Convergence Analysis of Iterative Learning Control for Stocastic Semilinear Generalized Distributed Parameter Systems

Zhaoqiang Ge

doi:doi:10.11648/j.sd.20251305.14

Research Article |

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Convergence Analysis of Iterative Learning Control for Stocastic Semilinear Generalized Distributed Parameter Systems

Zhaoqiang Ge^*

Published in Science Discovery (Volume 13, Issue 5)

Received: 6 September 2025 Accepted: 29 September 2025 Published: 29 October 2025

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Abstract

The iterative learning control problem of generalized distributed parameter systems is one of the important issues in the research of generalized distributed parameter systems. In this paper convergence analysis of P-type iterative learning control for stochastic semilinear generalized distributed parameter systems is discussed in the sense of integral solution by GE-semigroup theory in separable Hilbert spaces. Firstly, the basic concepts of GE-semigroup and integral solution are presented, along with a description of the iterative learning control problem for stochastic semilinear generalized distributed parameter systems under P-type learning law. Secondly, some sufficient conditions are obtained, by which the input tracking and output tracking errors can converge to be bounded in the mean square sense. At the same time, we have also obtained sufficient conditions for the input tracking and output tracking errors of the semilinear generalized distributed parameter system to converge uniformly to zero.

Published in	Science Discovery (Volume 13, Issue 5)
DOI	10.11648/j.sd.20251305.14
Page(s)	101-107
Creative Commons	This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.
Copyright	Copyright © The Author(s), 2025. Published by Science Publishing Group

Keywords

Convergence Analysis, Iterative Learning Control, Stochastic Semilinear Generalized Distributed Parameter Systems, Integral Solution, GE-semigroup

1．引言

广义分布参数系统是许多实际问题的固有性质。研究广义分布参数系统有重要的理论及应用价值。文献

[1-6]

表明广义分布参数系统对复合材料中的温度分布问题，经济学中的投入产出分析问题等的研究有重要意义。论文

[7, 8]

表明研究广义分布参数系统的迭代学习控制问题有重要的理论及应用价值。文献

[9-14]

表明渗流自由面的演化问题，DNA分子中纵波的传播问题等都可通过广义分布参数系统进行描述。随着高新技术的深入发展，对广义分布参数系统的研究显得更加重要。

广义分布参数系统的迭代学习控制是广义分布参数系统研究的重要问题之一。研究广义分布参数系统的迭代学习控制问题有重要的理论及应用价值。关于广义分布参数系统迭代学习控制问题的研究可以追溯到2016年，从那时起有关学者就开始研究由某种特殊的偏微分方程形式给出的广义分布参数系统的迭代学习控制问题。

2016年，文献

[7]

研究了一类时滞非线性广义分布参数系统的迭代学习控制问题，通过一种广义P-型算法给出了输出跟踪误差一致有界的充分条件。2017年，文献

[8, 15, 16]

研究了由时间导数项系数矩阵为奇异矩阵的抛物型和双曲型偏微分方程描述的特殊线性广义分布参数系统的迭代学习控制问题，应用P-型学习律给出了输出跟踪误差以L²范数收敛到零的充分条件。2018年，文献

[17]

研究了一类导数项系数矩阵为奇异矩阵的半线性方程描述的半线性广义分布参数的迭代学习控制问题。应用带有初值修改项的PD-型学习律，得到了输出跟踪误差以L²范数收敛于零的充分条件。2019年，文献

[18]

考虑了一类由时间导数项系数矩阵为奇异矩阵的抛物型方程描述的线性广义分布参数系统的迭代学习控制问题。应用PD-型学习律给出了输出跟踪误差以L²范数收敛于零的充分条件。2020年，文献

[19]

讨论了

[18]

中的广义分布参数系统离散化后所得到的离散广义分布参数系统的迭代学习控制问题。应用P-型学习律得到了系统收敛的充分条件。2021年，文献

[20]

研究了一类抛物型线性广义分布参数系统的混合PD-型迭代学习控制算法，给出了输出跟踪误差收敛的充分条件。同年，文献

[21]

