AN APPLICATION OF THE FOURIER TRANSFORM TO OPTIMIZATION OF CONTINUOUS 2-D SYSTEMS
V
ITALIDYMKOU
∗, M
ICHAELDYMKOV
∗∗∗
Graduate College, University of Erlangen-Nürnberg Cauerstreet 7, D–91058, Erlangen, Germany
e-mail:
dymkou@lnt.de∗∗
Institute of Mathematics, National Academy of Sciences, Surganov, 11, 220072, Minsk, Belarus
e-mail:
dymkov@im.bas-net.byThis paper uses the theory of entire functions to study the linear quadratic optimization problem for a class of continuous
2D systems. We show that in some cases optimal control can be given by an analytical formula. A simple method is alsoproposed to find an approximate solution with preassigned accuracy. Some application to the 1D optimization problem is presented, too. The obtained results form a theoretical background for the design problem of optimal controllers for relevant processes.
Keywords: 2D systems, optimization, entire functions, Fourier transform, approximation
1. Introduction
The research termed ‘multidimensional systems’ was ini- tially motivated by the need for a mathematical descrip- tion of some problems that had arisen in the area of cir- cuits and multidimensional signal, image and video pro- cessing (Bose, 1982; Fornasini and Marchesini, 1978).
The next studies showed that also many information pro- cesses in various fields posses such a unique mathemat- ical nature and they can be fully described in the form of multidimensional dynamical systems (Kaczorek, 1985;
Gałkowski and Wood, 2001). The unique key feature of an mD system is that the process dynamics depend on m indeterminates and hence information is propagated in many independent directions. A natural way is the repre- sentation of mD systems by a polynomial-based descrip- tion of the process dynamics. Although very promising, it is related to serious numerical problems. One of the prin- cipal advantages of a dynamic system formulation is that it provides a framework in which it is possible to examine traditional optimal control concepts. In the case of mD systems the propagation of dynamics in the independent directions can be realized by either (i) functions of dis- crete variables, (ii) continuous variables, or (iii) continu- ous variables in one direction and discrete variables in the other. Recently, close attention has been paid to discrete- continuous mD processes (Kaczorek, 1995; Dymkov, 2001) where at least along one direction system dynam- ics are defined in terms of continuous variables. On the
other hand, a few scientific works (Shankar and Willems, 2000; Idczak and Walczak, 2000) are devoted to continu- ous mD systems.
This paper reports an application of the theory of entire functions to control problems. This approach has been used, in particular, in optimization problems of some classes of continuous-discrete 2D models (Dymkov, 1999). It is shown that in some cases the optimization problem can be reduced to a linear programming prob- lem in the appropriate Hilbert space of entire functions.
This paper uses entire function theory to study the linear quadratic optimization problem for continuous 2D sys- tems. It is shown that in the scalar case the optimal con- trol can be given by an analytical formula. We discuss a method of finding an approximate solution with pre- assigned accuracy and also indicate some applications of entire functions to the 1D optimization problem. The ob- tained results provide a theoretical background for the de- sign problem of optimal controllers for relevant processes.
1.1. Preliminaries and Motivation
The simplest classes of linear 2D discrete systems used in applied problems and mathematical theory can be written as follows:
x(t + 1, s) = Ax(t, s) + Dx(t, s + 1) + g(t, s), (1)
or as a couple of equations
x(t + 1, s) = A
11x(t, s) + A
12y(t, s) +D
12x(t, s + 1) + g
1(t, s), y(t, s + 1) = A
21x(t, s) + A
22y(t, s)
+D
21y(t + 1, s) + g
2(t, s),
(2)
given on the space of the functions defined on the integer- valued lattice Z
+. Another state-space 2D objects were investigated by (Gaishun, 1983). In the simplest case they can be given in the form
( x(t + 1, s) = A
1x(t, s) + g
1(t, s),
x(t, s + 1) = A
2x(t, s) + g
2(t, s). (3) The main characteristic feature of such models is their overdetermination (in the sense that the number of equa- tions for this case is greater than that of the unknown functions) and, as a consequence, it is a problem to cor- rectly define the notion of the solution. In this sense, such a system is similar to a one-dimensional discrete- time system with parametric uncertainty. For this reason the classes of completely integrable systems for which the boundary Cauchy problem has a unique solution are of the strongest interest. These models can be also treated as discrete versions of Pfaff partial differential equations that have been used in elasticity theory, magnetohydrody- namics and other engineering problems (see, e.g., Perov, 1975).
