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(1)Doctoral Thesis. Control Control system and number number theory. Jun/30/ Jun/30/2012. Department of Electronics and Information Graduate School of Advanced Technology Kinki University. Fuhuo Li.

(2) CONTROL SYSTEMS AND NUMBER THEORY FUHUO LI. Abstract. In this paper we try to pave a smooth road to a proper understanding of control problems in terms of mathematical disciplines and we shall partially show how to number-theorize some practical problems. Our primary concern in the paper is linear systems from the point of view of our principle of visualization of the state, an interface between the past and the present. We view all the systems as embedded in the state equation, thus visualizing the state. Then we go on to treat the chain scattering representation of the plant a l` a [Kim], which includes the feedback connection in a natural way and we consider the H ∞ -control problem in this framework. We may view in particular the unit feedback system as accommodated in the chain scattering representation, giving a better insight into the structure of the system. Its homographic transformation works as the action of the symplectic group on the Siegel upper half-space (in the case of constant matrices) and we will have a new look at the control as a group action. In the unit feedback system included is the P ID-compensator. Both of H ∞ and PID-controllers are applied successfully in the EV control by Cao [Cao1], [Cao2], which we may unify in our framework. Finally, we mention some similarities between control theory and zeta-functions.. 1. Introduction and preliminaries It turns out there is great similarity in control theory and number theory in their treatment of the signals in time domain (t) and frequency domain (ω, s = σ + jω) which is conducted by the Laplace transform in the case of control theory while in the theory of zetafunctions, this role is played by the Mellin transform, both of which convert the signals in time domain to those in the right half-plane. For integral transforms, cf. §11. §5 introduces the Hardy space Hp which consists of functions analytic in RHP—right half-plane σ > 0. Date: October 23, 2012. This work is supported by the SMX SUDA CO.(No.SDJN1001) . 1.

(3) 2. FUHUO LI. 2. State space representation and the visualization principle Let x = x(t) ∈ Rn , u = u(t) ∈ Rr and y = y(t) ∈ Rm be the state function, input function and output function, respectively. We write d x˙ for dt x. The system of (differential equations) DEs ( x˙ = Ax + Bu, (2.1) y = Cx + Du is called a state equation for a linear system, where A ∈ Mn,n (R), B, C, D are given constant matrices.. The state x is not visible while the input and output are so, and the state may be thought of as an interface between the past and the present information since it contains all the information contained in the system from the past. The x being invisible, (2.1) would read (2.2). y = Du,. which appears in many places in literature in disguised form. All the subsequent systems e.g. (3.1) are variations of (2.2). And whenever we would like to obtain the state equation, we are to restore the state x to make a recourse to (2.1), which we would call the visualization principle. In the case of feedback system, it is often the case that (2.2) is given in the form of (3.8). It is quite remarkable that this controller S works for the matrix variable in the symplectic geometry (cf. §4). Using the matrix exponential function eAt , the first equation in (2.1) can be solved in the same way as for the scalar case: Z t At At (2.3) x = x(t) = e x(0) + Be e−Aτ u(t) dτ. 0. Definition 1. A linear system with the input u = o d x = Ax, dt called an autonomous system, is said to be asymptotically stable if for all initial values, x(t) approaches a limit as t → ∞. (2.4). x˙ =. Since the solution of (2.4) is given by (2.5). x = eAt x(0),.

(4) CONTROL SYSTEMS AND NUMBER THEORY. 3. the system is asymptotically stable if and only if (2.6). ||eAt || → 0 as t → ∞.. A linear system is said to be stable if (2.6) holds, which is the case if all the eigenvalues of A have negative real parts. Cf. §5 in this regard. It also amounts to saying that the step response of the system approaches a limit as time elapses, where step response means a response Z t (2.7) y(t) = eA(t−τ ) u(τ ) dτ, 0. with the unit step function u = u(t) as the input function, which is 0 for t < 0 and 1 for t ≥ 0.. Up here, the things are happening in the time domain. We now move to a frequency domain. For this purpose, we refer to the Laplace transform to be discussed in §11. It has the effect of shifting from the time domain to frequency domain and vice versa. For more details, see e.g. [Kim]. Taking the Laplace transform of (2.1) with x(0) = o, we obtain ( sX(s) = AX(s) + BU (s) (2.8) Y (s) = CX(s) + DU (s), which we solve as (2.9). Y (s) = G(s)U (s),. where (2.10). G(s) = C(sI − A)−1 B + D,. where I indicates the identity matrix, which is sometimes denoted In to show its size. In general, supposing that the initial values of all the signals in a system are 0, we call the ratio of output/input of the signal, the transfer function, and denote it by G(s), Φ(s), etc. We may suppose so because if the system is in equilibrium, then we may take the values of parameters at that moment as standard and may suppose the initial values to be 0..

(5) 4. FUHUO LI. (2.10) is called the state space representation (form, realization, description, characterization) of the transfer function G(s) of the system (2.1), and is written as A B G(s) = (2.11) . C D According to the visualization principle above, we have the embedding principle: Given a state space representation of a transfer function G(s), it is to be embedded in the state equation (2.1). Example 1. If (2.12). G(s) =. A B C D. . . 0 1  −2 −3 = −10 −2.  0 1 , 2. then it follows from (2.10) that −1 s 0 0 1 0 (2.13) G(s) = −10, −2 − +2 0 s −2 −3 1 s −1 −1 0 = −10, −2 +2 2 s+3 1 s + 3 1 −1 0 1 −10, −2 = +2 −2 s 1 (s + 1)(s + 2) 2(s + 3)(s − 1) s+5 +2= . = −2 (s + 1)(s + 2) (s + 1)(s + 2) The principle above will establish the most important cascade connection (concatenation rule) [Kim, (2.13),p.15]: Given two state space representations Ak Bk (2.14) Gk (s) = , k = 1, 2, Ck Dk their cascade connection G(s) = G1 (s)G2 (s) is given by . (2.15). A1 B1 C2  O A2 G(s) = G1 (s)G2 (s) = C1 D1 C2.  B1 D2 . B2 D1 D2. Proof of (2.15) We have the input/output relation (2.10) (2.16). Y (s) = G1 (s)U (s),. U (s) = G2 (s)V (s)..

