Abstract : We revisit the notion of discrete control by M. Asch considering the case where the control
is discreet. This point of view leads to problems of controllability with or without constraints on the
control. This article focuses on the case of hyperbolic equations although similar developments can be
done for other PDE’s. The main tool used is the HUM method which is "adapted" to the constraints.
Received: December 29, 2021. Revised: November 9, 2022. Accepted: December 7, 2022. Published: December 31, 2022.
Nomenclature
edo Ordinary differential equation.
edp Partial differential equation.
sd-o Semi-discrete observation.
HUM Hilbert Uniqueness Method.
Greek symbols
Laplacien operator.
The coefficient of heat transfer with a
constant rate.
Subscripts
ChW Semi-discrete observation
1. Control for a differential equation
Similar works with our study that are worth
mentioning are [1],[2],[3],[4],[5]. Specifically,
let () be the temperature of a small object,
controlled by the temperature of its
environment, (). Suppose that initially the
object is at temperature 0 and the heat transfer
takes place at a constant rate of . This system
can be described by an ordinary differential
equation
() = [()],
0=0.
If we can control the ambient temperature (),
we could ask that the object reach a given
temperature at time =, say =1. Is
there a control? Can it be calculated ?.
The equation admits an explicit solution,
=0+
0.
Replacing the solution =1 we obtain,
0=10.
A revision of M. Asch notion of Discrete control
BENBRAHIM ABDELOUAHAB, REZZOUG IMAD, NECIB ABDELHALIM
Department of Mathematics and Computer Science
Laboratory of dynamical systems and control
Larbi Ben M'hidi University of Oum El Bouaghi
"Oum El Bouaghi University, P.O.Box 358, OEB, Algeria."
ALGERIA
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DOI: 10.37394/232011.2022.17.29
Benbrahim Abdelouahab,
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Key-Words : Hyperbolic equations, Discrete control, HUM method, Mechanical Systems, Inverted Pendulum,
Mathematical Models in Mechanics.R
There is an infinite number of solutions, (),
to this equation. For example, for a constant
control, =0, one computes easily,
=0=10
1.
1.1 Optimal control for differential equation
ordinary
Let's find the control, (), which minimizes
the norm 2(0, ),
=()2
0.
Such control exists and it is unique. The control
function takes the form =0(),
Or 0= 2 10
12 .
We will see, later, how to compute such a
control in a more general framework for partial
differential equations.
Remark :
1/ A check exists for all > 0 ; the initial and
final states are arbitrary.
2/ The control found is the one that leads the
solution to =1 and minimizes .
1.2 Control of an EDO system
The case of a system, for a vector function of
dimension , is written :
=+(),
0=0,
where is a square matrix (×), is a
rectangular matrix (×) and is the control
of dimension . When is diagonalisable, each
eigenmode can be controlled arbitrarily.
Theorem : A system =+ is said to
be controllable if the controllability matrix
[ 1] is of rank equal to , the order
of the system. When this is the case, one can
control the system using a linear feedback
control =. This allows to write:
==,
and we can place the eigenvalues of the matrix
in the half-space < 0.
Example : Consider the equation for a simple
pendulum: +2= 0,
The equation is written in the form of a system
as follows:
0
=0 1
20
0
and the eigenvalues of matrix are obtained by
det= 0 and so = i,=i.
The system is marginally stable - it oscillates,
periodically, constantly.
In general, for non-linear oscillations,
+(y) = 0,
the energy , the amplitude and the period
are defined by
E = 1
2()2+=()
and
T = 4
2().
0
In order to obtain energy, we multiply the
equation by and we integrate. Energy is more
kinetic potential. When the kinetic energy is
zero, the oscillation is at full amplitude:
=.
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We are interested here in the feedback control
of an unstable system. Some examples of
application are: the autopilot of an airplane, the
monitoring of a nuclear reactor, the control of
chemical processes, and the thermostat of a
heating system.
1.3 The inverted pendulum - analytical
solution and stability
The physical problem considered is that of the
inverted pendulum, which can model for
example a rocket on a launch pad or a prosthetic
leg.
At continuous time, we have the following
equation: ²= 0,
we linearized with . This is rewritten
as a system of first-order equations:
=,
=2+,
where represents the angular position, the
angular velocity and the feedback control that
we want to calculate.
It can be shown that the solution (equation
without control) is given by
= 1
0
0
and that it grows without limit ...
lim
=.
Example : The matrices and so that we can
form the following system:
=+.
Are
=0 1
20
+0
1.
Eigenvalues are the solution of the
characteristic equationdet= 0and
so1=,2=.So the system is unstable.
The system is controllable since the matrix of
controllability :
==0 1
1 0
which is clearly of rank 2.
1.4 Stability in the phase plan
In the general case, the stability of a system of
ordinary differential equations
=()
is obtained from the examination of the
eigenvalues of the linear stability matrix,
=1
1
1
2
2
1
2
2
and we examine at the critical points
which satisfies = 0. Linear stability
implies the stability of the nonlinear system in a
neighborhood of the critical points.
