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Al-Mustansiriyah Journal of Science
ISSN: 1814-635X (print), ISSN:2521-3520 (online)
Volume 30, Issue 1, 2019
DOI: http://doi.org/10.23851/mjs.v30i1.464
143
Copyright © 2018 Authors and Al-Mustansiriyah Journal of Science. This work is licensed under a Creative Commons
Attribution-NonCommercial 4.0 International License.
Research Article
Open Access
The Solvability of the Continuous Classical Boundary
Optimal Control of Couple Nonlinear Elliptic Partial
Differential Equations with State Constraints
Jamil A. Ali Al-Hawasy*, Safaa J. Mohammed Al-Qaisi
Department of Mathematics, Faculty of Science, Mustansiriyah University, IRAQ
*Correspondent author email: jhawassy17@uomustansiriyah.edu.iq
A r t i c l e I n f o
Abstract
Received
19/02/2018
Accepted
12/06/2018
Published
15/08/2019
This paper concerns with, the proof of the existence and the uniqueness theorem for the
solution of the state vector of couple of nonlinear elliptic partial differential equations by
using the Minty-Browder theorem, where the continuous classical boundary control vector is
given. Also the existence theorem of a continuous classical boundary optimal control vector
governing by the couple of nonlinear elliptic partial differential equation with equality and
inequality constraints is proved. The existence of the uniqueness solution of the couple of
adjoins equations which are associated with the couple of the state equations with equality and
inequality constraints are studied. The necessary and sufficient conditions theorem for
optimality of the couple of nonlinear elliptic equations with equality and inequality constraints
are proved by using the Kuhn-Tucker-Lagrange multipliers theorems.
Keywords: Classical boundary optimal control, couple of nonlinear elliptic partial differential
equations, necessary and sufficient conditions.
لاةـصخ
Introduction
The optimal control problems play an
important role in many fields in the real life
problems, for examples in robotics [1], in an
electric power [2], in civil engineering [3], in
Aeronautics and Astronautics [4], in medicine
[5], in economic [6], in heat conduction [7], in
biology [8] and many others fields.
This importance of optimal control problems
encouraged many researchers interested to
study the optimal control problems of systems
are governed either by nonlinear ordinary
differential equations as in [9] and [10] or by
linear partial differential equations as in [11] or
are governed by nonlinear partial differential
equations either of a hyperbolic type as in [12]
or of a parabolic type as in [13] or by an
elliptic type as in [14], or optimal control
problem are governed either by a couple of
nonlinear partial differential equations of a
hyperbolic type as in [15] or of a parabolic type
as in [16] or by an elliptic type as in [17], or of
an elliptic type but involve a boundary control
as in [18]. While the optimal control problem
which, is considered in this work is an optimal
boundary (Neumann boundary conditions
NBCs) control problem governed by a couple
of nonlinear partial differential equations of
elliptic type.
This work is concerned at first with, the proof
of existence and the uniqueness theorem of the
state vector solution of a couple nonlinear
Al-Hawasy et al.
The Solvability of the Continuous Classical Boundary Optimal Control of Couple Nonlinear Elliptic Partial
Differential Equations with State Constraints
2019
144
elliptic partial differential equations "
CNLEPDEs" for a given continuous classical
boundary control vector (CCBCV) using the
Minty- Browder theorem. Second the existence
theorem of a continuous classical boundary
optimal control vector "CCBOCV" which is
governing by the considered couple of
nonlinear partial differential equation of elliptic
type with equality and inequality constraints is
proved. The existence and the uniqueness
solution of the couple of adjoint vector
equations associated with the couple of state
equations with equality and inequality
constraints are studied. The necessary
conditions theorem for optimality and the
sufficient conditions theorem for optimality of
CNLEPDEs with equality and inequality
constraints are proved via the Kuhn-Tucker-
Lagrange multipliers theorems.
Description of the problem
Let , with its boundary be
Lipschitz. Consider the following continuous
classical boundary optimal control consisting
of CNLEPDEs "state equations" with NBCs
, in Ω
)1(
, in
Ω
, in
Г
, in
Г
With
,
,
where
, and
is the classical boundary control
vector,
is the state vector, corresponding to the control
vector, and
and
are a vector of functions.
The constraint on the controls is given by
,
,
where
and
with
where
, and , is a
convex and compact set, and
The cost functional is
The state constraints are
The set of admissible control is
The CCBOCP is to find the minimum of (5)
such that "s.t." the state constraints (6) and (7),
i.e. to find
and
.
Let
. We denote
to the and
to be the inner product and the norm in
, by and the inner
product and the norm in , by
and
the inner product and the norm in
, by
and
the inner product
and the norm in
and
is the dual of
.