(6) CONTROL SYSTEMS AND NUMBER THEORY. 5. which means that ( x˙ = A1 x + B1 u, y = C1 x + D1 u. (2.17) and. ( ξ˙ = A2 ξ + B2 v, u = C2 ξ + D2 v.. (2.18). Eliminating u, we conclude that ( x˙ = A1 x + B1 C2 ξ + B1 D2 v, (2.19) y = C1 x + D1 C2 ξ + D1 D2 v. Hence. (2.20).  ! ! ! !  ˙ x A B C x B D  1 1 2 1 2  + v,   ξ˙ = O A2 B2 ξ ! x    + D1 D2 v,  y = C1 D1 C2 ξ. whence we conclude (2.15). Example 2. Given two state space representations (2.14), their parallel connection G(s) = G1 (s) + G2 (s) is given by . (2.21). A1 O G(s) = G1 (s) + G2 (s) =  O A2 C1 C2.  B1 . B2 D1 + D2. Indeed, we have (2.17) and for (2.18), we have ( ξ˙ = A2 ξ + B2 u, (2.22) y + z = C2 ξ + D2 u. Hence for (2.20), we have ( (x + ξ)· = A1 x + A2 ξ + (B1 + B2 )u, (2.23) y = C1 x + C2 ξ + (D1 + D2 )v, whence (2.21) follows. As an example. combining (2.15) and (2.21) we deduce.

(7) 6. FUHUO LI. . (2.24).  O −B1 D2  . −B2  V −1. I. O  −A1 −B1 C2 I − G1 (s)G2 (s) =   O O −A2 O − C1 − D1 C2. Example 3. For (2.1), we consider the inversion U (s) = G−1 (s)Y (s). Solving the second equality in (2.1) for u we obtain u = −D−1 Cx + D−1 y. Substituting this in the first equality in (2.1), we obtain x˙ = (A − BD−1 C)x + BD−1 y, whence (2.25). −1. G (s) =. . A − BD−1 C −BD−1 −D−1 C D−1. .. Example 4. If the transfer function Θ11 Θ12 (2.26) Θ(s) = Θ21 Θ22 has a state space representation   A B1 B2 A B Θ(s) = (2.27) =  C1 D11 D12  , C D C2 D21 D22 then we are to embed it in the linear system  ! b   1  ,  x˙ = Ax + B1 B2 b2 ! ! ! ! (2.28)  D D b C a  1 1 1 11 12  .   a =y = C x+ D b2 21 D22 2 2 3. Chain scattering representation Following [Kim, p. 7, p. 67], we first give the definition of a chain scattering representation of a system. Suppose a1 ∈ Rm , a2 ∈ Rq , b1 ∈ Rr and b2 ∈ Rp are related by a1 b1 (3.1) =P , a2 b2 where (3.2). P =. P11 P12 . P21 P22.

(8) CONTROL SYSTEMS AND NUMBER THEORY. 7. According to the embedding principle, this is to be thought of as y = Su corresponding to the second equality in (2.1). (3.1) means that (3.3). a1 = P11 b1 + P12 b2 ,. a2 = P21 b1 + P22 b2 .. Assume that P21 is a (square) regular matrix (whence q = r). Then from the second equality of (3.3), we obtain (3.4). −1 −1 −1 b1 = P21 (a2 − P22 b2 ) = −P21 P22 b2 + P21 a2 .. Substituting (3.4) in the first equality of (3.3), we deduce that −1 −1 (3.5) a1 = P12 − P11 P21 P22 b2 + P11 P21 a2 .. Hence putting (3.6). Θ = CHAIN (P ) = Θ11 Θ12 = , Θ21 Θ22. −1 −1 P12 − P11 P21 P22 P11 P21 −1 −1 −P21 P22 P21. . which is usually referred to as a chain scattering representation of P , we obtain an equivalent form of (3.1) a1 b2 Θ11 Θ12 a2 (3.7) = CHAIN (P ) = . b1 a2 Θ21 Θ22 b2 Suppose that a2 is fed back to b2 by (3.8). b2 = Sa2 ,. where S is a controller. Multiplying the second equality in (3.3) by S and incorporating (3.8), we find that b2 = Sa2 = SP21 b1 + SP22 b2 , whence b2 = (I − P22 K)−1 SP21 b1 . Let the closed-loop transfer function Φ be defined by (3.9). a1 = Φb1 .. Φ is given by (3.10). Φ = P11 + P12 (E − P22 S)−1 SP21 .. (3.10) is sometimes referred to as a linear fractional transformation and denoted by LF (P ; K)..

(9) 8. FUHUO LI. Substituting (3.8), (3.7) becomes a1 Θ11 S + Θ12 = a, b1 Θ21 S + Θ22 2 whence we deduce that (3.11). Φ = (Θ11 S + Θ12 ) (Θ21 S + Θ22 )−1 = ΘS,. the linear fractional transformation (which is referred to as a homographic transformation and denoted by HM (Φ; S)), where in the last equality we mean the action of Θ on the variable S. We must impose the non-constant condition |Θ| = 6 0. Then Θ ∈ GLm+r (R). If S is obtained from S 0 under the action of Θ0 , S = Θ0 S 0 , then its composition J with (3.11) yields JS 0 = ΦΦ0 = ΘΘ0 S 0 , i.e. (3.12). J = ΘΘ0 ,. HM (Θ; HM (Θ0 ; S)) = HM (ΘΘ0 ; S),. which is referred to as the cascade connection or the cascade structure of Θ and Θ0 . Thus the chain-scattering representation of a system allows us to treat the feedback connection as a cascade connection.. Suppose a closed-loop system is given with z = a1 ∈ Rm , y = a2 ∈ R , w = b1 ∈ Rr and u = b2 ∈ Rp and Φ given by (3.2). H ∞ -control problem Find a controller K such that the closed-loop system is internally stable and the transfer function Φ satisfies q. (3.13). ||Φ||∞ < γ. for a positive constant γ. For the meaning of the norm, cf. §5. 4. Siegel upper space Let ∗ denote the conjugate transpose of a square matrix: S ∗ = t S¯ and let the imaginary part of S defined by Im S = 2j1 (S − S ∗ ). Let Hn be the Siegel upper half-space consisting of all the matrices S (recall Eq. (3.8)) whose imaginary parts are positive definite (Im S > 0— imaginary parts of all eigen values are positive) and satisfies S = t S:

(10) (4.1) Hn = {S ∈ Mn (C)

(11) Im S > 0, S = t S}.

(12) CONTROL SYSTEMS AND NUMBER THEORY. 9. and let Sp(n, R) denote the symplectic group of order n: (4.2) Sp(n, R) ( −1 )

(13)

(14) Θ22 −t Θ12 Θ11 Θ12

(15) Θ11 Θ12 = . = Θ=

(16) −t Θ21 Θ11 Θ21 Θ22

(17) Θ21 Θ22 The action of Sp(n, R) on Hn is defined by (3.11) which we restate as ΘS = (Θ11 S + Θ12 ) (Θ21 S + Θ22 )−1 (= Φ),. (4.3). Theorem 4.1. For a controller S living in the Siegel upper space, its rotation Z = −jS lies in the right half-space RHS. i.e. stable having positive real parts. For the controller Z, the feedback connection −jb2 = Z(−ja2 ). (4.4). is accommodated in the cascade connection of the chain scattering representation Θ (3.12), which is then viewed as the action (3.12) of Θ ∈ Sp(n, R) on S ∈ Hn : (4.5) (ΘΘ0 )S = Θ(Θ0 S);. or HM (Θ; HM (Θ0 ; S)) = HM (ΘΘ0 ; S),. where Θ is subject to the condition t¯. (4.6). ΘU Θ = U,. O In , An FOPID controller (in §6), being a unity with U = −In O feedback connection, is also accommodated in this framework. . Remark 4.1. With action, we may introduce the orbit decomposition of Hn and whence the fundamental domain. We note that in the special case of n = 1, we have H1 = H and Sp(1, R) = SLn (R) and the theory of modular forms of one variable is well-known. Siegel modular forms are a generalization of the one variable case into several variables. As in the case of the sushmna principle in [Vista], there is a need to rotate the upper half-space into the right half-space RHS, which is a counter part of the right-half plane RHP. In the case of Siegel modular forms, the matrices are constant, while in control theory, they are analytic functions (mostly rational functions analytic in RHP). A general theory would be useful for controlling theory. See §7 for physically realizable cases. There are many research problems lying in this direction..

(18) 10. FUHUO LI. 5. Norm of the function spaces   x1 The norm x =  ...  ∈ Cn is defined to be the Euclidean norm xn v uX u n (5.1) ||x|| = ||x||2 = t |xj |2 j=1. or by the sup norm ||x|| = ||x||∞ = max{|x1 |, · · · .|xn |},. (5.2). or anything that satisfies the axioms of the norm. They introduce the same topology on Cn . The definition of the norm of a matrix should be given in a similar way by viewing its elements as an n2 -dimensional vector, i.e. embed2 ding it in Cn . If A = (aij ) , 1 ≤ i, j ≤ n, then v uX u n |aij |2 (5.3) ||A|| = ||A||2 = t i,j=1. or otherwise. The sup norm is a limit of the p-norm as p → ∞: For a = (a1 , · · · , an ), ! p1 n X = ||a||∞ = max {|ak |p }. (5.4) lim ||a||p = lim |ak |p p→∞. p→∞. 1≤k≤n. k=1. For suppose |a1 | = max1≤k≤n {|ak |p }. Then for any p > 0 |a1 | = 1 1 P (|a1 |p ) p ≤ ( nk=1 |ak |p ) p . On the other hand, since |a1 | ≥ |ak |, 1 ≤ k ≤ n, we obtain ! p1

(19) p ! p1 n n

(20) X X

(21)

(22) 1 a

(23) k

(24) (5.5) |ak |p = |a1 | 1 + ≤ |a1 |(1 + n − 1) p .

(25) a1

(26) k=1 k=2 1. For p > 1, the Bernoulli inequality gives (1 + n − 1) p ≤ 1 + as p → ∞. Hence the right-hand side of (5.5) tends to |a1 |. The proof of (5.4) can be readily generalized to give (5.6). lim ||f ||p = ||f ||∞ = sup |f (t)|.. p→∞. t≥0. n−1 p. →1.

(27) CONTROL SYSTEMS AND NUMBER THEORY. 11. The p-norm in (5.6) is defined by Z ∞ p1 p ||f ||p = ||f (t)|| dt , 0. where ||f (t)|| is any Euclidean norm. Note that the functions are not ordinary functions but classes of functions which are regarded as the same if they differ only at measure 0 set. Lp is a Banach space (i.e. a complete metric space) and in particular L2 is a Hilbert space. The 2-norm ||·||2 is induced from the inner product Z ∞ p (5.7) < f, g >= f ∗ (t)g(t) dt, ||f ||2 = < f, f >, 0. where ∗ refers to the transposed complex conjugation. The Parseval identity holds true if and only if the system be complete. However, the restriction that ||f (t)|| → 0 as t → ∞ excludes signals of infinite duration such as unit step signals or periodic ones from Lp . To circumvent the inconvenience, the notion of averaged norm, RT M2 (f ) = M2 (f, T ) = T1 0 ||f (t)||2 dt or similar, is important and the power norm has been introduced: 12 Z T 1 1 ||f (t)||2 dt . (5.8) power(f ) = lim M2 (f, T ) 2 = lim T →∞ T →∞ T 0 Remark 5.1. In mathematics and in particular in analytic number theory, studying the the mean square in the form of a sum or an integral is quite common. Especially, this idea is applied to finding out the true order of magnitude of the error term on average. Such an average result will give a hint on the order of the error term itself. Example 5. Let ζ(s) denote the Riemann zeta-function defined for σ > 1 (s = σ + it), in the first instance, where it is analytic and then continued meromorphically over the whole complex plane with a simple pole at s = 1. It is essential that it does not vanish on the line σ = 1 for the prime number theorem (PNT) to hold. The plausible best bound for the error term for the PNT is equivalent to the celebrated Riemann hypothesis (RH) to the effect that the Riemann zeta-function does not vanish on the critical line σ = 12 . Since the values on the critical line are expected to be small, the averaged norm M2 (ζ) or M4 (ζ), i.e. the RT mean value T1 0 |ζ 12 + it |2k dt for k = 1, 2 is of great interest and there have appeared a great deal of research on the subject. The first result for M4 (ζ) is due to Ingham who used the approximate functional.

(28) 12. FUHUO LI. equation for the Riemann zeta-function to obtain

(29) Z 1 T

(30)

(31) 1 1

(32) 4 (5.9) M4 (ζ) = + it

(33) dt = 2 log4 T (1 + o(1)).

(34) ζ T 0 2 4π for T → ∞. See e.g. [Bel]. The main interest in such estimates as (5.9) lies in the fact that estimates for all k ∈ N

(35) Z 1 1 T

(36)

(37)

(38) 2k + it

(39) dt = O(T a ) (5.10) M2k (ζ) =

(40) ζ T 0 2 would imply the weak Lindel¨ of hypothesis (LH) in the form a 1 (5.11) ζ + it = O(T 2k +ε ) 2 for every ε > 0. It is apparent that the RH implies the LH. The Hardy space Hp (cf. e.g. [Kim, p. 39]) is well-known. It consists of all f (s) which are analytic in RHP–right half-plane σ > 0 such that f (jω) ∈ Lp ; in particular, H∞ with sup norm. Thus H ∞ control problem is about those (rational) functions which are analytic in RHP, a fortiori stable, with regard to the sup norm. Thus the above-mentioned mean-value problem for the Riemann zeta-function is related to the H 2k -control problem with finite Dirichlet series (main ingredients in the approximate functional equation) or stated other way, the H 2k -control problem corresponds to the LH. The H ∞ -control problem asking for (almost) all individual values, it corresponds to the RH. 6. (Unity) feedback system The synthesis problem of a controller of the unity feedback system, depicted in Figure 1.2, refers to the sensitivity reduction problem, which asks for the estimation of the sensitivity function S = S(s) multiplied by an appropriate frequency weighting function W = W (s): (6.1). S = (I + P C)−1 ,. is a transfer function from r to e, where C = K is a compensator and P is a plant. The problem consists in reducing the magnitude of S over a specified frequency range Ω, which amounts to finding a compensator C stabilizing the closed-loop system such that (6.2) for a positive constant γ.. ||W S||∞ < γ.

(41) CONTROL SYSTEMS AND NUMBER THEORY. 13. To accommodate this in the H ∞ control problem (3.1), we choose the matrix elements Pij of P in such a way that the closed-loop transfer function Φ in (3.10) coincides with W S. First we are to choose P22 = −P . Then we would choose P12 P21 = W P . Then Φ becomes P11 + W P C(I + P C)−1 = P11 − W + W (I + P C)−1 . Hence choosing P11 = W , we have Φ = W S. Hence we may choose e.g. P11 P12 W WP (6.3) P = = . P21 P22 E −P Example 6. First we treat the case of general feedback scheme. Denoting the Laplace transforms by the corresponding capital letters, we have Y = P R + P U, U = KE, whence Y = P R + P KE. Now if it so happens that e = r − y and P is replaced by P K, i.e. in the case of unity FD, we derive (6.1) directly from Figure 1.2. We have E = R − Y , so that Y = P R + P K(R − Y ). Solving in Y , we deduce that (I + P K)−1 P KR. We take into account the disturbance d, we obtain since U = CE = C(R − Y ) Y = P U + P D = P C(R − Y ) + P D, whence Y = P C(R − Y ) + P D. It follows that Y = (I + P C)−1 P CR + (I + P C)−1 P D. In the case where d = 0, P C being the open loop transfer function, we have SR is the tracking error for the input R. Hence (6.1) holds true. 7. J-lossless factorization and Dualization In this section we mostly follow Helton ([Hel1], [Hel2], [HM]) who uses the unit ball inplace of RHP. They shift to each other under the complex exponential map. For conventional control theory, the unit ball is to be replaced by the critical line (σ = 0). In practice what appears is the algebra of functions ([Hel2, p.2]). (7.1) R ={functions defined on the unit ball having the rational continuation to the whole space} or still larger algebra ψ consisting of those functions which have (pseudo)meromorphic continuations ([Hel2, footenote 6, p.27]). The occurrence of the gamma function [Hel2, Fig. 2.5,p.17] justifies our incorporation of more advanced special functions and ultimately zeta-functions in control theory (see §13)..

(42) 14. FUHUO LI. Along with the algebra R, one considers (7.2) BH ∞ = {F |analytic on the unit ball having the supremum norm < 1}. Then the only mapping Θ ∈ RU (m, n) acting on BH ∞ must satisfy the J-lossless property: Let Θ denote an (m + n) × (m + n) matrix. Then (7.3). Θ∗ Jmn Θ ≤ Jmn ,. which is interpreted to be the power preservation of the system in the chain scattering representation (3.6) ([Kim, p.82]). We now briefly refer to the dual chain-scattering representation of the plant P in (3.2). We assume P12 is a square invertible matrix (whence m = p). Then the argument goes in parallel to that leading to (3.7). Defining the dual chain scattering matrix by −1 −1 P12 P11 P12 , (7.4) DCHAIN (P ) = −1 −1 −P12 P22 P21 − P22 P12 P11 we obtain (7.5). CHAIN (P ) · DCHAIN (P ) = E 8. FOPID. “FO” means “Fractional order and ”“PID” refers to “Proportional, Integral, Differential”, whence “Proportional” means just constant times the input function e(t), “Integral” means the fractional order integration Itλ Dt−λ of e(t) (λ > 0), and “Differential” the fractional order differentiation Dtδ of e(t) (δ > 0). The FO P I λ Dδ controller (control signal in the time domain) is one of the most refined feed-forward compensator defined by (8.1) u(t) = Kp + Ki Dt−λ + Kd Dtδ e(t), where u is the input function, e is the deviation and Kp , Ki , Kd are constant parameters which are to be specified (Kp − the position feedback gain, Kd − the velocity feedback gain). DE (8.1) translates into the state equation (8.2). Y (s) = C(s)E(s),. where U, Y indicate the Laplace transforms of u, y, respectively and G is the compensator continuous transfer function (8.3). C(s) = Kp + Ki s−λ + Kd sδ ..

(43) CONTROL SYSTEMS AND NUMBER THEORY. 15. The derivation of (8.3) from (8.1) depends on the following. The general fractional calculus operator a Dtα is symbolically stated as  α d  Re α > 0  dtα , α (8.4) 1, Re α = 0 a Dt =  R t 1 , Re α < 0, a dtα where a and t are the lower and upper limits of integration and α is the order of calculus. More precisely, the definition of the fractional differointegral is given by the Riemann-Liouville expression α−{α}+1 Z t d 1 α (t − τ )−{α} f (τ ) dτ (8.5) a Dt f (t) = Γ(1 − {α}) dt a where {α} = α − [α] indicates the fractional part of α, with [α] the integral part of α. Thus we are also led to the Riemann-Liouville fractional integral transform: Z y 1 RL[f ] = (8.6) (y − x)µ−1 f (x) dx. Γ(µ) 0 For applications, cf. §13. When α ∈ N, (8.5) reads α+1 Z t d α (8.7) f (τ ) dτ = f (α) (t), a Dt f (t) = dt a the α-th derivative of f . We shall see that the definition (8.5) is a natural outcome of the general formula for the difference operator of order α ∈ N with difference y≥0: α X α−ν α α (8.8) ∆y f (x) = (−1) f (x + νy). ν ν=0 If f has the α-th derivative f (α) , then Z Z x+y Z t1 +y α (8.9) ∆y f (x) = dt1 dt2 · · · x. t1. tα−1 +y. f (α) (tα ) dtα .. tα−1. The special case of (8.9) with tν = a, a + y → x (ϕ(t) = f (α) (t)) reads (8.10) Z x Z x Z x Z x 1 α ∆x−a ϕ(x) = dt dt · · · ϕ (t) dt = (x − t)α−1 ϕ(t) dt, Γ(α) a a a a whose far-right hand side is RL[ϕ]..

(44) 16. FUHUO LI. Let F (s) be the Laplace transform of the input function f (t). Then (8.11). L[0 Dtα f ](t) = sα F (s) − 0 Dtα−1 f (t) |t=0. and (8.12). L[0 Dt−α f ](t) = s−α F (s).. 9. Fourier, Mellin and (two-sided) Laplace transforms We state the Mellin, (two-sided) Laplace and the Fourier transforms. If f (x) = O(xα ), α ∈ R for x > 0, then its Mellin transform M [f ] is defined by Z ∞ dx (9.1) M [f ](s) = xs f (x) , σ > α, x 0 Under the change of variable x = e−t , the Mellin transform and the two-sided Laplace transform shift each other: Z ∞ ± (9.2) L [ϕ](s) = e−st ϕ(t) dt, σ > α, −∞. where we write ϕ(t) = f (e−t ). The ordinary Laplace transform (one-sided Laplace transform) is obtained by multiplying the integrand by the unit step function u = u(t) (cf. the passage immediately after (2.7)): Z ∞ ± e−st f (e−t ) dt, σ > α, (9.3) L[f ](s) = L [f u](s) = 0. Cf. Definition 2 below. If we fix κ > α and write s = κ + jω, G(y) = L± [f ](κ + jω), g(t) = e−κt f (e−t ) in (9.2), then it changes into Z ∞ e−jωt g(t) dt = L± [ϕ](jω), (9.4) G(ω) = F [g](ω) = −∞. the Fourier transform of g. We explain Plancherel’s theorem for functions in L2 (R). Let Z T 1 ˆ (9.5) fT (x) = √ e−ixt f (t) dt. 2π −T Then fˆT (x) is convergent to a function fˆ in L2 : (9.6). ||fˆT − fˆ|| → 0 T → ∞;. l.i.m.T →∞ fˆT (x) = fˆ(t),.

(45) CONTROL SYSTEMS AND NUMBER THEORY. 17. where l.i.m. is a short-hand for “limit in the mean”. The Parseval identity reads Z ∞ Z ∞ 2 ˆ ˆ |f (t)|2 dt. |f (t)| dt = (9.7) ||f ||2 = ||f ||2 , −∞. −∞. If we apply (9.7) to a causal function f , then it leads to [Kim, (3.19)] Z ∞ Z ∞ 2 ± (9.8) |L [f ](iω)| dω = |f (t)|2 dt. −∞. 0. Hence we see that [Kim, (3.19)] is indeed the Parseval identity for the Fourier (or Plancherel) transform for f ∈ L2 (R). 10. Examples of second-order systems • Electrical circuits The electric current i = i(t) flowing an electrical circuit which consists of four ingredients, electromotive-force e = e(t), resistance R, coil L and condenser C satisfies. (10.1). L. d2 i di 1 + R + i = e0 (t). 2 dt dt C. • Newton’s equation of motion(cf. [Gr]). (10.2). M. d2 y dy + R + Ky = e(t) = F, 2 dt dt. where M is the inertance of mass, R is the viscous resistance of the dashpot and K is the spring stiffness. Introducing the new parameters r K ωn = : natural angular frequence M R ζ=q : damping ratio, K 2M (10.2) becomes (10.3). 1 d2 y 2ζ dy 1 + + y = F. 2 2 ωn dt ωn dt K.

(46) 18. FUHUO LI. 11. Laplace transforms To solve (10.1), we use the Laplace transform which has been defined by (9.3) and we state its definition independently. Definition 2. Suppose y(t) = O(eat ), t → ∞ for an a ∈ R. The Laplace transform Y (s) = L[y](s) of y = y(t) is defined by Z ∞ e−st y(t) dt, Re s > a. (11.1) L[y](s) = 0. The integral converges absolutely in Re s > a and represents an analytic function there. Example 7. Let α ∈ C. Then 1 , s−α valid for Re s > Re α in the first instance. The right-hand side of (11.2) gives a meromorphic continuation of the left-hand side to the punctured domain C\{α}. Furthermore, (11.2) with α replaced by iα reads α (11.3) L[sin αt](s) = 2 s + α2 and s (11.4) L[cos αt](s) = 2 . s + α2 For α = ω ∈ R they reduce to familiar formulas: ω L[sin ωt](s) = 2 s + ω2 and s L[cos ωt](s) = 2 . s + ω2 L[eαt ](s) =. (11.2). Proof By definition, (11.2) clearly holds true. Since the right-hand side is analytic in C\{α}, the consistency theorem establishes the last assertion. Once (11.2) is established, we have L[eiαt ](s) =. (11.5). 1 , s − iα. L[e−iαt ](s) =. 1 s + iα. whence e.g. 1 s L[eiαt ](s) + L[e−iαt ](s) = 2 2 s + α2 by Euler’s identity, i.e. (11.4). L[cos αt](s) =.

(47) CONTROL SYSTEMS AND NUMBER THEORY. 19. 12. Partial fraction expansion and examples As long as the input function is a sinusiodal function, Example 7 will suffice to compute its Laplace transform. To go back to the time domain from the frequency domain, we need to solves the DE and for most purposes, the following partial fraction expansion will give the answer almost automatically. The following theorem, which is well-known, provides us with the partial fraction expansion. Theorem 12.1. If the denominator C(z) of the rational function S(z) = P (z) is given by C(z) (12.1). C(z) = c0. q Y. σi. (z − βi ) ,. i=1. q X. σi = deg C = L,. i=1. −1. where βi = σi . Then (12.2). S(z) =. q σi −j X X. ak,σk −j. i=1 j=0. 1 , (z − βi )σi −j. where the coefficients are given by (12.3). ai,σi −j =. dj 1 lim j ((z − βi )σi R(z)) . j! z→βi dz. Proof. By (12.1), for each i, 1 ≤ i ≤ q, we may write S(z) =. Pi (z) , Pi (z) ∈ C(z), (z − βi )σi. and Si (z) has no pole at z = βi . We write ((z − βi )σi S(z) =) Pi (z) =. σ i −1 X. ai,σi −j (z − βi )j + (z − βi )σi Hi (z),. j=0. where Hi (z) ∈ C(z) has no pole at z = βi . By successively differentiating and setting z = βi , we obtain (12.3). Now, the rational function F (z) := S(z) −. q σi −1 X X i=1 j=0. ak,rk −j. 1 (z − βi )σi −j. has no pole, so that it must be a polynomial. But, since limz→∞ F (z) = 0 (where we use the assumption deg P < deg C), it follows that F (z) must be zero. .

(48) 20. FUHUO LI. Now we shall give examples of (2.2) for the second-order systems which do not appear anywhere else save for [Vista]. Example 8. We find the output signal (current) y = y(t) described by the DE √ 3 00 0 − 21 t y +y +y =e sin t, 2 where the initial values are assumed to be 0: y(0) = 0, y 0 (0) = 0. Proof Let Y (s) = L[y](s) be the Laplace transform of y(t). Then we have √. (12.4). Y (s) = Φ(s)U (s),. U (s) = L[u](s) =. 3 2. s2 + s + 1. 2 1 √ L[y](s) = 2 3 (s + s + 1)2 and we may obtain the partial fraction expansion 2 √ − 3√2 3 i i − 13 − 13 1 3 3 + , = + + s−ρ (s2 + s + 1)2 (s − ρ)2 (s − ρ¯)2 s − ρ¯. where ρ = e. 2πi 3. =. √ −1+ 3i 2. is the first primitive cube root of 1. Hence √ √ 2 2 −1 2 −1t 3 3 2 −1t √ y(t) = √ L [L[y]](t) = − te 2 cos t + √ e 2 sin t. 3 2 2 3 3 3 3. As a transfer function, the function in (12.4) Φ(s) =. s2. 1 +s+1. is stable. Example 9. The following integral may be evaluated by the partial fraction expansion above or by the residue calculus. Z ∞ 1 2 4 √ i = √ π. 2 dx = 2πi − 2 3 3 3 3 −∞ (x + x + 1) Example 10. In the same vein as with Example 8, we may find the solution of the DE √ 1 3 00 0 t y − y + y = u(t) = e 2 sin t, 2 where the initial values are assumed to be 0: y(0) = 0, y 0 (0) = 0..

(49) CONTROL SYSTEMS AND NUMBER THEORY. 21. We have √. (12.5) or. Y (s) = Φ1 (s)U (s),. U (s) = L[u](s) =. 3 2. s2 − s + 1. 2 1 √ L[y](s) = . 2 3 (s − s + 1)2. Hence. √ √ √ 1t 1 2 2 −1 3 3 t √ y(t) = √ L [L[y]](t) = −te 2 cos t + 3e 2 sin t. 2 2 3 3 and the transfer function in (12.5) 1 Φ1 (s) = 2 s −s+1 is unstable. 13. The product of zeta-functions:Γ Γ-type In this section, we illustrate the use of fractional integrals by proving a slight generalization of the result of Chandrasekharan and Narasimhan ([ChN1]) involving the ΓΓ-type functional equation, which is the first instance beyond Hecke theory of the functional equation with a single gamma factor. First we state the basic settings. 13.1. Statement of the situation. Let {λk } , {µk } be increasing sequences of positive numbers tending to ∞, and let {αk } , {βk } be complex sequences. We form the Dirichlet series ∞ X αk (13.1) ϕ(s) = , λs k=1 k (13.2). ψ(s) =. ∞ X βk k=1. µsk. and suppose that they have finite abscissas of absolute convergence σϕ , σψ , respectively. We suppose the existence of the meromorphic function χ satisfying the functional equation (of ΓΓ-type) of the form with r a real number, and having a finite number of poles sk (1 ≤ k ≤ L). (13.3) ( Γ s + ν2 Γ s − ν2 ϕ(s), (Re s > σϕ ) χ(s) = Γ r − s + ν2 Γ r − s − ν2 ψ(r − s), (Re s < r − σψ )..

(50) 22. FUHUO LI. We introduce the processing gamma factor (13.4) n Γ {bj + Bj w}m j=1 Γ {aj − Aj w}j=1 ∆(w) = Γ {aj + Aj w}pj=n+1 Γ {bj − Bj w}qj=m+1. (Aj , Bj > 0),. and suppose that for any real numbers u1 , u2 (u1 < u2 ) (13.5). lim ∆(u + iv − s)χ(u + iv) = 0,. |v|→∞. uniformly in u1 ≤ u ≤ u2 . In the w-plane we take two deformed Bromwich paths L1 (s) : γ1 − i∞ → γ1 + i∞,. L2 (s) : γ2 − i∞ → γ2 + i∞ (γ2 < γ1 ). such that they squeeze a compact set S with boundary C for which sk ∈ S (1 ≤ k ≤ L) and all the poles of Γ {bj − Bj s + Bj w}m j=1 Γ {aj − Aj s + Aj w}pj=n+1 Γ {bj + Bj s − Bj w}qj=m+1 lie to the left of L2 (s), and those of Γ {aj + Aj s − Aj w}nj=1 Γ {aj − Aj s + Aj w}pj=n+1 Γ {bj + Bj s − Bj w}qj=m+1 lie to the right of L1 (s). Then we define the H-function by (0 ≤ n ≤ p, 0 ≤ m ≤ q, Aj , Bj > 0) (H-1)

(51)

(52) (1 − a1 , A1 ), . . . , (1 − an , An ), (an+1 , An+1 ), . . . , (ap , Ap ) m,n Hp,q z

(53)

(54) (b1 , B1 ), . . . , (bm , Bm ), (1 − bm+1 , Bm+1 ), . . . , (1 − bq , Bq ) Z 1 Γ(b1 + B1 s, . . . , bm + Bm s)Γ(a1 − A1 s, . . . , an − An s) = z −s ds. 2πi L Γ(an+1 + An+1 s, . . . , ap + Ap s)Γ(bm+1 − Bm+1 s, . . . , bq − Bq s) In the special case where Aj = Bj = 1, the H-function reduces to G-functions and denoted by G with other parameters remaining the same. We also define the χ-function X(z, s) by Z 1 (13.6) X(z, s) = ∆(w − s)χ(w)z −w dw, 2πi L1 (s) which is for χ = 1 one of H-functions. Hereafter we always assume that z > 0, which may be extended to Re z > 0. Then we have (13.7) Z Z 1 1 −w X(z, s) = ∆(w−s)χ(w)z dw+ ∆(w−s)χ(w)z −w dw, 2πi L2 (s) 2πi C.

(55) CONTROL SYSTEMS AND NUMBER THEORY. 23. which amounts to Theorem 13.1. ([MR]) We have the modular relation equivalent to (13.3): (13.8). X(z, s) = P∞ αk s  

(56) !  k=1 λk 

(57) p n   {(1 − a , A )} , {(a , A )}

(58) j j 1 j j n+1 m+2,n   z λk

(59) ×Hp,q+2  ν ν m 

(60) (s + 2 , 1), (s − 2 , 1), {(bj , Bj )}1 , {(1 − bj , Bj )}qm+1  P ∞ βk k=1 µr−s  k

(61) !  

(62) q m  , {(b , B )} {(1 − b , B )}  j j m+1 j j 1 n+2,m µk

(63)  ×Hq,p+2  p ν ν z

(64)

(65)  n  (r − s + , 1), (r − s − , 1), {(a , A )} j j 1 , {(1 − aj , Aj )}n+1  2 2   PL + k=1 Res (∆(w − s)χ(w)z s−w , w = sk ) . P Pq Pp Pm n j=m+1 Bj . j=n+1 Aj + j=1 Bj + 2 ≥ j=1 Aj + In the special case where Aj = Bj = 1, we have Theorem 13.2. (13.9). z s X(z, s)

(66)  !

(67)  P 1 − a , . . . , 1 − a , a , . . . , a

(68)  1 n n+1 p ∞ αk m+2,n  zλk

(69)  k=1 λsk Gp,q+2 ν ν 

(70)  s + , s − , b , . . . , b , 1 − b , . . . , 1 − bq m m+1  2 2 1  P  ∞ βk  k=1 µr−s k

(71) ! =

(72)   1 − b , . . . , 1 − b , b , . . . , b

(73) 1 m m+1 q µk   ×Gn+2,m  q,p+2 ν ν z

(74)

(75)  r − s + , r − s − , a , . . . , a , 1 − a ,  1 n n+1 . . . , 1 − ap 2 2    PL + k=1 Res (∆(w − s)χ(w) z s−w , w = sk ) , (2n + 2m + 2 ≥ p + q) . For many important applications, cf. [MR]..

(76) 24. FUHUO LI. 2,2 4,0 13.2. The Riesz sum: G4,4 ↔ G2,6 . Formula (13.9) in the special case of the title reads. (13.10) ∞ X αk. .

(77)

(78). . a, b, c, d G2,2 zλk

(79)

(80) ν s 4,4 , s − ν2 , e, f s + λ 2 k=1 k

(81) ∞ X βk 4,0 µk

(82)

(83) = G ν µr−s 2,6 z

(84) r − s + 2 , r − k=1. +. L X. 1 − e, 1 − f s − ν2 , 1 − a, 1 − b, 1 − c, 1 − d. . Res ∆(w − s)χ(w)z s−w , w = sk ,. k=1. where (13.11). ∆(w) =. Γ (1 − a − w) Γ (1 − b − w) . Γ (c + w) Γ (d + w) Γ (1 − e − w) Γ (1 − f − w). We treat the case r = 12 . Assuming λ is a non-negative integer, we put a = s + ν2 + λ2 + 21 , b = s + ν2 + λ2 + 1, c = s − ν2 , d = s + ν2 + λ + 1, e = s + ν2 + 21 , f = s + ν2 + λ + 1. Then (13.10) becomes (13.12) ∞ X αk.

(85)

(86) s + ν + λ + 1 , s + ν + λ + 1, s − ν , s + ν + λ + 1

(87) 2 2 2 2 2 2 2 zλk

(88) s + ν2 , s − ν2 , s + ν2 + 12 , s + ν2 + λ + 1

(89) µk

(90)

(91) −s − ν2 + 12 , −s − ν2 − λ 4,0 G2,6 1 ∗ z

(92) µ 2 −s. G2,2 s 4,4 λ k=1 k ∞ X βk =. k=1. +. L X. . Res ∆(w − s)χ(w)z s−w , w = sk ,. k=1. where ∗ indicates −s + ν2 + 21 , −s − ν2 + 21 , −s − ν2 − λ2 + 12 , −s − λ , −s + ν2 + 1, −s − ν2 − λ. 2 We note that the G-functions in (13.12) reduce to

(93)

(94) s + ν + λ + 1 , s + ν + λ + 1, s − ν , s + ν + λ + 1 2,2 2 2 2 2 2 2 2 G4,4 z

(95)

(96) s + ν2 , s − ν2 , s + ν2 + 12 , s + ν2 + λ + 1 = 2λ G1,0 1,1. . 

(97) √  2λ ν

(98) √ 2s + ν + λ + 1 z s+ 2 (1 − z)λ ,

(99) z

(100) = Γ(λ + 1) 2s + ν  0,. ν 2. −. (|z| < 1) (|z| > 1).

(101) CONTROL SYSTEMS AND NUMBER THEORY. (by the formula in [Erd]) and

(102)

(103)

(104) −s − ν + 1 , −s − ν − λ

(105) −s − 4,0 4,0 2 2 2

(106) G2,6 z

(107) = G2,6 z

(108)

(109) ∗∗. ν 2. 25. + 1, −s − †. ν 2. −λ. where ∗∗ indicates −s + ν2 + 12 , −s − ν2 + 12 , −s − ν2 − λ2 + 12 , −s − ν2 − λ , −s + ν2 + 1, −s − ν2 − λ and †, −s + ν2 + 12 , −s + ν2 + 1, −s − ν2 − 2 λ , −s − ν2 − λ2 + 12 , −s + ν2 + 1, −s − ν2 − λ. Hence it reduces further to 2 √ cos((ν + λ + 1)π)K2ν+λ+1 (4 4 z) π √ o √ + cos((ν + 1)π)Y2ν+λ+1 (4 4 z) + sin((ν + 1)π)J2ν+λ+1 (4 4 z) n √ λ 1 2 = −z −s− 2 + 4 (−1)λ cos(νπ)K2ν+λ+1 (4 4 z) π o √ √ 4 4 + cos(νπ)Y2ν+λ+1 (4 z) + sin(νπ)J2ν+λ+1 (4 z) √ λ 1 = z −s− 2 + 4 Gλ2ν+λ+1 (4 4 z), λ. 1. z −s− 2 + 4. n2. say, where, slightly more general than Wilton’s (1.22) [Wi], we put (13.13) Gλν (z). . ν−λ ν−λ π Kν (z) − sin π Yν (z) = − (−1) sin π 2 2 ν−λ + cos π Jν (z). 2 λ2. Hence (13.12) reads (13.14). ν p λ ν z s+ 2 2λ X 2 αk λk 1 − zλk Γ(λ + 1) 1. λk < z. (13.15). =z. − 14 s+ λ 4. ∞ X βk k=1. +. L X. r µk λ 44 λ 1 G2ν+λ+1 + z µ4 4 k. Res ∆(w − s)χ(w)z s−w , w = sk ,. k=1. which gives a more general form of Wilton’s Theorem 1 [Wi]. Rewriting (13.14) slightly, we deduce an analogue of Chandrasekharan and Narasimhan result ([ChN1, Theorem 7.1 (a)]),. ,.

(110) 26. FUHUO LI. Theorem 13.3. For x > 0, the functional equation (13.3) implies the identity X 1 (13.16) αk λνk (x − λk )λ Γ(λ + 1) λ <x k. =x. λ +ν+ 21 2. −λ. 2. ∞ X k=1. βk µk. λ + 12 2. √ Gλ2ν+λ+1 (4 µk x). + Pλ (x), where (13.17). Pλ (x) = x2s+λ+ν. L X. Res ∆(w − s)χ(w)x2(w−s) , w = sk. . k=1. and where (13.18) ∆(w) =. Γ s−. ν 2. Γ 1 − s − λ2 − ν2 − w Γ −s − λ2 − ν2 − w + w Γ s + λ + ν2 + 1 + w Γ 1 − s − ν2 − 12 − w Γ −s − λ −. with Gλ2ν+λ+1 being given by (13.13). Corollary 13.1. For x > 0, the functional equation (13.3) implies the identity X 1 (13.19) αk (x − λk )λ Aλ (x) := Γ(λ + 1) λ <x k. −λ. = −2. ∞ X k=1. βk. x µk. λ2 + 12. √ Fλ+1 (4 µk x) + Pλ (x),. where 2 Fλ+1 (z) = −Gλλ+1 (z) = Yλ+1 (z) + (−1)λ Kλ+1 (z). π We are now in a position to prove an analogue of [ChN1, Theorem 7.1 (b)] (although Theorem 13.3 contains [ChN1, Theorem 7.2], too) by the Riemann-Liouville fractional integral transform.. (13.20). Lemma 13.1. (Riemann-Liouville integral of Bessel functions) For the well-known Bessel functions J and Y , we have Z y 1 1 1 1 1 1 (13.21) (y − x)µ−1 x 2 ν Jν (ax 2 ) dx = 2µ a−µ y 2 µ+ 2 ν Jµ+ν (ay 2 ) Γ(µ) 0 (Re µ > 0, Re ν > 0).. ν 2. −w. ,.

(111) CONTROL SYSTEMS AND NUMBER THEORY. 27. (13.22) Z y 1 1 1 (y − x)µ−1 x 2 ν Yν (ax 2 ) dx Γ(µ) 0 1 1 1 1 Γ(ν + 1) ν+2 −µ 1 µ+ 1 ν 2 a y 2 2 Sµ−ν−1,µ+ν (ay 2 ) = 2µ a−µ y 2 µ+ 2 ν Jµ+ν (ay 2 ) + Γ(µ)π (Re µ > 0, Re ν > −1), where S stands for the Lommel function. (13.21) is [Erd1, (63),p.194] and (13.22) is [Erd1, p.196]. We only need (13.22) and (13.22) is for treating the J-Bessel function. Arguing in the same way as in [ChN1], we may prove Theorem 13.4. With a C ∞ -function Rρ and a certain constant c we have ρ2 + 12 ∞ X x √ βk Aρ (x) − Rρ (x) = c (13.23) Yρ+1 (4 µk x) , µk k=1 for integral λ, ρ = λ + α, 0 < α < 1, λ ≥ 2σψ − 23 . Table 1. Correspondence between control systems and zeta-functions. system functions action region of convergence critical line S rational symplectic σ>0 σ=1 ζ meromorphic modular σ>1 σ = 21 References [Bel] R. Bellman, Wigert’s approximate functional equation and the Riemann zetafunction, Duke Math. J. 16 (1949), 547-552. [Cao1] J.-Y. Cao and B.-G. Cao, Design of fractional order controller based on particle swarm optimization, Intern. J. Control, Automation, and Systems 4 (2006), 775-781. [ChN1] K. Chandrasekharan and R. Narasimhan, Functional equations with multiple gamma factors and the average order of arithmetical functions, Ann. Math. (2) 76 (1962), 93-136. [Erd] A. Erdélyi, W. Magnus, F. Oberhettinger and F. G. Tricomi, Higher transcendental functions, Vols 1-3, McGraw-Hill. New York 1953. [Erd1] A. Erdélyi, W. Magnus, F. Oberhettinger and F. G. Tricomi, Tables of integral transforms, Vols 1-3, McGraw-Hill. New York 1953. [Gr] F. S. Grodins, Control theory and biological systems, Columbia Univ. Press, New York and London 1963. [Hay] Y. Hayakawa, Systems and their controlling, Ohmsha, Tokyo 2008 (in Japanese). [Hel1] J. W. Helton, The distance of a function to H ∞ in the Poincaré metric; elctrical power transfer, J. Funct. Anal. 38 (1980), 273-314. [Hel2] J. W. Helton, Non-Euclidean funcitional analysis and electronics, Bull. Amer. Math. Soc. 7 (1982), 1-64..

(112) 28. FUHUO LI. [HM] J. W. Helton and O. Merino, Classical control using H ∞ methods, Theory, optimization and design, SIAM, Philadelphia 1998. [Vista] S. Kanemitsu and H. Tsukada, Vistas of special functions, World Scientific, Singapore etc. 2007. [MR] S. Kanemitsu and H. Tsukada, Contributons to the theory of zeta-functions The modular relation supremacy, World Scientific, Singapore etc. 2012. [Kim] H. Kimura, Chain scattering approach to H ∞ -control, Birkhäuser, BostonBasel-Berlin 1997. [Cao2] M. Ye, Z.-F.Bai and B.-G.Cao, Robust H2/Hinfinity control for regenerative braking of electric vehicles, ICCA 2007. IEEE International Conference on Control and Automation, 2007, 1366 - 1370. [Pod1] I. Podlubny, Fractional-order systems and P I λ Dδ controllers, IEEE Trans. Autom. control, 44, No.1 (1999), 208-213. [Pod2] I. Podlubny, Geometric and physical interpretation of fractional integration and fraction differentiation, Fractional calculus and applied analysis, 5, No. 4 (2002), 367-386. [Sie] C. L. Siegel, Symplectic geometry, Academic Press, New York 1964. [Tit] E. C. Titchmarsh, The theory of functions, 2nd ed. Oxford UP, Oxford 1937. [Wi] J. R. Wilton, An extended form of Dirichlet’s divisor problem, Proc. London Math. Soc. (2) 36 (1934), 391-426. Fuhuo Li,Director of Sanmenxia SuDa Communication Group, NO.1 Taiyang Road, Sanmenxia Economical Development Zone, Sanmenxia, Henan Province,P.R.China, 472000 E-mail address: [email protected].

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