=Paper=
{{Paper
|id=Vol-2837/paper6
|storemode=property
|title=Theoretical and numerical analysis of extremum problems for reaction-diffusion model
|pdfUrl=https://ceur-ws.org/Vol-2837/paper6.pdf
|volume=Vol-2837
|authors=Roman V. Brizitskii,Alexander Yu. Chebotarev,Victoriya S. Bystrova,Pavel A. Maksimov
}}
==Theoretical and numerical analysis of extremum problems for reaction-diffusion model==
https://ceur-ws.org/Vol-2837/paper6.pdf
Theoretical and numerical analysis of extremum
problems for reaction-diffusion model
Roman V. Brizitskiia , Alexander Yu. Chebotareva , Victoriya S. Bystrovab and Pavel
A. Maksimovb
a
Institute for Applied Mathematics FEB RAS, 7 Radio St, Vladivostok, 690041, Russian Federation
b
Far Eastern Federal University, Centre for Research and Education in Mathematics (CREM), 8 Sukhanova St, Vladivostok,
690950, Russian Federation
Abstract
Boundary and extremum problems for the reaction-diffusion equation, in which the reaction coefficient nonlin-
early depends on the concentration of the substance are studied. The maximum principle is stated for solving
boundary value problems. Optimality systems are derived for extreme problems. Based on the analysis of these
systems, local stability estimates of optimal solutions are derived, numerical algorithms for solving extreme
problems are developed, and a stationary analogue of the bang–bang principle is established.
Keywords
nonlinear reaction-diffusion equation, maximum principle, control problem, optility system, local stability
estimates, numerical algorithm, bang–bang principle
1. Introduction
In recent years, interest in the study of inverse and control problems for heat and mass transfer mod-
els has only increased. Note the works [1, 2, 3, 4, 5, 7, 6, 8, 9, 10, 11, 12, 13] devoted to the theoretical
analysis of these problems. In these papers, the solvability of boundary value, inverse and extremum
problems for the indicated models was studied, and the questions of uniqueness and stability of solu-
tions were studied. Related problems for models of complex heat transfer and ferroelectric hysteresis
are studied in [14, 15, 16, 17].
We also note that applications of control problems are not limited to the search for effective mech-
anisms for controlling physical fields in continuous media. Within the framework of the optimization
approach (see [10, 13]), problems of reconstructing unknown functions in the considered models are
reduced to control problems using additional information about solving the corresponding boundary
value problems.
This paper is devoted to the theoretical and numerical analysis of boundary value and extremum
problems for the reaction-diffusion equation, in which the reaction coefficient nonlinearly depends on
the concentration of the substance. In contrast to [7, 6, 8, 9, 10, 13], in this paper the optimality systems
obtained for control problems are used not only to study the stability (and uniqueness) of optimal
solutions. Using these systems, numerical algorithms for solving extreme problems are constructed
and new properties of their solutions are stated.
Far Eastern Workshop on Computational Technologies and Intelligent Systems, March 2–3, 2021, Khabarovsk, Russia
" mlnwizard@mail.ru (R.V. Brizitskii); cheb@iam.dvo.ru (A.Yu. Chebotarev); torirorik@gmail.ru (V.S. Bystrova);
maksimov.pa@dvfu.ru (P.A. Maksimov)
~ https://dblp.org/pid/128/7131.html (A.Yu. Chebotarev)
© 2021 Copyright for this paper by its authors.
Use permitted under Creative Commons License Attribution 4.0 International (CC BY 4.0).
CEUR
Workshop
Proceedings
http://ceur-ws.org
ISSN 1613-0073 CEUR Workshop Proceedings (CEUR-WS.org)
2. Boundary value problem
In bounded domain Ω ⊂ ℝ3 with boundary Γ the nonlinear reaction–diffusion equation is considered
− div(𝜆(𝐱)∇𝜑) + 𝑘(𝜑, 𝐱)𝜑 = 𝑓 on Ω, 𝜑 = 𝜓 on Γ. (1)
Here function 𝜑 means pollutant substance’s concentration, 𝑓 is a volume density of external sources
of substance, 𝜆(𝐱) is a diffusion coefficient, function 𝑘 = 𝑘(𝜑, 𝐱) is a reaction coefficient, 𝐱 ∈ Ω. This
problem (1) will be called Problem 1 below.
In this paper, we prove the global solvability of Problem 1 and the local uniqueness of its solution
in the case when the nonlinearity 𝑘(𝜑, 𝐱) 𝜑 is not monotone in the entire domain Ω, as was assumed
in [8]. This allows you to expand the range of mathematical models, the correctness of which we can
justify. For concentration 𝜑 the principle of minimum and maximum is stated.