研究了一类由时间导数项系数矩阵为奇异矩阵的抛物型方程给出的线性广义分布参数系统的迭代学习控制问题。应用D-型学习律给出了输出跟踪误差以L²范数收敛于零的充分条件。

就以上的研究来看，关于连续广义分布参数系统的迭代学习控制问题都是在系统强解

[6, 22]

意义下进行的，这样就限制了迭代学习控制理论的应用范围。此外，关于随机广义分布参数系统的迭代学习控制问题目前尚未见到有关研究结果，而随机广义分布参数分布参数系统对许多实际问题的研究也十分重要

[6, 22-25]

。鉴于以上，本文应用GE-半群理论

[24]

，在积分解

[25, 26]

（积分解比强解的适应范围更广）的意义下，研究一般随机半线性广义分布参数系统迭代学习控制收敛性问题。

本文结构如下。在第二节中主要给出随机半线性广义分布参数系统迭代学习控制问题的描述；在第三节中应用P-型学习律分析系统的收敛性问题；最后在第四节中给出结论。

符号说明：X，U，W，Y分别表示实的可析的Hilbert空间；S，C分别表示实数和复数全体所构成的集合；

(Ω, F, Γ)

表示一个完全概率空间；G 表示数学期望算子；

‖∙‖

表示范数；μ是一个正常数，

α (t)

是

[0, a]

上的一个有界函数，

α (t)

的μ范数为

{‖α‖}_{μ} = \sup \{e^{- μt} ‖α (t)‖, t \in [0, a]\}

;

D (A)

表示算子

A

的定义域。

2．问题的描述

考虑如下形式的重复运动随机半线性广义分布参数系统：

H {dh}_{i} (t) = [J h_{i} (t) + R r_{i} (t)] dt + l (h_{i} (t), r_{i} (t), t) dt + Md w_{i} (t),

s_{i} (t) = O h_{i} (t) + Z r_{i} (t)

,(1)

其中

t \in [0, a],

i \in \{0, 1, 2, \dots\}; h_{i} (t) \in X, r_{i} (t) \in U, s_{i} (t) \in Y

分别表示状态向量过程，输入向量过程和输出向量过程；

w_{i} (t) \in W

是定义在

(Ω, F, Γ)

上的标准Wiener 过程;

J : D (J) \to X

是一个闭线性算子;

H : X \to X, R : U \to X, O : X \to Y, Z : U \to Y, M : W \to X

都是有界线性算子；非线性算子

l : X \times U \times [0, + \infty] \to X

连续，且存在正常数

c_{l}

使

G {‖l (h_{1}, r_{1}, t) - l (h_{2}, r_{2}, t)‖}^{2} \leq c_{l} G ({‖h_{1} - h_{2}‖}^{2} + {‖r_{1} - r_{2}‖}^{2})

(2)

对任意

h_{1}, h_{2} \in X; r_{1}, r_{2} \in U

成立。

为了研究随机半线性广义分布参数系统(1)的收敛性，需下面的概念和假设。

定义1 设

V (t) : X \to X, t \geq 0

是一个有界线性算子族，如果

V (t + s) = V (t) HV (s), t, s \geq 0

，则称

V (t)

是由

H

引导的一个GE-半群，简称GE-半群。

如果

\lim_{t \to 0^{+}} ‖V (t) h - V (0) h‖ = 0

对任意的

h \in X

成立，则称

V (t)

是强连续的。

如果

V (t)

是强连续的，且在强收敛的意义下

Jh = \lim_{t \to 0^{+}} \frac{HV (t) H - HV (0) H}{t} h

对任意的

h \in D

成立，其中

D = \{h : h \in D (J), V (0) Hh = h, \exists \lim_{t \to 0^{+}} \frac{HV (t) H - HV (0) H}{t} h\}

则称

J

为

V (t)

的生成元。

定义2 如果

γH {(γH - J)}^{- 1}

在

γ \to + \infty

时强收敛，则随机半线性广义分布参数系统(1)的积分解定义为：

h_{i} (t) = V (t) H h_{i} (0) + \lim_{γ \to + \infty} γ [\int_{0}^{r} V (t - s) H {(γH - J)}^{- 1} ∙ R r_{i} (s) ds + \int_{0}^{r} V (t - s) H {(γH - J)}^{- 1} Md w_{i} (s)

