Turnpike in optimal shape design

11th IFAC Symposium on Nonlinear Control Systems 11th Symposium on Control 11th IFAC IFAC Symposium on Nonlinear Nonlinear Control Systems Systems Vienna, Austria, Sept. 4-6, 2019 11th IFAC Symposium on Nonlinear Control Systems Vienna, Austria, Sept. 4-6, 2019 Available online at www.sciencedirect.com Vienna, Austria, Sept. 4-6, 2019 11th IFAC Symposium on Nonlinear Control Systems Vienna, Austria, Sept. 4-6, 2019 Vienna, Austria, Sept. 4-6, 2019

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IFAC PapersOnLine 52-16 (2019) 496–501

Turnpike Turnpike in in optimal optimal shape shape design design Turnpike in optimal shape design Turnpike in optimal shape design ∗∗∗ G. Lance ∗∗ E. Tr´ elat ∗∗ ∗∗ E. Zuazua ∗∗∗

G. e G. Lance Lance ∗ E. E. Tr´ Tr´ elat lat ∗∗ E. E. Zuazua Zuazua ∗∗∗ G. Lance ∗∗ E. Tr´ elat ∗∗ E. Zuazua ∗∗∗ ∗∗ ∗∗∗ ∗ G. Lance E. Tr´ eelat E. Zuazua Sorbonne Universit´ e , Universit´ Paris-Diderot SPC, CNRS, Inria, ∗ ∗ Sorbonne Universit´ ee,, Universit´ ee Paris-Diderot SPC, CNRS, Inria, Universit´ Universit´ Paris-Diderot SPC, CNRS, Inria, ∗ Sorbonne Laboratoire Jacques-Louis Lions, ´ e quipe CAGE, F-75005 Paris Sorbonne Universit´ e , Universit´ e Paris-Diderot SPC, CNRS, Inria, Jacques-Louis Lions, ´ e quipe CAGE, F-75005 Paris ∗ Laboratoire Laboratoire Jacques-Louis Lions, ´ e quipe CAGE, F-75005 Paris Sorbonne Universit´ e , Universit´ e Paris-Diderot SPC, CNRS, Inria, ([email protected]) Laboratoire Jacques-Louis Lions, ´ e quipe CAGE, F-75005 Paris ([email protected]) ([email protected]) ∗∗ Laboratoire Jacques-Louis Lions, ´ e quipe CAGE, F-75005 Paris Sorbonne Universit´ e , Universit´ e Paris-Diderot SPC, CNRS, Inria, ∗∗ ([email protected]) ∗∗ Sorbonne Universit´ ee,, Universit´ ee Paris-Diderot SPC, CNRS, Inria, Universit´ Universit´ Paris-Diderot SPC, CNRS, Inria, ∗∗ Sorbonne ([email protected]) Laboratoire Jacques-Louis Lions, ´ e quipe CAGE, F-75005 Paris Sorbonne Universit´ e , Universit´ e Paris-Diderot SPC, CNRS, Inria, Jacques-Louis Lions, ´ e quipe CAGE, F-75005 Paris ∗∗ Laboratoire Laboratoire Jacques-Louis Lions, ´ e quipe CAGE, F-75005 Paris Sorbonne Universit´ e , Universit´ e Paris-Diderot SPC, CNRS, Inria, ([email protected]) Laboratoire Jacques-Louis Lions, ´ e quipe CAGE, F-75005 Paris ([email protected]) ([email protected]) ∗∗∗Laboratoire Jacques-Louis Lions, ´ e quipe CAGE, F-75005 Paris Fundaci´ o ∗∗∗ ([email protected]) ∗∗∗ DeustoTech, DeustoTech, Fundaci´ on n Deusto, Deusto, Avda Avda Universidades, Universidades, 24, 24, 48007, 48007, Fundaci´ o n Deusto, Avda Universidades, 24, 48007, ∗∗∗ DeustoTech, ([email protected]) Bilbao, Basque Country, Spain; DeustoTech, Fundaci´ o n Deusto, Avda Universidades, 24, 48007, Bilbao, Basque Country, Spain; ∗∗∗ Bilbao, Basque Country, Spain; DeustoTech, Fundaci´ oanticas, Deusto, Avda 24,Madrid, 48007, Departamento de Matem´ Universidad Aut´ o noma de Bilbao, Basque Country, Spain;Universidades, Departamento de a Universidad o Departamento de Matem´ Matem´ aticas, ticas, Universidad Aut´ onoma noma de de Madrid, Madrid, Bilbao, Basque Country, Spain; Aut´ 28049 Madrid, Spain; Departamento de Matem´ a ticas, Universidad Aut´ o noma de Madrid, 28049 Madrid, Spain; 28049 Madrid, Spain; Departamento de Matem´ aticas, Universidad Aut´ onoma de Madrid, Facultad Ingenier´ ıa, Universidad de Deusto, Avda. Universidades, 24, 28049 Madrid, Spain; Facultad ıa, Universidad de Avda. Facultad Ingenier´ Ingenier´ ıa, Universidad de Deusto, Deusto, Avda. Universidades, Universidades, 24, 24, 28049 Madrid, Spain; 48007 Bilbao, Basque Country, Spain; Facultad Ingenier´ ıa, Universidad de Deusto, Avda. Universidades, 24, 48007 Bilbao, Basque Country, Spain; 48007 Bilbao, Basque Country, Spain; Facultad Ingenier´ ıa, Universidad deeDeusto, Avda. Universidades, 24, Sorbonne Universit´ eeBasque ,, Universit´ SPC, CNRS, 48007 Bilbao, Country, Spain; Sorbonne Universit´ ee Paris-Diderot Paris-Diderot SPC, CNRS, Sorbonne Universit´ eBasque , Universit´ Universit´ Paris-Diderot SPC,France CNRS, 48007 Bilbao, Country, Spain; Paris, Laboratoire Jacques-Louis Lions, F-75005, Sorbonne Universit´ e , Universit´ e Paris-Diderot SPC, CNRS, Laboratoire Jacques-Louis Lions, F-75005, Paris, Paris, France Laboratoire Jacques-Louis Lions, F-75005, Sorbonne Universit´ e, Universit´ e Paris-Diderot SPC,France CNRS, ([email protected]). Laboratoire Jacques-Louis Lions, F-75005, Paris, France ([email protected]). ([email protected]). Laboratoire Jacques-Louis Lions, F-75005, Paris, France ([email protected]). ([email protected]). Abstract: We investigate the turnpike problem in optimal control, in the context of timeAbstract: We the problem in control, in the of timeAbstract: We investigate investigate the turnpike turnpike problem in optimal optimal control, inshape the context context ofsource timeevolving shapes. We focus here on the heat equation model where the acts as a Abstract: We investigate the turnpike problem in optimal control, in the context of timeevolving shapes. We focus here on the heat equation model where the shape acts as aaofsource evolving shapes. We focus here on the heat equation model where the shape acts as source Abstract: We investigate the turnpike problem in