact structure on W. The dissipative Reeb vector field is Rs=/s and the time Reeb vector field is Rt=/t. We also have that dη=dxdp. The Hamiltonian function associated to the Lagrangian function (5.2) is the function

    H=CL=pv12v2+12αx2+14βx4+δsγxcosωt C(W).

    We have that

    dH=γωxsin(ωt)dt+(αx+βx3γcosωt)dx+(pv)dv+vdp+δds,

    and hence

    dHRs(H)ηRt(H)dt=(αx+βx3+δpγcosωt)dx+(pv)dv+vdp.

    Given a vector field ZX(W) with local expression

    Z=At+Bx+Cv+Dp+Es,

    equations (3.1) give the conditions

    {A=1,B=v,D=αxβx3δp+γcosωt,pv=0,E=pBH=L.

    Thus, the vector field Z is a SODE and has the expression

    Z=t+vx+Cv+(αxβx3δp+γcosωt)p+Ls,

    and we have the constraint function

    ξ1=pv=0,

    which defines the first constraint submanifold W1W. The constraint algorithm continues by demanding the tangency of the vector field Z to W1. Hence,

    0=LZξ1=αxβx3δv+γcosωtC,

    determining the last coefficient of the vector field Z

    C=αxβx3δv+γcosωt,

    and no new constraints appear. Then, we have the unique solution

    Z=t+vx+(αxβx3δv+γcosωt)v+(αxβx3δv+γcosωt)p+Ls.

    Projecting onto each factor of W, namely using the projections ρ1:WR×TQ×R and ρ2:WR×TQ×R, we can recover both the Lagrangian and the Hamiltonian vector fields. In the Lagrangian formalism we obtain the holonomic vector field XX(R×TQ×R) given by

    X=t+vx+(αxβx3δv+γcosωt)v+Ls.

    We can see that the integral curves of X satisfy the Duffing equation

    ¨x+δ˙x+αx+βx3=γcosωt.

    On the other hand, projecting with ρ2, we obtain the Hamiltonian vector field YX(R×TQ×R) given by

    Y=t+px+(αxβx3δp+γcosωt)p+(12p212αx214βx4δs+γxcosωt)s.

    Notice that the integral curves of Y also satisfy the Duffing equation (5.1). Thus, we have shown that although the Duffing equation cannot be formulated as a standard Hamiltonian system, it can be described as a cocontact Lagrangian or Hamiltonian system.

    In this example we will consider a system of time-dependent mass with an engine providing an ascending force F>0 and subjected to a drag proportional to the square of the velocity.

    Let Q=R with coordinate (y) be the configuration manifold of our system and consider the Lagrangian function

    L:R×TQ×RR

    given by

    L(t,y,v,s)=12m(t)v2+m(t)g2γ(e2γy1)2γvs+12γF, (5.3)

    where γ is the drag coefficient and the mass is given by the monotone decreasing function m(t). Consider the extended Pontryagin bundle

    W=R×TQ×QTQ×R

    endowed with canonical coordinates (t,y,v,p,s), the coupling function C(t,y,v,p,s)=pv and the 1-forms τ=dt and η=dspdy. We have that (W,τ,η) is a precocontact manifold, with Reeb vector fields Rt=/t, Rs=/s. The Hamiltonian function H associated with the Lagrangian (5.3) is

    H=CL=pv12m(t)v2m(t)g2γ(e2γy1)+2γvs12γF C(W).

    In this case, we have

    dHRs(H)ηRt(H)dt==vdp+(pm(t)v+2γs)dv+(m(t)ge2γy+2γvp)dy+Htdt+2γvds.

    Consider a vector field ZX(W) with coordinate expression

    Z=At+By+Cv+Dp+Es.

    Then, equations (3.1) read

    {A=1,B=v,D=m(t)ge2γy2γvp,pm(t)v+2γs=0,E=pBH=L.

    Hence, the vector field Z is a SODE and has the expression

    Z=t+vy+Cv+(m(t)ge2γy2γvp)p+Ls,

    and we obtain the constraint function

    ξ1=pm(t)v+2γs,

    defining the first constraint submanifold W1W. Demanding the tangency of Z to W1, we obtain

    LZξ1=γm(t)v2m(t)C˙m(t)vm(t)g+F,

    which determines the remaining coefficient of the vector field Z:

    C=Fm(t)γv2˙m(t)m(t)vg

    and no new constraints appear. Thus, we have the unique solution

    Z=t+vy+(Fm(t)γv2˙m(t)m(t)vg)v+(m(t)ge2γy2γvp)p+Ls.

    We can project onto each factor of W by using the projections ρ1:WR×TQ×R and ρ2:WR×TQ×R thus recovering the Lagrangian and Hamiltonian formalisms. In the Lagrangian formalism we get the SODE XX(R×TQ×R) given by

    X=t+vy+(Fm(t)γv2˙m(t)m(t)vg)v+Ls.