Further, for Problem 1, a control problem is formulated, the role of controls in which is played by
the functions 𝜆 and 𝑓 , and its solvability is proved in the general case. Optimality systems are derived
for extreme problems with specific reaction coefficients. Based on the analysis of these systems,
local stability estimates of optimal solutions are derived, numerical algorithms for solving extreme
problems are developed.
A one-parameter control problem is considered separately, for which regularization is not used.
For this problem, the validity of a stationary analogue of the bang-bang principle is established (see
the meaning of this term below or in [14, 15]).
3. Solvability of the boundary value problem
While studying the considered problems we will use Sobolev spaces 𝐻 𝑠 (𝐷), 𝑠 ∈ ℝ. Here 𝐷 means
either the domain Ω or some subset 𝑄 ⊂ Ω, or the boundary Γ. By ‖ ⋅ ‖𝑠,𝑄 , | ⋅ |𝑠,𝑄 and (⋅, ⋅)𝑠,𝑄 we will
denote the norm, seminorm and the scalar product in 𝐻 𝑠 (𝑄). The norms and scalar products 𝐿2 (𝑄),
𝐿2 (Ω) or in 𝐿2 (Γ) will be denoted correspondingly by ‖ ⋅ ‖𝑄 and (⋅, ⋅)𝑄 , ‖ ⋅ ‖Ω and (⋅, ⋅) or ‖ ⋅ ‖Γ and (⋅, ⋅)Γ .
Let
𝐿+ (Ω) = {𝑘 ∈ 𝐿𝑝 (Ω) ∶ 𝑘 ≥ 0}, 𝑝 > 1, 𝑍 = {𝐯 ∈ 𝐿4 (Ω)3 ∶ div 𝐯 = 0 in Ω},
𝑝
𝐻𝜆𝑠0 (Ω) = {ℎ ∈ 𝐻 𝑠 (Ω) ∶ ℎ ≥ 𝜆0 > 0 in Ω}, 𝑠 > 3/2.
It will be assumed that the following conditions hold:
(i) Ω is a bounded domain in the space ℝ3 with boundary Γ ∈ 𝐶 0,1 ;
(ii) 𝑓 ∈ 𝐿2 (Ω), 𝜓 ∈ 𝐻 1/2 (Γ);
(iii) For any function 𝑣 ∈ 𝐻 1 (Ω) the embedding 𝑘(𝑣, ⋅) ∈ 𝐿+ (Ω) is true for some 𝑝 ≥ 5/3, which
𝑝
doesn’t depend on 𝑣, and on any sphere 𝐵𝑟 = {𝑣 ∈ 𝐻 1 (Ω) ∶ ‖𝑣‖1,Ω ≤ 𝑟} of radius 𝑟 the inequality
takes place
‖𝑘(𝑣1 , ⋅) − 𝑘(𝑣2 , ⋅)‖𝐿𝑝 (Ω) ≤ 𝐿1 ‖𝑣1 − 𝑣2 ‖𝐿5 (Ω) ∀𝑣1 , 𝑣2 ∈ 𝐵𝑟 .
Here 𝐿 is a constant, which depends on 𝑟, but doesn’t depend on 𝑣1 , 𝑣2 ∈ 𝐵𝑟 .
(iv) Let Ω1 ⊂ Ω be a subdomain of Ω, such that Ω1 ⊂ Ω. Put Ω2 = Ω ⧵ Ω1 .
The function 𝑘(𝜑, ⋅)𝜑 is monotone in the subdomain Ω2 in the following sense:
(𝑘(𝜑1 , ⋅)𝜑1 − 𝑘(𝜑2 , ⋅)𝜑2 , 𝜑1 − 𝜑2 )Ω2 ≥ 0 ∀𝜑1 , 𝜑2 ∈ 𝐻 1 (Ω) (2)
and bounded in the sense that there exist positive constants 𝐴1 , 𝐵1 , depending on 𝑘, such that
‖𝑘(𝜑, ⋅)‖𝐿𝑝 (Ω2 ) ≤ 𝐴1 ‖𝜑‖𝑡1,Ω + 𝐵1 , 𝑝 ≥ 5/3, 𝑡 ≥ 0. (3)
In subdomain Ω1 for function 𝑘(𝜑, ⋅) with constant 𝐶1 > 0 the following inequality is true:
‖𝑘(𝜑, ⋅)‖𝐿𝑝 (Ω1 ) ≤ 𝐶1 ∀𝜑 ∈ 𝐻 1 (Ω).