optimal control, in the context timeterm, and we search the optimal time-varying shape, minimizing a quadratic criterion. We first evolving shapes. We focus here on the heat equation model where the shape acts as a source term, and we search the optimal time-varying shape, minimizing aa quadratic criterion. We first term, and we search the optimal time-varying shape, minimizing quadratic criterion. We first evolving shapes. We focus here on the heat equation model where the shape acts as a source establish existence of optimal solutions under some appropriate sufficient conditions. We provide term, and we search the optimal time-varying shape, minimizing a quadratic criterion. We first establish existence of optimal solutions under some appropriate sufficient conditions. We provide establish existence of optimal solutions under some appropriate sufficient conditions. We provide term, and we search the optimal time-varying shape, minimizing a quadratic criterion. We first necessary conditions for optimality in terms of usual adjoint equations and then, thanks to strict establish existence of for optimal solutions underofsome appropriate sufficient conditions. We to provide necessary conditions optimality in terms usual adjoint equations and then, thanks strict necessary conditions optimality in terms ofsome usual adjointsatisfy equations andconditions. then, thanks to strict establish existence of for optimal solutions under appropriate sufficient Weproperty, provide dissipativity properties, we prove that state and adjoint a measure-turnpike necessary conditions for optimality in terms of usual adjoint equations and then, thanks to strict dissipativity properties, we prove state adjoint satisfy aa measure-turnpike dissipativity properties, wetime-varying prove that that state and adjoint satisfy measure-turnpike property, necessary conditions for optimality in terms ofand usual adjoint equations andtothen, thanksproperty, to strict meaning that the extremal solution remains essentially close an optimal solution dissipativity properties, wetime-varying prove that state and adjoint satisfy a measure-turnpike property, meaning that the extremal solution remains essentially close to an optimal solution meaning that the extremal time-varying solution remains essentially close to an optimal solution dissipativity properties, we prove that state and adjoint satisfy a measure-turnpike property, of an associated problem. We illustrate turnpike phenomenon in shape design with meaning that the static extremal time-varying solutionthe remains essentially close to an optimal solution of an problem. We turnpike phenomenon in shape design with of an associated associated static problem. We illustrate illustrate the turnpike phenomenon inan shape design with meaning that the static extremal time-varying solutionthe remains essentially close to optimal solution several numerical simulations. of an associated static problem. We illustrate the turnpike phenomenon in shape design with several numerical simulations. several numerical simulations. of an associated problem. We illustrate the turnpike phenomenon in shape design with several numericalstatic simulations. © 2019, numerical IFAC (International of Automatic Hosting direct by Elsevier Ltd. Allheat rights reserved. several simulations. Keywords: Optimal shape Federation design, turnpike, strictControl) dissipativity, methods, equation Keywords: Keywords: Optimal Optimal shape shape design, design, turnpike, turnpike, strict strict dissipativity, dissipativity, direct direct methods, methods, heat heat equation equation Keywords: Optimal shape design, turnpike, strict dissipativity, direct methods, heat equation Keywords: Optimal shape design, turnpike, strict dissipativity, direct methods, heat equation 1. INTRODUCTION According to the well known turnpike phenomenon, one 1. INTRODUCTION INTRODUCTION According to the known turnpike phenomenon, one 1. According to for theT well well known turnpike phenomenon, one expects that, large enough, optimal solutions of (1,2) 1. INTRODUCTION According to the well known turnpike phenomenon, one expects that, for T large enough, optimal solutions of (1,2) expects that, for T large enough, optimal solutions of (1,2) 1. INTRODUCTION According to the well known turnpike phenomenon, one remain most of the time “close” to an optimal (stationary) We start with an informal presentation of the turnpike expects that, for T large enough, optimal solutions of (1,2) remain most of the time “close” “close” tooptimal an optimal optimal (stationary) We start start with with an an informal informal presentation presentation of of the the turnpike turnpike remain most of the time to an (stationary) We expects that, for T large enough, solutions of (1,2) solution of the problem phenomenon general dynamical optimal shape probmost of static the time “close”(3). to an optimal (stationary) We start withfor informal presentation of the turnpike solution of problem phenomenon foran general dynamical optimal shape prob- remain solution of the the static problem (3). phenomenon for general dynamical optimal shape probremain most of static the time “close”(3). to an optimal (stationary) We start with anWe informal presentation