    The integral curves of X satisfy the second-order differential equation

    {˙y=v,˙v=Fm(t)γv2˙m(t)m(t)vg,

    which can be rewritten as

    {˙y=v,ddt(m(t)v)=Fm(t)gγm(t)v2.

    Consider a system where we have a charged particle with mass m and charge k in the plane immersed in a stationary electric field E=(E1,E2,E3)=ϕ and subjected to a time-dependent constraint given by f(t,q)=0, where denotes the Euclidean gradient and q=(x,y,z). Consider the phase space TR4, endowed with natural coordinates (x,y,z,λ;vx,vy,vz,vλ), and the contact Lagrangian function L:R×TR4×RR given by

    L(t,q,λ,v,vλ,s)=12mv2kϕ(q)+λf(t,q)γs, (5.4)

    where v=(vx,vy,vz) and v=v2x+v2y+v2z and γ is the friction coefficient. Since we have introduced the restriction f(t,q)=0 via a Lagrange multiplier, it is clear that the Lagrangian (5.4) is singular.

    Consider the extended Pontryagin bundle

    W=R×TR4×R4TR4×R

    endowed with natural coordinates (t,x,y,z,λ,vx,vy,vz,vλ,px,py,pλ,s). We have that (dt,η), where

    η=dspxdxpydypzdzpλdλ,

    is a precocontact structure on W. The coupling function is C=pxvx+pyvy+pzvz+pλvλ and the Hamiltonian function H=CL associated to the Lagrangian L is

    H=pxvx+pyvy+pzvz+pλvλ12mv2+kϕ(q)λf(t,q)+γs.

    Thus,

    dHRs(H)ηRt(H)dt=vxdpx+vydpy+vzdpz+vλdpλ+(pxmvx)dvx+(pymvy)dvy+(pzmvz)dvz+pλdvλ+(kϕxλfx+γpx)dx+(kϕyλfy+γpy)dy+(kϕzλfz+γpz)dz+(γpλf(t,q))dλ.

    Let Z be a vector field on W with local expression

    Z=At+Bxx+Byy+Bzz+Bλλ+Cxvx+Cyvy+Czvz+Cλvλ+Dxpx+Dypy+Dzpz+Dλpλ+Es,

    then, equations (3.1) yield the conditions

    A=1,Bx=vx,By=vy,Bz=vz,Bλ=vλ,Dx=λfxkϕxγpx,Dy=λfykϕyγpy,Dz=λfzkϕzγpz,Dλ=f(t,q)γpλ,pxmvx=0,pymvy=0,pzmvz=0,pλ=0,E=pxBx+pyBy+pzBz+pλBλH=L.

    Thus, the vector field Z is

    Z=t+vxx+vyy+vzz+vλλ+Cxvx+Cyvy+Czvz+Cλvλ+(λfxkϕxγmvx)px+(λfykϕyγmvy)py+(λfzkϕzγmvz)pzf(t,q)pλ+Ls,

    and we get the constraints

    ξx1pxmvx=0 ,ξy1pymvy=0 ,ξz1pzmvz=0 ,ξλ1pλ=0,

    defining the first constraint submanifold W1W. The constraint algorithm continues by demanding the tangency of the vector field Z to the submanifold W1:

    LZξx1=λfxkϕxγpxmCx=0LZξy1=λfykϕyγpymCy=0LZξz1=λfzkϕzγpzmCz=0LZξλ1=f(t,q)=0,

    thus determining the coefficients Cx,Cy,Cz. In addition, we have obtained the constraint function

    ξ2f(t,q)=0,

    defining the second constraint submanifold W2W1W. Then, the vector field Z has the form

    Z=t+vxx+vyy+vzz+vλλ+1m(λfxkϕxγmvx)vx+1m(λfykϕyγmvy)vy+1m(λfzkϕzγmvz)vz+Cλvλ+(λfxkϕxγmvx)px+(λfykϕyγmvy)py+(λfzkϕzγmvz)pz+Ls.

    Imposing the tangency to W2, we condition

    LZξ2=ft+vxfx+vyfy+vzfz=0,

    which is a new constraint function ξ3ft+vxfx+vyfy+vzfz=ft+vf=0. This process has to be iterated until no new constraints appear, depending on the function f.

    Now we are going to consider the particular case where f(t,q)=zt. In this case,

    ξ3vz1=0.

    Thus, imposing the tangency of Z to ξ3 we get the condition

    LZξ3=1m(λkϕzγmvz)=0,

    giving a new constraint function

    ξ4λkϕzγm=0.

    Then, the vector field Z is

    Z=t+vxx+vyy+z+vλλ+1m(kϕxγmvx)vx+1m(kϕyγmvy)vy+Cλvλ+(kϕxγmvx)px+(kϕyγmvy)py+Ls.

    Imposing the tangency condition with respect to ξ4, we obtain that

    vλ=k(2ϕxzvx+2ϕyzvy+2ϕy2).

