Galerkin approximations for the optimal control of nonlinear delay differential equations

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Galerkin approximations for the optimal control of nonlinear delay differential equations

Mickaël Chekroun, Axel Kröner, Honghu Liu

To cite this version:

Mickaël Chekroun, Axel Kröner, Honghu Liu. Galerkin approximations for the optimal control of nonlinear delay differential equations. [Research Report] INRIA. 2017. �hal-01534673�

DIFFERENTIAL EQUATIONS

MICKAËL D. CHEKROUN, AXEL KRÖNER, AND HONGHU LIU

ABSTRACT. Optimal control problems of nonlinear delay differential equations (DDEs) are considered for which we propose a general Galerkin approximation scheme built from Koornwinder polynomials.

Error estimates for the resulting Galerkin-Koornwinder approximations to the optimal control and the value function, are derived for a broad class of cost functionals and nonlinear DDEs. The approach is il- lustrated on a delayed logistic equation set not far away from its Hopf bifurcation point in the parameter space. In this case, we show that low-dimensional controls for a standard quadratic cost functional can be efficiently computed from Galerkin-Koornwinder approximations to reduce at a nearly optimal cost the oscillation amplitude displayed by the DDE’s solution. Optimal controls computed from the Pon- tryagin’s maximum principle (PMP) and the Hamilton-Jacobi-Bellman equation (HJB) associated with the corresponding ODE systems, are shown to provide numerical solutions in good agreement. It is fi- nally argued that the value function computed from the corresponding reduced HJB equation provides a good approximation of that obtained from the full HJB equation.

CONTENTS

1. Introduction 1

2. Optimal control of DDEs: Galerkin-Koornwinder approximations 3 2.1. Recasting a DDE into an infinite–dimensional evolution equation 3

2.2. Galerkin-Koornwinder approximation of DDEs 5

2.3. Galerkin-Koornwinder approximation of nonlinear optimal control problems associated

with DDEs 7

3. Application to an oscillation amplitude reduction problem 13

3.1. Optimal control of the Wright equation 13

3.2. Galerkin-Koornwinder approximation of the optimal control problem 16

3.3. Numerical results 17

4. Towards the approximation of HJB equations associated with DDEs 21

5. Concluding remarks 26

Acknowledgments 26

References 26

1. INTRODUCTION

Delay differential equations (DDEs) are widely used in many fields such as biosciences [20,41,45, 54], climate dynamics [15,24,40,56,58], chemistry and engineering [26,38,43,46,55]. The inclusion of time-lag terms are aimed to account for delayed responses of the modeled systems to either internal or external factors. Examples of such factors include incubation period of infectious diseases [41], wave propagation [10,56], or time lags arising in engineering [38] to name a few.

Key words and phrases. Delay differential equations, Galerkin approximations, Hamilton-Jacobi-Bellman equation, Hopf bifurcation, Koornwinder polynomials, Optimal control.

1

In contrast to ordinary differential equations (ODEs), the phase space associated even with a scalar DDE is infinite-dimensional, due to the presence of delay terms implying thus the knowledge of a function as an initial datum, namely the initial history to solve the Cauchy problem [18,26]; cf. Sec- tion2. The infinite-dimensional nature of the state space renders the related optimal control problems more challenging to solve compared to the ODE case. It is thus natural to seek for low-dimensional approximations to alleviate the inherent computational burden. The need of such approximations is particularly relevant when (nearly) optimal controls in feedback form are sought, to avoid solving the infinite-dimensional Hamilton-Jacobi-Bellman (HJB) equation associated with the optimal control problem of the original DDE.

For optimal control oflinearDDEs, averaging methods as well as spline approximations are often used for the design of (sub)optimal controls in either open-loop form or feedback form; see e.g. [1,4, 34,42] and references therein. The usage of spline approximations in the case of open-loop control fornonlinearDDEs has also been considered [44], but not in a systematic way. Furthermore, due to the locality of the underlying basis functions, the use of e.g. spline approximations for the design of feedback controls leads, especially in the nonlinear case, to surrogate HJB equations of too high dimension to be of practical interest.

In this article, we bring together a recent approach dealing with the Galerkin approximations for optimal control problems of general nonlinear evolution equations in a Hilbert space [11], with tech- niques for the finite-dimensional and analytic approximations of nonlinear DDEs based on Legendre- type polynomials, namely the Koornwinder polynomials [9]. Within the resulting framework, we adapt ideas from [11, Sect. 2.6] to derive—for a broad class of cost functionals and nonlinear DDEs—

error estimates for the approximations of the value function and optimal control, associated with the Galerkin-Koornwinder (GK) systems; see Theorem2.1and Corollary2.1. These error estimates are formulated in terms of residual energy, namely the energy contained in the controlled DDE solutions as projected onto the orthogonal complement of a given Galerkin subspace.

Our approach is then applied to a Wright equation¹ with the purpose of reducing (optimally) the amplitude of the oscillations displayed by the DDE’s solution subject to a quadratic cost; see Sect.3. For this model, the oscillations emerge through a supercritical Hopf bifurcation as the delay parameter τ crosses a critical value, τc, from below. We show, for τ above and close to τc, that a suboptimal controller at a nearly optimal cost can be synthesized from a 2D projection of the 6D GK system; see Sect.3.3. The projection is made onto the space spanned by the first two eigenvectors of the linear part of the 6D GK system; the sixth dimension constituting for this example the minimal dimension—using Koornwinder polynomials—to resolve accurately the linear contribution to the oscillation frequency of the DDE’s solution; see (3.29) in Sect.3.3.

Using the resulting 2D ODE system, the syntheses of (sub)optimal controls obtained either from application of the Pontryagin’s maximum principle (PMP) or by solving the associated HJB equation, are shown to provide nearly identical numerical solutions. Given the good control skills obtained from the controller synthesized from the reduced HJB equation, one can reasonably infer that the corresponding “reduced” value function provides thus a good approximation of the “full” value function associated with the DDE optimal control problem; see Sect.4.