Weak Formulation of the State
Equations
The weak form (WF) of problem (1- 4) is
obtained by multiplying both sides of (1- 2) by
and respectively, integrating
both sides and then by using the generalize
Green's theorem (in Hilbert Space) for the
terms which have the derivatives, once get.
ΩΩ
Ω
ΩГ
,
And
,
Al-Mustansiriyah Journal of Science
ISSN: 1814-635X (print), ISSN:2521-3520 (online)
Volume 30, Issue 1, 2019
DOI: http://doi.org/10.23851/mjs.v30i1.464
145
Copyright © 2018 Authors and Al-Mustansiriyah Journal of Science. This work is licensed under a Creative Commons
Attribution-NonCommercial 4.0 International License.
Adding (9) with (10), get that
where
with
,
,
, where ,
,
where , .
The following assumptions are useful to prove
the existence theorem of a unique solution of
the weak form (11).
Assumptions (A):
a) is coercive,
i.e.
,
b) , ,
c) and are of Carathéodory type " C.T."
on and satisfy the following conditions
with respect to " w.r.t. " and
respectively, i.e. for ,
and :
,
,
d) and are monotone for each
w.r.t. and respectively, and
, , .
e) and are of C.T. on Ω and satisfy for
, and , .
Proposition (1)[19]: Let is
of Carathéodory type, let be a functional, s.t.
, where Ω is a
measurable subset of , and suppose that
,
where
, and
, if , and , if .
Then is continuous on .
Proposition (2)[19]: Let ,
are of the Carathéodory type, let
be a functional, s.t. ,
where Ω is a measurable subset of , and
,
, where ,
,
, , if
, and , if .
Then the Fréchet derivative of exists for each
and is given by
.
Theorem (1) (Minty-Browder) [20]: "Let be
a reflexive Banach space, and be a
continuous nonlinear map s.t.
, ,
and
.
Then for every , there exists a unique
solution of the equation ".
Theorem (2) (Egorov's theorem) [18]: Let Ω
be a measurable subset of , and
, if the following inequality is
satisfied (or or ), for
each measurable subset , then
(or or ), a.e. in Ω.
Theorem (3): If the assumptions A are hold,
and if the function (or ) in (11) is strictly
monotone, then for a given control
, the
w.f. (11) has a unique solution
.
Proof: Let
. Then the w.f. (11) can
rewrite as
where
i) From assumptions A-(a & d), is
coercive.
ii) From assumptions A-(b & c) and using
iii) Proposition (1) then is continuous w.r.t.
.
From assumptions A-(a & d) and part (i) is
strictly monotone w.r.t. .
Al-Hawasy et al.
The Solvability of the Continuous Classical Boundary Optimal Control of Couple Nonlinear Elliptic Partial
Differential Equations with State Constraints
2019
146
Then by Theorem (1), the uniqueness
solution
of the w.f. (12) is obtained.
Existence of a Classical Optimal
Boundary Control of CCBOCV
This section deals with the state and proof the
existence theorem of CCBOCV with the
suitable assumptions. Therefore, the following
lemmas and assumptions are useful.
Lemma (1): If the assumptions (A) are hold,
the functions, are Lipschitz w.r.t. and
respectively, and if , are bounded.
Then the mapping
is Lipschitz
continuous from
into , i.e.
, with .
Proof: Let
be two given controls
vectors, and be their corresponding state
solutions vectors (of the weak form (11)).
Subtracting the above two obtained weak forms
from (11), setting
and
, with
, then adding the obtained two
equations, once get
ГГ
Using assumptions, A-(a, d), taking the
absolute value for both sides of (13), it
becomes
¹Ω²
¹Ω
¹Ω
ГГ
Using the Cauchy-Schwartz inequality and
then the trace operator in (14), to get
¹Ω²
²Г²
¹Ω²
¹Ω²
²Г²
where
,which gives
²Ω²
²Г²
,
with
Assumption (B):
Assume that and are of C.T. on
, , and
respectively, and , are satisfy
,
, and
where and
Lemma (2): If assumptions (B) are hold, then
( ) the functional
is continuous
on .
Proof: Set ,
, and
,
From assumptions (B), and by using
Proposition (1) on each of the functional
, and
are
continuous on and on
respectively. Hence
is continuous on.
Theorem (4): If the assumptions (A) and (B)
are hold,
, ( ) is independent of
(), and ) is bounded functions, s.t. for
, and ,
(
, ( )
, ( , ).
( is independent of (), (
is convex w.r.t. (w.r.t. ).
