Schulz M. Control Theory in Physics and Other Fields of Science: Concepts, Tools, and Applications

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4.1 Field Equations 95

Chap. 2.3, this profound principle requires that action (4.2) arrives the ex-

tremum S[X, T ] → extr. for the solution of the ﬁeld equations. The necessary

condition for this demand is that the ﬁrst variation with respect to all degrees

of freedom, i.e., for all ﬁeld components, vanishes:

δS =



Ω

rdt



∂L

∂Ψ

δΨ +

∂L

∂∇Ψ

δ∇Ψ +

∂L

∂Ψ

δΨ





Ω

rdt



∂L

∂Ψ

δΨ +

∂L

∂∇Ψ

∇δΨ +

∂L

∂Ψ

∂

∂t

δΨ



= 0 (4.4)



Ω

rdt



∂L

∂Ψ

−∇



∂L

∂∇Ψ



−

∂

∂t



∂L

∂Ψ



δΨ



Ω

rdt



∇



∂L

∂∇Ψ

δΨ



∂

∂t



∂L

∂Ψ

δΨ



. (4.5)

The last term is a generalized divergence with respect to the space–time con-

tinuum. Therefore, this contribution can be transformed into a generalized

surface integral. We assume again that the variations of the ﬁeld vanish at the

boundaries. But in contrast to the mechanical variations, where the boundary

conditions are formulated with respect to the initial and ﬁnal times, we now

have to consider also spatial boundary conditions.

The treatment of the ﬁrst term of (4.5) remains. Because this contribution

must vanish for all admissible variations δΨ of the ﬁeld components, one

obtains the Euler–Lagrange equations

∂

∂t



∂L

∂Ψ



+ ∇



∂L

∂∇Ψ



∂L

∂Ψ

(4.6)

of the classical ﬁeld theory. These equations, together with correctly deﬁned

boundary conditions, completely describe the evolution of the ﬁelds Ψ in the

space-time continuum.

Klein–Gordon Field Equation

Let us derive the ﬁeld equations in an explicit form for some standard ﬁelds.

The basic assumption that fundamental physical laws take places in the same

manner independently of the position and the current time requires that a

well formulated Lagrangian of a free ﬁeld need not depend explicitly on r and

t. The simplest physically motivated example is the Klein–Gordon ﬁeld. This

is a scalar ﬁeld, i.e., Ψ consists of only one real-valued

component φ(r,t).

The Lagrangian of the free Klein–Gordon ﬁeld is given by

The extension of the Klein–Gordon ﬁeld to a complex-valued ﬁeld is always pos-

sible and has a deep physical sense [3] when charged scalar ﬁelds are considered.

96 4 Control of Fields

Fig. 4.1. Bead spring model

L(Ψ,∇Ψ, Ψ



− (∇φ)

− m



. (4.7)

Inserting this into the Euler–Lagrange equation (4.6) we ﬁnd the standard

Klein–Gordon equation

∂

∂t

−∇

φ + m

φ =0. (4.8)

Such a free ﬁeld is not controllable because there is no coupling with an

external changeable source

. We may introduce such a coupling by the con-

sideration of the additional term Jφ in the Lagrangian. Thus, we obtain the

new Lagrangian

L(t, r,Ψ,∇Ψ,Ψ



− (∇φ)

− m



+ Jφ (4.9)

which can now be explicitly dependent on the time and space coordinates, r

and t, thanks to the general functional structure of J = J (r,t). The presence

of the source term changes the homogeneous ﬁeld equation (4.8)intothe

inhomogeneous ﬁeld equation

∂

∂t

−∇

φ + m

φ = J (r,t) . (4.10)

The Klein–Gordon equation is a very instructive example to study the above-

mentioned transition from a ﬁnite set to an inﬁnitely large set of degrees

of freedom which reﬂects the crossover from classical mechanics to the ﬁeld

theory. To this aim let us consider a classical harmonic chain (Fig. 4.1), a

series of N beads of mass µ arranged in a line of length , each connected to

the next via springs of strength k. The position of the i-th particle is then

given by i/N + x

, where x

is the displacement of the bead located at the

ground state position i/N . The Lagrangian of the classical system is therefore

given by

The possibility of controling the ﬁeld evolution by time-dependent boundary con-

ditions remains. However, this is rather a rare case.