The article is organized as follows. In Sect.2, we first introduce the class of nonlinear DDEs consid- ered in this article and recall in Sect.2.1how to recast such DDEs into infinite–dimensional evolution equations in a Hilbert space. We summarize then in Sect.2.2the main tools from [9] for the analytic determination of GK systems approximating DDEs. In Sect. 2.3, we derive error estimates for the approximations of the value function and optimal control, obtained from the GK systems. Our ap- proach is applied to the Wright equation in Sect.3. Section3.1introduces the optimal control problem

1Analogue to a logistic equation with delay; see Sect.3.1.

associated with this equation and its abstract functional formulation following Sect.2.1. The explicit GK approximations of the corresponding optimal control problem are derived in Sect.3.2. Numeri- cal results based on PMP from a projected 2D GK system are then presented in Sect.3.3. In Sect.4, we show that by solving numerically the associated reduced HJB equation, a (sub)optimal control in a feedback form can be synthesized at a nearly optimal cost. Finally, directions to derive reduced systems of even lower dimension from GK approximations are outlined in Sect.5.

2. OPTIMAL CONTROL OFDDES: GALERKIN-KOORNWINDER APPROXIMATIONS

In this article, we are concerned with optimal control problems associated with nonlinear DDEs of the form

(2.1) dm(t)

dt =am(t) +bm(t−τ) +c Z _t

t−τ

m(s)ds+F

m(t),m(t−τ), Z _t

t−τ

m(s)ds ,

where a, b and c are real numbers, τ > 0 is the delay parameter, and F is a nonlinear function.

We refer to Sect.2.3for assumption on F. To simplify the presentation, we restrict ourselves to this class of scalar DDEs, but nonlinear systems of DDEs involving several delays are allowed by the approximation framework of [9] adopted in this article.

Note that even for scalar DDEs such as given by Eq. (2.1), the associated state space is infinite- dimensional. This is due to the presence of time-delay terms, which require providing initial data over an interval [−τ, 0], where τ > 0 is the delay. It is often desirable, though, to have low- dimensional ODE systems that capture qualitative features, as well as approximating certain quanti- tative aspects of the DDE dynamics.

The approach adopted here consists of approximating the infinite-dimensional equation (2.1) by a finite-dimensional ODE system built from Koornwinder polynomials following [9], and then solve reduced-order optimal control problems aimed at approximating a given optimal control problem associated with Eq. (2.1).

To justify this approach, Eq. (2.1) needs first to be recast into an evolution equation, in order to apply in a second step, recent results dealing with the Galerkin approximations to optimal control problems governed by general nonlinear evolution equations in Hilbert spaces, such as presented in [11].

As a cornerstone, rigorous Galerkin approximations to Eq. (2.1) are crucial, and are recalled here- after. We first recall how a DDE such as Eq. (2.1) can be recast into an infinite-dimensional evolution equation in a Hilbert space.

2.1. Recasting a DDE into an infinite–dimensional evolution equation. The reformulation of a sys- tem of DDEs into an infinite-dimensional ODE is classical. For this purpose, two types of function spaces are typically used as state space: the space of continuous functionsC([−τ, 0];R^d), see [26], and the Hilbert spaceL²([−_{τ, 0})_;R^d)×_R^d, see [18]. The Banach space setting of continuous functions has been extensively used in the study of bifurcations arising from DDEs, see e.g. [8,17,19,21,37,47,60], while the Hilbert space setting is typically adopted for the approximation of DDEs or their optimal control; see e.g. [1–3,9,22,28,33,35,36,42].

Being concerned with the optimal control of scalar DDEs of the form (2.1), we adopt here the Hilbert space setting and consider in that respect our state space to be

(2.2) H _:= L²([−_{τ, 0})_;R)×_R,

endowed with the following inner product, defined for any(f₁,γ₁)and(f2,γ2)inHby:

(2.3) h(f₁,γ₁),(f2,γ2)i_H:= ¹ τ

Z ₀

−τ

f₁(θ)f2(θ)dθ+γ₁γ2.

We will also make use sometimes of the following subspace ofH:

(2.4) W := H¹([−τ, 0);R)×_R,

whereH¹([−τ, 0);R)denotes the standard Sobolev subspace ofL²([−τ, 0);R).

Let us denote bym_tthe time evolution of the history segments of a solution to Eq. (2.1), namely (2.5) mt(θ):=m(t+θ), t≥0, θ ∈[−τ, 0].

Now, by introducing the new variable

(2.6) y(t)_:= (m_t,m(t)) = (m_t,m_t(₀))_, t≥_0,

Eq. (2.1) can be rewritten as the following abstract ODE in the Hilbert spaceH:

(2.7) dy

dt =Ay+F(y), where the linear operatorA_: D(A)⊂ W → His defined as

(2.8) [A_Ψ](θ):=









 d⁺Ψ^D

dθ , θ ∈[−τ, 0),

aΨ^S+_bΨ^D(−τ) +c Z ₀

−τ

Ψ^D(s)ds, θ =0, with the domainD(A)ofAgiven by (cf. e.g. [33, Prop. 2.6])

(2.9) D(A) =ⁿ(_Ψ^D,Ψ^S)∈ H:Ψ^D ∈ H¹([−τ, 0);R), lim

θ→0⁻Ψ^D(θ) =_Ψ^So. The nonlinearityF: H → His here given, for allΨ= (_Ψ^D,Ψ^S)inH, by

(2.10) [F(_Ψ)](θ):=







0, θ ∈ [−τ, 0),

F

Ψ^S,Ψ^D(−τ),R0

−τΨ^D(s)ds

, θ =0,

WithD(A)such as given in (2.9), the operatorAgenerates a linearC₀-semigroup onHso that the Cauchy problem associated with the linear equation ˙y=Ayis well-posed in the Hadamard’s sense;

see e.g [18, Thm. 2.4.6]. The well-posedness problem for the nonlinear equation depends obviously on the nonlinear termF and we refer to [9] and references therein for a discussion of this problem;

see also [59].