Recently, in modern m-D theory, continuous and continuous-discrete versions of discrete multidimensional systems were actively investigated. Some of these, e.g.,
x(t + 1, s) = X
j∈Z+
A
jd
(j)x(t, s)
ds
j+ Bu(t, s), (4) dx(t, s)
ds = Ax(t, s) + Dx(t − 1, s) + Bu(t, s), (5) were considered in (Kaczorek, 1995; Dymkov, 1999). The continuous version of Roesser’s systems of the form
( ∂x(t, s)/∂t = A
11x(t, s)+A
12y(t, s)+B
1u(t, s),
∂y(t, s)/∂s = A
21x(t, s)+A
22y(t, s)+B
2u(t, s) (6) was investigated by Idczak and Walczak (2000), and others.
In this paper we consider a continuous version of the system (3). First applications of such equations were con- nected with differential geometry to find manifolds with a given tangential subspace (Rashevski, 1947). In elec- trodynamics, for example, this model describes the elec- tric potentials for the given electric field (Armand, 1977;
Perov, 1975). Some details concerning stability theory and related topics can be found in (Gaishun, 1983).
This paper reports an application of a subclass of en- tire functions, i.e. functions regular in the complex plane C except the point z = ∞ (Ibragimov, 1984), to control systems. This class has a complex topological structure but we only employ a simpler subclass of entire functions, i.e. the space of entire functions of exponential type and finite degree.
We say that a complex function f : C → C is an entire function of the exponential type and a finite degree σ if f is regular on C and for any ε > 0 there is a con- stant M = M (ε) such that the inequality M exp{(σ − ε)|z
s|} < |f (z)| < M exp{(σ + ε)|z|} holds for all z ∈ C and some z
s∈ C, z
s→ ∞, s → ∞. Let W
σdenote the set of entire functions of exponential type and a finite degree σ non-exceeding π such that its restriction to R consists of some functions from the space L
2(R).
Then it is known that W
σis a Hilbert space (also termed the Wiener-Paley space (Ibragimov, 1984)) where the in- ner product is defined by (f, g)
W= R
R
f (x)g(x) dx and the over-bar means the complex conjugate.
Some properties of the Wiener-Paley space are rel- evant to optimization theory. In particular, according to Wiener’s theorem, functions from this space admit the following description. The set W
σcoincides with the set of the analytical (regular) extension F (z) for the Fourier transformation of the functions f (t) from L
2([−σ, σ], R): F (z) = (1/ √
2π) R
σ−σ
f (t)e
−iztdt.
Moreover, the space W
σis compact in the sense that for any sequence {f
n(z)} of functions from W
σthere exists a subsequence {f
nk(z)} that is uniformly convergent on every compact set K from C (with respect to the L
2- norm) to some function from the space W
σ. Note that there is also another property of the Wiener-Paley space which can be used for solving optimization problems. In particular, according to the Kotelnikov theorem there is an isomorphism between W
σand the space of square summable sequences of complex numbers l
2:
f ∈ W
σ↔ {c
k} ∈ l
2, f (z) =
∞
X
k=−∞
(−1)
kc
ksin π(z − k) π(z − k) . Otherwise, the function f is determined by numbers c
k. This fact is used to give the complete solution to the 1D optimal control problem.
2. Linear Quadratic Optimization for Continuous 2D Systems
We consider the linear time-invariant continuous 2D sys- tem described by the equations
∂x(t
1, t
2)
∂t
1= A
1x(t
1, t
2) + B
1u(t
1, t
2),
∂x(t
1, t
2)
∂t
2= A
2x(t
1, t
2) + B
2u(t
1, t
2),
(7)
where (t
1, t
2) ∈ S = [−π, π] × [−π, π], x ∈ R
nis the state vector depending on parameters t
1and t
2, u ∈ R
mis the input control vector of the same parameters t
1, t
2; A
iand B
i, i = 1, 2 are constant matrices of dimen- sions (n × n) and (n × m), respectively. Also assume that u(t
1, t
2) is a function from the space C
1(Ω, R
m) of continuously differentiable functions defined on the set Ω, where Ω is some domain in R
2containing S . Definition 1. A function x: S → R
nis called the solution to (7) for a given function u(t
1, t
2) if x(·) ∈ C
1(Ω, R
n), where Ω is some domain in R
2including S, and this x(t
1, t
2) satisfies (7) for all (t
1, t
2) ∈ S.
Definition 2. We say that Eqns. (7) are completly solvable for a given function u(t
1, t
2) if for each point x
0∈ R
nthere exists a unique solution x = x(t
1, t
2, x
0) of (7) satisfying the initial condition x(−π, −π) = x
0.
It is well known that the following Frobenious com- mutativity relations (Gaishun, 1983):
A
1A
2= A
2A
1, A
1B
2u(t
1, t
2)+B
2∂u(t
1, t
2)
∂t
1= A
2B
1u(t
1, t
2)+B
1∂u(t
1, t
2)
∂t
2, (t
1, t
2) ∈ S, (8)
are necessary and sufficient conditions for the complete solvability of (7). For this reason we define the admissible control functions as follows:
Definition 3. A function u: S → R
mis called admis- sible if u(·) ∈ C
1(S, R
m) and u(·) satisfies (8) for all (t
1, t
2) ∈ S.