Let us mention that the conditions (iii), (iv) describe an operator, acting from 𝐻 1 (Ω) to 𝐿𝑝 (Ω), where
𝑝 ≥ 5/3, which give an opportunity to take into consideration the dependence of the reaction coeffi-
cient on either the solution 𝜑 or the spatial variable 𝐱. For example,
1
𝑘= in Ω1 and 𝑘 = 𝜑 2 in 𝑄 ⊂ Ω2 , 𝑘 = 𝑘0 (𝐱) ∈ 𝐿5/3
+ (Ω2 ⧵ 𝑄) in Ω2 ⧵ 𝑄,
1 + 𝜑2
where 𝑄 is a subdomain of Ω2 .
Let us also remind that on the strength of the Sobolev embedding theorem the space 𝐻 1 (Ω) is
embedded into the space 𝐿𝑠 (Ω) continuously at 𝑠 ≤ 6 and compactly at 𝑠 < 6, with some constant 𝐶𝑠 ,
which depend on 𝑠 and Ω, and the estimate is true
‖𝜑‖𝐿𝑠 (Ω) ≤ 𝐶𝑠 ‖𝜑‖1,Ω ∀𝜑 ∈ 𝐻 1 (Ω). (4)
The following lemmas hold (see., for example, [8]).
Lemma 1.1.
𝑝
If condition (i), (ii) hold and 𝜆 ∈ 𝐻𝜆𝑠0 (Ω), 𝑠 > 3/2, 𝑘1 ∈ 𝐿+ (Ω), 𝑝 ≥ 5/3 then there are such positive
constants 𝐶0 , 𝛿0 , 𝛾𝑝 , depending on Ω or Ω and 𝑝, such that the following inequalities hold:
|(𝜆∇𝜑, ∇𝜂)| ≤ 𝐶0 ‖𝜆‖𝑠,Ω ‖𝜑‖1,Ω ‖𝜂‖1,Ω ,
|(𝑘1 𝜑, 𝜂)| ≤ 𝛾𝑝 ‖𝑘1 ‖𝐿𝑝 (Ω) ‖𝜑‖1,Ω ‖𝜂‖1,Ω , ∀𝜑, 𝜂 ∈ 𝐻 1 (Ω), (5)
(𝜆∇𝜑, ∇𝜑) ≥ 𝜆∗ ‖𝜑‖21,Ω , (𝜆∇𝜑, ∇𝜑) + (𝑘1 𝜑, 𝜑) ≥ 𝜆∗ ‖𝜑‖21,Ω ∀𝜑 ∈ 𝐻01 (Ω), 𝜆∗ ≡ 𝛿0 𝜆0 . (6)
Lemma 3.1.
Let condition (i) holds. Then for any function 𝜓 ∈ 𝐻 1/2 (Γ) there exists function 𝜑0 ∈ 𝐻 1 (Ω) such that
𝜑0 = 𝜓 on Γ and with some constant 𝐶Γ , depending on Ω and Γ, the following estimate is true:
‖𝜑0 ‖1,Ω ≤ 𝐶Γ ‖𝜓 ‖1/2,Γ .
Let us multiply (1) by ℎ ∈ 𝐻01 (Ω) and integrate over Ω. Using the Green’s formula, we are coming
to the weak formulation of Problem 1. It consists in finding function 𝜑 ∈ 𝐻 1 (Ω) from condition
(𝜆∇𝜑, ∇ℎ) + (𝑘(𝜑)𝜑, ℎ) = (𝑓 , ℎ) ∀ℎ ∈ 𝐻01 (Ω), 𝜑|Γ = 𝜓 . (7)
The next theorem follows from results [11]
Theorem 3.1
If conditions (i)–(iv) hold. then a weak solution 𝜑 ∈ 𝐻 1 (Ω) of Problem 1 exists and the following estimate
takes place:
‖𝜑‖1,Ω ≤ 𝑀𝜑 ≡ 𝐶∗ (‖𝑓 ‖Ω + 𝐶Γ (𝐶0 ‖𝜆‖𝑠,Ω + 𝛾1 ‖𝐮‖𝐿4 (Ω)3 + 𝛾𝑝 𝐶1 )‖𝜓 ‖1/2,Γ +
+ 𝐶∗ 𝛾𝑝 𝐶Γ (𝐶Γ𝑟 𝐴1 ‖𝜓 ‖𝑟1/2,Γ + 𝐵1 )‖𝜓 ‖1/2,Γ + 𝐶Γ ‖𝜓 ‖1/2,Γ . (8)
If, besides, this condition is met
𝛾𝑝 𝐿𝑀𝜑 < 𝜆∗ , (9)
where constants 𝛾𝑝 and 𝜆∗ are specified in Lemma 1.1, constant 𝐿 in introduced in condition (iii), then
Problem 1’s solution is unique.