For later usage, we recall that amild solutionto Eq. (2.7) over[_0,T]with initial datumy(₀) =x in H, is an elementyinC([0,T],H)that satisfies the integral equation

(2.11) y(t) =T(t)x+

Z _t

0

T(t−s)F(y(s))ds, ∀t ∈[_0,T]_,

where(T(t))_t≥0 denotes the C0-semigroup generated by the operatorA: D(A) → H. This notion of mild solutions extend naturally when a control term C(u(t)) is added to the RHS of Eq. (2.7);

in such a case the definition of a mild solution is amended by the presence of the integral term Rt

0 T(t−s)C(u(s))dsto the RHS of Eq. (2.11).

2.2. Galerkin-Koornwinder approximation of DDEs. Once a DDE is reframed into an infinite- dimensional ODE in H, the approximation problem by finite-dimensional ODEs can be addressed in various ways. In that respect, different basis functions have been proposed to decompose the state spaceH; these include, among others, step functions [1,36], splines [2,3], and orthogonal polynomial functions, such as Legendre polynomials [28,33]. Compared with step functions or splines, the use of orthogonal polynomials leads typically to ODE approximations with lower dimensions, for a given precision [2,28]. On the other hand, classical polynomial basis functions do not live in the domain of the linear operator underlying the DDE, which leads to technical complications in establishing convergence results [28,33]; see [9, Remark 2.1-(iii)].

One of the main contributions of [9] consisted in identifying that Koornwinder polynomials [39]

lie in the domain D(A)of linear operators such as Agiven in (2.8), allowing in turn for adopting a classical Trotter-Kato (TK) approximation approach from theC₀-semigroup theory [25,49] to deal with the ODE approximation of DDEs such as given by Eq. (2.1). The TK approximation approach can be viewed as the functional analysis operator version of the Lax equivalence principle.²The work [9]

allows thus for positioning the approximation problem of DDEs within a well defined territory. In particular, as pointed out in [11] and discussed in Sect.2.3below, the optimal control of DDEs benefits from the TK approximation approach.

In this section, we focus on another important feature pointed out in [9] for applications, namely, Galerkin approximations of DDEs built from Koornwinder polynomials can be efficiently computed via simple analytic formulas; see [9, Sections 5-6 and Appendix C]. We recall below the main elements to do so referring to [9] for more details.

First, let us recall that Koornwinder polynomialsKnare obtained from Legendre polynomials Ln

according to the relation

(2.12) K_n(s):=−(1+s)^d

dsL_n(s) + (n²+n+1)L_n(s), s∈[−1, 1], n∈_N;

see [39, Eq. (2.1)].

Koornwinder polynomials are known to form an orthogonal set for the following weighted inner product with a point-mass on[−1, 1],

(2.13) µ(dx) = ¹

2dx+δ₁,

whereδ₁denotes the Dirac point-mass at the right endpointx =1; see [39]. In other words, (2.14)

Z ₁

−1K_n(s)K_p(s)dµ(s) = ¹ 2

Z ₁

−1K_n(s)K_p(s)ds+K_n(1)K_p(1)

=0, ifp6= n.

It is also worthwhile noting that the sequence given by

(2.15) {K_n := (Kn,Kn(1)):n∈_N} forms an orthogonal basis of the product space

(2.16) E := L²([−1, 1);R)×_R,

whereE is endowed with the following inner product:

(2.17) h(f,a),(g,b)i_E = ¹ 2

Z ₁

−1 f(s)g(s)ds+ab, (f,a),(g,b)∈ E. The norm induced by this inner product will be denoted hereafter byk · k_E.

2i.e., if “consistency” and “stability” are satisfied, then “convergence” holds, and reciprocally.

From the original Koornwinder basis given on the interval [−1, 1], orthogonal polynomials on the interval [−_{τ, 0}]for the inner product (2.3) can now be easily obtained by using a simple linear transformationT defined by:

(2.18) T : [−τ, 0]→[−1, 1], θ7→ 1+ ^2θ τ . Indeed, forK_ngiven by (2.12), let us define the rescaled polynomialK^τ_nby (2.19)

K_n^τ: [−τ, 0]→_R, θ 7→K_n

1+^2θ τ

, n∈ _N. As shown in [9], the sequence

(2.20) {K^τ_n:= (K_n^τ,K_n^τ(0)):n∈_N}

forms an orthogonal basis for the space H = L²([−τ, 0);R)×_R endowed with the inner product h·,·i_Hgiven in (2.3). Note that sinceK_n(1) =1 [9, Prop. 3.1], we have

(2.21) K^τ_n(0) =1.

As shown in [9, Sect. 5 & Appendix C], (rescaled) Koornwinder polynomials allow for analytical Galerkin approximations of general nonlinear systems of DDEs. In the case of a nonlinear scalar DDE such as Eq. (2.1), theN^thGalerkin-Koornwinder (GK) approximation,y_N, is obtained as

(2.22) y_N(t) =

N−1 j

∑

ξ_j^N(t)K^τ_j, t≥0, where theξ^N_j (t)solve theN-dimensional ODE system

(2.23)

dξ^N_j

dt = ¹

kK_jk²_E

N−1 n

∑

a+_bKn(−1) +_cτ(_2δn,0−1) + ²

τ

n−1 k

∑

a_n,k δ_j,kkK_jk²_E−₁ξ_n^N(t)

+ ¹

kK_jk²_E^F

N−1 n

∑

ξ^N_n(t),

N−1 n

∑

ξ_n^N(t)K_n(−1),τξ₀^N(t)−τ

N−1 n

∑

ξ_n^N(t)

! , 0≤ j≤ N−1.