+ \int_{0}^{t} V (t - s) H {(γH - J)}^{- 1} l (h_{i} (s), r_{i} (s), s) ds]

。

本文假设如下条件成立：

(H₁)

J

是

H

引导的

V (t)

的生成元，且常数

c_{V} > 0

满足

‖V (t)‖ \leq c_{V}, t \in [0, a]

。

(H₂) 存在常数

c > 0, ω \in S

使

(ω, + \infty)

包含在

ρ (H, J)

, 其中

ρ (H, J) = \{γ \in C : \exists {H (γH - J)}^{- 1}}

，

‖H {(γH - J)}^{- 1}‖ \leq \frac{c}{γ - ω}, γ > ω

且

γH {(γH - J)}^{- 1}

当

γ \to + \infty

时强收敛。

(H₃) 期望输出

s_{d} (t)

在

[0, a]

上连续，且存在唯一的期望输入

r_{d} (t)

在积分解的意义下满足

H \frac{d h_{d} (t)}{dt} = J h_{d} (t) + R r_{d} (t) + l (h_{d} (t), r_{d} (t), t)

s_{d} (t) = O h_{d} (t) + Z r_{d} (t)

,(3)

其中

h_{d} (t) \in X, r_{d} (t) \in U, s_{d} (t) \in Y

(H₄)

h_{i} (0) = h_{d} (0)

是确定的，且

s_{d} (0) = O h_{d} (0) + Z r_{d} (0)

Δ r_{i} (0) = r_{d} (0) - r_{i} (0)

且

∆ r_{i} (0)

有界。.

迭代学习控制是一种控制方法，它反复应用先前试验得到的信息，以获得能够产生期望输出轨迹的控制输入，改善控制质量。迭代学习控制的任务是：对于给定的期望目标，确定期望学习控制，使得迭代学习序列跟踪期望控制输入时，相应的被控系统在某种解的意义下对应的输出跟踪期望输出目标。

本文对随机半线性广义分布参数系统(1)采用P-型学习律，即

r_{i + 1} (t) = r_{i} (t) + PΔ s_{i} (t), t \in [0, a]

,(4)

其中

Δ s_{i} (t) = s_{d} (t) - s_{i} (t)

为输出跟踪误差，有界线性算子

P : Y \to U

为学习增益算子。

定义状态误差

Δ h_{i} (t)

，输入误差

Δ r_{i} (t)

和

{Δl}_{i} (t)

如下：

Δ h_{i} (t) = h_{d} (t) - h_{i} (t), Δ r_{i} (t) = r_{d} (t) - r_{i} (t)

Δ l_{i} (t) = l (h_{d} (t), r_{d} (t), t) - l (h_{i} (t), r_{i} (t), t)

。

由系统(1)和(3)可得：

Δ s_{i} (t) = OΔ h_{i} (t) + ZΔ r_{i} (t)

。(5)

根据P-型迭代学习控制率(4)，第

(i + 1)

次输入误差为

Δ r_{i + 1} (t) = r_{d} (t) - r_{i + 1} (t) = r_{d} (t) - r_{i} (t) - PΔ s_{i} (t)

= {Δr}_{i} (t) - P [OΔ h_{i} (t) + ZΔ r_{i} (t)]

= (I - PZ) Δ r_{i} (t) - POΔ h_{i} (t)

,(6)

其中

I

表示恒等算子。此外由系统(1)和(3)可得

HdΔ h_{i} (t) = Hd [h_{d} (t) - h_{i} (t)]

= [J h_{d} (t) + R r_{d} (t) + l (h_{d} (t), r_{d} (t), t)] dt - [J h_{i} (t) + R r_{i} (t) + l (h_{i} (t), r_{i} (t), t)] dt - Md w_{i} (t)

= [JΔ h_{i} (t) + RΔ r_{i} (t) + Δ l_{i} (t)] - Md w_{i} (t)

。(7)