Let, in addition to (i)–(iv), the following conditions be satisfied:
(v) 𝜓min ≤ 𝜓 ≤ 𝜓max on Γ, 𝜆0 ≤ 𝜆 ≤ 𝜆max on Ω, 𝑓min ≤ 𝑓 ≤ 𝑓max on Ω2 and 𝑓 = 0 on Ω1 (or Ω1 = ∅).
Here 𝜓min , 𝜓max , 𝑓min , 𝑓max are nonnegative numbers, 𝜆max > 𝜆0 > 0.
(vi) 𝑘(𝜑, 𝐱)𝜑 satisfies the inequality (2), while 𝑘(𝜑, 𝐱) = 𝑎(𝐱)𝑘1 (𝜑), where 0 < 𝑎min ≤ 𝑎(𝐱) ≤ 𝑎max < ∞
a.e. in Ω, 𝑘1 (⋅) ∶ ℝ → ℝ+ is a continuous function and the functional equations for 𝑀 and 𝑚:
𝑘1 (𝑀1 )𝑀1 = 𝑓max /𝑎min and 𝑘1 (𝑚1 )𝑚1 = 𝑓min /𝑎max (10)
have at least one solution.
Lemma 3.2.
Under conditions (i)–(vi) for the solution 𝜑 ∈ 𝐻 1 (Ω) of Problem 1 the following principle of maximum
and minimum is valid:
𝑚 ≤ 𝜑 ≤ 𝑀 a.e. in Ω, 𝑀 = max{𝜓max , 𝑀1 }, 𝑚 = min{𝜓min , 𝑚1 }. (11)
Here 𝑀1 is the minimum root of the first equation in (10) and 𝑚1 is the maximum root of the second
equation in (10).
Remark 3.1
For power-law reaction coefficients, the parameters 𝑀1 and 𝑚1 are easily calculated. For example, if
𝑘(𝜑) = |𝜑| then 𝑀1 = (𝑓max /𝑎min )1/2 and 𝑚1 = (𝑓min /𝑎max )1/2 .
4. Statement of optimal control problem
Let us formulate an optimal control problem for Problem 1. For this purpose the whole set of initial
data will be divided into two groups: the group of fixed functions, in which functions 𝜓 and 𝑘(𝜑, ⋅)
are included, and the group of controlling functions, in which 𝜆 and 𝑓 will be included. We assume
that 𝜆 and 𝑓 can be changed in subsets 𝐾1 and 𝐾2 , respectively, which satisfy the following condition:
(j) 𝐾1 ⊂ 𝐻𝜆𝑠0 (Ω), 𝑠 > 3/2, 𝐾2 ⊂ 𝐿2 (Ω) are nonempty convex closed sets.
Define functional space
𝑌 = 𝐻 −1 (Ω) × 𝐻 1/2 (Γ),
and set 𝑢 = (𝜆, 𝑓 ), 𝐾 = 𝐾1 × 𝐾2 .
Introduce an operator 𝐹 = (𝐹1 , 𝐹2 ) ∶ 𝐻 1 (Ω) × 𝐾 → 𝑌 by formulae:
⟨𝐹1 (𝜑, 𝑢), ℎ⟩ = (𝜆∇𝜑, ∇ℎ) + (𝑘(𝐱, 𝜑) 𝜑, ℎ) + (𝐮 ⋅ ∇𝜑, ℎ) − (𝑓 , ℎ),
𝐹2 (𝜑) = 𝜑|Γ − 𝜓
and rewrite a weak form (7) of Problem 1 in the form of the operator equation 𝐹 (𝐱, 𝑢) = 0.
Let 𝐼 ∶ 𝑋 → ℝ be a weakly lower semicontinuous functional. Consider the following multiplicative
control problem:
𝜇0 𝜇1 𝜇3
𝐽 (𝜑, 𝑢) ≡ 𝐼 (𝜑) + ‖𝜆‖2𝑠,Ω + ‖𝑓 ‖2Ω → inf,
2 2 2
𝐹 (𝜑, 𝑢) = 0, (𝜑, 𝑢) ∈ 𝐻 1 (Ω) × 𝐾 . (12)
The set of possible pairs for the problem (12) is denoted by
𝑍𝑎𝑑 = {(𝐱, 𝑢) ∈ 𝑋 × 𝐾 ∶ 𝐹 (𝐱, 𝑢) = 0, 𝐽 (𝐱, 𝑢) < ∞}.
Let, in addition to 4, the following condition hold:
(jj) 𝜇0 > 0, 𝜇𝑖 ≥ 0, 𝑖 = 1, 2 and 𝐾 = 𝐾1 × 𝐾2 is a bounded set in 𝐻 𝑠 (Ω) × 𝐿2 (Ω), 𝑠 > 3/2, or 𝜇𝑖 > 0,
𝑖 = 0, 1, 2 and a functional 𝐼 is bounded from below.