Here the Kronecker symbol δ_j,k has been used, and the coefficients a_n,k are obtained by solving a triangular linear system in which the right hand side has explicit coefficients depending onn; see [9, Prop. 5.1].

In practice, an approximationm_N(t)ofm(t)solving Eq. (2.1) is obtained as the state part (at timet) ofy_N given by (2.22) which, thanks to the normalization propertyK^τ_n(0) =1 given in (2.21), reduces to

(2.24) m_N(t) =

N−1 j

∑

ξ_j^N(t).

For later usage, we rewrite the above GK system into the following compact form:

(2.25) dξ_N

dt = Mξ_N+G(ξ_N),

where Mξ_N denotes the linear part of Eq. (2.23), and G(ξ_N)the nonlinear part. Namely, M is the N×Nmatrix whose elements are given by

(2.26)

(M)_j,n= ¹ kK_jk²_E

a+bK_n(−1) +cτ(2δ_n,0−1) + ²

τ

n−1 k

∑

a_n,k δ_j,kkK_jk²_E−1 ,

wherej,n = 0,· · · ,N−1, and the nonlinear vector fieldG: R^N → _R^N, is given component-wisely by

(2.27) G_j(ξ_N) = ¹ kK_jk²_E^F

N−1 n

∑

ξ_n^N(t),

N−1 n

∑

ξ_n^N(t)K_n(−1),τξ₀^N(t)−τ

N−1 n

∑

ξ^N_n(t)

! , for each 0≤ j≤ N−1.

For rigorous convergence results of GK systems (2.25) for Eq. (2.1) whenFdoes not depend on the delay termm(t−τ), we refer to [9, Sect. 4]. When Fdoes depend onm(t−τ), convergence is also believed to hold, as supported by the striking numerical results shown in [9, Sect. 6].

2.3. Galerkin-Koornwinder approximation of nonlinear optimal control problems associated with DDEs.

2.3.1. Preliminaries. In this section, we outline how the recent results of [11] concerned with the Galerkin approximations to optimal control problems governed by nonlinear evolution equations in Hilbert spaces, apply to the context of nonlinear DDEs when these approximations are built from the Koornwinder polynomials. We provide thus here further elements regarding the research program outlined in [11, Sect. 4].

We consider here, given a finite horizonT >0, the following controlled version of Eq. (2.7), (2.28)

dy

ds =Ay+F(y) +C(u(s)), s∈(t,T], u ∈ U_ad[t,T], y(t) =x ∈ H.

The (possibly nonlinear) mappingC : V → His assumed to be such thatC(0) = 0, and the control u(s)lies in a bounded subsetU of a separable Hilbert spaceVpossibly different fromH. Assump- tions about the set,U_ad[t,T], of admissible controls,u, is made precise below; see (2.36).

Here our Galerkin subspaces are spanned by the rescaled Koornwinder polynomials, namely (2.29) H_N =span{K₀^τ,· · · ,K^τ_N−1}, N =1, 2,· · · ,

whereK^τ_N is defined in (2.20). Denoting byΠN the projector ofH ontoH_N, the corresponding GK approximation of Eq. (2.28) is then given by

(2.30)

dy_N

ds =A_Ny_N+_ΠNF(y_N) +_ΠNC(u(s)), s∈ (t,T], u∈ U_ad[t,T], yN(t) =_ΠNx∈ H_N,

whereA_N :=_ΠNA_ΠN.

Given a cost functional assessed along a solutiony_t,x(·;u)of Eq. (2.28) driven byu (2.31) J_t,x(u)_:=

Z _T

t

[G(y_t,x(s;u)) +E(u(s))]ds, t ∈[_0,T)_, u∈ U_ad[t,T]_,

(withG andE to be defined) the focus of [11] was to identify simple checkable conditions that guar- antee in—but not limited to—such a context, the convergence of the corresponding value functions, namely

(2.32) lim

N→_∞ sup

t∈[0,T]

|v_N(t,ΠNx)−v(t,x)|=0, where

v(t,x):= inf

u∈U_ad[t,T]Jt,x(u), (t,x)∈ [0,T)× H and v(T,x) =0, (2.33a)

v_N(t,x_N):= inf

u∈U_ad[t,T]J_t,x^N_N(u), (t,x_N)∈[0,T)× H_N and v_N(T,x_N) =0.

(2.33b)

and J_t,x^N_N(u) denotes the cost functional J assessed along the Galerkin approximation y_t,x^N_N(·;u) to (2.28) driven byuand emanating at timetfrom

(2.34) x_N :=_ΠNx.

As shown in [11], conditions ensuring the convergence (2.32) can be grouped into three categories.

The first set of conditions deal with the operatorAand its Galerkin approximationA_N. Essentially, it boils down to show thatAgenerates aC₀-semigroup onH, and that the Trotter-Kato approximation conditions are satisfied; see Assumptions(A0)–(A2)in [11, Section 2.1]. The latter conditions are, as mentioned earlier, satisfied in the case of DDEs and GK approximations as shown by [9, Lemmas 4.2 and 4.3].

The second group of conditions identified in [11] is also not restrictive in the sense that only local Lipschitz conditions³onF, G, andC are required as well as continuity ofE and compactness ofU whereU_ad[t,T]is taken to be

(2.35) U_ad[t,T]:= {u|_[t,T] : u ∈ U_ad}, with

(2.36) U_ad:={u∈ L^q(0,T;V) : u(s)∈Ufor a.e. s∈[0,T]}, q≥1.

Also required to hold, are a priori bounds—uniform inuinU_ad—for the solutions of Eq. (2.28) as well as of their Galerkin approximations. Depending on the specific form of Eq. (2.28) such bounds can be derived for a broad class of DDEs; in that respect the proofs of [9, Estimates (4.75)] and [9, Corollary 4.3] can be easily adapted to the case of controlled DDEs.