Then there exists a CCBOCV.
Proof: The set is convex and bounded
, since it is, then so is . On
the other hand, and by Egorov's theorem,
is closed since it is, then
is closed, hence it is weakly compact " w.c.".
From the assumption on
, there is an
element
with
,
and a minimum sequence
, for each , s.t.
.
But
is w.c., this means that
has a
subsequence say again
which
converges weakly to
in
.
Then from the proof of Theorem (3),
corresponding to this sequence
there is a
sequence of solutions of the sequence of
weak form:
Al-Mustansiriyah Journal of Science
ISSN: 1814-635X (print), ISSN:2521-3520 (online)
Volume 30, Issue 1, 2019
DOI: http://doi.org/10.23851/mjs.v30i1.464
147
Copyright © 2018 Authors and Al-Mustansiriyah Journal of Science. This work is licensed under a Creative Commons
Attribution-NonCommercial 4.0 International License.
s.t. is bounded, for each . Then
has a subsequence say again s.t.
weakly in
(Alaoglu theorem [22]).
To prove that (17) converges to
Let
, and first for the left
hand sides, since weakly in , i.e.
weakly in , for each .
Then from the left hand sides of (17), (18) and
by using Cauchy- Schwarz inequality, one has
ΩΩ
Ω
Ω
ΩΩ
Ω
Ω
ΩΩ
ΩΩ
ΩΩ
ΩΩ
ΩΩ
ΩΩ
From assumptions (B), and proposition (1) the
functional and
are continuous with
respect to and respectively. But
weakly in (since
weakly in
), then by using the compactness
theorem (Rellich-Kondrachov theorem) in [21],
once get that strongly in ,
and
, we have
,
i.e. the left-hand side of (17) the left-hand
side of (18)
Second, but weakly in and so
as , then
Г Г
From (20a) and (20b) give us that (17)
converges to (18).
But
is dense in
, then these
convergences hold
, which gives
satisfies the w.f. of the state equations.
From Lemma (2), the functional
is
continuous on , .
From the assumptions on , ,
is
continuous, and the strongly converges of
, in , once get
Also, from the assumptions on and
( ) and Lemma (2), the
integrals and
are continuous w.r.t. and
respectively, but , (for each )
is convex w.r.t. , then is
weakly lower semicontinuous "w.l.sc." w.r.t.
, and then
By the same way one can get
, (for each )
From the above inequalities, one gets
( ) is w.l.sc. with respect to
.
But
,, then
Finally,
Al-Hawasy et al.
The Solvability of the Continuous Classical Boundary Optimal Control of Couple Nonlinear Elliptic Partial
Differential Equations with State Constraints
2019
148
Which implies that
is a CCBOCV.
The NCFO "necessary conditions for
optimality " of CCBOCV
To find the derivatives of the Hamiltonian"
Fréchet derivatives” The following assumption
is useful.
Assumptions(C):
a) , are of the C.T. on , and
satisfy for the conditions
, , with .
and .
b) , are of the C. T. on Ω and for
and satisfy
, , .
c) , , , are of
the C. T. on and satisfy
, ,
,
, .
Theorem (5): If the assumptions (A), (B) and
(C) are hold, the Hamiltonian is defined by:
"
"
The adjoint equations of state equations (1- 4)
are given by
, in
Ω
, in Ω
, in
Г
, in
Г
Then the Fréchet derivatives of are given by
, where
and
is the
adjoint of the state
.
Proof: Writing the couple of the adjoint
equations (21-24) by their w.f., then adding
them, and then substituting
in the
obtained equation to get
One can easily prove that the w.f. (25) "for a
given control
" has a unique solution
using a similar way which is used in
proof of theorem (3).
Now, substituting once the solutions in the
weak form of the state equations (9) and once
again the solution , then subtracting
the 1st obtained weak form from the other one,
to obtain
ΩΩ
Ω
Г
,
The above substituting and subtracting are
repeated with the solutions and ,
and in the weak form of the state equations
(10), to obtain
,
Adding (26) with (27), then substituting
in the resulting equation, to get
,
From the assumptions on ( , and by using
Proposition (2), the Fréchet derivative of ()
exists, and hence from Lemma (1) and the
Minkowski inequality, (28) becomes
where
as
.
Subtracting (25) from (29), to get
ΩΩ
Г
Г
ГГ
Now, from the assumptions on , ,
and , the definition of the Fréchet derivative
Al-Mustansiriyah Journal of Science
ISSN: 1814-635X (print), ISSN:2521-3520 (online)
Volume 30, Issue 1, 2019
DOI: http://doi.org/10.23851/mjs.v30i1.464
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and then using the result of Lemma (1), we
have
Г
Г
Г²
where
But from the definition of the Fréchet
derivative of , once get
.