4.1 Field Equations 97

L =



i=1

˙x

−

N−1



i=1

k (x

i+1

− x

)

. (4.11)

Now let us assume that the number of mass points diverges while the total

mass M and the total length  of the chain is conserved, i.e., we write N = /

and µ =  where  is a small length scale,  → 0and is the mass density.

In the limit  → 0, the Lagrangian becomes

L =



i=1

 ˙x

−

k

N−1



i=1

i+1

− x

)



→









∂φ(r, t)

∂t



− κ



∂φ(r, t)

∂r





(4.12)

where we have taken the limit via

 → dx

k → κ

i/N = i → r

(t) → φ(r, t) .

(4.13)

Here, the ﬁeld φ(r, t) is now the displacement of an inﬁnitesimally small par-

ticle with respect to its equilibrium position r,andκ is the Young’s modulus.

If we use the Euler–Lagrange equation, we arrive at

∂

φ(r, t)

∂t

−



∂

φ(r, t)

∂r

= 0 (4.14)

which is just the familiar wave equation for a one-dimensional system traveling

with the velocity (κ/)

1/2

. This equation is a special case of (4.8) with m =0

and a rescaled time scale t → t (κ/)

1/2

The physical importance of the Klein–Gordon equation is mainly of quan-

tum ﬁeld theoretical character which is not the topic of this book. However,

the degenerated case of a simple wave equation is especially often used also

for modeling several control theoretical problems. Furthermore, the Klein–

Gordon equation is a suitable candidate for fundamental studies of the behav-

ior of controlled partial diﬀerential equations of a hyperbolic type [5, 6, 7, 8].

Schr¨odinger Equation

Another interesting ﬁeld equation is the Schr¨odinger equation. Although the

main property of this equation is the foundation of the non-relativistic quan-

tum mechanics, the Schr¨odinger equation itself is a ﬁeld equation for the

complex wave function ψ(r, t). Here, we have the Lagrangian

L(t, r,Ψ,∇Ψ,Ψ

)=−





|∇ψ|



(ψψ

∗

− ψ

∗

)



− V (r,t) |ψ|

(4.15)

98 4 Control of Fields

with the potential V (r,t). This Lagrangian is one of the few physically rea-

sonable candidates of canonical ﬁeld theories with an explicite dependence on

space and time due to the potential. The application of the Euler–Lagrange

equation (4.6) with respect to the two independent ﬁeld components ψ

∗

and

ψ yield the Schr¨odinger equation,

iψ

= −



∇

ψ + V (r,t)ψ, (4.16)

and the complex conjugated equation. In contrast to the Klein–Gordon

equation which is a partial diﬀerential equation of a hyperbolic type, the

Schr¨odinger equation is essentially a partial diﬀerential equation of the par-

abolic type. The control of the Schr¨odinger ﬁeld takes place via the time

dependent potential V (r,t). It should be remarked that the origin of the po-

tential is often the interaction with a classical electromagnetic ﬁeld. In this

case, V (r,t) is deﬁned by the scalar electromagnetic potential, while further

terms containing the vector potential additionally occur in the Schr¨odinger

equation. However, in contrast to the Klein–Gordon equation, the coupling

between the Schr¨odinger ﬁeld and external sources does not change the ho-

mogeneous character of the corresponding ﬁeld equation.