The optimization problem is to minimize the cost functional
J (u) = Z Z
S
(|x(t
1, t
2)|
2+ |u(t
1, t
2)|
2) dt
1dt
2, (9)
where x(t
1, t
2) is the solution of (7) corresponding to the given admissible control u(t
1, t
2) and satisfying to fol- lowing boundary conditions:
x(−π, −π) = x
0, x(π, π) = x
π, (10) where x
π, x
0∈ R
nare given points. For simplicity, we set x
π= 0.
Remark 1. To guarantee the existence of admissible controls which solve the controllability problem (10), we have to formulate some additional conditions. The lemma given below presents the conditions which guarantee the existence of the admissible controls defined on some time segment of the form [−π, π] × [t
∗1, t
∗2]. The proper zero
controllability conditions with fixed time segment are not known till now. Nevertheless, we assume that the analysed control system has a nonempty set of admissible controls.
The controllability problem for Pfaff differential equations can be stated in a differ manner. In fact, more than one distinct concepts of controllability can be defined for this case (Chramtzov, 1985). The simplest one is as follows:
Definition 4. The system (7) is called controllable if for each x
0, x
∗∈ R
nthere are a moment T
∗= (t
∗1, t
∗2) ∈ R
2and an admissible control function u(t
1, t
2), (0 ≤ t
1≤ t
∗1, 0 ≤ t
2≤ t
∗2) such that the solution x = x(t
1, t
2, x
0) of (7) corresponding to this control satisfies the conditions x(−π, −π) = x
0, x(T
∗) = x(t
∗1, t
∗2) = x
∗.
Denote by Θ the subclass of systems (7) for which the conditions (8) and
rank [B
1, B
2] = rank [B
1, B
2, P ] = m, P = A
1B
2− A
2B
1,
∃ α ∈ R
1: rank [αB
1+ (1 − α)B
2] = m hold. Then the following result gives the required control- lability conditions (Chramtzov, 1985):
Lemma 1. The system (7) of the class Θ is controllable if, and only if, rank F (α) = n for some α ∈ R
1, where
F (α) = B(α), A(α)B(α), . . . , A
n−1(α)B(α) , B(α) = αB
1+ (1 − α)B
2,
A(α) = αA
1+ (1 − α)A
2.
The previous studies of the structural properties of discrete 2D systems were often realized on their repre- sentations in the form of 1D dynamical systems (For- nasini and Marchesini, 1978; Dymkov, 1999). Such a kind of representation based on the Fourier transform is applied to the model under consideration. To realize this approach for (7) we use the class of finite functions, whose Fourier transforms belong to the class of entire functions (Ibragimov, 1984). We suppose that the control function in (7) is finite on S in the following sense: for each t
2∈ [−π, π] the function u(t
1, t
2) ≡ 0, ∀t
16∈ [−π, π].
In accordance with the Wiener-Paley theorem the analytic extension ˜ u(z, t
2) to the complex plane C of the follow- ing function (i.e. of the Fourier transform of the function u(t
1, t
2)):
˜
u(ω, t
2) = 1
√ 2π
π
Z
−π
u(t
1, t
2)e
−iωt1dt
1(11)
is an element of the Wiener-Paley space W
πfor each
t
2∈ [−π, π]. Applying the Fourier transform to (8) for
each fixed t
2∈ [−π, π] yields the following singular dif- ferential equation:
B
1d˜ u(ω, t
2)
dt
2+ B(ω)˜ u(ω, t
2) = 0,
ω ∈ R, t
2∈ [−π, π], (12) where B(ω) = A
2B
1− A
1B
2− iωB
2. It is known that the solvability of singular systems (12) is determined, in general, by the properties of the pencil L(λ, ω) = λB
1+ B(ω). In this paper we consider the special case of the regular pencil L(λ, ω) when n = m and the matrix B
1has the inverse B
−11. In this case the solution of (12) is as follows:
˜
u(ω, t
2) = e
− ˆB(ω)(t2+π)v(ω), (13) where ˆ B(ω) = ˆ A + iω ˆ B = B
−11(A
2B
1− A
1B
2) + iω(−B
1−1B
2), v(ω) = ˜ u(ω, −π).
Thus the Fourier transforms of the control func- tions u(t
1, t
2) that are finite on [−π, π] for a fixed t
2and satisfy the differential equality of (8) are described by (13), where v(z) is an arbitrary entire function from the Wiener-Paley space W
π.
Remark 2. Note that, in general, the inverse Fourier trans- formation of the function (13) with v(z) from W
πis not a function from the class C
1(S, R
m), which is required for the admissible control functions. It is well known that the class L
2[−π, π] of square integrable functions is in- variant under the Fourier transform. In this case we deter- mine first functions ˜ u(t
1, t
2), ˜ u(·, t
2) ∈ L
2[−π, π], t
2∈ [−π, π], which together with the corresponding solution
˜
x(t
1, t
2) of (7), (10) minimize the cost functional (9).