Finally, the last condition to ensure (2.32), requires that the residual energy of the solution s 7→

y(s;x,u)of (2.28) (withy(0) =x) driven byu, satisfies

(2.37) k_Π^⊥_Ny(s;x,u)k_H −→0,

uniformly with respect to both the control u in U_ad and the time s in [0,T]; see Assumption (A7) in [11, Section 2.4]. Here,

(2.38) Π^⊥_N :=IdH−_ΠN.

Easily checkable conditions ensuring such a double uniform convergence are identified in [11, Section 2.7] for a broad class of evolution equations and their Galerkin approximations but unfortunately do not apply to the case of the GK approximations considered here. The scope of this article does not allow for an investigation of such conditions in the case of DDEs, and results along this direction will be communicated elsewhere.

3Note however that in the case of DDEs, nonlinearitiesFthat include discrete delays may complicate the verification of a local Lipschitz property inHas given by (2.2). The case of nonlinearitiesFdepending on distributed delays and/or the current state can be handled in general more easily; see [9, Sect. 4.3.2].

Nevertheless, error estimates can be still derived in the case of DDEs by adapting elements pro- vided in [11, Section 2.6]. We turn now to the formulation of such error estimates.

2.3.2. Error estimates. Our aim is to derive error estimates for GK approximations to the following type of optimal control problem associated with DDEs (2.1) framed into the abstract form (2.7), (P) min J(x,u) _s.t. (y,u)∈ L²(_0,T;H)× U_adsolves (2.28),

in whichJ is given by (2.31) witht =0, andysolves (2.28) withy(0) =xinH. To do so, we consider the following set of assumptions

(i) The mappingsF :H → H,G :H →_R⁺are locally Lipschitz, andE :V →_R⁺is continuous.

(ii) The linear operatorA_N :H_N → H_N is diagonalizable overCand there existsαinRsuch that for eachN

(2.39) Reλ^N_j ≤α, ∀j∈ {1,· · · ,N}, where{λ^N_j }denotes the set of (complex) eigenvalues ofA_N.

(iii) LetU_adbe given by (2.36) withUbounded inV. For eachT>0,(x,u)inH × U_ad, the problem (2.28) (witht=0) admits a unique mild solutiony(·;x,u)inC([0,T],H), and for eachN ≥0, its GK approximation (2.30) admits a unique solutiony_N(·_;x_N,u)_inC([_0,T]_,H). Moreover, there exists a constantC :=C(T,x)such that

(2.40) ky(t;x,u)k_H≤ C, ∀t∈[0,T], u∈ U_ad,

ky_N(t;x_N,u)k_H ≤ C, ∀t∈ [0,T], N∈_N^∗, u∈ U_ad. (iv) The mild solutiony(·;x,u)belongs toC([0,T],D(A))ifxlives inD(A). Remark 2.1.

(i) Compared with the case of eigen-subspaces considered in [11, Section 2.6], the main difference here lies in the regularity assumption (iv) above. This assumption is used to handle the termΠNA_Π^⊥_Ny arising in the error estimates; see e.g.(2.48)below. Note that this term is zero whenAis self-adjoint andH_N is an eigen-subspace ofA, which is the setting considered in [11, Section 2.6].

(ii) A way to ensure Condition (iv) to hold consists of proving that the mild solution belongs to C¹([_0,T]_,H). For conditions onF ensuring such a regularity see e.g. [6, Theorem 3.2].

Finally, we introduce the cost functionalJ_N to be the cost functionalJassessed along the Galerkin approximationy_Nsolving (2.30) withy_N(0) =x_N ∈ H_N, namely

(2.41) J_N(x_N,u):=

Z _T

0

[G(y_N(s;x_N,u)) +E(u(s))]ds, along with the optimal control problem

(P_N) min J_N(x_N,u) s.t. (y_N,u)∈ L²(0,T;H_N)× U_adsolves (2.30), withy_N(0) =x_N ∈ H_N.

We are now in position to derive error estimates as formulated in Theorem2.1 below. For that purpose Table1 provides a list of the main symbols necessary to a good reading of the proof and statement of this theorem. The proof is largely inspired from that of [11, Theorem 2.3]; the main amendments being specified within this proof.

Theorem 2.1. Assume that assumptions (i)-(iv) hold. Assume furthermore that for each(t,x)in [0,T)× D(A), there exists a minimizer u^∗_t,x(resp. u^N,_t,x^∗_N) for the value function v (resp. vN) defined in(2.33).

TABLE1. Glossary of principal symbols Symbol Terminology

(y^∗,u^∗) An optimal pair of the optimal control problem (P) u^∗_t,x Minimizer of the value functionvdefined in (2.33) u^N,_t,x^∗

N Minimizer of the value functionv_Ndefined in (2.33) y_t,x(·;u^∗_t,x) Mild solution to (2.28) driven byu^∗_t,x

y_t,x(·;u_t,x^N,∗_N) Mild solution to (2.28) driven byu_t,x^N,∗_N

B The closed ball inHcentered at 0Hwith radiusCgiven by (2.40) kAk_op Norm ofAas a linear bounded operator fromD(A)toH

Then for any(t,x)in[0,T)×D(A)it holds that (2.42)

|v(t,x)−v_N(t,x_N)|

≤_Lip(G|_B) √

T−t+ (T−t) q

f(T)

R(u^∗_t,x,u^N,_t,x^∗

N)_, with

(2.43) f(T):=

Lip(F |_B) +kAk²_op

exp

2h α+ ³

2Lip(F |_B) + ¹ 2 i

T

, and

(2.44) R(u^∗_t,x,u_t,x^N,∗_N):= k_Π^⊥_Ny_t,x(·;u^∗_t,x)k_L2(t,T;D(A))+k_Π^⊥_Ny_t,x(·;u^N,_t,x^∗_N)k_L2(t,T;D(A)).

Proof. We provide here the main elements of the proof that needs to be amended from [11, Theorem 2.3]. First note that mild solutions to (2.28) lie in C([0,T],D(A))due to Assumption (iv) and since the initial datumxis taken inD(A).