Maxwell Equations

The third example which should be mentioned is the classical ﬁeld theory

of electromagnetism. Here, we have to consider a four-component ﬁeld Ψ =

(A,ϕ) consisting of the scalar potential ϕ and the vector potential A.The

Lagrangian is given by

L(E, B)=



− B



+ jA − ϕ , (4.17)

where the electric ﬁeld, E, and the magnetic ﬁeld, B, are given by

E = −∇ϕ −

∂A

∂t

and B = curl A . (4.18)

The external sources are given by the arbitrarily changeable charge density

(r,t) and the electric current j(r,t). We remark that the three-dimensional

ﬁelds B and E are not independent from each other. It is simple to check that

(4.18) immediately requires

curl E +

∂B

∂t

= 0 and div B = 0 (4.19)

i.e., the ﬁrst group of the Maxwell equations. The second group of ﬁeld equa-

tions follows from the application of the Euler–Lagrange equations (4.6)on

the Lagrangian (4.17), namely

div E =  and curl B −

∂E

∂t

= j . (4.20)

4.1 Field Equations 99

Another important phenomenon belongs to the construction of a control due

to the external sources (r,t)andj(r,t). These quantities are not completely

independent from each other, because the second group of the Maxwell equa-

tions automatically requires the balance equation

∂

∂t

+div j = 0 (4.21)

as a constraint for the sources.

Maxwell equations are of great importance for the control of transmitting

and receiving areals [9, 10], for the ﬁeld controlled phase transition [10, 11]in

magnetic materials, or for the optimization of laser beams [12, 13].

In addition to the few classical examples presented here, the modern

physics knows a large set of further canonical ﬁeld theories, e.g., the gen-

eral gravitation theory or the theory of Dirac ﬁelds. But a detailed discussion

of all these interesting theories would lead us too far. Here, we refer to the

large fundus of textbooks and monographs [14, 15, 16, 17, 18].

4.1.2 Hydrodynamic Field Equations

There exists a large class of ﬁeld theories besides the canonical ones. A char-

acteristic property of these theories is that they cannot generally be reduced

to a physically reasonable Lagrangian. A standard example of these types of

ﬁeld equations is the evolution equations for liquids and gases. The state of

a liquid is mathematically described by the three-dimensional velocity ﬁeld

v(r,t) and two other thermodynamic quantities, for example, the local pres-

sure p(r,t) and the density (r,t). Thus, the ﬁeld state is deﬁned by the ﬁve

components of Ψ =(v,p,). Obviously, the use of these state variables as ba-

sic elements of a ﬁeld theory requires a continuously smeared liquid. In other

words, such a theory fails for too short length scales of an order of magni-

tude of the liquid molecules. On the other hand, the concept of a continuous

smearing of a discrete structure to a ﬁeld has a widespread application also

for other ﬁeld theoretic descriptions. Therefore, the transition from a discrete

structure of a many-body problem to a continuous description is often called

the hydrodynamic limit.

The ﬁve components of the hydrodynamic ﬁeld Ψ require ﬁve ﬁeld equa-

tions. The ﬁrst equation is the mass balance

∂

∂t

+div v = Q, (4.22)

where the right-hand side describes the eﬀects of possible positive and neg-

ative liquid sources injecting and removing matter. But these sources must

be located at the surface of the hydrodynamic system so that the right-hand

side contributions can be installed in the boundary conditions. The remaining

bulk material balance always requires Q =0.

The second ﬁeld equation concerns the velocity ﬁeld. This vector equation

represents the balance of the momenta and consists of the components

100 4 Control of Fields







∂v

∂t



β=1

∂v

∂r





= −

∂p

∂r

+ f



β=1

∂

∂r





∂v

∂r

∂v

∂r



∂

∂r



ζ −



div v



(4.23)

These equations are also denoted as Navier–Stokes equations. They contain

two material parameters, the shear viscosity η and the bulk viscosity ζ. These

are generally dependent on the pressure p and the density . Furthermore,

the force density components

represent the inﬂuence of external forces,

e.g., of the gravity ﬁeld or also partially of electromagnetic ﬁelds. A control of

the liquid dynamics via these forces is strongly restricted from a technological

point of view.