ofofthe turnpike lems. Let T > 0. consider the problem determining solution of the static problem (3). phenomenon for general dynamical optimal shape problems. Let T T > >for 0. We We consider the problem problem of shape determining The turnpike phenomenon was first lems. Let 0. consider the of determining solution of the static problem (3). phenomenon general dynamical optimal probThe turnpike phenomenon was first observed observed and and invesinvesa time-varying shape t →  ω(t) (viewed as a control) The turnpike phenomenon was first observed and inveslems. Let T > 0.shape We consider the problem ofasdetermining alems. time-varying t →  ω(t) (viewed a control) tigated by economists for discrete-time optimal The turnpike phenomenon was first observed andcontrol invesaminimizing time-varying shape t →  ω(t) (viewed as a control) Let T the > 0.cost Wefunctional consider the problem ofasdetermining tigated by economists for discrete-time optimal control tigated by economists for discrete-time optimal control aminimizing time-varying shape t →  ω(t) (viewed a control) The turnpike phenomenon was et firstal.observed and investhe cost cost functional problems e.g., Dorfman (1958); McKenzie tigated by(see, economists for discrete-time optimal control minimizing the functional a time-varying shape t →  ω(t) (viewed as a control) problems (see, e.g., Dorfman et al. (1958); McKenzie problems (see, e.g., several Dorfman et al.notions (1958); McKenzie minimizing the cost functional T tigated by economists for discrete-time optimal control (1963)). There are possible of turnpike T problems (see, e.g., Dorfman et al. (1958); McKenzie  T minimizing the functional (1963)). There are several possible of turnpike     cost 1 (1963)). There are several possible notions of turnpike problems (see, e.g., Dorfman et al.notions (1958); McKenzie 1  dt + g y(T ), ω(T ) properties, some of them being stronger than the others. 1 TT ff 000 y(t), (1963)). There are several possible notions of turnpike ω(t) (1) = J T (ω) properties, some of them being stronger than the others.    y(t), ω(t) dt + g y(T ), ω(T ) (1) (ω) = J properties, some of them being stronger than the others. 1  y(t), ω(t) dt + g y(T ), ω(T ) (1) f JTT (ω) = T (1963)). There are several possible notions of turnpike 0 For continuous-time problem, exponential turnpike propsome of them being stronger than the others. (1) properties, JT (ω) = T T1 0 f 0 y(t), ω(t) dt + g y(T ), ω(T ) For continuous-time problem, exponential turnpike propFor continuous-time problem, exponential turnpike propsome established of them being stronger than the (2015); others. (1) properties, JT (ω) = T 00 f y(t), ω(t) dt + g y(T ), ω(T ) erties have been in Tr´ eelat and Zuazua For continuous-time problem, exponential turnpike properties have been established in Tr´ lat and Zuazua (2015); T 0 erties have been established in Tr´ e lat and Zuazua (2015); For continuous-time problem, exponential turnpike propPorretta Zuazua (2013, eelat et al. (2018) erties haveand been established in2016); Tr´elatTr´ and Zuazua (2015); under the constraints 0 Porretta and Zuazua (2013, 2016); Tr´ lat et al. (2018) Porretta and Zuazua (2013,in2016); elatZuazua et al. of (2018) under under the the constraints constraints erties have been established Tr´ ethe latTr´ and (2015);     for the optimal triple resulting of application PonPorretta and Zuazua (2013, 2016); Tr´ e lat et al. (2018) under y(t) constraints for the triple resulting of application of Pon, y(0), y(T ) = 0 ˙˙the = f y(t), ω(t) R (2) for the optimal optimal tripleprinciple, resulting of the theTr´ application of(2018) Ponand Zuazua (2013, 2016); ethat lat et al. = R (2) tryagin’s maximum ensuring the extremal under y(t) for the optimal triple resulting of the application of Pony(t) ˙the constraints = ff y(t), y(t), ω(t) ω(t),, Ry(0), y(0), y(T y(T )) = =0 0 (2) Porretta maximum principle, ensuring that the extremal tryagin’s maximum principle, ensuring that the extremal y(t) ˙ = f y(t), ω(t), Ry(0), y(T ) = 0 (2) tryagin’s for the optimal tripleprinciple, resulting of the application of Ponsolution (state, adjoint and control) remains exponentially tryagin’s maximum ensuring that the extremal y(t) ˙ = f y(t), ω(t) , R y(0), y(T ) = 0 (2) solution (state, adjoint and control) remains exponentially where (2) may be a partial differential equation. solution (state, adjoint and control) remains tryagin’s maximum principle, ensuring that exponentially the extremal where close to an optimal solution of the corresponding static where (2) (2) may may be be a a partial partial differential differential equation. equation. solution (state, adjoint and control) remains exponentially close to an optimal solution of the corresponding static close to an optimal solution of the corresponding static where (2) may be a partial differential equation. solution (state, adjoint and control) remains exponentially controlled problem, except at the beginning and at the We associate to the dynamical problem (1,2) a static close to an optimal solution of the corresponding static where (2) may to be the a partial differential equation. controlled problem, except at the beginning and at the We associate dynamical problem (1,2) aa static