We use the following cost functionals:
𝐼1 (𝜑) = ‖𝜑 − 𝜑 𝑑 ‖2𝑄 = ∫ |𝜑 − 𝜑 𝑑 |2 𝑑𝐱, 𝐼2 (𝜑) = ‖𝜑 − 𝜑 𝑑 ‖21,𝑄 , (13)
𝑄
Here a function 𝜑 𝑑 ∈ 𝐿2 (𝑄) denotes a desired concentration field, which is given in a subdomain
𝑄 ⊂ Ω.
Theorem 4.1.
Assume that the assumptions (i)–(vi) and 4, 4 take place. Let 𝐼 ∶ 𝑋 → ℝ be a weakly semicontinuous
below functional and let 𝑍𝑎𝑑 ≠ 0. Then there is at least one solution (𝜑, 𝑢) ∈ 𝑋 × 𝐾 of the control problem
(12)
Let us note that he solution (𝜑̂ , 𝑢̂ ) of the extremum problem (12) can be treated as an approximate
solution of the inverse problem of recovering unknown functions 𝜆, 𝑓 and 𝜒 with the help of measured
concentration 𝜑𝑑 ∈ 𝑄 ⊂ Ω in case, when 𝐼1 (𝜑̂ )/‖𝜑𝑑 ‖2𝑄 is small enough. For example, it can be within
the limits of the measurment error of 𝜑𝑑 . However, the only possibility to show this is numerical in a
general case. Sometimes it is possible to construct the exact solutions of the corresponding boundary
value problem and to get the upper estimate of the value 𝐼1 with the their help (see, for example, the
papers [19], which are dedicated to the use of the optimization method for solving inverse problems
of heat cloaking).
5. Optimality system and stability estimates
The next step in the study of the extreme problem is the derivation of the optimality system, which
provides valuable information about additional properties of optimal solutions. Based on its analysis,
one can establish, in particular, the uniqueness and stability of optimal solutions, and also to construct
numerical algorithms for finding optimal solutions to extreme problems.
We will further assume that 𝑘(𝜑) = |𝜑|, which corresponds to a slowly decaying pollutant.
We introduce a dual space 𝑌 ∗ = 𝐻01 (Ω) × 𝐻 −1/2 (Γ) to 𝑌 . It is easy to show that Fr𝑒́ chet derivative of
an operator
𝐹 = (𝐹1 , 𝐹2 ) ∶ 𝐻 1 (Ω) × 𝐾 → 𝑌
with respect to 𝜑 at any point
(𝜑̂ , 𝑢̂ ) = (𝜑̂ , 𝜆̂, 𝑓̂ , 𝜒̂ )
is a linear continuous operator
𝐹𝜑′ (𝜑̂ , 𝑢̂ ) ∶ 𝐻 1 (Ω) → 𝑌 ,
that maps each element 𝜏 ∈ 𝐻 1 (Ω) into an element 𝐹𝜑′ (𝜑̂ , 𝑢̂ )(𝜏 ) = (𝑦̂ 1 , 𝑦̂ 2 ) ∈ 𝑌 . Here the elements
𝑦̂ 1 ∈ 𝐻 −1 (Ω) and 𝑦̂ 2 ∈ 𝐻 1/2 (Γ) are defined by 𝜑̂ and 𝜏 and by the following relations:
⟨𝑦̂ 1 , 𝜏 ⟩ = (𝜆̂∇𝜏 , ∇ℎ) + 2(|𝜑̂ | 𝜏 , ℎ) ∀ℎ ∈ 𝐻 1 (Ω), 𝑦2 = 𝜏 |Γ𝐷 . (14)
By 𝐹𝜑′ (𝜑̂ , 𝑢̂ )∗ ∶ 𝑌 ∗ → 𝐻 1 (Ω)∗ we denote an operator adjoint to 𝐹𝜑′ (𝜑̂ , 𝑢̂ ).
In accordance with a general theory of smooth-convex extremum problems [18], we introduce an
element 𝐲∗ = (𝜃, 𝜁 ) ∈ 𝑌 ∗ , to which we will refer as to an adjoint state and will define the Lagrangian
∶ 𝐻 1 (Ω) × 𝐾 × ℝ × 𝑌 ∗ → ℝ by formula
(𝜑, 𝑢, 𝐲∗ ) = 𝐽 (𝜑, 𝑢)+⟨𝐲∗ , 𝐹 (𝜑, 𝑢)⟩𝑌 ∗ ×𝑌 ≡ 𝐽 (𝜑, 𝑢) + ⟨𝐹1 (𝜑, 𝑢), 𝜃⟩ + ⟨𝜁 , 𝐹2 (𝜑, 𝑢)⟩Γ𝐷 , (15)
where ⟨𝜁 , ⋅⟩Γ𝐷 = ⟨𝜁 , ⋅⟩𝐻 −1/2 (Γ𝐷 )×𝐻 1/2 (Γ𝐷 ) .