The last equation which is necessary to complete the set of hydrodynamic

equations is the entropy balance

T





∂s

∂t



β=1

∂s

∂r







α,β=1





∂v

∂r



∂v

∂r

∂v

∂r





ζ −



(div v)

+div (κ∇T ) (4.24)

with the temperature T and the temperature-dependent heat conductivity κ.

The temperature is not an independent quantity, because the thermodynamic

state equations always require a well-deﬁned representation T = T (s, p). The

entropy density s is a thermodynamic potential which may be expressed as a

function of p and . Similar to the balance equations of mass and momenta,

an additional input or output of entropy is only possible via the boundaries

The hydrodynamic ﬁeld equations are strongly nonlinear equations. Hence,

they are treated almost by numerical techniques, e.g., by ﬁnite element meth-

ods [19, 20, 21, 22, 23, 24]. Furthermore, the viscosity moduli and the heat con-

ductivity indicate that the processes described by the hydrodynamic equations

(4.22), (4.23), and (4.24) are irreversible ones. The control of hydrodynamic

equations is an important, but largely still open, problem for chemical engi-

neering and several related scientiﬁc ﬁelds. Some important problems are the

control of turbulent eﬀects during the process of combustion [25, 26, 27, 28],

The force density is deﬁned as force per unit volume.

A creation or annihilation of entropy inside the bulk may be possible by chem-

ical reactions. But in this case, the hydrodynamic system must be extended by

transport and balance equations for the reacting components. In other words, the

equation for entropy now contains new source terms, but these contributions are

controlled by further ﬁeld equations with no bulk source terms but a possible

input at the output via the surface of the hydrodynamic system.

4.1 Field Equations 101

the optimal control of mixing procedures, and reaction processes [29] in chem-

ical plants or the optimal control of ﬂow-transport [30, 31] in contaminated

natural and artiﬁcial lakes and rivers.

4.1.3 Other Field Equations

Vibrational Fields

The classical elasticity theory can be interpreted as the hydrodynamic limit of

solids. An initial deformation of a given solid is followed by vibrations spread-

ing the whole body. In contrast to the hydrodynamics, the basic equations

describing this phenomena are almost linear equations. The basic quantity

is the three-dimensional local deviation h(r,t). This is simply the diﬀerence

between the time-dependent position of an observed point of the solid from

its equilibrium position r. In other words, the general ﬁeld function Ψ is now

given by Ψ = h. The corresponding ﬁeld equations depend on the symme-

try of the underlying solid. Therefore, the complete ﬁeld equations for the

vibrational problem contain a set of diﬀerent moduli. We restrict, for the sake

of simplicity, our brief overview on an isotropic solid. In this case, the ﬁeld

equations are given by



∂

∂t

−

2(1 + ν)



∇

h +

(1 − 2ν)

∇div h



= f , (4.25)

where the Young modulus, E, and the Poisson modulus, ν, are material con-

stants. Although a real vibrating system can be usually inﬂuenced only via

its surface, it is, from a technical point of view, reasonable to introduce these

eﬀects as an external force ﬁeld f = f (r,t). The ﬁeld equation is a partial

diﬀerential equation of the hyperbolic type. In fact, simple transverse and

longitudinal plane waves are the basic solutions of (4.25). We remark that the

consideration of additional viscosity terms in (4.25) leads to damped vibra-

tions. Such generalizations are considered in several model equations, e.g., the

Kelvin model, the Maxwell model, or the Maxwell–Kelvin model [32, 33, 34].

The optimal control of undamped or damped vibrations is very important

for the stabilization of buildings and bridges [35, 36, 37], but also for the

reduction of energy losses of machines, motors, and generators [38].