Note: In the prove of the above theorem, we
have found the Fréchet derivative for the
functional , so the same technique is used to
find the Fréchet derivative for and .
Theorem (6):
(a) If assumptions (A), (B) and (C) are
hold,
is convex, and if
is a classical
optimal, then , there exists
multipliers , with , are
nonnegative,
, s.t. the following
Kuhn- Tucker- Lagrange(K.T.L.)conditions are
satisfied:
Г
′ᵀ
,
For each
, with
where
and
,
in (Theorem (5)),
,
(b) (Minimum Principle in point wise weak
form): The inequality (32a) is equivalent to
′ᵀ
′ᵀ
Г
Proof: (a) From Theorem (5),
(for each
and at each
) has a continuous
Fréchet derivative, since the control (classical)
is optimal, then using the K.T.L.
theorem , there exists multipliers
, with , are nonnegative, and
, s.t.
,
Then from Theorem 5, (34a) with setting
, , can be
written
as
where
,
, for
,
,
,
.
(b) Let
be a dense sequence in
, and
be a measurable set s.t.
Then (32a), gives
,
Then using Theorem (2) once get that
,
The above inequality holds on the boundary
of the region except in a subset with
, for each , where is a Lebesgue
measure, then this equality satisfies on the
boundary except in the union of
with
, but
is a dense sequence in
the control set
, then there exists
s.t.
, a.e.on Г.
The converse of the proof is obtained directly.
SCFO "Sufficient Conditions for
Optimality" of CCBOCV
Theorem (7): If assumptions (A), (B) and (C)
are hold, if and (, ) are affine w.r.t.
(), ( ) is affine w.r.t. (, and
are bounded functional for , and if
, , , are convex w.r.t.
,, , respectively. Then the NCFO in
Theorem 6, are also sufficient if is positive.
Proof: Assume
,
is satisfied the
conditions(32a) and(32b).
Let
.
Since
,
Al-Hawasy et al.
The Solvability of the Continuous Classical Boundary Optimal Control of Couple Nonlinear Elliptic Partial
Differential Equations with State Constraints
2019
150
,
Let
,
be a given
controls then ,,
are
their corresponding solutions, substituting the
pair
in (1-4) and multiplying all the
obtained equations by once, and then
substituting the pair
in (1-4) and
multiplying all the obtained equations by
, finally adding each pair from the
corresponding equations together one gets:
And
Now, if we have the control vector
with
and
Then from (35a, 35b) , (36a, 36b), once get that
the state vector
with
and
,
are their corresponding solution, i.e. are
satisfied (1-4) respectively. So, the operators
, are convex- linear
w.r.t. and respectively.
Now, from this result and since , , ,
are affine w. r. t. , , ,
respectively, on , once get that ,
is convex-linear w.r.t.
,.
Also, since , , ,
, are convex w.r.t. , , , and
respectively, i.e.
is convex w.r.t. and
.
Then
is convex w.r.t. ,and
, in "the
convex set"
, and has a " continuous" Fréchet
derivative satisfies
minimize
, i.e. for any
in
, we have
Now, let
is also admissible and satisfies the
Transversality condition, then (37) becomes
,
, i.e.
is a CCBOCV
continuous classical optimal control for the
problem.
Conclusions
The Minty-Browder theorem can be used
successfully to prove the existence and
uniqueness solution of the continuous state
vector of the couple nonlinear elliptic partial
differential equations when the continuous
classical boundary control vector is given. The
existence theorem of a continuous classical
boundary optimal control vector which is
governing by the considered couple of
nonlinear partial differential equation of elliptic
type with equality and inequality constraints is
proved. The existence and uniqueness solution
of the couple of adjoint equations which is
associated with the considered couple
equations with equality and inequality
constraints of the state are studied. The
necessary conditions theorem so as the
sufficient conditions theorem of optimality of a
CNLEPDEs with equality and inequality
constraints are proved via Kuhn- Tucker-
Lagrange's Multipliers theorems.
References
[1] D. J. Braun, F. Petit, F. Huber, S. Haddadin, A.
Albu-Schaffer, and S. Vijayakumar, "Robots Driven by
Compliant Actuators: Optimal Control under Actuation
Constraints", Ieee Transactions On Robotics, VOL. 29,
NO.5, 2013.
[2] Y. Wang, X. Lin, S. Park, and N. Chang,
"Optimal Control of a Grid-Connected Hybrid Electrical
Energy Storage System for Homes", 978-3-9815370-0-0,
Date 13, Edaa, 2013.