Since |𝜑̂ | ∈ 𝐿6+ (Ω), then from [8] it follows that for any 𝑓 ∈ and 𝜓 ∈ 𝐻 1/2 (Γ𝐷 ) there is a unique
solution 𝜏 ∈ 𝐻 1 (Ω) of linear problem
(𝜆̂∇𝜏 , ∇ℎ) + 2(|𝜑̂ | 𝜏 , ℎ) = ⟨𝑓 , ℎ⟩ ∀ℎ ∈ , 𝜏 |Γ = 𝜓 . (16)
Then operator 𝐹𝜑′ (𝜑̂ , 𝑢̂ ) ∶ 𝐻 1 (Ω) → 𝑌 is an isomorphism and from [18] it follows
Theorem 5.1
Assume that assumptions (i)–(iv), (vi) and 4, 4 take place and let an element (𝜑̂ , 𝑢̂ ) ∈ 𝑋 × 𝐾 be a local
minimizer for the problem (12). Suppose also that a cost functional 𝐼 ∶ 𝑋 → ℝ is continuously Frechet
differentiable with respect to the state 𝐱 in a point 𝐱̂ . Then there is a unique nonzero Lagrange multiplier
𝐲∗ = (𝜃, 𝜁 ) ∈ 𝑌 ∗ such that the Euler–Lagrange equation takes place
𝐹𝜑′ (𝜑̂ , 𝑢̂ )∗ 𝐲∗ = −𝐽𝜑′ (𝜑̂ , 𝑢̂ ) in 𝐻 1 (Ω)∗ ,
which is equivalent to the relation
(𝜆̂∇𝜏 , ∇𝜃) + 2(|𝜑̂ | 𝜏 , 𝜃) + ⟨𝜁 , 𝜏 ⟩Γ𝐷 = −(𝜇0 /2)⟨𝐼𝜑′ (𝜑̂ ), 𝜏 ⟩ ∀𝜏 ∈ 𝐻 1 (Ω), (17)
and a minimum principle
(𝜑̂ , 𝑢̂ , 𝐲∗ ) ≤ (𝜑̂ , 𝑢, 𝐲∗ ) ∀𝑢 ∈ 𝐾 ,
which is equivalent to the inequalities
𝜇1 (𝜆̂, 𝜆 − 𝜆̂)𝑠,Ω + ((𝜆 − 𝜆̂)∇𝜑̂ , ∇𝜃) ≥ 0 ∀𝜆 ∈ 𝐾1 , (18)
𝜇2 (𝑓̂ , 𝑓 − 𝑓̂ )Ω − (𝑓 − 𝑓̂ , 𝜃) ≥ 0 ∀𝑓 ∈ 𝐾2 . (19)
Let us formulate a theorem on the local stability of optimal solutions to problem (12) for 𝐼 (𝜑) =
‖𝜑 − 𝜑 𝑑 ‖2𝑄 , which is proved according to the scheme suggested in [7].
Theorem 5.2.
Let in addition to the conditions (i), (ii) and 4, 𝐾 be a bounded set and let the pair (𝜑𝑖 , 𝑢𝑖 ) ∈ 𝐻 1 (Ω)×𝐾 be
the solution of problem (12), which corresponds to the specified function 𝜑𝑖𝑑 ∈ 𝐿2 (𝑄), 𝑖 = 1, 2, where
𝑄 ⊂ Ω is an arbitrary open bounded set. Let us suppose that 𝜇0 > 0 and the following conditions
𝛽12 𝜇0 ≤ (1 − 𝜀)𝜇1 , 𝛽22 𝜇0 ≤ (1 − 𝜀)𝜇2 , (20)
are satisfied, where 𝜀 ∈ (0, 1) is an arbitrary number and parameters 𝛽1 and 𝛽2 , monotonically depend
on the norms of the initial data of the problem (12).
