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3.4 Control of Linear Oscillations and Relaxations 85

and derivatives can capture parsimoniously long-run structures [45] that de-

cay very slowly. The advantage of fractional derivatives is that these involve

only one parameter, the fractional order α of the derivation. This makes frac-

tional derivatives and integrals an interesting candidate for several theoretical

approaches.

The fractional integral can be introduced in the following way. First of all

we deﬁne the nth order integral over a given function as the inverse n-order

derivative

=(d/dt)

−n

f(t)=



dξ



dξ

...

n−1



dξ

f(ξ

)

(n − 1)!



dξ (t − ξ)

n−1

f(ξ) . (3.107)

In fact, the application of

−n

f(t) yields

−n

f(t)=f(t), or more

general we have the group property

−m

n−m

. Now, we generalize the

integral to a noninteger n and obtain the Riemann–Liouville fractional integral

of order a which corresponds to a derivative of order −a

−a

f(t)=

Γ (a)



dξ (t − ξ)

a−1

f(ξ) , (3.108)

where Γ (a) is the Gamma function. The operator

is called the fractional

diﬀerential operator. We remark that the classical distinction between inte-

grals and derivatives disappears in the fractional calculus. Obviously, (3.108)is

a representation of the memory term with the long-run memory K(t) ∼ t

a−1

Let us ﬁnally point out the role of fractional derivations in the description

of the time evolution of dynamical systems. By no means, the fractional cal-

culus applied to physical problems provides other physical basic laws. But it

is indeed possible to use the fractional calculus as a completely consistent the-

ory of time evolution of dynamical systems associated with algebraic memory

terms.

3.4.2 Optimal Control of Generalized Linear Evolution Equations

As mentioned above, oscillations and linear relaxations correspond to small

changes of the state variables X against a stable equilibrium or ground state

of the system. Therefore, we may use the same arguments as in Sect. 3.1.2

to introduce a quadratic or linear quadratic performance functional. Then we

are able to construct a generalized action

S = J[X, u]+S

constr



i=1



i=1

(3.109)

86 3 Linear Quadratic Problems

with the contributions of constraints (3.101)

constr



dtP (t)





X(t) − X

(t) −



H(t − τ)f(τ )dτ





(3.110)

containing the momentum P (t) as a functional Lagrange multiplier. The ho-

mogeneous solution X

(t) is given by

(t)=



k=1

(t)

k−1

X(t)

k−1



t=0

. (3.111)

The third term of (3.109) may consider I isoperimeter links



dtg

(t, X(t)) = 0 i =1,...,I , (3.112)

while the last term is a consequence of the possible point constraints

= g

,X(t

)) = 0 0 <t

<T i=1,...,K , (3.113)

which are coupled to the generalized action via the Lagrange multipliers λ

and λ

. The structure of the generalized action allows us to determine the

Lagrangian L, which is given by

L =

[X(t)Q(t)X(t)+u(t)R(t)u(t)] +

Y (T )ΩY (T )δ(t − T )

+ P (t)





X(t) −



k=1

(t)

k−1

X(t)

k−1



t=0

−



H(t − τ)f(τ )dτ







i=1

(t, X(t)) +



i=1

(t, X(t)) δ(t − t

) . (3.114)

The main diﬀerence with the Lagrangian discussed in Sect. is the lack

of the velocities

X(t). On the other hand, we have the disadvantage that

the Lagrangian is no longer local in time. For the sake of simplicity, we now

focus on a problem without constraints of type (3.112) and (3.113) and with

vanishing initial conditions for the state vector X

. Then, the variation of

the generalized action with respect to the momentum P leads again to the

dynamic equation

∗

(t)=



H(t − τ)[ψ(τ )+Bu

∗

(τ)] dτ (3.115)

We remark that the consideration of isoperimetric constraints, point constraints,

and nonvanishing initial conditions for the state vector does not change the below-

discussed general relation between the state dynamics and the control functions.

2.4.1

3.4 Control of Linear Oscillations and Relaxations 87

for the optimum trajectory. Note that we have substituted the forces by the

superposition (3.97) of external sources and control forces. The variation with

respect to the state X yields

∗

(t)=−Q(t)X

∗

(t) , (3.116)

while the variation with respect to the control functions gives

∗

(t)=R

−1

(t)



(τ − t)P

∗

(τ)dτ . (3.117)

Finally, the substitution of (3.116)in(3.117) eliminates the momentum from

the control equations, and we obtain

∗

(t)=−R

−1

(t)



(τ − t)Q(τ)X

∗

(τ)dτ. (3.118)