Such control functions are called generalized optimal con- trols for the problem (7), (9)–(10). Then the approximate optimal control u
ap(t
1, t
2) from the required class of ad- missible functions is determined as a proper approxima- tion of the obtained function v
0(t
1) = ˜ u(t
1, t
2), t
1∈ [−π, π] from L
2[−π, π] for fixed t
2∈ [−π, π] by the functions from the space C
1[−π, π]. Hence, the solu- tion x
ap(t
1, t
2) of (7) corresponding to u
ap(t
1, t
2) satis- fies approximately the boundary conditions (10) and they provide the approximate optimal cost value. It is shown that the accuracy of this approximation can be easily evaluated.
It is easy to determine the solution of (7) along the two edges of the rectangle S:
x(−π, t
2) = e
A2(t2+π)x
0,
x(π, −π) = e
2πA1x
0+
π
Z
−π
e
(π−τ )A1B
1u(τ, −π) dτ,
x(π, t
2) = e
A2(t2+π)x(π, −π), t
2∈ [−π, π]. (14)
Since x(π, π) = e
2πA2x(π, −π) = 0, we have x(π, −π) = 0. We suppose that the matrix A
1has n single eigenvalues {λ
1, . . . , λ
n}. In this case
e
A1t=
n−1
X
i=0
α
i(t)A
i1,
where the α
i(t)’s are the coefficients of the Lagrange- Sylvester interpolation polynomial corresponding to A
1. Moreover, for the eigenvalues {λ
1, . . . , λ
n} of the matrix A
1we have
e
λkt=
n−1
X
i=0
λ
ikα
i(t), k = 1, 2, . . . , n.
Denote by Λ the Vandermond matrix of the n- th degree defined by the eigenvalues {λ
1, . . . , λ
n} of A
1, and let D = diag {e
λ1π, . . . , e
λnπ}, G = {B
1, A
1B
1, . . . , A
n−11B
1}. Then from (14) we have
−e
2πA1x
0=
π
Z
−π
e
A1(π−τ )B
1u(τ, −π) dτ
=
π
Z
−π n−1
X
i=0
A
i1B
1α
i(π − τ )u(τ, −π) dτ
= G
π
Z
−π
α
0(π − τ ) .. . α
n−1(π − τ )
u(τ, −π) dτ
= GΛ
−1π
Z
−π
n−1
P
i=0
λ
i1α
i(π − τ ) .. .
n−1
P
i=0
λ
inα
i(π − τ )
u(τ, −π) dτ
= GΛ
−1D
π
R
−π
e
−λ1τu(τ, −π) dτ .. .
π
R
−π
e
−λnτu(τ, −π) dτ
= GΛ
−1D
˜
u(−iλ
1, −π) .. .
˜
u(−iλ
n, −π)
,
which yields
e
2πA1x
0+ GΛ
−1D ˆ U = 0, where
U = ˜ ˆ u(−iλ
1, −π), . . . , ˜ u(−iλ
n, −π)
and ˜ u(ω, −π) is given by (11).
Extend now the function v(ω) = ˜ u(ω, −π) to the complex plane as an entire function of the exponential type from the space W
π. Then the following interpola- tion problem arises: Find a function v(z) from the space W
πsuch that the equalities
F ˆ v = f, (15)
where ˆ v = (v(−iλ
1), . . . , v(−iλ
n)), hold at the given points z
1= −iλ
1, . . . , z
n= −iλ
nof the complex plane C, and F = GΛ
−1D, f = −e
−2πA1x
0.
In general, the interpolation problem (15) does not have a unique solution. Let F
Hbe a nonsingular subma- trix defined by the (i
1, . . . , i
p)-th rows and (j
1, . . . , j
p)- th columns of the matrix F where p = rank F . Then (15) yields
ˆ
v
H= F
H−1f − F
H−1F
rv ˆ
r, (16) where F
ris determined by those rows and columns of the matrix F which are not used in F
H, and the vector v is composed in accordance with this partition as v = (v
H, v
r). The latter can be written in coordinate form
v|
z=zk= β
k, k = j
1, . . . , j
p. (17) Note that the components of the (n − p)-vector ˆ v
rand, hence, the p-vector β are free variables. The set of all solutions to (15) can be written as
u(z) = v
1(z) + Q(z)v(z), (18) where v
1(z) is a particular solution to (15), Q(z) is some polynomial of the p-th degree, whose roots are given numbers z
jk, k = 1, 2, . . . , p, and v(z) is an arbitrary function such that Q(z)v(z) ∈ W
π. The set of such func- tions is denoted by V . By the Lagrange formula, a partic- ular solution to (15) can be chosen as
v
1(z) =
p
X
i=1
β
iϕ(z) ϕ
0(z)(z − z
i) ,
ϕ(z) =
p
Y
i=1
sin π
p (z − z
i). (19) Note that ϕ(z) cannot be chosen as the simplest in- terpolation polynomial of the form ψ(z) = Q
pi=1
(z − z
i) since ψ(z)/(z − z
i) 6∈ W
πfor every i. Thus the set of all admissible controls (their Fourier transforms) driving the point x
0to the point x
πis given by the formula
˜
u(ω, t
2) = e
−( ˆA+iω ˆB)(t2+π)(v
1(ω) + Q ω)v(ω). (20) The problem is now how to find the function v(z) that minimizes the functional (8). Applying the Fourier transform to the first equation of (7) with respect to the variable t
1yields
e
−iωπ(π, t
2) + (iωI − A
1)˜ x(ω, t
2) = B
1u(ω, t ˜
2).
Suppose now that the eigenvalues of the matrix A
1are located in the unit disc of the complex plane. In this case there exists the inverse (iωI − A
1)
−1for each ω ∈ R
1which allows writing
˜
x(ω, t
2) = (iωI − A
1)
−1h
B
1e
−( ˆA+iω ˆB)(t2+π)× v
1(ω) + Q(ω)v(ω) + e
iωπe
A2(t2+π)− e
−iωπe
2πA1+A2(t2+π)x
0− e
−iωπe
(t2+π)A2f i
. (21)
We consider now the particular case when n = m = 1, a
1b
16= 0, where we give the complete solution to the problem. In this case, from the formula above we have
˜
u(ω, t
2) = e
−b(ω)(t2+π)v
1(ω) + Q(ω)v(ω),
˜
x(ω, t
2) = (iω − a
1)
−1b
1e
−b(ω)(t2+π)× v
1(ω) + Q(ω)v(ω) + (iω − a
1)
−1e
iωπe
a2(t2+π)x
0, where
b(ω) = a
1b
1− a
1b
2b
1+ iω b
2b
1= ˆ . a + iωˆ b.
The isometric property of the Fourier transform in the space L
2[−π, π] implies that the functional (8) can be rewritten via the function v(ω) as
J (u) = Z Z
S
|x(t
1, t
2)|
2+ |u(t
1, t
2)|
2dt
1dt
2=
∞
Z
−∞
dω
π
Z
−π
|˜ x(ω, t
2)|
2+ |˜ u(ω, t
2)|
2dt
2=
∞
Z
−∞
ϕ
2(ω)
v(ω) + ψ(ω) ϕ
2(ω)
2
dω
+
∞
Z
−∞
ν
2(ω) −
ψ(ω) ϕ
2(ω)
2
! dω,
where ϕ
2(ω) =
b
21|iω − a
1|
2+ 1 1 − e
−4πˆa2
|Q(ω)|
2,
ψ(ω) = 1 − e
−4πˆa2 v
1(ω)Q(ω) + b
21(e
−2πb(ω)− 1)
2(iω − a
1)
2v
0(ω)Q(ω)
+ b
1x
0e
iωπ(e
2πa2− 1)(e
−2πb(ω)− 1)
(iω − a
1)
2Q(ω),
ν
2(ω) =
b
21|iω − a
1|
2+ 1 1 − e
−4πˆa2
|v
1(ω)|
2+ x
20(e
4πa2− 1) 2|iω − a
1|
2+ 2Re b
1(e
−2πb(ω)− 1)e
−iωπ(e
2πa2− 1)x
0(iω − a
1)
2. Since the second integral above is not dependent on v(ω), the problem is to minimize the functional
J (u) =
∞
Z
−∞
ϕ
2(ω)|v(ω) − l(ω)|
2dω (22)
in the class V where l(ω) = −ψ(ω)/ϕ
2(ω) is a known function. Now, introduce the Hilbert space of the func- tions that are square integrable on R
1with the weight function ϕ
2(ω) and call it L
2,ϕ. In this space the inner product is given by (f, g) = R
∞−∞
ϕ
2(ω)f (ω)g(ω) d(ω).
Hence the minimization of (22) is reduced to the follow- ing problem: Find a function v from the class V that pro- vides the best approximation to the known function l(ω) in the space L
2,ϕ.
Since the set V is a closed subspace from the space L
2,ϕ, there exists a unique best approximation to l(ω) and this approximation is the projection of the function l(ω) onto V . This projection can be written as a linear combi- nation v = P
∞k=1
c
ke
kof the vectors of some orthonor- malized basis e
1, e
2, . . . , chosen in V, where the Fourier coefficients c
kare calculated by the formula c
k= (l, e
k), k = 1, 2, . . . . The basis in V can also be chosen in a different manner. First, use the Kotelnikov theorem to choose the required basis, (Hurgin and Yakovlev, 1971).
To highlight this, note that each function f (z) ∈ W
πcan be expanded into the following power series:
f (z) = 1 π
∞
X
k=−∞
(−1)
ksin πz z − k ,
∞
X
k=−∞
|f (k)|
2< ∞.
If v(z) ∈ V and Q(z) is an arbitrary polynomial of the p-th degree, then v(z)Q(z) ∈ W
π(see the definition of V ). Hence
v(z) = 1 π
∞
X
k=−∞
a
ksin πz
Q(z)(z − k) , (23) where a
k= (−1)
kv(k)Q(k). Since the function (23) is the entire function if the numbers k = 0, 1, . . . , p − 1 are roots of the polynomial Q(z), we have Q(z) = z. Thus the collection of the functions
g
k(z) = sin πz
Q(z)(z − k) , k ∈ P = Z\{0, 1, . . . , p − 1}
forms a basis in the space V . Let f
1, f
2, . . . , f
n, . . . de- note the re-numbered orthonormalized vectors of the basis {g
k}, k ∈ P . Now the required vectors {e
k} can be de- termined from the formula
e
k= y
k(Γ
kΓ
k−1)
−1/2, k = 1, 2, . . . , (24) where
y
k=
(f
1, f
2) · · · (f
1, f
k−1)f
1. . . . (f
k, f
1) . . . (f
k, f
k−1)f
k
,
Γ
k=
(f
1, f
1) . . . (f
1, f
k) . . . . (f
k, f
1) . . . (f
k, f
k)
, k = 1, 2, . . . (25) Hence we have proven the following result:
Theorem 1. Let n = m = 1 and a
1b
16= 0. Then the generalized optimal control for the problem (7), (9)–(10) is given as
u
0(t
1, t
2) = 1
√ 2π
∞
Z
−∞
Re (e
s(ω)v
0(ω)) dω, (26)
where
v
0(ω) = ω
∞
X
k=1
C
ke
k(ω) + v
1(ω),
s(ω) = a
1b
2− a
2b
1b
1+ iω t
1− b
2b
1(t
2+ π), (27)
C
k= (l, e
k) = (Γ
kΓ
k−1)
−1/2×
(f
1, f
1) . . . (f
1, f
k−1)(l, l
1) . . . . (f
k, f
1) . . . (f
k, f
k−1)(l, l
k)
, (28)
and v
1(ω) is some particular solution of (23) to the inter- polation problem (15).
Note that the inner product (f
i, f
j) can be easily cal- culated from the residual theory as
(f
k, f
l) =
∞
Z
−∞
ϕ
2(ω) sin
2πω Q
2(ω)(ω − k)(ω − l) dω
= π
2p−1
X
j=0
ϕ
2(j)
(j!(p−1−j)!)
2(j −k)(j −l) , k 6= l,
(f
k, f
l) = π
2p−1
X
j=0
ϕ
2(j)
(j!(p − 1 − j)!)
2(j − l)
2+ π
2ϕ
2(k)
k
2(k − 1)
2· · · (k − p − 1)
2, k = l.
Based on the inverse Fourier transform v
0(t) ∈ L
2[−π, π] of a given function v
0(z) ∈ W
πwe are able to determine the generalized optimal control function for the problem under consideration. The approximate optimal control from the class C
1∈ [−π, π] can be established as an approximation to the given function v
0(t). In par- ticular, this approximation can be obtained by cutting the power series (27), where we consider the finite sum
v
(s)(ω) = v
1(ω) + Q(ω)
s
X
k=1
C
ke
k(ω).
It should be noted that the inverse Fourier transforms for the functions v
1(ω), e
k(ω), k = 1, . . . , s are contin- uously differentiable functions. The accuracy of this ap- proximation can be evaluated from the following inequal- ities:
kv
(s)(t) − v
0(t)k
2L2
= Q(ω)
∞
X
k=s+1
C
ke
k(ω)
2 W
≤
∞
Z
−∞
|Q(ω)|
2∞
X
k=s+1
C
ke
k(ω)
2
dω
=
∞
Z
−∞
ϕ
2(ω)
∞
X
k=s+1
C
ke
k(ω)
2
dω ≤
∞
X
k=s+1
|C
k|
2.
3. Optimal Control of 1D Systems with Energy Performance Criteria
In this section, based on the proposed method, we give a complete solution to the following continuous 1D opti- mization problem:
π
Z
−π
|u(t)|
2dt → min, x = Ax + bu, ˙
t ∈ [−π, π], x(−π) = x
0, Hx(π) = 0.
(29)
Here x is an n-phase vector, A is an (n × n)-matrix, b and x
0are given n-vectors, u(t), t ∈ [−π, π] is a control function from the space L
2[−π, π] of measure- able and square summable functions on [−π, π], H is a given (m × n)-matrix. We suppose that the system is controllable and hence the set of admissible controls is nonempty. The existence of an optimal control for this op- timization problem can be stated on the analogy of (Vasil- jev, 1981). In addition to that, we suppose that A has sin- gle eigenvalues. Write G = [Hb, HAb, . . . , HA
n−1b], R = −[Hx
0, HAx
0, . . . , HA
n−1x
0]. V is the (n × n) Vandermonde matrix, generated by the eigenvalues λ
1, . . . , λ
nof A; F = GV
−1Λ, f = (2π)
−1/2RV
−1g, g = (e
2λ1π, . . . , e
2λnπ)
0.
Theorem 2. The Fourier transform of the optimal control in (29) is given by
u
0(z) =
m
X
s=1
β
s nX
j=1
F
sjD(πz
j− πz),
where the numbers β
s= ν
s+ iγ
s, (here i
2= −1), s = 1, 2, . . . , are determined as
m
X
s=1
β
s nX
j=1 n
X
i=1
F
ljF
sjD(πz
i− πz
j) = f
l,
for l = 1, 2, . . . , m. Here D(z) = sin z/z, D(0) = 1, F
lj, f
l, l = 1, 2, . . . , m, j = 1, 2, . . . , n are elements of the (m × n)-matrix F and the n-vector f , respectively.
Proof. The solution of (29) for a given control function can be written as follows:
x(t) = e
A(t+π)x
0+
t
Z
−π
e
A(t−τ )bu(τ ) dτ, t ∈ [−π, π]. (30)
The matrix function e
Atcan be represented in the form e
At=
n−1
X
i=0
α
i(t)A
i,
where the α
i(t)’s are the coefficients of the Lagrange- Silvester interpolation polynomial r(A) that is deter- mined by the matrix A. Then from (29) we have that the admissible control functions satisfy
n−1
X
i=0
HA
ib
π
Z
−π
α
i(π − τ )u(τ ) dτ
= −
n−1
X
i=0
α
i(2π)HA
ix
0. (31) Set
G = [Hb, HAb, . . . , HA
n−1b]
and
R = −[Hx
0, HAx
0, . . . , HA
n−1x
0].
Similarly as in the previous section, it can be estab- lished that the Fourier transform u(w) of the admissible e control, which solves the controllability problem (29), can be extended to the complex plane as the entire function of the form
e u(z) = (2π)
−1/2π
Z
−π
u(τ )e
−iztdt. (32)
Denote by w = ( e w e
1, . . . , w e
n) the n-vector, whose coordinates w e
k= u(−iλ e
k), k = 1, . . . , n are the val- ues of the function (32) at the points z
k= −iλ
k, k = 1, 2, . . . , n of the complex plane C. Then (31) yields
F w = f, e (33)
where F = GV
−1Λ, f = (2π)
1/2RV
−1g, g = (e
2λ1π, . . . , e
2λnπ)
0. The Kotelnikov theorem implies that each u(z) ∈ W
πcan be represented as
u(z) =
∞
X
k=−∞
u
kD(πz − kπ),
∞
X
k=−∞
|u
k|
2< ∞, (34)
where u
k= u(k) .
= x
k+ iy
k, D(z) = z
−1sin z, D(0) = 1. Since
∞
Z
−∞
sin π(w − k) π(w − k)
sin π(w − n) π(w − n) dw =
( 1, k = n, 0, k 6= n, k, n = 0, ±1, . . . , we get
J (u) =
∞
Z
−∞
|u(w)|
2dw =
∞
X
k−∞
|u
k|
2=
∞
X
k=−∞
(x
2k+ y
2k).
Finally, the following optimization problem appears:
Minimize the functional J (u) =
∞
X
k−∞
u
ku ¯
k−→ min
uk
(35)
in the space W
π, subject to the constraint F
∞
X
k−∞
u
kD ˆ
k= f, (36)
where ˆ D
k= (D(πz
1− kπ), . . . , D(πz
n− kπ))
0, and
¯
u
k= x
k− iy
kdenotes the complex conjugate for u
k. Next, set D(πz
j− πk) .
= a
k(z
j) + ib
k(z
j), j = 1, 2, . . . , n, where a
k(z
j) and b
k(z
j) are some real num- bers. Then the problem (35), (36) can be rewritten as
∞
X
k=−∞
(x
2k+ y
2k) −→ min, (37)
subject to the constraints
∞
X
k=−∞
n
X
l=1
F
slx
ka
k(z
l) − y
kb
k(z
l) = f
s,
∞
X
k=−∞
n
X
l=1
F
sly
ka
k(z
l) + x
kb
k(z
l) = 0, s = 1, 2, . . . , m.