Then the following stability estimates hold:
√
‖𝜆1 − 𝜆2 ‖𝑠,Ω ≤ 𝜇0 /(𝜀𝜇1 )(0.5 + 𝛽3 )‖𝜑1𝑑 − 𝜑2𝑑 ‖𝑄 , (21)
√
‖𝑓1 − 𝑓2 ‖Ω ≤ 𝜇0 /(𝜀𝜇2 )(0.5 + 𝛽3 )‖𝜑1𝑑 − 𝜑2𝑑 ‖𝑄 , (22)
√ √
‖𝜑1 − 𝜑2 ‖1,Ω ≤ 𝐶∗ (𝐶0 𝑀𝜑 𝜇0 /(𝜀𝜇1 ) + 𝜇0 /(𝜀𝜇2 ))(0.5 + 𝛽3 )‖𝜑1𝑑 − 𝜑2𝑑 ‖𝑄 . (23)
Here parameter 𝛽3 depends on the initial data of the problem (12), 𝐶∗ , 𝐶0 are constants from Lemma
1.1 and 𝑀𝜑 is introduced in (8).
The stability estimates (21)–(23) are interesting by themselves, as they clearly characterize the lo-
cal stability of multiplicative control problem’s solution, and the problem has a strong nonlinearity.
Moreover, the optimization approach gives an opportunity to reduce the inverse coefficient problems
to the problems of multiplicative control (see [10]). Let us note that the method, which was used to
obtain these estimates, can be applied also for the studying of convergence of numerical algorithms,
which are used for obtaining an approximate solution of extremum problems and which are based on
using of optimality systems as in [20].
6. Numerical algorithm
The optimality system (7), (17)–(19) can be used to design efficient numerical algorithms for solving
control problem (12). The simplest one for 𝐼1 can be obtained by applying the fixed point iteration
method to the optimality system. The 𝑚-th iteration of this algorithm consist of finding unknown
values 𝜑 𝑚 , 𝜃 𝑚 , 𝜁 𝑚 , 𝜆𝑚+1 and 𝑓 𝑚+1 for given (𝜆𝑚 , 𝑓 𝑚 ), 𝑚 = 0, 1, 2, ... beginning with given initial values
𝜆0 and 𝑓 0 by sequentially solving following problems:
(𝜆∇𝜑 𝑚 , ∇ℎ) + (|𝜑 𝑚 |𝜑 𝑚 , ℎ) = (𝑓 𝑚 , ℎ) ∀ℎ ∈ 𝐻01 (Ω), 𝜑 𝑚 |Γ = 𝜓 , (24)
(𝜆̂∇𝜏 , ∇𝜃 𝑚 ) + 2(|𝜑 𝑚 | 𝜏 , 𝜃 𝑚 ) + ⟨𝜁 𝑚 , 𝜏 ⟩Γ = −𝜇0 (𝜑 𝑚 − 𝜑 𝑑 , 𝜏 )𝑄 ∀𝜏 ∈ 𝐻 1 (Ω), (25)
𝜇1 (𝜆 𝑚+1 𝑚
, 𝜆 − 𝜆 )𝑠,Ω + ((𝜆 − 𝜆 𝑚+1 𝑚 𝑚
)∇𝜑 , ∇𝜃 ) ≥ 0 ∀𝜆 ∈ 𝐾1 , (26)
𝜇2 (𝑓 𝑚+1 , 𝑓 − 𝑓 𝑚 )Ω − (𝑓 − 𝑓 𝑚+1 , 𝜃 𝑚 ) ≥ 0 ∀𝑓 ∈ 𝐾2 . (27)
For discretization and solving variational problem (24), (25), one can use open software freeFEM++
(www.freefem.org) based on using the finite element method. For discretization of variation inequal-
ities (26), (27), it is comfortable to look for solutions 𝜆𝑚+1 and 𝑓 𝑚+1 as
𝑁 𝑁
𝜆 𝑚+1
(𝐱) = ∑ 𝜆𝑗𝑚+1 𝑙𝑗 (𝐱), 𝑓 𝑚+1
(𝐱) = ∑ 𝑓𝑗𝑚+1 𝑔𝑘 (𝐱), 𝐱 ∈ Ω. (28)
𝑗=1 𝑘=1
Here, 𝑁 is an integer, 𝑙𝑗 (𝐱) ∈ 𝐻𝜆𝑠0 (Ω), are nonnegative basic functions in 𝐻𝜆𝑠0 (Ω), 𝑠 > 3/2, 𝑔𝑘 (𝐱) ∈ 𝐿2 (Ω)
are basic functions in 𝐿2 (Ω), 𝜆𝑗𝑚+1 ≥ 0 and 𝑓𝑘𝑚+1 ∈ ℝ are unknown coefficients.
7. Bang–bang principle for a one parameter control problem
In this section, we will state additional properties of the optimal solution to the following control
problem:
𝐽 (𝜑) ≡ (1/2)𝐼 (𝜑) → inf, (𝜑, 𝑓 ) = 0, (𝜑, 𝑓 ) ∈ 𝐻 1 (Ω) × 𝐾2 . (29)
The role of control in the problem (29) is played only by the function 𝑓 , which can change in the
subset 𝐾2 . Whereas the function 𝜆 is considered to be given.