We want to prove that for eachN, there exists a constantβ > 0 such that for any(t,x) ∈ [0,T)× D(A), andu∈ U_ad[t,T],

(2.45) k_ΠNyt,x(s;u)−y^N_t,xN(s;u)k²_H≤ f(T)

Z _s

t

k_Π^⊥_Nyt,x(s⁰;u)k²_D(A)ds⁰, s∈[t,T]. Let us introduce

(2.46) w(s):= _ΠNyt,x(s;u)−y^N_t,xN(s;u).

By applyingΠNto both sides of Eq. (2.28), we obtain thatΠNy_t,x(·;u)(denoted byΠNyto simplify), satisfies:

dΠNy

ds =A_NΠNy+_ΠNA_Π^⊥_Ny+_ΠNF(_ΠNy+_Π^⊥_Ny) +_ΠNC(u(s)). This together with (2.30) implies thatwsatisfies the following problem:

(2.47)

dw

ds =A_Nw+_ΠNA_Π^⊥_Ny+_ΠNF(_ΠNy+_Π^⊥_Ny)−_ΠNF(y_N), w(t) =0.

By taking theH-inner product on both sides of (2.47) withw, we obtain:

(2.48)

1 2

dkwk²_H

ds =hA_Nw,wi_H+h_ΠNA_Π^⊥_Ny,wi_H

+h_ΠN F(_ΠNy+_Π^⊥_Ny)− F(y_N),wi_H.

We estimate now the termhA_Nw,wi_Hin the above equation using Assumption (ii). For this purpose, note that for anypinH_N, the following identity holds:

(2.49) hA_Np,pi_H =hMξ,ξi_Rⁿ,

where M is the matrix representation of A_N under the Koornwinder basis given by (2.26), and ξ denotes the column vector(ξ₀,· · · ,ξ_N−1)^trwhose entries are given by

ξ_j := ¹

kK_jk_Ehp,K^τ_ji_H, j=_0,· · · _,N−_1.

Note that theN×NmatrixMhas the same eigenvalues{λ^N_j : j=1,N}asA_N. Thanks to Assump- tion (ii),Mis diagonalizable overC. Denoting by f_jthe normalized eigenvector ofMcorresponding to eachλ^N_j , we have

Mξ = M

∑

ξe_jf_j =

∑

λ^N_j ξe_jf_j, whereξe_j =hξ, f_ji_Cⁿ. It follows then

hMξ,ξi_Rⁿ = h

∑

λ^N_j ξe_jf_j,ξi_Cn =

∑

λ^N_j |ξe_j|²=

∑

Reλ^N_j |ξe_j|²,

for which the latter equality holds sinceξ 7→ hMξ,ξi_Rⁿ is real-valued. Due to the bound (2.39), we have thus

(2.50) hMξ,ξi_Rn ≤α

∑

|ξe_j|²_. On the other hand, by noting that

(2.51) |ξe_j|²= hξ,eξei_Cn = hξ,ξi_Rn = hp,pi. we obtain from (2.49)–(2.51), by takingp =w, that

(2.52) hA_Nw(s),w(s)i_H≤αkw(s)k²_H, s∈[t,T]. We infer from (2.48) that

1 2

dkw(s)k²_H

ds ≤

α+³

2Lip(F |_B) +¹ 2

kw(s)k²_H

+ ¹

2kAk²_opk_Π^⊥_Nyk²_D(A)+ ¹

2Lip(F |_B)k_Π^⊥_Nyk²_H,

in which we have also used Assumption (iii) and the local Lipschitz property ofF on the closed ball BinHcentered at 0_Hwith radiusC given by (2.40).

By Gronwall’s lemma and recalling thatw(t) =0, we obtain thus k_ΠNy_t,x(s;u)−y^N_t,xN(s;u)k²_H≤ f(T)

Z _s

t

k_Π^⊥_Ny_t,x(s⁰;u)k²_D(A)ds⁰, with

f(T) =

Lip(F |_B) +kAk²_op

exp

2h α+ ³

2Lip(F|_B) + ¹ 2 i

T

.

Then by noting that

|Jt,x(u)−J_t,x^N_N(u)| ≤Lip(G|_B)

Z _T

t

k_ΠNyt,x(s;u)−y_t,x^N_N(s;u)k_D(A)

+k_Π^⊥_Nyt,x(s;u)k_D(A)ds, we have

(2.53)

|J_t,x(u)−J_t,x^N_N(u)|

≤Lip(G|_B) √

T−t+ (T−t) q

f(T)

k_Π^⊥_Nyt,x(·;u)k_L2(t,T;D(A)), for allu∈ U_ad[t,T].

Finally by noting that

v(t,x) =J_t,x(u^∗_t,x)≤ J_t,x(u^N,_t,x^∗_N) and v_N(t,x_N) = J_t,x^N_N(u^N,_t,x^∗_N), we obtain

v(t,x)−v_N(t,x_N)≤ J_t,x(u^N,_t,x^∗_N)−J_t,x^N_N(u_t,x^N,∗_N). It follows then from (2.53) that

(2.54)

v(t,x)−vN(t,xN)

≤Lip(G|_B) √

T−t+ (T−t) q

f(T)

k_Π^⊥_Ny_t,x(·;u^N,_t,x^∗_N)k_L2(t,T;D(A)). Similarly,

(2.55)

v_N(t,x_N)−v(t,x)

≤Lip(G|_B) √

T−t+ (T−t) q

f(T)

k_Π^⊥_Nyt,x(·;u^∗_t,x)k_L2(t,T;D(A)). The estimate (2.42) results then from (2.54) and (2.55).

We conclude this section with the following corollary providing the error estimates between the optimal control and that obtained from a GK approximation.