Continuous Reaction–Diﬀusion Processes

Another very important class of ﬁeld equations belongs to the theoretical

description of diﬀusion and reaction processes. Usually, systems in which re-

actants are transported by diﬀusion [39, 40] are denoted as reaction–diﬀusion

systems. Two fundamental time scales characterize these systems, namely the

diﬀusion time as a typical time between collisions of reacting particles and the

reaction time deﬁning the inverse reaction rate of neighboring particles. When

102 4 Control of Fields

the reaction time is much larger then the diﬀusion time, the process follows ap-

proximately the classical kinetic equations. Such a process is reaction-limited

while the opposite case is called a diﬀusion-limited processes [41, 42, 43, 44, 45,

46]. But slow reaction processes may also be diﬀusion-limited. For instance,

heterogeneous reaction systems form increasing product layers between the

educt regions, so that a particle has to overcome this barrier by diﬀusion

processes in order to ﬁnd an appropriate reaction partner. Such a system

shows initially an reaction-limited behavior. But with increasing thickness of

the product layer the further evolution is dominated by a diﬀusion limited

law. In particular, the structure of the educt-product interfaces at this late

stage of the heterogeneous chemical reaction is strongly determined by local

diﬀusion coeﬃcients [47].

The corresponding ﬁelds of an N-component diﬀusion-reaction system are

the local concentrations c

(r,t) which form the ﬁeld vector Ψ =(c

,...,c

Diﬀusion and reaction processes are considered in continuous mean ﬁeld equa-

tion for the particle concentrations which are of the type

∂c

(r,t)

∂t



β=1

∇D

αβ

(Ψ)∇c

(r,t)+



β,γ=1

βγ

(r,t)c

(r,t)

−



δ=1

δα

(r,t)c

(r,t) (4.26)

with the concentration-dependent diﬀusion coeﬃcients D

αβ

(Ψ) and the kinetic

coeﬃcients K

βγ

and K

δα

characterizing the reaction rate of creation processes

[β]+[γ] → [α]+[δ] and annihilation processes [δ]+[α] → [β]+[γ] with respect

to the component α. Obviously, (4.26) considers only pair reactions which may

be straightforwardly extended to multicomponent reactions.

Such diﬀusion–reaction equations are very important for understanding

and modeling of an optimal control of contaminated soils by venting pro-

cedures [48, 49, 50], but also for the maximum suppression of undesirable

interface reactions between chemically active solids [47].

A very-often analyzed special case is diﬀusion–reaction systems in which

the concentrations of all components with the exception of only one are ﬁxed

or eliminated due to the presence of fast processes restoring the chemical

equilibrium for the involved components

. In this case, one get the parabolic

partial diﬀerential equation

∂c(r,t)

∂t

= D∇

c(r,t)+ac(r,t) −



k≥2

(r,t) , (4.27)

where the coeﬃcients a and b

depend on the underlying chemical processes.

This equation is the starting point of a broad class of theoretical investigations

concerning the time evolution of reaction–diﬀusion processes [51, 52, 53, 54].

This approximative procedure is also known as adiabatic elemination.

4.2 Control by External Sources 103

Nonlocal Field Theories

Field theories does not necessarily correspond to local partial diﬀerential equa-

tions. There are several physical reasons to formulate a physical ﬁeld theory

with nonlocal terms. The basic idea is always the ﬁnite velocity of propaga-

tion of excitations in materials. That means the response on a disturbance

at a certain point r



and at time t



reaches another point r at a later time t.

Evolution equations of the following type are well known:

∂ϕ(r,t)

∂t

= F (r,t,ϕ(r,t)) +









K(r − r



,t− t



)ϕ(r,t) (4.28)

with a space–time-dependent memory term. Such equations are used for the

modeling of active walker motions [55, 56, 57, 58] of diﬀusion processes in

glasses or supercooled liquids [60] or of the evolution of biological ﬁelds and

related quantities [61, 62, 63]. A typical control problem analyzed repeatedly

by several authors [64, 65, 66] is the heat conductivity in materials with mem-

ory terms. However, a systematic discussion of ﬁeld equations with memory

is still open.

