[PDF] The bang-bang property in some parabolic bilinear optimal control

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hQ +Bi2 i?Bb p2`bBQM, The bang-bang property in some parabolic bilinear optimal control problemsviatwo-scale asymptotic expansions

Idriss Mazari

November 8, 2021

Abstract

We investigate the bang-bang property for fairly general classes ofL1L1constrained bilinear optimal control problems in two cases: that of the one-dimensional torus, in which case we consider parabolic equations, and that of generalddimensional domains for time-discrete parabolic models. Such a study is motivated by several applications in applied mathematics, most importantly in the study of reaction-diusion models. The main equation in the one- dimensional case writes@tumum=mum+f(t;x;um), wherem=m(x) is the control, which must satisfy someL1bounds (06m61 a.e.) and anL1constraint (Rm=m0is xed), and wherefis a non-linearity that must only satisfy that any solution of this equation is positive at any given time. The time-discrete models are simply time-discretisations of such equations. The functionals we seek to optimise are rather general; in the case of the torus, they writeJ(m) =RR (0;T)Tj1(t;x;um) +R

Tj2(x;um(T;)). Roughly speaking we prove in

this article that, ifj1andj2are increasing, then any maximisermofJis bang-bang in the sense that it writesm=1Efor some subsetEof the torus. It should be noted that such a result rewrites as an existence property for a shape optimisation problem. We prove an analogous result for time-discrete systems in any dimension. Our proofs rely on second order optimality conditions, combined with a ne study of two-scale asymptotic expansions. In the conclusion of this article, we oer several possible generalisations of our results to more involved situations (for instance for controls of the formm'(um)), and we discuss the limits of our methods by explaining which diculties may arise in other contexts. Keywords:Bilinear optimal control, reaction-diusion equations, shape optimisation, qualitative properties of optimisation problems, bang-bang property, shape optimisation. AMS classication:35K55,35K57, 49J30, 49K20, 49N99, 49Q10. Acknowledgment.This work was partially supported by the French ANR Project ANR-18- CE40-0013 - SHAPO on Shape Optimization and by the Project "Analysis and simulation of optimal shapes - application to life sciences" of the Paris City Hall. The author warmly thanks G. Allaire, B. Geshkovski, G. Nadin and Y. Privat for scientic exchanges. Contact Information:CEREMADE, UMR CNRS 7534, Universite Paris-Dauphine, Universite PSL, Place du Marechal De Lattre De Tassigny, 75775 Paris cedex 16, France. mazari@ceremade.dauphine.fr 1

Contents

1 Introduction3

1.1 Scope of the paper, informal presentation of our results

3

1.2 Main model and result for parabolic problems

4

1.2.1 The parabolic equation

4

1.2.2 Main result for the parabolic problem

6

1.3 Main model and result for time-discrete problems

9

1.3.1 The time-discrete model

9

1.3.2 Main result for time-discrete models

10

1.4 Comments on the proofs of Theorems

I and I I 10

1.5 Relationship with some shape optimisation problems

11

1.6 Bibliographical references and comments

12

1.6.1 Elliptic bilinear optimal control problem

12

1.6.2 Optimal bilinear control of parabolic equations

13

2 Proofs of theorems

I 13

2.1 Computation of rst and second order Gateaux derivativesviaan adjoint state. . 13

2.2 Lower estimate on the second order Gateaux derivative ofJ. . . . . . . . . . . .16

2.3 Construction of an admissible perturbation

19

2.4 Computations for single-mode perturbations

20

2.5 Formal estimate of the leading order term

21

2.6 Formal estimate of the lower order term

25

2.7 Strategy and comment for the proof of the asymptotic expansion

26

2.8 Proof of proposition

19 28

3 Proof of Theorem

I I 33

3.1 Preliminary analysis of the system

33

3.2 Computation and estimate on the derivatives of the functional

34

4 Conclusion39

4.1 Possible generalisations of theorem

I 39

4.1.1 General comment about generalisations

39

4.1.2 Approximations of time dependent controls

40

4.1.3 Other types of interactions: generalisation and obstruction

42

4.1.4 Some interactions not covered by our method

44

4.2 Generalisations and obstructions for Theorem

I 45

4.2.1 Higher dimensional tori

45

4.2.2 Possible obstructions in other domains

45

4.2.3 Possible obstructions for other diusion operators

45

4.2.4 Some possible open questions

46

4.3 The diculty with general time dependent controls

46

A Study of the parabolic model

50

A.1 Proof of lemma

13 50

A.2 Regularity results: proof of proposition

16 50

A.3 Proof of lemma

21
53

B Study of the time-discrete model

53

B.1 Proof of Lemma

22
53
2

1 Introduction

1.1 Scope of the paper, informal presentation of our results

In this paper, we oer a theoretical analysis of an ubiquitous constrained optimal control problem, in which one aims at optimising a criteria by acting in a bilinear way on the state of the PDE. Prototypically, the model under consideration reads as follows: for a given non-linearityf= f(t;x;u) and a controlm=m(t;x), we letumbe the solution of tumum=mum+f(t;x;um) with variablesx2 andt2[0;T] under certain boundary conditions. For a certain time horizon