2. Control of an EDP
Consider a chord on the interval [0, 1]. For
small oscillations, its motion can be described
by the wave equation,
2
2=22
2,
0, =0,
0, =1,
, 0= 0, , 1=,
for0 and 01. We apply here a
border control, (), at the right end, = 1.
We must, of course, specify all the functional
spaces...
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Control problem: find () such that, at time
=,
,=0
,
,=1
.
2.1 Existence, uniqueness, causality,
geometry
Is it possible to build such a control? It will be
necessary to take into account:
- the functional spaces for0, 1 and ;
- the minimum time for a wave to cross
the string,2
;
- the geometry (in 2 and 3 dimensions)
A robust and constructive way to find the
solution is the HUM (Hilbert Uniqueness
Method).
2.2 The control system
2.2.1 Notation
- an open domain, bounded
with borderand time interval,]0, [, > 0,
- the space-time cylinder,=0, ×
- the control border,0 and 0=
0, ×0corresponding
- Note: in 1D,is the interval]0, 1[and
the edgecontrol0is the point = 1.
-=
and is the Laplacian operator
2.2.2 The control system
Consider the wave equation with a control over
a part of the edge
= 0 =×0, ,
0, =0,(0, ) = 1 U
,=, 0=0, ×0,
0 0=0, ×0,
The problem of controllability by the edge is
then:
for , (0,1)2() × 1()given is what
we can find a control2(0)such as the
solution of () verifies,=,=
0 ?
The answer is "yes" if we take large enough,
and if we control on a set that is large enough
and that satisfies certain geometric conditions.
2.3 Existence of a solution
For all 0,1=2() × 1()and
all=2(0), there is only one weak
solution
(,)(0, ;)
and the application0,1,,is linear;
moreover, there exists a constant () > 0 such
that
(,)(0,;)
0,1+.
Remark: The wave equation is reversible in
time and the regularity is valid in both
directions.
2.4 Types of Controllability
Is ; 0,1
=, . ,, . ; ()
set of reachable states with initial data0,1
and control.
Definition: The system is exactly controllable
in time if; 0,1= for all
0,1;approximately controllable in
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time if; 0,1is dense in; null
controllable if the state(0,0) ; 0,1.
Remark: For linear PDEs systems, null
controllability and exact controllability are
equivalent.
2.5 An auxiliary system
= 0 =×0, ,
,= 0, (,) = 0 P
,=, 0=0, ×0,
0 0=0, ×0.
According to the existence
theorem,(, . ), (, . )and the problem
of finding a control that drives this system
back to(0, . ), (0, . )=0,1is
equivalent to solving the original control
problem.
2.6 The adjoint system
The HUM method is built on the relationship
between the direct system (), which is self-
adjoint, and its adjoint system,
= 0 =×0, ,
0, =0,(0, ) = 1 A
,= 0 =0, ×,
with initial data0,1=0
1() × 2().
Theorem: For all 0,1 the adjoint
system admits a single weak solution
,0, ;.
Furthermore
2()
and there is a constant () > 0 such that
(,)(0,;)0,1.
Remark: The regularity-2 is stronger than the
standard trace result (a half more ...) that could
be obtained from (, . ) 0
1(). This result is
known as the "hidden regularity" of the wave
equation.
3. Discrete Control
In order to calculate an approximate control, we
must discretize the system ... We discretize the
wave equation in 2 steps:
1. in space, which produces a semi-
discrete model an ODE system!
2. in time, which produces the complete
discrete system. We can use the control theory
for linear ODE, but we lose the very rich
Hilbertian structure, as well as the notion of
control time, .
3.1 HUM semi-discreet
- the wave equation is approximated by a
system of ordinary differential equations
- or , of dimension 2, the approximation of
space =0
1() × 2()
and , of dimension 2, the approximation of
space =1() × 2()
- let,,,, and ,,
,their approximations by vectors
- norms vectoriels are
=1+0
=1+0
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where .1,.0,.1, are approximations of
the norms 0
1,2,1()
- the product of duality.,.,is approached by
,
,=,1,1 +,0
- discrete norms and their product of duality
depending on the choice of semi-discretization
(finite differences, finite elements,
discontinuous elements, etc.)
3.2 Approximation
We introduce an approximation ()of
dimension 2the control system (),
=+, 0, ,
0=0,
or
- the initial data (0,1) were projected
ongiving0= (0,1)
- the function () is a border control
applied at the right extremity of the
domain
- the matrix is an approximation of the
spatial derivative
- matrix affects the scalar boundary
condition, () to the system
3.3 Semi-discrete control problem
We can define the problem of semi-discrete
control: For 0given, find such that
the discrete system () is led to zero at time
=, that is to say, () = 0.