[3] F. Amini and M. Afshar, "Modified Predictive
Optimal linear Control of Structures in Seismic Region",
Iranian Journal of Science & Technology, Transaction B,
Engineering, Vol.32, No.B2, 2008, pp: 91-106.
[4] L. Lessard, L and S. Lall, "Optimal Controller
Synthesis for the Decentralized Two-Player Problem
with output Feedback", American Control Conference,
2012, pp. 6314–6321.
[5] A. Di. Liddo, "Optimal Control and Treatment
of Infectious Diseases", the case of huge treatment
Costs, Mathematics doi: 10.3390/math4020021, 2016.
[6] M. Derakhshan, "Control Theory and Economic
Policy Optimization: The Origin, Achievements and the
Al-Mustansiriyah Journal of Science
ISSN: 1814-635X (print), ISSN:2521-3520 (online)
Volume 30, Issue 1, 2019
DOI: http://doi.org/10.23851/mjs.v30i1.464
151
Copyright © 2018 Authors and Al-Mustansiriyah Journal of Science. This work is licensed under a Creative Commons
Attribution-NonCommercial 4.0 International License.
Fading Optimism from a Historical Standpoint",
International Journal of Business and Development
Studies Vol. 7, No. 1, 2015, pp 5-29.
[7] A. Yilmaz, I. Mahariq, and F. Yilmaz,
"Numerical Solutions of Optimal Control Problems for
microwave heating", International Journal of Advances
in Science Engineering and Technology, ISSN: 2321-
9009 Vol.4, Issue3, 2016.
[8] M. Chalak, "Optimal Control for a Dispersing
Biological Agent", Journal of Agricultural and Resource
Economics, 39(2):271–289, 2014, ISSN 1068-5502.
[9] J. Warga, "Optimal Control of Differential and
Functional Equations", Academic Press: New York, and
London, 1972.
[10] A. Orpel, "Optimal Control Problems with
Higher order Constraints", Folia Mathematical, Vol.16,
No.1, 2009, pp: 31-44.
[11] J. L. Lions, "Optimal Control of Systems
Governed by partial Differential Equations", Springer-
Verlag, New York, 1972.
[12] J. Al-Hawasy, "The Continuous Classical
Optimal Control of a nonlinear Hyperbolic Equation
(CCOCP)", Mustansiriyah Journal of Science., Vol.19,
No.8, 2008, pp.96-110.
[13] I. Chryssoverghi and J. Al-Hawasy, "The
Continuous Classical Optimal Control Problem of a semi
linear Parabolic Equations (CCOCP)", Journal of
Karbala University, Vol.8, No.3, 2010, pp: 57-70.
[14] D. Bors and S. Walczak, "Optimal control
elliptic system with distributed and boundary controls",
Nonlinear Analysis 63, 2005, e1367- e1376.
[15] J. Al-Hawasy, "The Continuous Classical
Optimal Control of a Couple Nonlinear Hyperbolic
Partial Differential Equations with equality and
inequality Constraints", Iraqi Journal of Science, Vol.57,
No.2C, 2016, pp: 1528-1538.
[16] J. Al-Hawasy and G. Kadhem, "The Continuous
Classical Optimal Control of a coupled of nonlinear
parabolic Equations", Al-Nahrain Journal of Sciences,
Vol.19, No.1, 2016, pp: 173-186.
[17] J. Al-Hawasy and E. Al-Rawdanee, "The
Continuous Classical Optimal Control of a Coupled of
non-linear Elliptic Equations", Mathematical Theory and
modeling, Vol.4, No.14, 2014, pp: 211-219.
[18] B. Vexler, "Finite Element Approximation of
elliptic Dirichlet Optimal Control Problems", Numer,
Funct, Anal., Optim., 28, 2007, 957–973.
[19] I. Chryssoverghi and A. Bacopoulos,
"Approximation of Relaxed Nonlinear Parabolic
Optimal Control Problems", Journal of Optimization
Theory and Applications, Vol.77, No.1, 1993.
[20] A. H. Borzabadi, A. V. Kamyad, and M. H. G.
Farahi, "Optimal control of the heat equation in an
inhomogeneous body", J. Appl. Math., and Computing,
Vol.15, No.1-2, 2004, pp.127-146.
[21] R. Temam, "Navier-Stokes Equations", North-
Holand Publishing Company, 1977.
[22] A. Bacopoulos and I. Chryssoverghi,
"Numerical solutions of partial differential equations by
finite elements methods", Symeom publishing co,
Athens, 2003.