    The tangency condition with respect to this constraint determines the last coefficient of the vector field,

    Cλ=k(v2x3ϕx2z+v2y3ϕy2z+(1+2vx)vy3ϕxyz++2vx3ϕxz2+3ϕyz2+3ϕz3(kmϕx+γvx)2ϕxz(kmϕy+γvy)2ϕyz),

    and no new constraints arise. In conclusion, the vector field Z has local expression

    Z=t+vxx+vyy+z+k(2ϕxzvx+2ϕyzvy+2ϕy2)λ+1m(kϕxγmvx)vx+1m(kϕyγmvy)vy+Cλvλ+(kϕxγmvx)px+(kϕyγmvy)py+Ls.

    The integral curves of the vector field Z satisfy the system of differential equations

    m¨x=kϕxγmvx,m¨y=kϕyγmvy,z=t.

    In Figure 2 one can see the trajectory of a charged particle with charge k=2104 and mass m=1 in the electric field induced by a charge fixed in the origin with charge 2104 and in absence of gravity. The friction coefficient is γ=0.3 and the initial configuration of the system is q(0)=(2,0,0), v(0)=(0,10,0). As indicated above, the particle is subjected to the restriction z=t.

    Figure 2.  Trajectory of the charged particle (blue) and position of the fixed charge (red).

    We have generalized the Skinner–Rusk unified formalism for time-dependent contact systems. This framework allows to skip the second-order problem, since this condition is recovered in the first step of the constraint algorithm for both regular and singular Lagrangians. This makes this formalism especially interesting when working with systems described by singular Lagrangians.

    The key tool of this formalism is the Pontryagin bundle W=R×TQ×TQ×R and its canonical precocontact structure. Imposing the compatibility of the dynamical equations on W we obtain a set of constraint function defining a submanifold W1, which coincides with the graph of the Legendre map, the second-order conditions and the Herglotz–Euler–Lagrange equations. We have also shown that the Skinner–Rusk formalism for cocontact systems is equivalent to both the Hamiltonian and the Lagrangian formalisms (in this last case when imposing the second order condition).

    In addition, we have described in full detail three examples in order to illustrate this method: the Duffing equation, an ascending particle with time-dependent mass and quadratic drag, and a charged particle in a stationary electric field with a time-dependent constraint.

    The formulation introduced in this paper will permit to extend the k-contact formalism for field theories with damping introduced in [44,45] to non-autonomous field theories. This new formulation will permit to describe many field theories, such as damped vibrating membranes with external forces, Maxwell's equations with charges and currents, etc.

    XR acknowledges the financial support of the Ministerio de Ciencia, Innovación y Universidades (Spain), project PGC2018-098265-B-C33 and of Novee Idee 2B-POB II project PSP: 501-D111-20-2004310 funded by the "Inicjatywa Doskonałości - Uczelnia Badawcza" (IDUB) program.

    All authors declare no conflicts of interest in this paper.

  • This article has been cited by:

    1. Manuel de León, Manuel Lainz, Asier López-Gordón, Xavier Rivas, Hamilton–Jacobi theory and integrability for autonomous and non-autonomous contact systems, 2023, 187, 03930440, 104787, 10.1016/j.geomphys.2023.104787
    2. Manuel de León, Jordi Gaset, Miguel C Muñoz-Lecanda, Xavier Rivas, Narciso Román-Roy, Multicontact formulation for non-conservative field theories, 2023, 56, 1751-8113, 025201, 10.1088/1751-8121/acb575
    3. Xavier Rivas, Nonautonomous k-contact field theories, 2023, 64, 0022-2488, 033507, 10.1063/5.0131110
    4. Javier de Lucas, Xavier Rivas, Contact Lie systems: theory and applications, 2023, 56, 1751-8113, 335203, 10.1088/1751-8121/ace0e7
    5. R. Azuaje, A. Bravetti, Scaling symmetries and canonoid transformations in Hamiltonian systems, 2024, 21, 0219-8878, 10.1142/S0219887824500774
    6. Jordi Gaset, Manuel Lainz, Arnau Mas, Xavier Rivas, The Herglotz variational principle for dissipative field theories, 2024, 01, 2972-4589, 153, 10.1142/S2972458924500060
    7. Begüm Ateşli, Oğul Esen, Manuel de León, Cristina Sardón, On non-autonomous Hamiltonian dynamics, dual spaces, and kinetic lifts, 2024, 01672789, 134504, 10.1016/j.physd.2024.134504
  • Reader Comments
  • © 2007 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
通讯作者: 陈斌, bchen63@163.com
  • 1. 

    沈阳化工大学材料科学与工程学院 沈阳 110142

  1. 本站搜索
  2. 百度学术搜索
  3. 万方数据库搜索
  4. CNKI搜索

Metrics

Article views(2485) PDF downloads(487) Cited by(9)

Other Articles By Authors

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog