Control Systems and Number Theory

Abstract

We try to pave a smooth road to a proper understanding of control problems in terms of mathematical disciplines, and partially show how to number-theorize some practical problems. Our primary concern 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 of Kimura 1997, 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. Both of 𝐻^∞- and PID-controllers are applied successfully in the EV control by J.-Y. Cao and B.-G. Cao 2006 and Cao et al. 2007, 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 zeta-functions, 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, compare Section 11.

Section 5 introduces the Hardy space 𝐻_𝑝 which consists of functions analytic in ℛℋ𝒫—right half-plane 𝜎>0.

2. State Space Representation and the Visualization Principle

Let 𝐱=𝐱(𝑡)∈ℝ^𝑛, 𝐮=𝐮(𝑡)∈ℝ^𝑟, and 𝐲=𝐲(𝑡)∈ℝ^𝑚 be the state function, input function, and output function, respectively. We write 𝐱̇ for (𝑑/𝑑𝑡)𝐱. The system of (differential equations) DEs𝐱𝐲̇==𝐴𝐶𝐱𝐱++𝐵𝐷𝐮𝐮,(2.1) is called a state equation for a linear system, where 𝐴∈𝑀_𝑛,𝑛(ℝ), 𝐵,𝐶,𝐷 are given constant matrices.

The state 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 read𝐲=𝐷𝐮,(2.2) which appears in many places in the literature in disguised form. All the subsequent systems, for example, (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 (compare Section 4).

Using the matrix exponential function 𝑒^𝐴𝑡, the first equation in (2.1) can be solved in the same way as for the scalar case:𝐱=𝐱(𝑡)=𝑒^𝐴𝑡𝐱(0)+𝐵𝑒^𝐴𝑡^𝑡0𝑒^−𝐴𝜏𝐮(𝑡)d𝜏.(2.3)

Definition 2.1. A linear system with the input 𝐮=𝐨𝐱̇=𝑑𝑑𝑡𝐱=𝐴𝐱,(2.4) called an autonomous system, is said to be asymptotically stable if for all initial values, 𝐱(𝑡) approaches a limit as 𝑡→∞.

Since the solution of (2.4) is given by𝐱=𝑒^𝐴𝑡𝐱(0),(2.5) the system is asymptotically stable if and only if||||||𝑒_𝐴𝑡||||||⟶0as𝑡⟶∞.(2.6)

A linear system is said to be stable if (2.6) holds, which is the case if all the eigenvalues of 𝐴have negative real parts. Compare Section 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 response𝐲(𝑡)=^𝑡0𝑒^{𝐴(𝑡−𝜏)}𝑢(𝜏)𝑑𝜏,(2.7) 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 Section 11. It has the effect of shifting from the time domain to frequency domain and vice versa. For more details, see, for example, [1]. Taking the Laplace transform of (2.1) with 𝐱(0)=𝐨, we obtain 𝑠𝑌𝑋((𝑠𝑠))==𝐶𝐴𝑋𝑋(𝑠(𝑠))++𝐷𝐵𝑈𝑈((𝑠𝑠)),,(2.8) which we solve as𝑌(𝑠)=𝐺(𝑠)𝑈(𝑠),(2.9) where𝐺(𝑠)=𝐶(𝑠𝐼−𝐴)⁻¹𝐵+𝐷,(2.10) where 𝐼indicates the identity matrix, which is sometimes denoted by 𝐼_𝑛to 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), and so forth. 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.

Equation (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 as𝐺(𝑠)=⎛⎜⎜⎝𝐴𝐵𝐶𝐷⎞⎟⎟⎠.(2.11)

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 2.2. If 𝐺(𝑠)=⎛⎜⎜⎝𝐴𝐵𝐶𝐷⎞⎟⎟⎠=⎛⎜⎜⎜⎜⎝01−2−301−10−22⎞⎟⎟⎟⎟⎠,(2.12) then it follows from (2.10) that 𝐺(𝑠)=−10,−2⎛⎜⎜⎝⎛⎜⎜⎝0𝑠0𝑠⎞⎟⎟⎠−⎛⎜⎜⎝−02−13⎞⎟⎟⎠⎞⎟⎟⎠⁻¹=−10,−2⎛⎜⎜⎝2𝑠𝑠−+13⎞⎟⎟⎠₋₁⎛⎜⎜⎝01⎞⎟⎟⎠+2=1⎛⎜⎜⎝01⎞⎟⎟⎠+2(𝑠+1)(𝑠+2)−10,−2⎛⎜⎜⎝𝑠−+231𝑠⎞⎟⎟⎠⁻¹=−2𝑠+5⎛⎜⎜⎝01⎞⎟⎟⎠+2(𝑠+1)(𝑠+2)+2=2((𝑠𝑠++13))((𝑠𝑠+−21)).(2.13)

The principle above will establish the most important cascade connection (concatenation rule) [1, (2.13), page 15]. Given two state space representations𝐺_𝑘(𝑠)=⎛⎜⎜⎝𝐴^𝑘𝐵_𝑘𝐶_𝑘𝐷_𝑘⎞⎟⎟⎠,𝑘=1,2,(2.14) their cascade connection 𝐺(𝑠)=𝐺₁(𝑠)𝐺₂(𝑠) is given by𝐺(𝑠)=𝐺₁(𝑠)𝐺₂(𝑠)=⎛⎜⎜⎜⎜⎝𝐴¹𝐵₁𝐶₂𝐵₁𝐷₂𝑂𝐴₂𝐵₂𝐶₁𝐷₁𝐶₂𝐷₁𝐷₂⎞⎟⎟⎟⎟⎠.(2.15)

Proof of (2.15). We have the input/output relation (2.10) 𝑌(𝑠)=𝐺₁(𝑠)𝑈(𝑠),𝑈(𝑠)=𝐺₂(𝑠)𝑉(𝑠),(2.16) which means that 𝐲𝐱̇==𝐶𝐴₁₁𝐱+𝐵₁𝐮,𝝃̇=𝐴₂𝐱+𝐷₁𝐮,(2.17)𝐮=𝐶₂𝝃+𝐵₂𝐯,𝝃+𝐷₂𝐯.(2.18)
Eliminating 𝐮, we conclude that𝐲𝐱̇==𝐶𝐴₁₁𝐱+𝐵₁𝐶₂𝜉+𝐵₁𝐷₂𝐯,𝐱+𝐷₁𝐶₂𝜉+𝐷₁𝐷₂𝐯.(2.19) Hence ⎛⎜⎜⎝𝝃𝐱̇̇⎞⎟⎟⎠=⎛⎜⎜⎝𝐴₁𝐵₁𝐶₂𝑂𝐴₂⎞⎟⎟⎠⎛⎜⎜⎝𝝃𝐱⎞⎟⎟⎠+⎛⎜⎜⎝𝐵₁𝐷₂𝐵₂𝐲=𝐶₁⎞⎟⎟⎠𝐯,𝐷₁𝐶₂⎛⎜⎜⎝𝝃𝐱⎞⎟⎟⎠+𝐷¹𝐷₂𝐯,(2.20) whence we conclude (2.15).

Example 2.3. Given two state space representations (2.14), their parallel connection 𝐺(𝑠)=𝐺₁(𝑠)+𝐺₂(𝑠) is given by 𝐺(𝑠)=𝐺₁(𝑠)+𝐺₂(𝑠)=⎛⎜⎜⎜⎜⎝𝐴¹𝑂𝐵₁𝑂𝐴₂𝐵₂𝐶₁𝐶₂𝐷₁+𝐷₂⎞⎟⎟⎟⎟⎠.(2.21)
Indeed, we have (2.17), and for (2.18), we have 𝐲+𝝃̇=𝐳=𝐴₂𝐶𝝃₂+𝐵₂𝐮,𝝃+𝐷₂𝐮.(2.22)
Hence for (2.20), we have(𝐱+𝐲𝝃=)^⋅𝐶=₁𝐴₁𝐱+𝐴₂𝜉+𝐵₁+𝐵₂𝐮,𝐱+𝐶₂𝜉+𝐷₁+𝐷₂𝐯,(2.23) whence (2.21) follows.