controlled problem, except at the beginning and at the We associate to the dynamical problem (1,2) static close to an optimal solution of the corresponding static end of the time interval, as soon as T is large enough. This problem, not depending on time, controlled problem, except at the beginning and at the We associate to the dynamical problem (1,2) a static end of the time interval, as soon as T is large enough. This problem, not depending on time, end of the time interval, as soon as T is large enough. This problem, not depending on time, controlled problem, except at feature the and atThis the We associate to the dynamical problem (1,2) a static follows from the hyperbolicity the Hamiltonian 0 end of the time interval, as soon as Tbeginning isof large enough. problem, not min depending on time, follows from the hyperbolicity feature of the Hamiltonian f (y, ω), f (y, ω) = 0 (3) 0 follows from theinterval, hyperbolicity of the Hamiltonian 0 (y, ω), ff (y, ω) = 0 (3) end ofFor the time as soonfeature as T ishas large enough. This problem, not min depending on time, ω f min f (y, ω), (y, ω) = 0 (3) flow. discrete-time problems it been shown in follows from the hyperbolicity feature of the Hamiltonian 0 ω For discrete-time problems it has been shown in min f (y, ω) = 0 (3) flow. ω f 0 (y, ω), flow. For discrete-time problems ituller has been shown in follows from the hyperbolicity feature of the Hamiltonian Damm et al. (2014); Gr¨ u ne and M¨ (2016) that ex Partially funded ω by flow. For discrete-time problems it has been shown in min f (y, ω), f (y, ω) = 0 (3) the Advanced Grant DyCon of the European Damm et al. (2014); Gr¨ u ne and M¨ u ller (2016) that ex Partially funded Damm et al. (2014); Gr¨ u ne and M¨ u ller (2016) that ex the Advanced Grant DyCon of the European ω by flow. For discrete-time problems it has been shown in Partially funded by the Advanced Grant DyCon of the European ponential turnpike is closely related to strict dissipativity. Damm et turnpike al. (2014); Gr¨ une related and M¨ uller (2016) that ex Research Council (ERC) (Advanced EuropeanGrant Unions Horizon 2020, grant Partially funded by the DyCon of the European ponential is closely to strict dissipativity. Research Council (ERC) (( European Unions Horizon 2020, grant ponential is closely to strict dissipativity. Research Council (ERC) European Unions Horizon 2020, grant Damm et turnpike al. (2014); Gr¨ une related and M¨ uller (2016) that ex agreement No. 694126-DyCon). The work of the second author Partially funded by the Advanced Grant DyCon of the European ponential turnpike is closely related to strict dissipativity. Research Council (ERC) ( European Unions Horizon 2020,author grant agreement No. 694126-DyCon). The work the second Measure-turnpike a weaker notion of turnpike, meaning agreement No.supported 694126-DyCon). The Unions work of ofHorizon the second author ponential turnpikeis related to strict dissipativity. Measure-turnpike is weaker notion of turnpike, meaning was partially MINECO (Spain), Research Council (ERC)by( MTM2017-92996 European 2020, grant Measure-turnpike isisaaclosely weaker notion of turnpike, meaning agreement No.supported 694126-DyCon). The work ofof the second author was partially by MTM2017-92996 of MINECO (Spain), that any optimal solution, along the time frame, remains was partially supported by MTM2017-92996 of MINECO (Spain), Measure-turnpike is a weaker notion of turnpike, meaning FA9550-18-1-0242 of AFOSR, ICON of the French ANR and the agreement No. 694126-DyCon). The work of the second author that any optimal solution, along the time frame, remains was partially supported by MTM2017-92996 of MINECO (Spain), that any optimal solution, along the time frame, remains FA9550-18-1-0242 of AFOSR, ICON of the French ANR and the Measure-turnpike is a weaker notion of turnpike, meaning FA9550-18-1-0242 of AFOSR, ICON of the French ANR and the close to an optimal solution of the associated static optithat any optimal solution, along the time frame, remains ROAD2DC ofAFOSR, the 2018 Program of the (Spain), Basque was partiallyproject supported by Elkartek MTM2017-92996 of MINECO close to an optimal solution of the associated static optiFA9550-18-1-0242 of ICON of the French ANR and the close to an optimal solution of the associated static optiROAD2DC project of the Elkartek 2018 Program of the Basque that any optimal solution, along the time frame, remains ROAD2DC project of the Elkartek 2018 Program of the Basque mization problem except along a subset of times that is close to an optimal solution of the associated static optiGovernment. FA9550-18-1-0242 of AFOSR, ICON of the French ANR and the ROAD2DC project of the Elkartek 2018 Program of the Basque mization problem except along a subset of times that is of of Government. mization problem except along a subset of times that is of Government. close to an optimal solution of the associated static ROAD2DC mization problem except along a subset of times thatoptiis of Government.project of the Elkartek 2018 Program of the Basque mization except along a subset of times that is of Government. 2405-8963 © 2019, IFAC (International Federation of Automatic Control) Hosting by problem Elsevier Ltd. All rights reserved.