4.2 Control by External Sources

4.2.1 General Aspects

Before we start the discussion of ﬁeld control problems, we should brieﬂy

summarize the general concept of formulation of a control problem. The basic

consideration is the existence of a set of evolution equations or equivalent

relations describing the dynamical state of the system. These equations have a

clearly physical origin independent wether these equations describe the motion

of particles or the movable components of a machine. But it is important that

the evolution equations are controllable, i.e., they must contain terms which

couple the system with its external environment. From a philosophical point

of view, these equations characterize the objective part of the control problem.

The second important quantity is the performance or cost functional. This

quantity deﬁnes the control aim and is primarily not of a physical origin. The

performance is more or less a measure of penalty for the deviation from a

desired ideal result of an experiment or from the desired ideal output of a

process. Therefore, the cost functional is the subjective part of the control

problem.

The procedures determining the optimal solution for a system under con-

trol generally yield three types of equations. The ﬁrst group is the originally

formulated evolution equations of the system state. These equations are iden-

tically reproduced by all the techniques discussed in the previous chapters.

The second group is the set of the adjoint evolution equations describing the

104 4 Control of Fields

evolution of the generalized momenta

of the system. The third group is the

set of control equations which connect the control functions with the dynamics

of the state variables and the generalized momenta.

We expect for the control of ﬁelds an analogous situation. The control

problem extends the ﬁeld equations by the dynamics of the adjoint ﬁelds and

the control ﬁelds. The main diﬀerence, except that the evolution equations

are now partial diﬀerential equations, with the control of systems with a ﬁ-

nite degree of freedom is the importance of the boundary conditions. While

these conditions are deﬁned as initial and ﬁnal conditions in the mechanical

approach to a control problem, the ﬁeld theoretical approach also allows spa-

tial boundary conditions. These conditions ﬁx a general ﬁeld solution to a

certain condition at the surface of the considered volume.

This situation opens a second type of ﬁeld control. While the control of a

ﬁeld by the source terms is the counterpart to the control of a system with

a ﬁnite number of degrees of freedom by external forces, ﬁelds may also be

controlled by an appropriate change of the boundary conditions.

4.2.2 Control Without Spatial Boundaries

Optimal Control Field Equations

We assume, that the ﬁeld equations for a given N-component ﬁeld Ψ(r,t)can

be written as a set of partial diﬀerential equations of the form

∂Ψ

∂t

= F



t, r,Ψ,∇Ψ,∇

Ψ,...,u,∇u, ∇

u,...,J



. (4.29)

Obviously, there is no symmetry between the time scale and the spatial co-

ordinates. Each evolution equation contains only ﬁrst-order derivatives with

respect to the time while higher-order spatial derivatives are allowed. All

above-discussed local ﬁeld theories can be transformed into this representa-

tion

. The external sources are explicitely separated into the n-component

control ﬁeld u(r,t), see Fig. 4.2, which may be used to change the dynami-

cal behavior of Ψ(r,t), and further external, but unchangeable and therefore

noncontrollable sources J(r,t) with n



components. The ﬁeld equations (4.29)

are declared for all sites r ∈ R

of a d-dimensional space and for all times

t ∈ (−∞, ∞).

The performance of the ﬁeld theoretical control problem may be a func-

tional of the type

In order to avoid confusion, we stress again that the generalized momenta are not

the physical momenta. The latter are usually a part of the state vector.

It may be possible that these representation requires an extension of the number

of ﬁeld components. For example, the inhomogeneous wave equation, ¨ϕ−∇

ϕ = u

may be rewritten as a system of two partial diﬀerential equations with ﬁrst-order

derivatives with respect to the time. In other words, we obtain the two-component

ﬁeld Ψ =(ϕ, ψ) and the components satisfy the equations ˙ϕ = ψ and

ψ = ∇

ϕ+u.