As an example, combining (2.15) and (2.21) we deduce𝐼−𝐺₁(𝑠)𝐺₂(𝑠)=⎛⎜⎜⎜⎜⎜⎜⎜⎜⎝𝐼𝑂−𝐴₁𝑂−𝐵₁𝐶₂−𝐵₁𝐷₂𝑂𝑂−𝐴₂−𝐵₂𝑂−𝐶₁−𝐷₁𝐶₂𝑉⁻¹⎞⎟⎟⎟⎟⎟⎟⎟⎟⎠.(2.24)

Example 2.4. For (2.1), we consider the inversion𝑈(𝑠)=𝐺⁻¹(𝑠)𝑌(𝑠). Solving the second equality in (2.1) for 𝐮  we obtain 𝐮=−𝐷⁻¹𝐶𝐱+𝐷⁻¹𝐲.(2.25)
Substituting this in the first equality in (2.1), we obtain𝐱̇=𝐴−𝐵𝐷⁻¹𝐶𝐱+𝐵𝐷⁻¹𝐲.(2.26) Whence 𝐺⁻¹(𝑠)=⎛⎜⎜⎝𝐴−𝐵𝐷⁻¹𝐶−𝐵𝐷⁻¹−𝐷⁻¹𝐶𝐷⁻¹⎞⎟⎟⎠.(2.27)

Example 2.5. If the transfer function Θ(𝑠)=⎛⎜⎜⎝Θ¹¹Θ₁₂Θ₂₁Θ₂₂⎞⎟⎟⎠(2.28) has a state space representation Θ(𝑠)=⎛⎜⎜⎝𝐴𝐵𝐶𝐷⎞⎟⎟⎠=⎛⎜⎜⎜⎜⎝𝐴𝐵₁𝐵₂𝐶₁𝐷₁₁𝐷₁₂𝐶₂𝐷₂₁𝐷₂₂⎞⎟⎟⎟⎟⎠,(2.29) then we are to embed it in the linear system 𝐱̇=𝐴𝐱+𝐵₁𝐵₂⎛⎜⎜⎝𝐛¹𝐛₂⎛⎜⎜⎝𝐚_𝟏⎞⎟⎟⎠,𝐚_𝟐⎞⎟⎟⎠=𝐲=⎛⎜⎜⎝𝐶₁𝐶₂⎞⎟⎟⎠𝐱+⎛⎜⎜⎝𝐷₁₁𝐷₁₂𝐷₂₁𝐷₂₂⎞⎟⎟⎠⎛⎜⎜⎝𝐛₁𝐛₂⎞⎟⎟⎠.(2.30)

3. Chain-Scattering Representation

Following [1, pages 7 and 67], we first give the definition of a chain-scattering representation of a system.

Suppose 𝐚₁∈ℝ^𝑚, 𝐚₂∈ℝ^𝑞, 𝐛₁∈ℝ^𝑟, and 𝐛₂∈ℝ^𝑝 are related by⎛⎜⎜⎝𝐚₁𝐚₂⎞⎟⎟⎠=𝑃⎛⎜⎜⎝𝐛₁𝐛₂⎞⎟⎟⎠,(3.1) where𝑃=⎛⎜⎜⎝𝑃¹¹𝑃₁₂𝑃₂₁𝑃₂₂⎞⎟⎟⎠.(3.2)

According to the embedding principle, this is to be thought of as 𝐲=𝑆𝐮 corresponding to the second equality in (2.1).

Equation (3.1) means that𝐚₁=𝑃₁₁𝐛₁+𝑃₁₂𝐛₂,𝐚₂=𝑃₂₁𝐛₁+𝑃₂₂𝐛₂.(3.3)

Assume that 𝑃₂₁ is a (square) regular matrix (whence 𝑞=𝑟). Then from the second equality of (3.3), we obtain𝐛₁=𝑃⁻¹²¹𝐚₂−𝑃₂₂𝐛₂=−𝑃⁻¹²¹𝑃₂₂𝐛₂+𝑃⁻¹²¹𝐚₂.(3.4) Substituting (3.4) in the first equality of (3.3), we deduce that𝐚₁=𝑃₁₂−𝑃₁₁𝑃⁻¹²¹𝑃₂₂𝐛₂+𝑃₁₁𝑃⁻¹²¹𝐚₂.(3.5)

Hence puttingΘ=CHAIN(𝑃)=⎛⎜⎜⎜⎝𝑃¹²−𝑃₁₁𝑃⁻¹²¹𝑃₂₂𝑃₁₁𝑃⁻¹²¹−𝑃⁻¹²¹𝑃₂₂𝑃⁻¹²¹⎞⎟⎟⎟⎠=⎛⎜⎜⎝Θ₁₁Θ₁₂Θ₂₁Θ₂₂⎞⎟⎟⎠,(3.6) which is usually referred to as a chain-scattering representation of 𝑃, we obtain an equivalent form of (3.1)⎛⎜⎜⎝𝐚₁𝐛₁⎞⎟⎟⎠=CHAIN(𝑃)⎛⎜⎜⎝𝐛₂𝐚₂⎞⎟⎟⎠=⎛⎜⎜⎝Θ₁₁Θ₁₂Θ₂₁Θ₂₂⎞⎟⎟⎠⎛⎜⎜⎝𝐚₂𝐛₂⎞⎟⎟⎠.(3.7)

Suppose that 𝐚₂ is fed back to 𝐛₂ by𝐛₂=𝑆𝐚₂,(3.8) where 𝑆 is a controller. Multiplying the second equality in (3.3) by 𝑆 and incorporating (3.8), we find that𝐛₂=𝑆𝐚₂=𝑆𝑃₂₁𝐛₁+𝑆𝑃₂₂𝐛₂,(3.9)

whence 𝐛₂=(𝐼−𝑃₂₂𝐾)⁻¹𝑆𝑃₂₁𝐛₁.

Let the closed-loop transfer function Φ be defined by𝐚₁=Φ𝐛₁.(3.10)

Φ is given byΦ=𝑃₁₁+𝑃₁₂𝐸−𝑃₂₂𝑆⁻¹𝑆𝑃₂₁.(3.11) Equation (3.11) is sometimes referred to as a linear fractional transformation and denoted by𝐿𝐹(𝑃;𝐾).(3.12) Substituting (3.8), (3.7) becomes⎛⎜⎜⎝𝐚₁𝐛₁⎞⎟⎟⎠=⎛⎜⎜⎝Θ₁₁𝑆+Θ₁₂Θ₂₁𝑆+Θ₂₂⎞⎟⎟⎠𝐚₂,(3.13)

whence we deduce thatΦ=Θ₁₁𝑆+Θ₁₂Θ₂₁𝑆+Θ₂₂⁻¹=Θ𝑆,(3.14) 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 nonconstant condition |Θ|≠0. Then Θ∈𝐺𝐿_𝑚+𝑟(ℝ).

If 𝑆 is obtained from 𝑆′ under the action of Θ′, 𝑆=Θ′𝑆′, then its composition 𝐽 with (3.14) yields 𝐽𝑆′=ΦΦ′=ΘΘ′𝑆′, that is,𝐽=ΘΘ′,HMΘ;HMΘ^;𝑆=HMΘΘ^;𝑆,(3.15) which is referred to as the cascade connection or the cascade structure of Θ and Θ′.

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∈ℝ^𝑚, 𝐲=𝐚₂∈ℝ^𝑞, 𝐰=𝐛₁∈ℝ^𝑟, and 𝐮=𝐛₂∈ℝ^𝑝 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‖Φ‖_∞<𝛾,(3.16) for a positive constant 𝛾. For the meaning of the norm, compare Section 5.