3.4 Adjoint system and semi-discrete
observation
We introduce an approximation () of
dimension2 of the adjoint system (),
=, 0, ,
0=0,
for which the initial data0=
(0,1)correspond to the (0,1) of () and
let () be a discrete approximation of the
normal derivative to = 1,
, 1.
Finding the observation() from the initial
data 0 is called semi-discrete observation.
Definition: Let 0 be the initial data of the
system () and let () be its solution.
Calculating the output of Neumann () is
called the semi-discrete observation and the
corresponding operator,
:
defined by :0
is called the operator semi-discrete observation
“observation-sd (o-sd)”.
3.5 Retrograde system and semi-discrete
reconstruction
We also define a semi-discrete version () of
the auxiliary system ()
=+, 0, ,
0= 0,
which is resolved retrograde in time.
Definition: For a given function ,
suppose that ()admits a solution. Resolves
()to get the output 0,(0) and
call this operation the reconstruction-sd.
The corresponding operator,
:
defined by
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: (0)
(0)
is called the operator of reconstruction-sd.
3.6 Discrete control function
We are looking for a specific function,
which checks the condition
=1
0.
Such a function will, by construction, solve
the semi-discrete control system () and is
therefore called a control. A control that leads
the semi-discrete system () to zero in time
= is called a HUM control, if it is
calculated by = 0
1,
where0=0,1 is a set of initial data for
the semi-discrete adjoint ().
3.7 Semi-discrete HUM operator
The equation for the semi-discrete HUM
operator then becomes
0
1
=1
0, ()
where the operator
:
defined by =
approach the continuous operator ,
: =.
with the observation operator
:
defined by 0,1=
0.
and the auxiliary system for is resolved
backward in time. Introduce the reconstruction
operator, , associated with this system
:
defined by
:(0, . ,0, . ).
Thus=1,0.
3.8 Summary
1. The operator associates with the discrete
initial data, 0,1, the edge data of
Neumann approached .
2. Then, takes these data as a boundary
condition of Dirichlet () = and
associates with it the state at = 0,
0,(0).
3. If the solution
0of the semi-discrete HUM
equation (*) exists, then it provides the desired
control by =
0.
3.9 Complete discretization of HUM
We introduce:
- uniform discretization in time of the interval
[0, ] by = for = 0, 1, . . . , 1.
- the set of discrete operators that correspond to
the choice of the integration scheme in time.
Definition 6: For an initial data
0,(0)we define the discrete
observation operator
:
defined by :0
1,
where=0,,(),=
with=is the solution to
() at the time step .
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Definition: For given, we define the
discrete reconstruction operator
:
defined by : 1
2,
where1,0 is the state of () at = 0
after itsintegration of to 0.
3.10 Discreet lambda
We can now define the discrete approximation
of , :X
defined by =.
As for the semi-discrete HUM operator, we
introduce the equation for the discrete operator
0
1
=1
0. ()
Its solution, 0,1, if it exists, provides the
control sought by
=0
1.
The continuous operator depends only on
(for 0 fixed), but its approximation also
depends on:
1. The semi-discretization scheme (element
size , order of approximation ).
2. The approximation of the normal derivative
.
3. The assignment of the Dirichlet condition
with .
4. Time integration: schema and .
3.11 HUM numerical
The discrete HUM equation (**) can be solved
directly by constructing as a matrix, or
iteratively.
Finding : The problem discreet and ill-posed!
Indeed Stability + consistency convergence.
Solutions:
0. Filtering by two grids.
1. Mixed finished elements.
2. Regulation of Tychonov.
3. Schemes uniformly controllable.
3.12 Iterative HUM by conjugate gradient
The works by [6], [7], [8], [9] proposed a
preconditioned conjugate gradient algorithm
to solve the HUM numerically. The conjugate
gradient method is an iterative algorithm
for solving the linear system
= ,
where is a matrix ( × ), symmetric,
positive definite. This algorithm is the natural
choice for HUM since the underlying
operator,, is self-adjoint and positive. We use
a preconditioning, with a matrix which is
easy to reverse, so that the new problem,
1 = 1
be easier to solve. The ideal preconditioning
is = 1.
3.13 Algorithm
For discrete initial conditions [1, 0] given
for the control problem ( ), we aim to solve
the preconditioned HUM problem,
10
012
340
1
=
10
01
0
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where the preconditioned is an
approximation of Laplacian.
Acknowledgements :
The authors thank the referees for their
careful reading and their precious
comments. Their help is much appreciated.
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Contribution of Individual Authors to the
Creation of a Scientific Article (Ghostwriting
Policy)
The author(s) contributed in the present research,
at all stages from the formulation of the problem
to the final findings and solution.
Sources of Funding for Research Presented in a
Scientific Article or Scientific Article Itself
No funding was received for conducting this study.
Conflict of Interest
The author(s) declare no potential conflicts of
interest concerning the research, authorship, or
publication of this article.
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