Copyright © 2019 IFAC 790 Copyright 2019 790 Copyright © under 2019 IFAC IFAC 790 Control. Peer review© responsibility of International Federation of Automatic Copyright © 2019 IFAC 790 10.1016/j.ifacol.2019.12.010 Copyright © 2019 IFAC 790

2019 IFAC NOLCOS Vienna, Austria, Sept. 4-6, 2019

G. Lance et al. / IFAC PapersOnLine 52-16 (2019) 496–501

small Lebesgue measure. It has been proved in Faulwasser et al. (2017); Tr´elat and Zhang (2018) that measureturnpike follows from strict dissipativity or from strong duality. Applications of turnpike in practice are numerous. Indeed, the knowledge of a static optimal solution is a way to reduce significantly the complexity of the dynamical optimal control problem. For instance it has been shown in Tr´elat and Zuazua (2015) that this is a way to successfully initialize a shooting method, when trying to compute numerically an optimal solution. In shape design and despite of the industrial progress, it is easier to design pieces which do not evolve with time. Turnpike can legitimate such decisions for large-time evolving systems.

Throughout the paper, we denote by: N

• |Q| the Lebesgue measure of a subset Q ⊂ R , N  ≥1. • p, q for p, q in L2 (Ω) is the scalar product in L2 (Ω). • y for y ∈ L2 (Ω) is the L2 -norm. • χω is the indicator (or characteristic) function of a subset ω ∈ RN . 2.1 Problem Let Ω ⊂ RN (N ∈ N∗ ) be a bounded domain with a smooth boundary ∂Ω. Let L ∈ (0, 1). We define the set of admissible shapes (4)

where the set of subsets ω ⊂ Ω is a measured space, endowed with the Lebesgue measure | · |. Dynamical optimal shape design problem. Let y0 ∈ L2 (Ω) be arbitrary. We consider the Dirichlet heat equation controlled by a (measurable) time-varying map t → ω(t) of subdomains ∂y − y = χω(·) , ∂t

y|∂Ω = 0,

y(0) = y0

2.2 Preliminaries Convexification. Given any measurable subset ω ⊂ Ω, we identify ω with the function χω ∈ L∞ (Ω; {0, 1}) and we identify as well UL with a subset of L∞ (Ω). The convex closure of UL in L∞ star topology is      U L = a ∈ L∞ Ω; [0, 1] | a(x) dx ≤ L|Ω| Ω

which is also weak star compact. We define the convexified optimal control problem (ocp)T of determining a control t → a(t) such that a.e. t ∈ [0, T ], a(t) ∈ U L and minimizing T 1 y(t) − yd 2 dt JT (a) = T 0

2. SHAPE TURNPIKE FOR THE HEAT EQUATION

UL = {ω ⊂ Ω measurable | |ω| ≤ L|Ω|}

497

(5)

Given T > 0 and yd ∈ L2 (Ω), we consider the dynamical optimal shape design problem (OSD)T of determining a measurable map of shapes t → ω(t) ∈ UL that minimizes the cost functional T 1 y(t) − yd 2 dt (6) JT (ω(·)) = T

under the constraints ∂y − y = a, ∂t

y|∂Ω = 0,

y(0) = y0

(8)

The corresponding convexified static optimization problem (sop) is min y − yd 2 ,

y + a = 0,

a∈U L

y|∂Ω = 0

(9)

We recall some useful inequalities to study existence and turnpike. First the energy inequality. There exists C > 0 such that for any solution y of (8), for a.e. t ∈ [0, T ], 1 y(t)2 + 2

t 0

1 ∇y(s) ds ≤ y0 2 +C 2 2

t 0

a(s)2 ds (10)

We can improve this inequality by using the Poincar´e inequality and the Gronwall lemma to get for a.e. t ∈ [0, T ], 2

t 2 −C

y(t) ≤ y0  e

+C

t 0

e−

t−s C

a(s)2 ds

(11)

The constant C > 0 depends only on the domain Ω and comes from the Poincar´e inequality. Taking a minimizing sequence and by classical arguments of functional analysis (see, e.g., Lions (1968)), it is not dif¯ respectively ficult to prove existence of solutions aT and a of (ocp)T and (sop). Necessary optimality conditions for (ocp)T . Applying the Pontryagin maximum principle in (Lions, 1968, Chapitre 3, Th´eor`eme 2.1), for any optimal solution (yT , aT ) of (ocp)T there exists an adjoint state pT ∈ L2 (0, T ; Ω) such that

0

where y is the solution of (5) corresponding to ω(·). Besides, for the same target function yd ∈ L2 (Ω), we consider an associated static shape design problem (SSD): Static problem. min y − yd 2 ,

ω∈UL

y + χω = 0,

y|∂Ω = 0

(7)

We want to compare the solution of (OSD)T and (SSD). 791

∂yT − yT = aT , yT|∂Ω = 0, yT (0) = y0 ∂t (12) ∂pT + pT = 2(yT − yd ), pT|∂Ω = 0, pT (T ) = 0 ∂t   ∀a ∈ U L , for a.e. t ∈ [0, T ], pT (t), aT (t) − a ≥ 0 (13)

Necessary optimality conditions for (sop). Similarly, applying (Lions, 1968, Chapitre 2, Th´eor`eme 1.4), for any optimal solution (¯ y, a ¯) of (sop) there exists an adjoint state p¯ ∈ L2 (Ω) such that

2019 IFAC NOLCOS 498 Vienna, Austria, Sept. 4-6, 2019

¯ y+a ¯ = 0,

¯ p = 2(¯ y − yd ),

G. Lance et al. / IFAC PapersOnLine 52-16 (2019) 496–501

y¯|∂Ω = 0 p¯|∂Ω = 0

(14)

(¯ p, a ¯ − a) ≥ 0 (15) ∀a ∈ U L , Using the bathtub principle (see, e.g., (Lieb and Loss, 2001, Theorem 1.14)), (13) and (15) give aT (·) = χ{pT (·)>sT (·)} + cT (·)χ{pT (·)=sT (·)} with

¯χ{p=¯ a ¯ = χ{p>¯ ¯ s} + c ¯ s}

(16) (17)

a.e. t ∈ [0, T ], cT (t) ∈ L∞ (Ω; [0, 1]) and c¯ ∈ L∞ (Ω; [0, 1])   sT (·) = inf σ ∈ R | |{pT (·) > σ}| ≤ L|Ω|   s¯ = inf σ ∈ R | |{¯ p > σ}| ≤ L|Ω|   It is important to note that, if | p¯ = s¯ | = 0, then it follows from (17) that the static optimal control a ¯ is actually the characteristic function of a shape ω ¯ ∈ UL . 2.3 Main results Existence of solutions. Proving existence of solutions for (OSD)T and (SSD) is not an easy task. We can find cases where there is no existence for (SSD) in (Henrot and Pierre, 2005, Section 4.2, Example 2): this is the relaxation phenomenon. This is why some assumptions are required on the target function yd . First, using maximum principle for elliptic (see Evans (1998) sec. 6.4) and parabolic equations (see Evans (1998) sec. 7.1.4) we introduce : • y T,0 and y T,1 are solutions of (8) with respectively a(·) = 0 and a(·) = 1 • y s,0 and y s,1 solutions of (9) with respectively a = 0 and a = 1  • y 0 = min y s,0 , min y T,0 (t) t∈(0,T )   1 s,1 • y = max y , max y T,1 t∈(0,T )

Theorem 1. If either yd verifies yd < y 0 or yd > y 1 or yd convex then we have existence and uniqueness of optimal solutions for both (SSD) and (OSD)T . Thanks to Theorem 1, hereafter we denote by • (yT , pT , ωT ) an optimal triple of (OSD)T . • (¯ y , p¯, ω ¯ ) an optimal triple of (SSD). T 1 yT (t) − yd 2 and J¯ = ¯ y − yd 2 . • JT = T 0

Measure-turnpike. Definition 1. We say that (yT , pT ) satisfies the stateadjoint measure-turnpike property if for every  > 0 there exists Λ() > 0, independent of T , such that |P,T | < Λ(), ∀T > 0 where   P,T = t ∈ [0, T ] | yT (t) − y¯ + pT (t) − p¯ > 

We refer to Carlson et al. (1991); Faulwasser et al. (2017); Tr´elat and Zhang (2018) (and references therein) for

792

similar definitions. Here P,T is the set of times  at which the optimal couple state-adjoint solution yT (·), pT (·) stays outside an -neighborhood of (¯ y , p¯) in L2 topology. We next recall the notion of dissipativity (see Willems (1972)). Definition 2. We say that (OSD)T is strictly dissipative at an optimal stationary point (¯ y, ω ¯ ) of (7) with respect to the supply rate function y − yd 2 w(y, ω) = y − yd 2 − ¯ if there exists a storage function S : E → R locally bounded and bounded from below and a K-class function α(·) such that, for any T > 0 and any 0 < τ < T , the strict dissipation inequality τ τ   S(y(τ ))+ α(y(t)− y¯)dt < S(y(0))+ w y(t), ω(t) dt (18) 0

0





is satisfied for any couple y(·), ω(·) solution of (5). Theorem 2. (i) (OSD)T is strictly dissipative in the sense of Definition 2. (ii) If yd is convex then the unique optimal solution of (OSD)T satisfies the measure-turnpike property. The measure-turnpike property is here a nice-to-have. We nonetheless get the stronger internal turnpike property which implies the previous one. Integral turnpike. Theorem 3. If yd is convex then there exists M > 0 such that T   ∀T > 0, yT (t) − y¯2 + pT (t) − p¯2 dt < M 0

Exponential turnpike. The exponential turnpike property is a stronger property and can be either on the state, the adjoint or the control or even the three together. Based on the numerical simulations presented in Section 3 we conjecture: Conjecture 4. If yd is convex then there exist C1 > 0 and C2 > 0 independent of T such that, for a.e. t ∈ [0, T ],   yT (t) − y¯ ≤ C1 e−C2 t + e−C2 (T −t)   pT (t) − p¯ ≤ C1 e−C2 t + e−C2 (T −t)   χωT (t) − χω¯  ≤ C1 e−C2 t + e−C2 (T −t) 2.4 Sketch of proof

Sketch of proof of Theorem 1. static problem (SSD).