Corollary 2.1. Assume that the conditions given in Theorem2.1 hold. Given x in H, let us introduce the notations, u^∗ := u_0,x^∗ and u^∗_N := u_0,x^N,∗_N. Assume furthermore that there existsσ > 0such that the following local growth condition is satisfied for the cost functional J (with t=0) given by(2.31):

(2.56) σku^∗−vk^q_Lq(0,T;V) ≤ J(x,v)−J(x,u^∗),

for all v in some neighborhoodO ⊂ U_ad of u^∗, with U_ad given by(2.36). Assume finally that u^∗_N lies in O. Then,

(2.57)

ku^∗−u^∗_Nk^q_Lq(0,T;V)

≤ ¹

σLip(G|_B) √

T+T q

f(T)

k_Π^⊥_Ny^∗(·;u^∗)k_L2(0,T;D(A))

+2k_Π^⊥_Ny(·;u^∗_N)k_L2(_0,T;D(A))

, where the constant f(T)is given by(2.43).

Proof. By the assumptions, we have

ku^∗−u^∗_Nk^q_Lq(0,T;V)≤ ¹

σ (J(x,u^∗_N)−J(x,u^∗)). Note also that

J(x,u^∗_N)−J(x,u^∗) = J(x,u^∗_N)−J_N(x_N,u^∗_N) +J_N(x_N,u^∗_N)−J(x,u^∗)

= J_0,x(u^∗_N)−J_0,x^N _N(u^∗_N) +v_N(0,x_N)−v(0,x), where we used the fact that

J(x,u^∗_N) =J_0,x(u^∗_N), J_N(x_N,u^∗_N) = J_0,x^N_N(u^∗_N), J_N(x_N,u^∗_N) =v_N(0,x_N) and J(x,u^∗) =v(0,x). The result follows by applying the estimate (2.53) to J_0,x(u^∗_N)− J_0,x^N

N(u^∗_N)and the estimate (2.42) to vN(0,xN)−v(0,x).

3. APPLICATION TO AN OSCILLATION AMPLITUDE REDUCTION PROBLEM

3.1. Optimal control of the Wright equation. As an application, we consider the Wright equation

(3.1) dm

dt = −m(t−τ)(1+m(t)),

wheremis the unknown scalar function, andτis a nonnegative delay parameter. This equation has been studied by numerous authors, and notably among the first ones, are Jones [29,30], Kakutani and Markus [31], and Wright [61]. This equation can also be transformed via a simple change of variable [53] into the well-known Hutchinson equation [27] arising in population dynamics. It cor- responds then to the logistic equation with a delay effect introduced into theintraspecific competition term, namely with the change of variablem=−1+p, Eq. (3.1) becomes

(3.2) dp

dt = p(t)(1−p(t−τ)).

Essentially the idea of Hutchinson [27] was to point out that negative effects that high population density p has on the environment, influences birth rates at later times due to developmental and maturation delays, justifying thus Eq. (3.2).

As a result, the solutions of Eq. (3.1) are obtained from a simple shift of the solutions of Eq. (3.2), and reciprocally. Since Eq. (3.2) benefits of a more intuitive interpretation, we will often prefer to think about Eq. (3.1) in terms of Eq. (3.2).

Anyway, it is known that Eq. (3.1) (and thus Eq. (3.2)) experiences a supercritical Hopf bifurcation as the delay parameter is varied and crosses a critical value, τ_c, from below; see e.g. [17, Sect. 9]

and [57].⁴

Figure1shows—in the embedded reduced phase space (m(t),m(t−τ))—the corresponding bi- furcation of the limit cycle unfolding from the trivial steady state, asτis varied⁵. As Fig.1illustrates, the amplitude of the oscillation that takes place via the Hopf bifurcation, increases asτis increased away fromτ_c.

4The critical valueτcat which the trivial steady state of the Wright equation (3.1) loses its linear stability is given by τc= ^π₂. This can be found by analyzing the associated linear eigenvalue problem−φ(θ−τ) =βφ(θ),θ∈[−τ, 0]. By using the ansatzφ(_θ) =e^βθ, we obtain−e^−βτ =β. Assuming thatβ=±iω, we get−cos(_ωτ) +isin(_ωτ) =iω, leading thus to ω=1 andτ= ^(2n+1)π₂ ,n=0, 1, 2,· · ·. Consequently, the critical delay parameter isτc= ^π₂.

5Such an embedded phase space is classically used to visualize attractors associated with DDEs; see e.g. [15].

1.57 1.575

1.58 1.585

1.59 1.595

1.6 1.605 1.61

−0.4

−0.2 0

0.2 0.4

0.6

−0.5

−0.4

−0.3

−0.2

−0.1 0 0.1 0.2 0.3 0.4 0.5

m(t) τ

m(t−τ)

FIGURE 1. Unfolding of the Hopf bifurcation taking place for Eq.(3.1) asτ is varied. The limit cycles are represented in the the embedded reduced phase space(m(t),m(t−τ)). The limit cycle in heavy black indicates the “τ-secion” considered for the numerical results of Sectns.3.3and4below. The trivial steady state is represented via a dotted line when unstable and a plain one when stable.

Since fluctuating populations are susceptible to extinction due to sudden and unforeseen environ- mental disturbances [51], a knowledge of the conditions for which the population is resilient to these disturbances is of great interest in planning and designing control as well as management strategies;

see e.g. [14,50,52].

Given a convex functional depending onmand a control to be designed, our goal is to show (for Eq. (3.1)) that relatively close to the criticality (i.e.τ ≈ τ_c), the oscillation amplitude can be reduced at a nearly optimal cost, by solving efficiently low-dimensional optimal control problems.

For that purpose, we consider, givenu in L²(0,T;R)andT > 0, the following forced version of Eq. (3.1)

(3.3) dm

dt =−m(t−τ)(1+m(t)) +u(t), t ∈(0,T),

supplemented by the initial historym(t) =φ(t)(withφ ∈ L²(−τ, 0;R)) fortin the interval[−τ, 0), and bym(0) =m₀(inR) att =0. In Eq. (3.3),u(t)can be thought as an environmental management strategy. Our goal is to understand the management strategies that may lead to a reduction of the oscillation amplitude of the population (and possibly a stabilization); i.e. to determine ufor which m≈0 and thusp≈1, while dealing with limited resources to plan such strategies.