This relation and (3.115) are a complete set of integral equations determining

the corresponding control problem. It is interesting to remark that (3.115)

requires the history of all controls in order to determine the current state

while the current control follows from the future evolution of the state; see

(3.118) and Fig. 3.8. This does not contradict the physically necessary causal-

ity because the optimal trajectory and the optimal control are deﬁned by the

complete knowledge of the evolution of the environment, given by ψ(t), and

the control parameters, Q(t)andR(t), over the whole period [0,T]. Finally,

the substitution of (3.115)in(3.118) leads to a closed relation between the

external forces and the optimum answer via the control functions

u( )τ

X( )τ

Fig. 3.8. Schematic representation of causality and anticausality in the relations

between the optimal trajectory and the optimal control function

88 3 Linear Quadratic Problems

R(t)u

∗

(t)+



dτB

U (t, τ) Bu

∗

(τ)=−



dτB

U (t, τ) ψ(τ) (3.119)

with

U (t, τ)=



max(t,τ)

dτ



(τ



− t)Q(τ



)H(τ



− τ) . (3.120)

The integral equation (3.119) is of the Fredholm type. Although the theory

of the class of integral equations is well established, a quantitative solution of

(3.119) usually needs a numerical support.

We end this chapter with an important remark concerning the estimation

of the performance. As we have stressed repeatedly, the performance or cost

functional has no strict physical meaning, in contrast to the constraints, i.e.,

the equations of motion for the system state. That means in the case of a

quadratic performance, the weight matrices, Q and R, must be chosen em-

pirically with respect to the control aim. But this is, with the exception of

the formulation of the classical mechanics as a special control problem, by no

means a pure physical task. Economical and technical factors may be as impor-

tant as physical arguments. An often used performance is the time-averaged

energy of the system [46], but the power of heat loss [47] or the injected energy

from a certain external source are also reasonable cost functionals.

3.4.3 Perturbation Theory for Weakly Nonlinear Dynamics

If a suﬃciently complex system contains nonlinear terms in its equations of

motion, then it is, as a rule, often impossible to determine a closed analytical

solution. In this case, an approximative solution may be helpful. The standard

method is a perturbation expansion, which is also denoted by the concept of

successive approximations. Let us add a small nonlinear term to the evolution

equation (3.98) of the state X,

dX(t)+εΦ(t, X(t)) = f (t) . (3.121)

For the sake of simplicity, we have set

M = 1. The modiﬁcation of (3.121)

leads to slightly changed integral equations for the optimum control, namely

∗

(t)=



H(t − τ)[ψ(τ ) − εΦ(τ, X

∗

(τ)) + Bu

∗

(τ)] dτ (3.122)

for the optimum trajectory, and

∗

(t)=−Q(t)X

∗

(t)+ε

∂Φ

(t, X)

∂X



X=X

∗

(t)



(τ −t)P

∗

(τ)dτ (3.123)

3.4 Control of Linear Oscillations and Relaxations 89

for the momentum. The equation for the control function remains unchanged

∗

(t)=R

−1

(t)



(τ − t)P

∗

(τ)dτ. (3.124)

The set of equations (3.122), (3.123), and (3.124) can now be treated in

terms of a perturbation theory. The ﬁrst step is the calculation of the ref-

erence solution (P

∗

) corresponding to ε = 0. In a subsequent step,

these solutions are substituted into the right-hand side of the three equations

considering now the small perturbation. Then, the left-hand side of (3.122),

(3.123), and (3.124) yield the ﬁrst approximation

∗

). The succes-

sive repetition of this procedure leads eventually to a convergence of the series

∗

)(i =0, 1, 2,....) to the solution of systems (3.122), (3.123), and

(3.124). The convergence of this successive approximation method depends on

the underlying system. It is still an open problem whether a series converges

or not. Some necessary and suﬃcient conditions can be found in the literature

[48, 49, 50, 51]. However, the ﬁrst-order correction is often a suﬃcient quan-

titative estimation of the eﬀects due to a small nonlinearity in the evolution

equations of the system.

Finally, we remark that a large class of weak nonlinear control problems

is related to bilinear systems. They represent large number of real world phe-

nomena [53, 54, 55]. Bilinear control systems are described by the following

evolution equation:



β=1

αβ



β=1





γ=1

αβ

= ψ

(t)+





γ=1

αγ

. (3.125)

The bilinear contributions, X

, distinguish (3.125) from the above-

discussed linear ones. Although bilinear systems are also a special case of

nonlinear interacting systems, bilinear systems were brought to attention. A

ﬁrst comprehensive study was given by Wiener [52], who believed that such

equations are the essence of understanding the behavior of neural networks.

Very import problems, such as the control of nuclear reactors [56], of the

dynamics of heat exchange [57], or of the induction motor systems [58], and

rotary multi motor systems [59] were successfully described by bilinear sys-

tems.