We give the idea for the

We suppose yd > y 1 (we proceed similarly for yd < y 0 ). Having in mind (14) and (17) we get −¯ y = c¯ on {¯ p = s¯}. By contradiction, if c¯ ≤ 1 on {¯ p = s¯}, let us consider the solution y ∗ of (9) with the same a ¯ verifying (17) except that c¯ = 1 on {¯ p = s¯}. Then, by application of maximum principle (see Evans (1998) sec. 6.4), we get yd ≥ y ∗ ≥ y¯ and so y ∗ − yd  ≤ ¯ y − yd . That means a ¯ verifying

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(17) with c¯ = 1 is an optimal control. We conclude after with the uniqueness. We use similar reasoning for (OSD)T solution’s existence. Now if yd is convex, we have yd ≥ 0 on Ω. Having in mind (14) and (17), we assume by contradiction that |{¯ p = s¯}| > 0. By (Le Dret, 2013, Theorem 3.2), we have ¯ p = 0 on {¯ p = s¯}. We infer that yd = −¯ a on {¯ p = s¯}, which contradicts yd ≥ 0. Hence |{¯ p = s¯}| = 0 and thus a ¯ = χω¯ for some ω ¯ ∈ UL . Existence of solution for (SSD) is proved. Uniqueness of a ¯=ω ¯ comes from the fact that the problem (sop) is strictly convex. Uniqueness of y¯ and p¯ follows by application of (11). Remark 1. Proving existence for (OSD)T is more difficult. Anyway, if one replaces the Lagrange cost functional (6) with the Mayer cost functional JT (ω) = y(T ) − yd 2 then the optimality system becomes ∂yT − yT = aT , yT|∂Ω = 0, yT (0) = y0 ∂t   (19) ∂pT + pT = 0, pT|∂Ω = 0, pT (T ) = 2 yd − yT (T ) ∂t

with (13) unchanged. It follows that pT is analytic on (0, T ) × Ω and that all level sets {pT (t) = α} have zero Lebesgue measure. We conclude that the optimal control aT satisfying (13,19) is such that aT (t) = χωT (t) with ωT (t) = {pT (t) > sT (t)} for a.e. t ∈ (0, T ). Hence, for a Mayer problem, existence of an optimal time-shape is proved. Proof of Theorem 2. We follow Tr´elat and Zhang (2018) and the idea that strict dissipativity implies measureturnpike.

(i) Strict dissipativity  is established thanks to the storage function S(y) = y, p¯ where p¯ is the static optimal adjoint. Indeed, we consider an admissible pair (y(·), χω (·)) satisfying (5), we multiply it by p¯ and we integrate over Ω. Then we integrate in time on (0, T ), we use the static optimality conditions (14) and we get a strict dissipation inequality (18) with α : s → s2 .

(ii) Following the argument of Tr´elat and Zhang (2018), we prove that strict dissipativity implies measure-turnpike. Applying (18) to the optimal solution (yT , ωT ) we get 1 T

T 0

(y(0) − y(T ), p¯) yT (t) − y¯ dt ≤ JT − J¯ + T 2

(20)

¯ and Considering then the solution ys of (5) with ω(·) = ω  1 T 2 Js = T 0 ys (t) − yd  , we have JT −Js < 0 and we show −CT that Js − J¯ ≤ 1−eCT , then 1 T

T 0

yT (t) − y¯2 dt ≤

M T

(21)

1 2C

T 0

499

2

pT (t) − p¯ dt ≤ C

T 0

yT (t) − y¯2 dt

pT (0) − p¯2 − pT (T ) − p¯2 2 Using again the strict dissipativity equation (18) we get 2 |P,T | ≤ M T T . Hence we can find a constant M > 0 which does not depend on T such that |P,T | ≤ M 2 . +

Proof of Theorem 3. We consider the triples (yT , pT , χωT) and (¯ y , p¯, χω¯ ) satisfying the optimality conditions (12) and (14). Since χωT is bounded at each time t ∈ [0, T ] and by application of (11) to yT and pT we can find a constant C depending on y0 , yd , Ω, L such that yT (T )2 ≤ C

∀T > 0,

and

pT (0)2 ≤ C

(22)

˜ = χωT − χω¯ which verify We set y˜ = yT − y¯, p˜ = pT − p¯, a ∂ y˜ − ˜ y=a ˜, y˜|∂Ω = 0, y˜(0) = y0 − y¯ ∂t ∂ p˜ − ˜ p = 2˜ y , p˜|∂Ω = 0, p˜(T ) = −¯ p ∂t First, using (12) and (14) one can show that   a.e.t ∈ [0, T ], p˜(t), a ˜(t) ≥ 0

(23) (24) (25)

Multiplying then (23) by p˜, (24) by y˜ and then adding them we get T T    p˜(t), a ˜(t) dt + ˜ y (t)2 dt y¯ − y0 , p˜(0) − y˜(T ), p¯ = 





0

0

We apply then Cauchy-Schwarz inequality and (22) to find C > 0 such that T T   C 1 1 2 (26) p˜(t), a ˜(t) dt ≤ ˜ y (t) dt + T T T 0

0

The two terms at the left-hand side are positive and using the inequality (10) with p˜(T − t) we finally get 1 T

T 0



 M yT (t) − y¯2 + pT (t) − p¯2 dt ≤ T

(27)

Note again that the integral turnpike property is stronger than the measure-turnpike property. 3. NUMERICAL SIMULATIONS: OPTIMAL SHAPE DESIGN FOR THE 2D HEAT EQUATION We set Ω = [−1, 1]2 , L = 18 , T = 5, yd = Cst = 0.1 and y0 = 0. We consider the minimization problem 5  |y(t, x) − 0.1|2 dx dt (28) min ω(.)

0 [−1,1]2

To add the adjoint dependence, we apply (10) to the quantity ψ(·) = pT (T −·)− p¯ combined with the optimality conditions (12,14) and get 793

under the constraints ∂y − y = χω , ∂t

y(0, ·) = 0,

y|∂Ω = 0

(29)

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We compute numerically a solution by solving the equivalent convexified problem (ocp)T thanks to a direct method in optimal control (see Tr´elat (2005)). We discretize here with an implicit Euler method in time and with a decomposition on a finite element meshing of Ω using FREEFEM++ (see Hecht (2012)). We express the problem as a quadratic programming problem in finite dimension. We use then the routine IpOpt (see W¨ achter and Biegler (2006)) on a standard desktop machine. Fig. 3. Error between time optimal triple and static one In Fig. 3 we observe that the function is exponentially close to 0. This behavior lets us think that the exponential turnpike property should be verified in our case.

Fig. 1. Time optimal shape’s evolution cylinder We plot in Fig. 1 the evolution in time of the shape t → ω(t) which appears like a cylinder whose section at time t represents the shape ω(t). At the beginning (t = 0) we notice that the shape concentrate at the middle of Ω in order to warm as soon as possible near to yd . Once it is acceptable the shape stabilizes during a long time. Finally close to the final time the shape moves to the boundary of Ω in order to flatten the state yT because yd is here taken as a constant.

(a)

(b)

(c)

(d)

(e)

(f)

To complete this work, we need to clarify the existence of optimal shapes for (OSD)T when yd is convex. We see numerically in fig. 2 the time optimal shape’s existence for yd convex on Ω. Otherwise we can sometimes observe a relaxation phenomenon due to the presence of c¯ and cT (·) in the optimality conditions (12,14). We consider the same problem (ocp)T in 2D with Ω = [−1, 1]2 , L = 18 , T = 5 and the static one associated (sop). We take a target 1 function yd (x, y) = − 20 (x2 + y 2 − 2).

(a)

(b)

(d)

(e)

(c)

(f)

Fig. 4. Relaxation phenomenon : (a) t = 0; (b) t = 0.5; (c) t ∈ [1, 4]; (d) t = 4.5; (e) t = T ; (f) static shape

Fig. 2. Time optimal shape - Static shape: (a) t = 0; (b) t = 0.5; (c) t ∈ [1, 4]; (d) t = 4.5; (e) t = T ; (f) static shape We plot in Fig. 2 the comparison between the optimal shape at several times (in red) and the optimal static shape (in yellow). We see the same behavior when t = T2 . Now in order to mirror the turnpike phenomenon we plot the evolution in time of the distance between the optimal dynamic triple and the optimal static one t → yT (t) − y¯ + pT (t) − p¯ + χωT (t) − χω¯ . 794

Fig. 5. Error between time optimal triple and static one (Relaxation case) ¯) of (ocp)T In Fig. 4 we see that optimal control (aT , a and (sop) are in (0, 1) in the middle of Ω. This illustrates that relaxation occurs for some yd . It was choosen to verify −yd ∈ (0, 1). Here we calibrate the previous parameter L to observe this phenomenon, but for same yd and smaller L, optimal solutions are both shapes. Despite the relaxation we see Fig. 5 that turnpike still occurs.

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4. COMMENTS AND FURTHER WORKS

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Hecht, F. (2012). New development in freefem++. J. Numer. Math., 20(3-4), 251–265. Numerical simulations when yd is convex motivates us to Henrot, A. and Pierre, M. (2005). Variation et optimisation de formes, volume 48 of Math´ematiques & conjecture the existence of optimal shape for (OSD)T , Applications (Berlin) [Mathematics & Applications]. because we have never observed relaxation in that case. Springer, Berlin. doi:10.1007/3-540-37689-5. URL Moreover our simulations and particularly Fig. 3 indicate https://doi.org/10.1007/3-540-37689-5. Une analthe occurence of the exponential turnpike property. yse g´eom´etrique. [A geometric analysis]. ´ The work that we presented here is focused on the heat Le Dret, H. (2013). Equations aux d´eriv´ees partielles equation. It seems reasonable to extend our results to elliptiques non lin´eaires, volume 72 of Math´ematiques general parabolic operators, because we did not use any & Applications (Berlin) [Mathematics & Applications]. of the specific properties of the Laplacian operator. We Springer, Heidelberg. doi:10.1007/978-3-642-36175-3. consider here a linear partial differential equation which URL https://doi.org/10.1007/978-3-642-36175-3. gives us the uniqueness of the solution thanks to the strict Lieb, E.H. and Loss, M. (2001). Analysis, convexity of the criterion. As in Tr´elat and Zhang (2018), volume 14 of Graduate Studies in Mathematics. the notion of measure-turnpike seems to be a good and American Mathematical Society, Providence, soft way to obtain turnpike results. RI, second edition. doi:10.1090/gsm/014. URL https://doi.org/10.1090/gsm/014. To go further with the numerical simulations, our objective Lions, J.L. (1968). 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