The operator
= (1 , 𝐹2 ) ∶ 𝐻 1 (Ω) × 𝐾2 → 𝑌
is defined by formulas:
⟨1 (𝜑, 𝑢), ℎ⟩ = (𝜆∇𝜑, ∇ℎ) + (|𝜑|𝜑, ℎ) − (𝑓 , ℎ),
𝐹2 (𝜑) = 𝜑|Γ − 𝜓 .
Let us denote by
𝑎𝑑 = {(𝜑, 𝑓 ) ∈ 𝐻 1 (Ω) × 𝐾2 ∶ (𝜑, 𝑓 ) = 0, 𝐽 (𝜑, 𝑓 ) < ∞}
the set of admissible pairs for the problem (29) and assume that the condition
(jjj) 𝐾2 ⊂ 𝐿2 (Ω) is a nonempty convex, closed and bounded set.
Theorem 7.1.
Assume that the assumptions (i), (ii) and 7 take place. Let 𝐼 ∶ 𝑋 → ℝ be a weakly semicontinuous below
functional and let 𝑎𝑑 ≠ 0. Then there is at least one solution (𝜑, 𝑓 ) ∈ 𝐻 1 (Ω) × 𝐾2 of the control problem
(29).
It clear, for the problem (29) an analog of Theorem 3.1 and the minimum principle takes the fol-
lowing form:
(𝑓 − 𝑓̂ , 𝜃) ≤ 0 ∀𝑓 ∈ 𝐾2 . (30)
Let a more stringent condition be satisfied instead of 7:
(jjj‘) 𝑓min ≤ 𝑓 ≤ 𝑓max a.e. in Ω for all 𝑓 ∈ 𝐾2 , where 𝑓min and 𝑓max are positive numbers.
It is clear that conditions 7 define a special case of a convex, bounded, and closed set 𝐾2 introduced
in 7.
Let us show that the optimal control 𝑓̂ (𝐱) of the problem (29) has the bang-bang property, according
to which it takes one of two values 𝑓min or 𝑓max , respectively, depending on the sign of the function
𝜃(𝐱) at the point 𝐱 ∈ Ω.
Lemma 7.1.
Under the conditions 7 the inequality (30) is equivalent to the following inequality
(𝑓 − 𝑓̂ )𝜃 ≤ 0 a.e. in Ω ∀𝑓 ∈ 𝐾2 . (31)
Proof.
Let us show that (30) implies (31). Suppose that there is a function 𝑓1 ∈ 𝐾2 , with which on the set
𝐷0 ⊂ Ω, meas 𝐷0 > 0, the inequality holds
(𝑓1 − 𝑓̂ ) 𝜃 > 0 a.e. in 𝐷0 .
Consider a 𝑓2 , such that 𝑓2 = 𝑓̂ if 𝐱 ∉ 𝐷0 and 𝑓2 = 𝑓1 if 𝐱 ∈ 𝐷0 . It clear, that 𝑓2 ∈ 𝐾2 and the inequality is
true for it
(𝑓2 − 𝑓̂ , 𝜃) = (𝑓1 − 𝑓̂ , 𝜃)𝐷0 > 0,
which contradicts (30).
Corollary 7.1.
From (31) it follows that if 𝜃 < 0 in 𝐷1 , then 𝑓̂ = 𝑓min in 𝐷1 and 𝑓̂ = 𝑓max in 𝐷2 , if 𝜃 > 0 in 𝐷2 . Note
that interest in the bang–bang property is due to the study of control problems in which, for practical
reasons, regularization is not used. In particular, such a formulation of control problems is used in
the study of applied problems of thermal and electromagnetic cloaking (see, for example, [19]).
8. Conclusion
It is interesting to note that, on the one hand, a well-developed numerical algorithm for solving the
extremal problem should show that the maximum principle for the concentration 𝜑 and the bang–
bang principle for the optimal control 𝑓 are satisfied. On the other hand, these properties can serve as
a criterion for checking numerical algorithms, since they have been correctly proven theoretically. Of
particular interest is the study of the convergence of a numerical algorithm based on the optimality
system from Section 6. In this case, the method for deriving estimates of the local stability of optimal
solutions from Section 5, which is also based on the analysis of the optimality system, can be applied
(see [20]).
Acknowledgments
The first authors was supported by the state assignment of Institute of Applied Mathematics FEB RAS
(Theme No. 075-01095-20-00), the second author was supported by the Russian Foundation for Basic
Research (project no. 20–01–00113 (a)), the third and fourth authors was supported by the Ministry of
Science and Higher Education of the Russian Federation (project no. 075-02-2020-1482-1, additional
agreement of 21.04.2020).
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