T >0, we aim at optimising criteria of the form

J(m) =ZZ

(0;T) j

1(t;x;um) +Z

j

2(x;um(T;))

under some constraints on the controlm. Throughout the paper, the constraints onmwill be of L

1andL1type; in other words, one constraint takes the form

8t2[0;T];Z

m(t;x)dx=V0xed orZ m(x)dx=V0(ifmdoes not depend on time) while the other is of the type

6m6a.e.

In this type of setting, one of the salient qualitative features of optimisers is thebang-bang property.

In other words, is it true that any maximiser writesm=+ ()1Efor some measurable subsetEof ? This property is linked to (non-)existence results for shape optimisation problems. There were, in recent years, several ne qualitative studies of this property in the elliptic case or in the space-discretised case; we refer to Section 1.6 . However, in the context of parabolic models and despite the current activity in the study of parabolic bilinear optimal control problems, this property does not seem to be reachable by the available techniques; we refer to section 1.4 and section 1.6 In the rst part of this paper, we prove that, under reasonable assumptions on the non-linearity fthat ensure the well-posedness of the parabolic system, and on the cost functionsj1;j2(roughly speaking, they must both be non-decreasing, and one has to be increasing), the bang-bang property holdsif we assume that admissible controls are constant in timeand that the domain is one- dimensional. This is the main contribution of this article. It hinges on the methods of [ 39
coupled with two-scale asymptotic techniques previously used in [ 37
] in the context of the optimal control of initial conditions in reaction-diusion equations. The reason why we tackle the one- dimensional periodic case will be explained later on. It should be noted that we explain in the conclusion how we may cover, with the same type of arguments, higher-dimensional orthotopes. The main explanation behind having to work with time constant controls is a technical one; this allows to gain further regularity on the solutions of the parabolic PDE under consideration. For this reason, section 4.1.2 of the co nclusioncon tainsa discussion of p ossiblegeneralisations and obstructions to generalisations; we explain, for instance, how to deal with the case of controlsm writingPN i=1i(t)mi(x). As a rst side comment, it should also be noted that our analysis cover the case of sometracking-type functionals. This is not the main topic, and we refer to remark5 . As a second side comment, our analysis can encompass more intricate interactions between the control and the state. For instance, we provide, in section 4.1.3 of the conclusion, a generalisation of our results to the case where the control and the state are coupledviaa term of the formm'(um) for a large class of'. 3 Our second contribution deals with a semi-discretised (in time) parabolic model, where the main system of equation is given by w m;k+1wm;kt wm;k+1=mk+1wm;k+1+fk+1(x;wm;k+1);k= 0;:::;N1 for some time stept, wherem= (m1;:::;mN) and eachmisatisesL1andL1constraints. The optimisation problem is rather, in this case

J(m) =NX

i=1j i(x;um;i): For this semi-discrete parabolic model, we prove that, provided the functionsjiare increasing, any optimisermis of bang-bang type. Here, our analysis holds in any dimension, in any smooth bounded domain for Neumann or Robin boundary conditions. The reason we deal with this semi- discretised version is twofold: rst, there has recently been some interest in the discretisation of bilinear optimal control problems [ 43
]. Second, this allows us to give more perspective on the proof of the bang-bang property for the parabolic model in general domains.

Our introduction is divided accordingly: section

1.2 is dev otedto the study of p arabolicprob- lems, while section 1.3 t acklestime-discrete parab olicmo dels.

1.2 Main model and result for parabolic problems

1.2.1 The parabolic equation

Admissible controls in parabolic modelsIn the case of parabolic models, we are working in the (one-dimensional) torusT. In section4.2.1 , we explain how our methods may extend to the case of higher dimensional tori. Regarding the time regularity of admissible resources distribution, we shall make a strong assumption: the admissible controls are constant in time. The reason is that the method we introduce and develop hinges on ne regularity properties of solutions of the associated evolution equation that can not be obtained in the case where the controlmalso depends on time. We also refer to remark 8 for furth ercommen ts. In this setting, denoting byumthe state of the equation and bymthe control, the only type of interaction we are interested in is bilinear; in other words, the control appears in the model viathe termmum(see Remark7 and section 4.1.3 for consideration son the case of in teractions of the formm'(um)). In terms of constraints, we impose two on the controls, anL1and anL1 one. Each of these constraints has a natural interpretation in dierent elds of applications. In spatial ecology for instance, one may think ofmas a resources distributions, in which case theL1 constraint simply models the fact that, at any given point, there can only be a maximum amount of resources available, while theL1constraint accounts for the limitation of the global quantity of resources involved. For theL1constraints, without loss of generality (we also refer to remark6 ), we shall consider controls satisfying