Although the structure of (3.125), does not seem to be very complicated,

the control equations of bilinear systems are very hard to handle. In general,

there exists no analytical solution. However, it has been shown [60]thatthe

method of successive approximations converges to a stable solution under

relatively weak conditions.

That means the reference state plus the ﬁrst-order correction in the language of

the perturbation theory.

90 3 Linear Quadratic Problems

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Control of Fields

4.1 Field Equations

4.1.1 Classical Field Theory

Euler–Lagrange Equations

The theoretical representation of classical mechanics and ﬁeld theory is very

similar in some ways. This also applies to the axiomatic structure. It is well

known that essential aspects of the classical mechanics can be translated into

the language of ﬁeld theory. Especially, there exist equivalent relations be-

tween the canonical formalism of the ﬁeld theory and of the classical mechanics

with respect to the Hamilton principle and the Euler–Lagrange equations. Let

us brieﬂy study the basic ideas of the variational principle for classical ﬁelds,

in particular, with respect to some speciﬁc features occurring in ﬁeld theoret-

ical equations. This is not the place to discuss the mathematical and physical

details of this well-established theoretical concept. The interested reader may

ﬁnd more information in the literature [1, 2, 3, 4]. But this very short overview

should give at least a guideline for the following chapter, namely for the trans-

lation of the previously demonstrated variational method for the control of

systems with a ﬁnite set of degrees of freedom to the variational method for

the control of ﬁelds.

Of course, we must extend the terms of classical mechanics in such a

way that they become suitable for a ﬁeld theory. The basic quantities of the

Newtonian mechanics are the positions of the particles, or more general the

set of degrees of freedom which are collected in the state vector X(t). All

admissible states are embedded in a ﬁnite-dimensional phase space P.

The state vector is now generalized to a ﬁeld function

X(t) → Ψ (r,t) , (4.1)

which is deﬁned for each time t and each site r of a given d-dimensional

space R

. The knowledge of the geometry and topology of this space and any

M. Schulz: Control Theory in Physics and other Fields of Science

STMP 215, 93–121 (2006)

 Springer-Verlag Berlin Heidelberg 2006

94 4 Control of Fields

information about a possible fusion with the time coordinate to a continuous

space–time structure as they are known from all modern ﬁeld theories are not

important for the variational principle. These questions become relevant if we

are interested in solving a special ﬁeld theoretical problem. Formally, the ﬁeld

function Ψ (r,t)isanN-dimensional vector describing the N components of

the ﬁeld at the spatial position r at time t. The physical meaning of the ﬁeld

components Ψ

(i =1,...,N) and their relation to the space R

or the space–

time continuum depends on the theory in mind. Scalar ﬁeld theories have only

one ﬁeld component

, while the Maxwell theory requires a four-dimensional

vector with well-deﬁned symmetry relations and transformation rules between

the ﬁeld components considering the geometry of the underlying space–time

continuum.

The main diﬀerence between the classical mechanics and a ﬁeld theory is

the dependence of the ﬁelds on the spatial coordinates. That means, a ﬁeld

theory consists of an uncountable inﬁnitely large set of degrees of freedom

in contrast to the mechanics with a ﬁnite number of degrees of freedom.

The ﬁeld theoretical action is deﬁned as the integral

S =



Ω

rdtL(t, r,Ψ,∇Ψ, Ψ

) (4.2)

over a so-called Lagrange density L with Ψ

= ∂Ψ/∂t. Here, Ω is a certain

region of the whole space–time continuum which is deﬁned by possibly time-

dependent boundary conditions. The appearance of gradient terms ∇Ψ is a

typical feature of a physical ﬁeld theory reﬂecting the Nahwirkungsprinzip

Furthermore, we may interpret the integral

L =



Ω

rL(t, r,Ψ,∇Ψ, Ψ

) (4.3)

as Lagrangian in order to obtain another analogy to the classical mechanics.

However, in the framework of a ﬁeld theory it is much more conventional to

denote the Lagrange density as the Lagrangian.

In the classical mechanics, the Euler–Lagrange equations are the equa-

tions of motion of the underlying system. This statement is also true for a

ﬁeld theory. Usually, the Euler–Lagrange equations are called the ﬁeld equa-

tions. The derivation of these equations follows from Hamiltonian’s varia-

tional principle. Just as we have pointed out for the classical mechanics in

Or two, in the case of a complex ﬁeld.

In the case of a lattice ﬁeld theory, the degrees of freedom form a countable

inﬁnite large set or sometimes a ﬁnite set (for example on a lattice with a spher-

ical topology). In this sense, lattice ﬁeld theories may be interpreted as the link

between mechanics and ﬁeld theory.

We remark that several ﬁeld theories also contain higher derivatives in space

and time. As an example we refer to Einstein’s general relativity theory with a

Lagrangian containing second-order derivatives of the ﬁeld components.