(38)
The Lagrange function for the problem (37), (38) is Φ(u, ν, γ)
= 1 2
∞
X
k=−∞
(x
2k+ y
2k)
+
m
X
s=1
ν
sh f
s−
∞
X
k=−∞
n
X
l=1
F
slx
ka
k(z
l) − y
kb
k(z
l) i
−
m
X
s=1
γ
sh X
∞k=−∞
n
X
l=1
F
sly
ka
k(z
l) + x
kb
k(z
l) i ,
and the stationarity conditions
x
k=
m
X
s=1 n
X
l=1
F
slν
sa
k(z
l) + γ
sb
k(z
l),
y
k=
m
X
s=1 n
X
l=1
F
slγ
sa
k(z
l) − ν
sb
k(z
l),
k = 0, ±1, ±2, . . . ,
hold. Substituting this into the first equation from (38), we have
n
X
l=1 m
X
r=1 n
X
t=1
F
slF
rt∞
X
k−∞
h
ν
ra
k(z
t)a
k(z
l) + b
k(z
t)b
k(z
l)
+γ
rb
k(z
t)a
k(z
l) − a
k(z
t)b
k(z
l) i
= f
s, (39) s = 1, 2, . . . , m. Applying (34) to the function u(z) = D(πz − πz
t) at z = z
lyields
D(πz
l− πz
t) =
∞
X
k=−∞
D(πz
t− kπ)D(πz
l− kπ).
Since [a
k(z
t) + ib
k(z
t)][a
k(z
l) − ib
k(z
l)] = D(πz
t− kπ)D(πz
l− kπ), from (39) we have
n
X
l=1 m
X
r=1 n
X
t=1
F
slF
rth ν
rRe
D(πz
l− πz
t)
−γ
rIm
D(πz
l− πz
t) i
= f
s. (40) On the analogy with the above calculations, the sec- ond equation of (38) leads to
n
X
l=1 m
X
r=1 n
X
t=1
F
slF
rth ν
rIm
D(πz
l− πz
t)
+γ
rRe
D(πz
l− πz
t) i
= 0. (41)
Next, set β
r= ν
r+ iγ
r, r = 1, 2, . . . , m. Combining (40) and (41) leads to the required relations. Substituting the given values u
k= x
k+ iy
kinto (34) gives
u
0(z) =
∞
X
k=−∞
(x
k+ iy
k)D(πz − kπ)
=
∞
X
k=−∞
m
X
s=1 n
X
l=1
F
slh
ν
sa
k(z
l) + γ
sb
k(z
l)
+ +i γ
sa
k(z
l) − ν
sb
k(z
l) i
D(πz − kπ)
=
m
X
s=1 n
X
l=1
F
sl∞
X
k=−∞
h
ν
sD(πz
l− kπ)D(πz − kπ)
+ iγ
sD(πz
l− kπ)D(πz − kπ) i
.
Using the representation (34) for u(z) = D(πz − πz
l) yields the required optimal control
u
0(z) =
m
X
s=1 n
X
l=1
F
slh
ν
sD(πz − πz
l)+iγ
sD(πz − πz
l) i
=
m
X
s=1
β
sn
X
l=1
F
slD(πz − πz
l),
which completes the proof.
It is also possible to prove that the Fourier transform of the optimal controls can be represented by the series expansion for the basis l
k= Q
−1(z)(z − k)
−1sin πz, k = 0, 1, . . . , where Q(z) is some polynomial of a fi- nite degree. Therefore the approximate solution can be obtained by the cutting of this power series in much the same way as in the previous section.
4. Example
To illustrate the proposed method we consider the simple optimal control problem
˙
x = u, t ∈ [−π, π], x(−π) = x
0,
x(π) = 0, J (u) =
π
Z
−π
u
2(t) dt → min.
Here A = 0, b = 1 and H = 1. The notation required for this case is as follows:
R = −Hx
0= x
0, G = Hb = 1, V = 1, Λ = e
0= 1, F = 1, g = 1, f = −x
0/ √
2π.
From Theorem 2 we get βD(0) = −x
0/ √
2π. Since D(0) = 1, we have β = −x
0/ √
2π and
u
0(z) = −x
0D(−πz)/ √
2π = −x
0sin πz πz
/ √
2π.
The optimal control function is the Fourier image of the function u
0(z):
u
0(t) = 1
√ 2π
∞
Z
−∞
u
0(ω)e
iωtdω = − x
02π
∞
Z
−∞
sin πω πω e
iωtdω
= − x
02π
2∞
Z
−∞
e
iπω− e
−iπω2iω e
iωtdω
= − x
04π
2i
Z
∞−∞
e
iω(π+t)ω dω −
∞
Z
−∞
e
iω(t−π)ω dω
.
Applying residual theory to the improper integrals yields
∞
Z
−∞
e
iω(π+t)ω dω =
( iπ, π + t > 0,
−iπ, π + t < 0,
∞
Z
−∞