06m61 a.e. inT:

For theL1constraint, we x a volume constraintV02(0;Vol(T)), and we shall consider controls satisfyingZ T m=V0:

This leads to considering the admissible class

4

M(T) :=

m2L1(T) : 06m61 a.e. inT;Z T m=V0 :(Adm) Of notable interest inM(T) arebang-bangfunctions; as they are the central theme of this paper we isolate their denition here. Denition 1.A functionm2 M(T)is calledbang-bangif there existsETsuch thatm=1E. Nonlinearities under considerationOur choice of nonlinearity in the parabolic model also derives from considerations in mathematical biology or chemistry. Namely, we want the solutions not only to exist but to be uniformly bounded (in time) in theL1norm, as well as to enjoy a strong maximum property (in the sense that, starting from a non-zero initial condition, the solution is positive at any arbitrary positive time). The latter is not only important from a modelling point of view but also in the course of the proof, as it in uences the monotonicity of the functional under consideration. As the right hand side of the reaction-diusion equation writesmu+f(t;x;u) we shall make the following assumptions that guarantee the well-posedness of the ensuing system: fisC1in time,L1inx, andC2inu, and, for anyK2IR, sup x2T;u2[0;K];t2[0;K] @f@t +@f@u +@2f@u 2 <1:(H1) Assumption (H1) serves to derive the proper regularity of the solutions of the equation. The next assumption is used to obtain upper and lower bounds on the solution: f(;;0)>0,f(;;0)2L1((0;T)T) and there exists >0 such that for anyu>, for anyt2IR+, for a.e.x2T,f(t;x;u)6u:(H2) In the rst condition, if we hadf(;;0) = 0 this would simply model that when no individuals are present no reaction is happening. Assuming the general inequality allows to consider non-negative source terms (i.e.one may takef(t;x;u) =ug(t;x;u) +y(t;x) for a certaingand a non-negative source termy). It should be noted that, had we taken06m61asL1constraints, the nal inequality in (H2) would rewritef(t;x;u)6supfj0j;j1jgu. Finally the last assumption is seemingly the most restrictive one, but we explain, in Remark 2 why it is not problematic for the type of problems we have in mind. fis, uniformly in (t;x)2(0;T)T, uniformly Lipschitz inu2IR: there existsAsuch that, for any (t;x)2(0;T)T;for anyu;u02IR,jf(t;x;u)f(t;x;u0)j6Ajuu0j:(H3) Remark 2(Comment on (H3)).(H3)may seem restrictive, as the typical monostable logistic diusive equation would involve the non-linearityf(u) =u2, which grossly violates the Lips- chitz condition of(H3). However, assumption(H2)ensures that, if we start from a positive bounded initial conditionu0, then the solution remains positive and bounded uniformly in time by maxku0kL1;(see lemma13 ) so that it suces to extendf(t;x;)outside(0;maxku0kL1;) to a globally uniformly Lipschitz function onIR. Initial conditionWe simply take an initial condition independent ofm, sayu0, satisfying inf

Tu0>0;u02C2(T):(1.1)

5 Parabolic modelWe dene, for anym2 M(T),umas the unique solution of (@u m@t um=mum+f(t;x;um) in (0;T)T; u m(0;) =u0inT:(1.2) By [ 45
, Theorem 5.2, Chapter 1] there exists a unique solutionumof (1.2) (we also refer to lemma 16 for furth erregularit yinformation ab outum). Optimisation problem in the parabolic context: time-constant controlsWe consider fairly general functionals that we seek to optimise. To dene this functional, we consider two functionsj1;j2, a time horizonT >0 and we dene

J:M(T)3m7!ZZ

(0;T)Tj

1(t;x;um) +Z

T j

2(x;um(T;)):(1.3)

We mentioned earlier the crucial role of the monotonicity of the functionalJ, which hinges on that ofj1andj2; we refer to section1. 6for further commen ts.W eth usassume that j1;j2satisfy j

1andj2are non-decreasing in the second variable on IR+,

j

1isC1in its two rst variable andC2in its third variable,

j

2isC2in its two variables,

8(t;x)2(0;T)T;8K2IR+;

sup (t;x)2(0;T)T;u2[0;K]sup =0;1;2 @j 1@t (t;x;u)+@j1@u (t;x;u)+@j2@u (x;u)<1; and either for any (t;x)2(0;T)T,@uj1(t;x;)>0 in (0;+1) orquotesdbs_dbs45.pdfusesText_45

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