Naturally, we introduce thus the following cost functional

(3.4) J(m,v):=

Z _T

0

1

2m(t)²+ ^µ 2u(t)²

dt, µ>0.

We are interested in solving optimal control problem associated with Eq. (3.3) for this cost functional.

To address this problem, we recast first this optimal control problem into the abstract framework of Sect.2.1.

In that respect, we introduce the variables

(3.5) y:= (mt,mt(0)), withmsolving (3.3),

and we takeU_addefined in (2.36) withq=2 and withUto be (possibly) a bounded subset ofV:=_R.

The control operatorC:V → Happearing in Eq. (2.28) is taken here to be linear and given by

(3.6) [Cv](θ)_:=

(0, θ ∈[−τ, 0)

v, θ =0 , v∈V.

As explained in Sect.2.1, Eq. (3.3) can be recast into the following abstract ODE posed onH

(3.7) dy

dt = Ay+F(y) +Cu(t),

with the linear operatorAgiven in (2.8) that, in the case of Eq. (3.3), takes the form

(3.8) [A_Ψ](θ):=





 d⁺Ψ^D

dθ , θ ∈[−τ, 0),

−_Ψ^D(−τ), θ =0,

with domain given in (2.9). The nonlinearityF given in (2.10) takes here the form, (3.9) [F(_Ψ)](θ):=

(0, θ∈ [−τ, 0),

−_Ψ^D(−τ)_Ψ^S, θ=0, ∀_Ψ= (_Ψ^D,Ψ^S)∈ H. LetΦbe inHgiven by

(3.10) Φ:= (φ,m₀), withφ∈L²(−τ, 0;R), m₀ ∈_R.

We rewrite now the cost functional defined in (3.4) by using the state-variableygiven in (3.5), and define thus

(3.11) J(y,u):=

Z _T

0

1

2 mt(0)²+ ^µ 2u(t)²

dt;

still denoted by J.

The optimal control problem associated with Eq. (3.7) becomes then

(3.12) min J(_Φ,u)s.t.(y,u)∈ L²(0,T;H)× U_adsolves Eq. (3.7) withy(0) =_Φ∈ H,

withU_addefined in (2.36) withq=2 and with eitherU=_RorUto be a bounded subset ofV=_R.

Remark 3.1. Note that in the case u(t)in Eq.(3.3)is replaced by u(t−r)(with0 < r), one can still recast Eq.(3.3)into an abstract ODE of the form(3.7); i.e. to deal with control with delay. In that case by taking V:= H¹(−r, 0;R)×_R, the operatorCbecomes

(3.13) [C(φ^D,φ^S)](θ):=

(0, θ ∈[−τ, 0)

φ^D(−r), θ =0 , (φ^D,φ^S)∈V.

This reformulation is compatible with the framework of [9] and thus allow us to derive GK approximations of such abstract ODEs. We leave the associated optimal control problems to the interested reader.

3.2. Galerkin-Koornwinder approximation of the optimal control problem. First, we specify for Eq. (3.3), the (concrete) ODE version (2.23) of the (abstract) N^th GK approximation (2.30). For this purpose, we first computeΠNCu(s).

Note that since{K^τ_j : j ∈ _N}forms a Hilbert basis of H, for anyΨ = (_Ψ^D,Ψ^S)in H, we have (see also [9, Eq. (3.25)])

(3.14)

Ψ=

∑

h_Ψ,K^τ_ji_H kK^τ_jk²_H K^τ_j

=

∑

1

τh_Ψ^D,K^τ_ji_L2+_Ψ^SK^τ_j(0) K^τ_j kK^τ_jk²_H^. By taking nowΨ=CvwithCgiven by (3.6) andvinR, we have

(3.15) Cv=

∑

vK^τ_j(0) K^τ_j kK^τ_jk²_H^, which leads to

(3.16) ΠNCv=v

N−1 j

∑

K^τ_j kK^τ_jk²_H^, recalling thatK^τ_j(0) =1; see (2.21).

Then by definingC_N: V→_R^N to be

(3.17) C_Nv:= ¹

kK₀k²_E^,· · · _, ¹ kK_N−1k²_E

tr

v,

one can write the Galerkin approximation (2.30) in the Koornwinder basis as the ODE system (3.18)

dξ_N

dt = Mξ_N+G(_ξN) +C_Nu(t)_, ξ_N(0) =ζ_N ∈_R^N.

Hereξ_N = (ξ₀^N,· · · ,ξ_N^N₋₁)^tr; theN×N-matrixMgiven in (2.26) has its elements given here by (3.19) (M)_j,n = ¹

kK_jk²_E

−Kn(−1) + ² τ

n−1 k

∑

a_n,k δ_j,kkK_jk²_E−1

, j,n=0,· · · ,N−1;

Recall that here the Kronecker symbolδ_j,k has been used, and the coefficients a_n,k are obtained by solving a triangular linear system in which the right hand side has explicit coefficients depending on n; see [9, Prop. 5.1].

The initial datumζ_N := (ζ^N₀,· · · ,ζ^N_N−1)^trhas its components defined by

(3.20) ζ^N_j = ¹

kK_jk²_Eh_Φ,K^τ_ji_H, j=0,· · · ,N−1,

whereΦinHdenotes the initial data given by (3.10) for the abstract ODE (3.7).

The nonlinearlityGgiven in (2.27) takes, in the case of Eq. (3.1), the form (3.21) G_j(ξ_N) =− ¹

kK_jk²_E

"

N−1 n

∑

ξ_n^N(t)

#"

N−1 n

∑

ξ_n^N(t)K_n(−1)

#

, 0≤ j≤N−1.