4. Siegel Upper Space

Let ∗ denote the conjugate transpose of a square matrix: 𝑆^∗=^𝑡𝑆, and let the imaginary part of 𝑆 defined by Im𝑆=(1/2𝑗)(𝑆−𝑆^∗). Let ℋ_𝑛 be the Siegel upper half-space consisting of all the matrices 𝑆 (recall (3.8)) whose imaginary parts are positive definite (Im𝑆>0—imaginary parts of all eigen values are positive) and satisfies 𝑆=^𝑡𝑆:ℋ_𝑛=𝑆∈M_𝑛(ℂ)∣Im𝑆>0,𝑆=^𝑡𝑆,(4.1) and let Sp(𝑛,ℝ) denote the symplectic group of order 𝑛:Sp(𝑛,ℝ)=⎧⎪⎨⎪⎩Θ=⎛⎜⎜⎝Θ¹¹Θ₁₂Θ₂₁Θ₂₂⎞⎟⎟⎠∣⎛⎜⎜⎝Θ₁₁Θ₁₂Θ₂₁Θ₂₂⎞⎟⎟⎠⁻¹=⎛⎜⎜⎝Θ²²−^𝑡Θ₁₂−^𝑡Θ₂₁Θ₁₁⎞⎟⎟⎠⎫⎪⎬⎪⎭.(4.2)

The action of Sp(𝑛,ℝ) on ℋ_𝑛 is defined by (3.14) which we restate asΘ𝑆=Θ₁₁𝑆+Θ₁₂Θ₂₁𝑆+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, that is, 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 (3.15), which is then viewed as the action (3.15) of Θ∈Sp(𝑛,ℝ) on 𝑆∈ℋ_𝑛: ΘΘ^𝑆=ΘΘ^𝑆;𝑜𝑟HMΘ;HMΘ^;𝑆=HMΘΘ^;𝑆,(4.5) where Θ is subject to the condition ^𝑡Θ𝑈Θ=𝑈,(4.6) with 𝑈=^𝑂𝐼𝑛_{−𝐼𝑛𝑂}. An FOPID controller (in Section 6), being a unity feedback connection, is also accommodated in this framework.

Remark 4.2. With action, we may introduce the orbit decomposition of ℋ_𝑛 and whence the fundamental domain. We note that, in the special case of 𝑛=1, we have ℋ₁=ℋ and Sp(1,ℝ)=SL_𝑛(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 [2], 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 Section 7 for physically realizable cases. There are many research problems lying in this direction.

5. Norm of the Function Spaces

The norm 𝐱=^𝑥1⋮𝑥_𝑛∈ℂ^𝑛 is defined to be the Euclidean norm‖𝐱‖=‖𝐱‖₂=⎷^𝑛_𝑗=1||𝑥_𝑗||²,(5.1) or by the sup norm ‖𝐱‖=‖𝐱‖_∞=max||𝑥₁||,….||𝑥_𝑛||,(5.2) or anything that satisfies the axioms of the norm. They introduce the same topology on ℂ^𝑛.

The definition of the norm of a matrix should be given in a similar way by viewing its elements as an 𝑛²-dimensional vector, that is, embedding it in ℂ^𝑛2. If 𝐴=(𝑎_𝑖𝑗),1≤𝑖,𝑗≤𝑛, then‖𝐴‖=‖𝐴‖₂=⎷^𝑛_𝑖_,𝑗=1||𝑎_𝑖𝑗||²,(5.3) or otherwise.

The sup norm is a limit of the 𝑝-norm as 𝑝→∞. For 𝐚=(𝑎₁,…,𝑎_𝑛), lim_𝑝→∞‖𝐚‖_𝑝=lim_𝑝→∞^𝑛_𝑘=1||𝑎_𝑘||^𝑝^1/𝑝=‖𝐚‖_∞=max_{1≤𝑘≤𝑛}||𝑎_𝑘||^𝑝.(5.4)

Suppose  |𝑎₁|=max_{1≤𝑘≤𝑛}{|𝑎_𝑘|^𝑝}. Then for any 𝑝>0|𝑎₁|=(|𝑎₁|^𝑝)^1/𝑝≤(∑^𝑛𝑘=1|𝑎_𝑘|^𝑝)^1/𝑝.

On the other hand, since |𝑎₁|≥|𝑎_𝑘|,1≤𝑘≤𝑛, we obtain^𝑛_𝑘=1||𝑎_𝑘||^𝑝^1/𝑝=||𝑎₁||1+^𝑛_𝑘=2||||𝑎_𝑘𝑎₁||||^𝑝^1/𝑝≤||𝑎₁||(1+𝑛−1)^1/𝑝.(5.5)

For 𝑝>1, the Bernoulli inequality gives (1+𝑛−1)^1/𝑝≤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 givelim_𝑝→∞‖𝑓‖_𝑝=‖𝑓‖_∞=sup_𝑡≥0||𝑓(𝑡)||.(5.6)

The 𝑝-norm in (5.6) is defined by‖𝑓‖_𝑝=^∞0‖𝑓(𝑡)‖^𝑝𝑑𝑡^1/𝑝,(5.7)

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. 𝐿^𝑝 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⟨𝑓,𝑔⟩=^∞0𝑓^∗(𝑡)𝑔(𝑡)𝑑𝑡,‖𝑓‖₂=√⟨𝑓,𝑓⟩,(5.8) where ∗ refers to the transposed complex conjugation.

The Parseval identity holds true if and only if the system is complete.

However, the restriction that ‖𝑓(𝑡)‖→0 as 𝑡→∞ excludes signals of infinite duration such as unit step signals or periodic ones from 𝐿_𝑝. To circumvent the inconvenience, the notion of averaged norm, 𝑀₂(𝑓)=𝑀₂(𝑓,𝑇)=(1/𝑇)∫^𝑇0‖𝑓(𝑡)‖²𝑑𝑡 or similar, is important and the power norm has been introduced: power(𝑓)=lim_𝑇→∞𝑀₂(𝑓,𝑇)^1/2=lim_𝑇→∞𝑇1^𝑇0‖𝑓(𝑡)‖²𝑑𝑡^1/2.(5.9)

Remark 5.1. In mathematics and in particular in analytic number theory, studying 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 5.2. 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=1/2. Since the values on the critical line are expected to be small, the averaged norm 𝑀₂(𝜁) or 𝑀₄(𝜁), that is, the mean value (1/𝑇)∫^𝑇0|𝜁((1/2)+𝑖𝑡)|^2𝑘𝑑𝑡 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 equation for the Riemann zeta-function to obtain 𝑀₄(𝜁)=1𝑇^𝑇0|||𝜁12+𝑖𝑡|||⁴𝑑𝑡=14𝜋²log⁴𝑇(1+𝑜(1)),(5.10) for 𝑇→∞. See, for example, [3]. The main interest in such estimates as (5.10) lies in the fact that estimates for all 𝑘∈ℕ𝑀_2𝑘(𝜁)=1𝑇^𝑇0|||𝜁12+𝑖𝑡|||^2𝑘𝑑𝑡=𝑂(𝑇^𝑎)(5.11) would imply the weak Lindelöf hypothesis (LH) in the form 𝜁12+𝑖𝑡=𝑂𝑇^{(𝑎/2𝑘)+𝜀},(5.12) for every 𝜀>0. It is apparent that the RH implies the LH.

The Hardy space 𝐻_𝑝 (cf. e.g., [1, page 39]) is well known. It consists of all 𝑓(𝑠) which are analytic in ℛℋ𝒫—right half-plane 𝜎>0 such that 𝑓(𝑗𝜔)∈𝐿_𝑝, 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). Since the 𝐻^∞-control problem asks for all individual values, it flows afar from the 𝐻^2𝑘-control problem and goes up to the LH or the RH.

6. (Unity) Feedback System

The synthesis problem of a controller of the unity feedback system, depicted in  Figure 1, 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):𝑆=(𝐼+𝑃𝐶)⁻¹(6.1) 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‖𝑊𝑆‖_∞<𝛾(6.2) for a positive constant 𝛾.

Figure 1 

Unity feedback system.

To accommodate this in the 𝐻^∞ control problem (3.1), we choose the matrix elements 𝑃_𝑖𝑗 of 𝑃 in such a way that the closed-loop transfer function Φ in (3.11) coincides with 𝑊𝑆. First we are to choose 𝑃₂₂=−𝑃. Then we would choose 𝑃₁₂𝑃₂₁=𝑊𝑃. Then Φ becomes 𝑃₁₁+𝑊𝑃𝐶(𝐼+𝑃𝐶)⁻¹=𝑃₁₁−𝑊+𝑊(𝐼+𝑃𝐶)⁻¹. Hence choosing 𝑃₁₁=𝑊, we have Φ=𝑊𝑆. Hence we may choose, for example,𝑃=⎛⎜⎜⎝𝑃¹¹𝑃₁₂𝑃₂₁𝑃₂₂⎞⎟⎟⎠=⎛⎜⎜⎝𝑊𝐸𝑊−𝑃𝑃⎞⎟⎟⎠.(6.3)

Example 6.1. First we treat the case of general feedback scheme. Denoting the Laplace transforms by the corresponding capital letters, we have 𝑌=𝑃𝑅+𝑃𝑈,𝑈=𝐾𝐸,(6.4) whence 𝑌=𝑃𝑅+𝑃𝐾𝐸. Now if it so happens that 𝑒=𝑟−𝑦 and 𝑃 is replaced by 𝑃𝐾, that is, in the case of unity FD, we derive (6.1) directly from  Figure 2. We have 𝐸=𝑅−𝑌, so that 𝑌=𝑃𝑅+𝑃𝐾(𝑅−𝑌). Solving in 𝑌, we deduce that (𝐼+𝑃𝐾)⁻¹𝑃𝐾𝑅.
We take into account the disturbance 𝑑,  and  we obtain since 𝑈=𝐶𝐸=𝐶(𝑅−𝑌)𝑌=𝑃𝑈+𝑃𝐷=𝑃𝐶(𝑅−𝑌)+𝑃𝐷,(6.5) whence 𝑌=𝑃𝐶(𝑅−𝑌)+𝑃𝐷. It follows that 𝑌=(𝐼+𝑃𝐶)⁻¹𝑃𝐶𝑅+(𝐼+𝑃𝐶)⁻¹𝑃𝐷. In the case where 𝑑=0, 𝑃𝐶 being the open-loop transfer function, we have 𝑆𝑅 is the tracking error for the input 𝑅. Hence (6.1) holds true.

Figure 2 

7. 𝐽-Lossless Factorization and Dualization

In this section we mostly follow Helton ([4–6]) who uses the unit ball in place 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 ([5, page 2]), (Table 1)ℛ={funtchteiornastiodneafilnceodnotinnuthaetiounnittobtahllehwahvoinlegspace},(7.1) or still larger algebra 𝜓 consisting of those functions which have (pseudo)meromorphic continuations ([5, footnote 6, page 27]). The occurrence of the gamma function [5, Figure  2.5, page 17] justifies our incorporation of more advanced special functions and ultimately zeta-functions in control theory (see Section 13).

Table 1 

Correspondence between control systems and zeta-functions.

Along with the algebra ℛ, one considersℬ𝐻^∞={𝐹∣analyticontheunitballhavingthesupremumnorm<1}.(7.2)

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.

ThenΘ^∗𝐽_𝑚𝑛Θ≤𝐽_𝑚𝑛,(7.3) which is interpreted to be the power preservation of the system in the chain-scattering representation (3.6) ([1, page 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 byDCHAIN(𝑃)=⎛⎜⎜⎜⎝𝑃⁻¹¹²𝑃₁₁𝑃⁻¹¹²−𝑃⁻¹¹²𝑃₂₂𝑃₂₁−𝑃₂₂𝑃⁻¹¹²𝑃₁₁⎞⎟⎟⎟⎠,(7.4) we obtainCHAIN(𝑃)⋅DCHAIN(𝑃)=𝐸.(7.5)

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 𝐼^𝜆𝑡𝐷^𝑡−𝜆 of 𝑒(𝑡) (𝜆>0), and “Differential” the fractional order differentiation 𝐷^𝛿𝑡 of 𝑒(𝑡) (𝛿>0).

The FO 𝑃𝐼^𝜆𝐷^𝛿 controller (control signal in the time domain) is one of the most refined feed-forward compensators defined by𝑢(𝑡)=𝐾_𝑝+𝐾_𝑖𝐷^𝑡−𝜆+𝐾_𝑑𝐷^𝛿𝑡𝑒(𝑡),(8.1) where 𝑢 is the input function, 𝑒 is the deviation, and 𝐾_𝑝,𝐾_𝑖,𝐾_𝑑 are constant parameters which are to be specified (𝐾_𝑝: the position feedback gain, 𝐾_𝑑: the velocity feedback gain). DE (8.1) translates into the state equation 𝑌(𝑠)=𝐶(𝑠)𝐸(𝑠),(8.2) where 𝑈,𝑌 indicate the Laplace transforms of 𝑢,𝑦, respectively, and 𝐺 is the compensators continuous transfer function 𝐶(𝑠)=𝐾_𝑝+𝐾_𝑖𝑠^−𝜆+𝐾_𝑑𝑠^𝛿.(8.3)

The derivation of (8.3) from (8.1) depends on the following. The general fractional calculus operator_𝑎𝐷^𝛼𝑡 is symbolically stated as_𝑎𝐷^𝛼𝑡=⎧⎪⎪⎨⎪⎪⎩𝑑^𝛼𝑑𝑡^𝛼1,,RRee𝛼𝛼=>00,_𝑡𝑎,1𝑑𝑡^𝛼,Re𝛼<0,(8.4) 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 differintegral is given by the Riemann-Liouville expression_𝑎𝐷^𝛼𝑡𝑓(𝑡)=1Γ(1−{𝛼})𝑑𝑑𝑡^{𝛼−{𝛼}+1}^𝑡𝑎(𝑡−𝜏)^−{𝛼}𝑓(𝜏)𝑑𝜏,(8.5) 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:ℛ𝐿[𝑓]=1Γ(𝜇)^𝑦0(𝑦−𝑥)^𝜇−1𝑓(𝑥)𝑑𝑥.(8.6)

For applications, compare Section 13.

When 𝛼∈ℕ, (8.5) reads_𝑎𝐷^𝛼𝑡𝑓(𝑡)=𝑑𝑑𝑡^𝛼+1^𝑡𝑎𝑓(𝜏)𝑑𝜏=𝑓^(𝛼)(𝑡),(8.7) the 𝛼th derivative of 𝑓.

We will 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:Δ^𝛼𝑦𝑓(𝑥)=^𝛼_𝜈=0(−1)^𝛼−𝜈⎛⎜⎜⎝𝛼𝜈⎞⎟⎟⎠𝑓(𝑥+𝜈𝑦).(8.8)

If 𝑓 has the 𝛼-th derivative 𝑓^(𝛼), thenΔ^𝛼𝑦𝑓(𝑥)=^{𝑥𝑥+𝑦}𝑑𝑡₁^{𝑡1𝑡+𝑦}₁𝑑𝑡₂⋯^{𝑡𝛼−1𝑡+𝑦}_𝛼−1𝑓^(𝛼)𝑡_𝛼𝑑𝑡_𝛼.(8.9)

The special case of (8.9) with 𝑡_𝜈=𝑎,𝑎+𝑦→𝑥(𝜑(𝑡)=𝑓^(𝛼)(𝑡)) readsΔ^{𝛼𝑥−𝑎}𝜑(𝑥)=^𝑥𝑎d𝑡^𝑥𝑎d𝑡⋯^𝑥𝑎𝜑(𝑡)𝑑𝑡=1Γ(𝛼)^𝑥𝑎(𝑥−𝑡)^𝛼−1𝜑(𝑡)𝑑𝑡,(8.10) whose far-right hand side is ℛ𝐿[𝜑].

Let 𝐹(𝑠) be the Laplace transform of the input function 𝑓(𝑡). Then 𝐿₀𝐷^𝛼𝑡𝐿𝑓₀(𝑡)=𝑠^𝛼𝐹(𝑠)−₀𝐷^𝑡𝛼−1𝑓(𝑡)∣_𝑡=0,𝐷^𝑡−𝛼𝑓(𝑡)=𝑠^−𝛼𝐹(𝑠).(8.11)

9. Fourier, Mellin, and (Two-Sided) Laplace Transforms

We state the Mellin, (two-sided) Laplace, and the Fourier transforms. If 𝑓(𝑥)=𝑂(𝑥^𝛼),𝛼∈ℝ for 𝑥>0, then its Mellin transform 𝑀[𝑓] is defined by𝑀[𝑓](𝑠)=^∞0𝑥^𝑠𝑓(𝑥)𝑑𝑥𝑥,𝜎>𝛼(9.1)

Under the change of variable 𝑥=𝑒^−𝑡, the Mellin transform and the two-sided Laplace transform shift each other:𝐿^±[𝜑](𝑠)=^∞−∞𝑒^−𝑠𝑡𝜑(𝑡)𝑑𝑡,𝜎>𝛼,(9.2) 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)): 𝐿[𝑓](𝑠)=𝐿^±[𝑓𝑢](𝑠)=^∞0𝑒^−𝑠𝑡𝑓𝑒^−𝑡𝑑𝑡,𝜎>𝛼,(9.3) compare Definition 11.1.

If we fix 𝜘>𝛼 and write 𝑠=𝜘+𝑗𝜔, 𝐺(𝑦)=𝐿^±[𝑓](𝜘+𝑗𝜔),𝑔(𝑡)=𝑒^−𝜘𝑡𝑓(𝑒^−𝑡) in (9.2), then it changes into𝐺(𝜔)=𝐹[𝑔](𝜔)=^∞−∞𝑒^{−𝑗𝜔𝑡}𝑔(𝑡)𝑑𝑡=𝐿^±[𝜑](𝑗𝜔),(9.4) the Fourier transform of 𝑔.

We explain Plancherel’s theorem for functions in 𝐿₂(ℝ). Let𝑓_𝑇(𝑥)=1√2𝜋^𝑇−𝑇𝑒^{−𝑖𝑥𝑡}𝑓(𝑡)𝑑𝑡.(9.5) Then 𝑓_𝑇(𝑥) is convergent to a function 𝑓 in 𝐿²:‖‖𝑓_𝑇−𝑓‖‖⟶0,𝑇⟶∞,lim_𝑇→∞𝑓_𝑇(𝑥)=𝑓(𝑡),(9.6) where  lim is a short-hand for “limit in the mean.” The Parseval identity reads‖‖𝑓‖‖₂=‖𝑓‖₂,^∞0||𝑓(𝑡)||²𝑑𝑡=^∞0||𝑓(𝑡)||²𝑑𝑡.(9.7)

If we apply (9.7) to a causal function 𝑓, then it leads to [1, (3.19)]^∞−∞||𝐿^±[𝑓](𝑖𝜔)||²𝑑𝜔=^∞0||𝑓(𝑡)||²𝑑𝑡.(9.8)

Hence we see that [1, (3.19)] is indeed the Parseval identity for the Fourier (or Plancherel) transform for 𝑓∈𝐿₂(ℝ).

10. Examples of Second-Order Systems

10.1. 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, satisfies𝐿𝑑²𝑖𝑑𝑡²+𝑅𝑑𝑖𝑑𝑡+𝐶1𝑖=𝑒^(𝑡).(10.1)

10.2. Newton’s Equation of Motion (cf. [7])

One has𝑀𝑑²𝑦𝑑𝑡²+𝑅𝑑𝑦𝑑𝑡+𝐾𝑦=𝑒(𝑡)=𝐹,(10.2) where 𝑀 is the inertance of mass, 𝑅 is the viscous resistance of the dashpot, and 𝐾 is the spring stiffness.

Introducing the new parameters𝜔_𝑛=𝐾𝜁=𝑀∶𝑅naturalangularfrequence,√2𝐾/𝑀∶dampingratio,(10.3)

(10.2) becomes1𝜔^2𝑛𝑑²𝑦𝑑𝑡²+2𝜁𝜔_𝑛𝑑𝑦𝑑𝑡+𝑦=𝐾1𝐹.(10.4)

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 11.1. Suppose 𝑦(𝑡)=𝑂(𝑒^𝑎𝑡),𝑡→∞ for an a∈ℝ. The Laplace transform 𝑌(𝑠)=𝐿[𝑦](𝑠) of 𝑦=𝑦(𝑡) is defined by 𝐿[𝑦](𝑠)=^∞0𝑒^−𝑠𝑡𝑦(𝑡)𝑑𝑡,Re𝑠>𝑎.(11.1) The integral converges absolutely in Re𝑠>𝑎 and represents an analytic function there.

Example 11.2. Let  𝛼∈ℂ. Then 𝐿𝑒^𝛼𝑡(𝑠)=1𝑠−𝛼,(11.2) valid for Re𝑠>𝑅𝑒𝛼 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 𝐿[sin𝛼𝑡](𝑠)=𝛼𝑠²+𝛼²𝐿[cos𝛼𝑡](𝑠)=𝑠,(11.3)𝑠²+𝛼².(11.4)
For 𝛼=𝜔∈ℝ they reduce to familiar formulas: 𝐿[sin𝜔𝑡](𝑠)=𝜔𝑠²+𝜔²𝐿[cos𝜔𝑡](𝑠)=𝑠,𝑠²+𝜔².(11.5)

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𝑒^𝑖𝛼𝑡(𝑠)=1𝑠−𝑖𝛼,𝐿𝑒^{−𝑖𝛼𝑡}(𝑠)=𝑠+1𝑖𝛼,(11.6) whence, for example, 𝐿[cos𝛼𝑡](𝑠)=12𝐿𝑒^𝑖𝛼𝑡(𝑠)+𝐿𝑒^{−𝑖𝛼𝑡}(𝑠)=𝑠²+𝑠𝛼²,(11.7)by Euler’s identity, that is, (11.4).

12. Partial Fraction Expansion and Examples

As long as the input function is a sinusoidal function, Example 11.2 will suffice to compute its Laplace transform. To go back to the time domain from the frequency domain, we need to solve 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)/𝐶(𝑧) is given by 𝐶(𝑧)=𝑐_0𝑞_𝑖=1𝑧−𝛽_𝑖^𝜎𝑖,^𝑞_𝑖=1𝜎_𝑖=deg𝐶=𝐿,(12.1) where 𝛽_𝑖=𝜎_𝑖−1, then 𝑆(𝑧)=^𝑞_𝜎𝑖=1^𝑖−𝑗_𝑗=0𝑎_{𝑘,𝜎𝑘−𝑗}1𝑧−𝛽_𝑖^{𝜎𝑖−𝑗},(12.2) where the coefficients are given by 𝑎_{𝑖,𝜎𝑖−𝑗}=1𝑗!lim^𝑧→𝛽_𝑖d^𝑗d𝑧^𝑗𝑧−𝛽_𝑖^𝜎𝑖𝑅(𝑧).(12.3)

Proof. By (12.1), for each 𝑖, 1≤𝑖≤𝑞, we may write 𝑆(𝑧)=𝑃^𝑖(𝑧)𝑧−𝛽_𝑖^𝜎𝑖,𝑃_𝑖(𝑧)∈ℂ(𝑧),(12.4) and 𝑆_𝑖(𝑧) has no pole at 𝑧=𝛽_𝑖. We write 𝑧−𝛽_𝑖^𝜎𝑖𝑆(𝑧)=𝑃_𝑖(𝑧)=^𝜎𝑖−1_𝑗=0𝑎_{𝑖,𝜎𝑖−𝑗}𝑧−𝛽_𝑖^𝑗+𝑧−𝛽_𝑖^𝜎𝑖𝐻_𝑖(𝑧),(12.5) where 𝐻_𝑖(𝑧)∈ℂ(𝑧) has no pole at 𝑧=𝛽_𝑖. By successively differentiating and setting 𝑧=𝛽_𝑖, we obtain (12.3).
Now, the rational function 𝐹(𝑧)∶=𝑆(𝑧)−^𝑞_𝜎𝑖=1^𝑖−1_𝑗=0𝑎_{𝑘,𝑟𝑘−𝑗}1𝑧−𝛽_𝑖^{𝜎𝑖−𝑗}(12.6) has no pole, so that it must be a polynomial. But, since lim_𝑧→∞𝐹(𝑧)=0 (where we use the assumption deg𝑃<deg𝐶), it follows that 𝐹(𝑧) must be zero.

Now we will give examples of (2.2) for the second-order systems which do not appear anywhere else save for [2].

Example 12.2. We find the output signal (current) 𝑦=𝑦(𝑡) described by the DE 𝑦^+𝑦^+𝑦=𝑒^−(1/2)𝑡sin√32𝑡,(12.7) where the initial values are assumed to be 0: 𝑦(0)=0,𝑦^(0)=0.

Proof. Let 𝑌(𝑠)=𝐿[𝑦](𝑠) be the Laplace transform of 𝑦(𝑡). Then we have 𝑌(𝑠)=Φ(𝑠)𝑈(𝑠),𝑈(𝑠)=𝐿[𝑢](𝑠)=√3/22𝑠²+𝑠+1,√3𝐿[𝑦](𝑠)=𝑠²+𝑠1+1²,(12.8) and we may obtain the partial fraction expansion 1𝑠²+𝑠+1²=−1/3(𝑠−𝜌)²+−2/3√3𝑖𝑠−𝜌+−1/3𝑠−𝜌²+2/3√3𝑖𝑠−𝜌,(12.9)where 𝜌=𝑒^{(2𝜋𝑖/3)}=(−1+√3𝑖)/2 is the first primitive cube root of 1. Hence 2√3𝑦(𝑡)=√23𝐿⁻¹[𝐿[𝑦]](𝑡)=−23𝑡𝑒^−(1/2)𝑡cos√32𝑡+3√23𝑒^−(1/2)𝑡sin√32𝑡.(12.10) As a transfer function, the function in (12.8) Φ(𝑠)=1𝑠²+𝑠+1(12.11) is stable.

Example 12.3. The following integral may be evaluated by the partial fraction expansion above or by the residue calculus: ^∞−∞1𝑥²+𝑥+1²𝑑𝑥=2𝜋𝑖−23√3𝑖=3√43𝜋.(12.12)

Example 12.4. In the same vein as with Example 12.2, we may find the solution of the DE 𝑦^−𝑦^+𝑦=𝑢(𝑡)=𝑒^(1/2)𝑡sin√32𝑡,(12.13)where the initial values are assumed to be 0: 𝑦(0)=0,𝑦^(0)=0.

We have𝑌(𝑠)=Φ₁(𝑠)𝑈(𝑠),𝑈(𝑠)=𝐿[𝑢](𝑠)=√3/2𝑠²−𝑠+1,(12.14) or2√3𝐿[𝑦](𝑠)=𝑠²−1𝑠+1².(12.15)

Hence2√3𝑦(𝑡)=√23𝐿⁻¹[𝐿[𝑦]](𝑡)=−𝑡𝑒^(1/2)𝑡cos√32𝑡+√3𝑒^(1/2)𝑡sin√32𝑡,(12.16)

and the transfer function in (12.14)Φ₁(𝑠)=1𝑠²−𝑠+1(12.17)

is unstable.

13. The Product of Zeta-Functions: ΓΓ-Type

In this section, we illustrate the use of fractional integrals by proving a slight generalization of the result of Chandrasekharan and Narasimhan ([8]) 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 {𝜆_𝑘},{𝜇_𝑘} be increasing sequences of positive numbers tending to ∞, and let {𝛼_𝑘},{𝛽_𝑘} be complex sequences. We form the Dirichlet series𝜑(𝑠)=^∞_𝑘=1𝛼_𝑘𝜆^𝑠𝑘𝜓(𝑠)=_∞,_𝑘=1𝛽_𝑘𝜇^𝑠𝑘(13.1) and suppose that they have finite abscissas of absolute convergence 𝜎_𝜑, 𝜎_𝜓, 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 𝑠_𝑘(1≤𝑘≤𝐿): 𝜒(𝑠)=⎧⎪⎨⎪⎩Γ𝑠+𝜈Γ𝑟−𝑠2+Γ𝜈𝑠−𝜈2𝜑(𝑠),Re𝑠>𝜎^𝜑,2Γ𝑟−𝑠−𝜈2𝜓(𝑟−𝑠),Re𝑠<𝑟−𝜎^𝜓(13.2) We introduce the processing gamma factorΔ(𝑤)=Γ𝑏^𝑗+𝐵_𝑗𝑤^𝑚𝑗=1Γ𝑎^𝑗−𝐴_𝑗𝑤^𝑛𝑗=1Γ𝑎_𝑗+𝐴_𝑗𝑤^{𝑝𝑗=𝑛+1}Γ𝑏^𝑗−𝐵_𝑗𝑤^{𝑞𝑗=𝑚+1}𝐴^𝑗,𝐵_𝑗>0(13.3) and suppose that for any real numbers 𝑢₁,𝑢₂(𝑢₁<𝑢₂)lim_|𝑣|→∞Δ(𝑢+𝑖𝑣−𝑠)𝜒(𝑢+𝑖𝑣)=0,(13.4) uniformly in 𝑢₁≤𝑢≤𝑢₂.

In the 𝑤-plane we take two deformed Bromwich paths𝐿₁(𝑠)∶𝛾₁−𝑖∞⟶𝛾₁+𝑖∞,𝐿₂(𝑠)∶𝛾₂−𝑖∞⟶𝛾₂+𝑖∞𝛾₂<𝛾₁(13.5)

such that they squeeze a compact set 𝒮 with boundary 𝒞 for which 𝑠_𝑘∈𝒮(1≤𝑘≤𝐿) and all the poles ofΓ𝑏_𝑗−𝐵_𝑗𝑠+𝐵_𝑗𝑤^𝑚𝑗=1Γ𝑎_𝑗−𝐴_𝑗𝑠+𝐴_𝑗𝑤^{𝑝𝑗=𝑛+1}Γ𝑏^𝑗+𝐵_𝑗𝑠−𝐵_𝑗𝑤^{𝑞𝑗=𝑚+1}(13.6)

lie to the left of 𝐿₂(𝑠) and those ofΓ𝑎_𝑗+𝐴_𝑗𝑠−𝐴_𝑗𝑤^𝑛𝑗=1Γ𝑎_𝑗−𝐴_𝑗𝑠+𝐴_𝑗𝑤^{𝑝𝑗=𝑛+1}Γ𝑏^𝑗+𝐵_𝑗𝑠−𝐵_𝑗𝑤^{𝑞𝑗=𝑚+1}(13.7)

lie to the right of 𝐿₁(𝑠).

Then we define the 𝐻-function by (0≤𝑛≤𝑝,0≤𝑚≤𝑞,𝐴_𝑗,𝐵_𝑗>0)𝐻^{𝑚,𝑛𝑝,𝑞}⎛⎜⎜⎝𝑧∣𝑏1₁−𝑎₁,𝐴₁,…,1−𝑎_𝑛,𝐴_𝑛,𝑎_𝑛+1,𝐴_𝑛+1,…,𝑎_𝑝,𝐴_𝑝=1,𝐵₁,…,𝑏_𝑚,𝐵_𝑚,1−𝑏_𝑚+1,𝐵_𝑚+1,…,1−𝑏_𝑞,𝐵_𝑞⎞⎟⎟⎠2𝜋𝑖_𝐿Γ𝑏₁+𝐵₁𝑠,…,𝑏_𝑚+𝐵_𝑚𝑠Γ𝑎₁−𝐴₁𝑠,…,𝑎_𝑛−𝐴_𝑛𝑠Γ𝑎_𝑛+1+𝐴_𝑛+1𝑠,…,𝑎_𝑝+𝐴_𝑝𝑠Γ𝑏_𝑚+1−𝐵_𝑚+1𝑠,…,𝑏_𝑞−𝐵_𝑞𝑠𝑧^−𝑠𝑑𝑠.(H-1)

In the special case, where 𝐴_𝑗=𝐵_𝑗=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) byX(𝑧,𝑠)=12𝜋𝑖_𝐿1(𝑠)Δ(𝑤−𝑠)𝜒(𝑤)𝑧^−𝑤𝑑𝑤,(13.8) which is for 𝜒=1 one of 𝐻-functions. Hereafter we always assume that 𝑧>0, which may be extended to Re𝑧>0.

Then we haveX(𝑧,𝑠)=12𝜋𝑖_𝐿2(𝑠)Δ(𝑤−𝑠)𝜒(𝑤)𝑧^−𝑤𝑑𝑤+12𝜋𝑖_𝒞Δ(𝑤−𝑠)𝜒(𝑤)𝑧^−𝑤𝑑𝑤,(13.9) which amounts to the following.

Theorem 13.1 ([9]). One has the modular relation equivalent to (13.2):X=(𝑧,𝑠)_∞_𝑘=1𝛼_𝑘𝜆^𝑠𝑘×𝐻^{𝑚+2,𝑛𝑝,𝑞+2}⎛⎜⎜⎝𝑧𝜆_𝑘∣1−𝑎^𝑗,𝐴_𝑗^𝑛1,𝑎_𝑗,𝐴_𝑗^𝑝𝑛+1𝑠+𝜈2,1,𝑠−𝜈2,1,𝑏^𝑗,𝐵_𝑗^𝑚1,1−𝑏_𝑗,𝐵_𝑗^𝑞𝑚+1=_∞⎞⎟⎟⎠_𝑘=1𝛽_𝑘𝜇^{𝑘𝑟−𝑠}×𝐻^{𝑛+2,𝑚𝑞,𝑝+2}⎛⎜⎜⎝𝜇_𝑘𝑧∣1−𝑏^𝑗,𝐵_𝑗^𝑚1,𝑏_𝑗,𝐵_𝑗^𝑞𝑚+1𝑟−𝑠+𝜈2,1,𝑟−𝑠−𝜈2,1,𝑎^𝑗,𝐴_𝑗^𝑛1,1−𝑎_𝑗,𝐴_𝑗^𝑝𝑛+1+_𝐿⎞⎟⎟⎠_𝑘∑₌₁ResΔ(𝑤−𝑠_𝑛)𝜒(𝑤)𝑧^𝑠−𝑤,𝑤=𝑠_𝑘._𝑗∑₌₁𝐴_𝑗+∑^𝑚_𝑗=1𝐵_𝑗+2≥∑^𝑝_𝑗=𝑛+1𝐴_𝑗+∑^𝑞_𝑗=𝑚+1𝐵_𝑗.(13.10)

In the special case, where 𝐴_𝑗=𝐵_𝑗=1, we have the following.

Theorem 13.2. One has 𝑧^𝑠X(𝑧,𝑠)=^∞_𝑘=1𝛼_𝑘𝜆^𝑠𝑘𝐺^{𝑚+2,𝑛𝑝,𝑞+2}⎛⎜⎜⎝𝑧𝜆_𝑘∣1−𝑎¹,…,1−𝑎_𝑛,𝑎_𝑛+1,…,𝑎_𝑝𝑠+𝜈2,𝑠−𝜈2,𝑏¹,…,𝑏_𝑚,1−𝑏_𝑚+1,…,1−𝑏_𝑞=_∞⎞⎟⎟⎠_𝑘=1𝛽_𝑘𝜇^{𝑘𝑟−𝑠}×𝐺^{𝑛+2,𝑚𝑞,𝑝+2}⎛⎜⎜⎝𝜇_𝑘𝑧∣1−𝑏¹,…,1−𝑏_𝑚,𝑏_𝑚+1,…,𝑏_𝑞𝑟−𝑠+𝜈2,𝑟−𝑠−𝜈2,𝑎¹,…,𝑎_𝑛,1−𝑎_𝑛+1,…,1−𝑎_𝑝+_𝐿⎞⎟⎟⎠_𝑘=1ResΔ(𝑤(−2𝑛𝑠+)𝜒2(𝑚𝑤+)𝑧2≥^𝑠−𝑤,𝑝𝑤+=𝑞)𝑠._𝑘,(13.11)

For many important applications, compare [9].

13.2. The Riesz Sum (𝐺^2,24,4↔𝐺^4,02,6).

Formula (13.11) in the special case of the title reads^∞_𝑘=1𝛼_𝑘𝜆^𝑠𝑘𝐺^2,24,4⎛⎜⎜⎝𝑧𝜆_𝑘∣𝑠+𝜈𝑎,𝑏,𝑐,𝑑=2,𝑠−𝜈2,𝑒,𝑓_∞⎞⎟⎟⎠_𝑘=1𝛽_𝑘𝜇^𝑟−𝑠𝐺^4,02,6⎛⎜⎜⎝𝜇_𝑘𝑧∣𝑟−𝑠+𝜈1−𝑒,1−𝑓+2,𝑟−𝑠−𝜈2,1−𝑎,1−𝑏,1−𝑐,1−𝑑_𝐿⎞⎟⎟⎠_𝑘=1ResΔ(𝑤−𝑠)𝜒(𝑤)𝑧^𝑠−𝑤,𝑤=𝑠_𝑘,(13.12) whereΔ(𝑤)=Γ(1−𝑎−𝑤)Γ(1−𝑏−𝑤)Γ(𝑐+𝑤)Γ(𝑑+𝑤)Γ(1−𝑒−𝑤)Γ(1−𝑓−𝑤).(13.13)

We treat the case 𝑟=1/2. Assuming 𝜆 is a nonnegative integer, we put 𝑎=𝑠+(𝜈/2)+(𝜆/2)+(1/2), 𝑏=𝑠+(𝜈/2)+(𝜆/2)+1, 𝑐=𝑠−(𝜈/2), 𝑑=𝑠+(𝜈/2)+𝜆+1, 𝑒=𝑠+(𝜈/2)+(1/2), 𝑓=𝑠+(𝜈/2)+𝜆+1. Then (13.12) becomes ^∞_𝑘=1𝛼_𝑘𝜆^𝑠𝑘𝐺^2,24,4⎛⎜⎜⎝𝑧𝜆_𝑘∣𝑠+𝜈2+𝑠+𝜆2𝜈+12,𝑠+𝜈2+𝜆2+1,𝑠−𝜈2,𝑠+𝜈2+𝜆+1=2,𝑠−𝜈2,𝑠+𝜈2+21,𝑠+𝜈2+𝜆+1_∞⎞⎟⎟⎠_𝑘=1𝛽_𝑘𝜇^(1/2)−𝑠𝐺^4,02,6⎛⎜⎜⎝𝜇_𝑘𝑧∣−𝑠−𝜈+2+21∗,−𝑠−𝜈2−𝜆_𝐿⎞⎟⎟⎠_𝑘=1ResΔ(𝑤−𝑠)𝜒(𝑤)𝑧^𝑠−𝑤,𝑤=𝑠_𝑘,(13.14) where ∗ indicates −𝑠+𝜈/2+1/2, −𝑠−𝜈/2+1/2, −𝑠−𝜈/2−𝜆/2+1/2, −𝑠−𝜈/2−𝜆/2, −𝑠+𝜈/2+1, −𝑠−𝜈/2−𝜆.

We note that the 𝐺-functions in (13.14) reduce to𝐺^2,24,4⎛⎜⎜⎝𝑧∣𝑠+𝜈2+𝑠+𝜆2𝜈+12,𝑠+𝜈2+𝜆2+1,𝑠−𝜈2,𝑠+𝜈2+𝜆+1=2_𝜆2,𝑠−𝜈2,𝑠+𝜈2+21,𝑠+𝜈2+𝜆+1⎞⎟⎟⎠𝐺^1,01,1⎛⎜⎜⎝√𝑧∣2𝑠+2𝜈𝑠++𝜆𝜈+1⎞⎟⎟⎠=⎧⎪⎨⎪⎩Γ(𝜆2^𝜆+1)𝑧^{𝑠+(𝜈/2)}1−√0,𝑧^𝜆,((||𝑧𝑧||<>11))(13.15)

(by the formula in [10]) and𝐺^4,02,6⎛⎜⎜⎝𝑧∣−𝑠−𝜈2+21∗,∗−𝑠−𝜈2−𝜆⎞⎟⎟⎠=𝐺^4,02,6⎛⎜⎜⎝𝑧∣−𝑠−𝜈2+1†,−𝑠−𝜈2−𝜆⎞⎟⎟⎠,(13.16)

where ∗∗ indicates −𝑠+𝜈/2+1/2, −𝑠−𝜈/2+1/2, −𝑠−𝜈/2−𝜆/2+1/2, −𝑠−𝜈/2−𝜆/2, −𝑠+𝜈/2+1, −𝑠−𝜈/2−𝜆 and †, −𝑠+𝜈/2+1/2, −𝑠+𝜈/2+1, −𝑠−𝜈/2−𝜆/2, −𝑠−𝜈/2−𝜆/2+1/2, −𝑠+𝜈/2+1, −𝑠−𝜈/2−𝜆. Hence it reduces further to𝑧^{−𝑠−(𝜆/2)+(1/4)}2𝜋cos((𝜈+𝜆+1)𝜋)𝐾^2𝜈+𝜆+14⁴+cos((𝜈+1)𝜋)𝑌_2𝜈+𝜆+1√𝑧=−𝑧_{−𝑠−(𝜆/2)+(1/4)}4√⁴𝑧+sin((𝜈+1)𝜋)𝐽_2𝜈+𝜆+14√⁴𝑧(−1)^𝜆2𝜋cos(𝜈𝜋)𝐾^2𝜈+𝜆+14⁴+cos(𝜈𝜋)𝑌_2𝜈+𝜆+1√𝑧=𝑧_{−𝑠−(𝜆/2)+(1/4)}4√⁴𝑧+sin(𝜈𝜋)𝐽_2𝜈+𝜆+14√⁴𝑧𝐺^{𝜆2𝜈+𝜆+1}4√⁴𝑧,(13.17)

say, where, slightly more general than Wilton’s (1.22) [11], we put𝐺^𝜆𝜈(𝑧)=−(−1)^𝜆2𝜋sin𝜈−2𝜆𝜋𝐾^𝜈(𝑧)−sin𝜈−𝜆2𝜋𝑌^𝜈(𝑧)+cos𝜈−𝜆2𝜋𝐽^𝜈(𝑧).(13.18) Hence (13.14) reads𝑧^𝑠+𝜈/22^𝜆Γ(𝜆+1)_{𝜆𝑘<1/𝑧}=𝑧_{∞𝑠+𝜆/4−1/4}𝛼_𝑘𝜆^𝑘𝜈/21−√𝑧𝜆_𝑘^𝜆(13.19)_𝑘=1𝛽_𝑘𝜇^{𝑘𝜆/4+1/4}𝐺^{𝜆2𝜈+𝜆+1}4⁴𝜇_𝑘+_𝐿𝑧_𝑘=1ResΔ(𝑤−𝑠)𝜒(𝑤)𝑧^𝑠−𝑤,𝑤=𝑠_𝑘,(13.20) which gives a more general form of Wilton’s Theorem  1 [11].

Rewriting (13.19) slightly, we deduce an analogue of Chandrasekharan and Narasimhan result [8, Theorem  7.1(a)],

Theorem 13.3. For 𝑥>0, the functional equation (13.2) implies the identity 1Γ(𝜆+1)_{𝜆𝑘<𝑥}𝛼_𝑘𝜆^𝜈𝑘𝑥−𝜆_𝑘^𝜆=𝑥^{𝜆/2+𝜈+1/2}2^∞−𝜆_𝑘=1𝛽_𝑘𝜇_{𝑘𝜆/2+1/2}𝐺^{𝜆2𝜈+𝜆+1}4√𝜇_𝑘𝑥+P_𝜆(𝑥),(13.21) where P_𝜆(𝑥)=𝑥^{𝐿2𝑠+𝜆+𝜈}_𝑘=1Re𝑠Δ(𝑤−𝑠)𝜒(𝑤)𝑥^{2(𝑤−𝑠)},𝑤=𝑠_𝑘(13.22) and where Δ(𝑤)=Γ(1−𝑠−𝜆/2−𝜈/2−𝑤)Γ(−𝑠−𝜆/2−𝜈/2−𝑤)Γ(𝑠−𝜈/2+𝑤)Γ(𝑠+𝜆+𝜈/2+1+𝑤)Γ(1−𝑠−𝜈/2−1/2−𝑤)Γℨ,(13.23) where ℨ denotes (−𝑠−𝜆−𝜈/2−𝑤), with 𝐺^{𝜆2𝜈+𝜆+1} being given by (13.18).

Corollary 13.4. For 𝑥>0, the functional equation (13.2) implies the identity 𝐴^𝜆(𝑥)∶=1Γ(𝜆+1)_{𝜆𝑘<𝑥}𝛼_𝑘𝑥−𝜆_𝑘^𝜆=−2^∞−𝜆_𝑘=1𝛽_𝑘𝜇𝑥_𝑘^𝜆/2+1/2𝐹_𝜆+14√𝜇_𝑘𝑥+P_𝜆(𝑥),(13.24) where 𝐹_𝜆+1(𝑧)=−𝐺^𝜆𝜆+1(𝑧)=𝑌_𝜆+1(𝑧)+(−1)^𝜆2𝜋𝐾^𝜆+1(𝑧).(13.25)

We are now in a position to prove an analogue of [8, Theorem 7.1(b)] (although Theorem 13.3 contains [8, Theorem 7.2], too) by the Riemann-Liouville fractional integral transform.

Lemma 13.5 (Riemann-Liouville integral of Bessel functions). For the well-known Bessel functions 𝐽 and 𝑌, one has 1Γ(𝜇)^𝑦0(𝑦−𝑥)^𝜇−1𝑥^(1/2)𝜈𝐽R_𝜈e𝜇𝑎𝑥>^1/20,𝑑R𝑥e=𝜈2>^𝜇𝑎0^−𝜇𝑦^{(1/2)𝜇+(1/2)𝜈}𝐽_𝜇+𝜈𝑎𝑦^1/2(13.26)1Γ(𝜇)^𝑦0=2_𝜇(𝑦−𝑥)^𝜇−1𝑥^(1/2)𝜈𝑌_𝜈𝑎𝑥^1/2𝑑𝑥𝑎^−𝜇𝑦^{(1/2)𝜇+(1/2)𝜈}𝐽_𝜇+𝜈𝑎𝑦^1/2+Γ(𝜈+1)Γ(𝜇)𝜋2^𝜈+2Re𝜇>0,Re𝜈>−1𝑎,^−𝜇𝑦^{(1/2)𝜇+(1/2)𝜈}𝑆_{𝜇−𝜈−1,𝜇+𝜈}𝑎𝑦^1/2(13.27) where 𝑆 stands for the Lommel function.

Equation (13.26) is [12, (63), page 194] and (13.27) is [12, page 196]. We only need (13.27) and (13.27) is for treating the 𝐽-Bessel function.

Arguing in the same way as in [8], we may prove the following.

Theorem 13.6. With a 𝐶^∞-function 𝑅_𝜌 and a certain constant 𝑐 one has 𝐴^𝜌(𝑥)−𝑅_𝜌(𝑥)=𝑐^∞_𝑘=1𝛽_𝑘𝜇𝑥_𝑘^𝜌/2+1/2𝑌_𝜌+14√𝜇_𝑘𝑥,(13.28) for integral 𝜆, 𝜌=𝜆+𝛼, 0<𝛼<1, 𝜆≥2𝜎_𝜓−3/2.

Acknowledgment

This work is supported by the SMX SUDA CO. (No. SDJN1001).