由定义2和（H₄），系统(7)的积分解为

Δ h_{i} (t) = \lim_{γ \to + \infty} γ [\int_{0}^{t} V (t - s) H {(γH - J)}^{- 1} R Δ r_{i} (s) ds + \int_{0}^{T} V (t - s) H {(γH - J)}^{- 1} Δ l_{i} (s) ds

- \int_{0}^{r} V (t - s) H {(γH - J)}^{- 1} Md w_{i} (s)]

。(8)

3．收敛性分析

本节研究输入跟踪误差和输出跟踪误差的收敛性问题。

定理1 对于随机半线性广义分布参数系统(1)和P-型学习律(4)，假设(H₁)-(H₄) 成立。如果学习增益算子 P 满足

{‖I - PZ‖}^{2} = β < 1 / 2

,(9)

则存在常数

c_{1} > 0

，使

\lim_{i \to + \infty} G {‖Δ r_{i} (t)‖}^{2} \leq c_{1}, \forall t \in [0, a]

。(10)

证明由(8)可得

G {‖Δ h_{i} (t)‖}^{2} \leq 3 c_{V}^{2} c^{2} {‖R‖}^{2} G (\int_{0}^{t} {‖Δ r_{i} (s)‖ ds)}^{2} {+ G (\int_{0}^{t} ‖Δ l_{i} (s)‖ ds)}^{2} + 3 c_{V}^{2} c^{2} a {‖M‖}^{2}

\leq 3 c_{V}^{2} c^{2} a [{‖R‖}^{2} \int_{0}^{t} G {‖Δ r_{i} (s)‖}^{2} ds + c_{l} \int_{0}^{t} G ({‖∆ h_{i} (s)‖}^{2} + {‖∆ r_{i} (s)‖}^{2}) ds + 3 c_{V}^{2} c^{2} a {‖M‖}^{2}

.(11)

在(11)式两边同乘以

e^{- 2 μt}

，有

e^{- 2 μt} G {‖Δ h_{i} (t)‖}^{2} \leq 3 c_{V}^{2} c^{2} a [{‖R‖}^{2} \int_{0}^{t} G {‖Δ r_{i} (s)‖}^{2} e^{- 2 μt} ds

+ c_{l} \int_{0}^{t} {G ({‖∆ h_{i} (s)‖}^{2} + {‖∆ r_{i} (s)‖}^{2}) e}^{- 2 μt} ds] + 3 e^{- 2 μt} {‖M‖}^{2} c_{V}^{2} c^{2} a

.(12)

由于

μ > 0

，故

e^{- 2 μt} \in (0, 1], t \in [0, a]

。从而

3 c_{V}^{2} c^{2} {‖M‖}^{2} a e^{- 2 μt} \leq 3 c_{V}^{2} c^{2} a {‖M‖}^{2}

.(13)

把(13)代入(12)，由

μ -

范数的定义，可得

G {‖∆ h_{i}‖}_{μ}^{2} \leq 3 c_{V}^{2} c^{2} a [\frac{{‖R‖}^{2}}{2 μ} G {‖∆ r_{i}‖}_{μ}^{2} + c_{l} \frac{G ({‖∆‖}_{μ}^{2} + {‖∆ r_{i}‖}_{μ}^{2})}{2 μ}] + 3 c_{V}^{2} c^{2} {‖M‖}^{2} a

.(14)

由(14)，有

G {‖{∆ h}_{i}‖}_{μ}^{2} \leq 3 c_{V}^{2} c^{2} a \frac{({‖R‖}^{2} + c_{l}) G {‖∆ r_{i}‖}_{μ}^{2}}{2 μ - 3 c_{V}^{2} c^{2} a c_{l}} + \frac{6 c_{V}^{2} c^{2} {‖M‖}^{2} aμ}{2 μ - 3 c_{V}^{2} C^{2} a c_{l}}

.(15)

又由(6)，可得

{‖∆ r_{i + 1} (t)‖}^{2} = {‖(I - PZ) ∆ r_{i} (t) - PO ∆ h_{i} (t)‖}^{2}

\leq 2 [{‖(I - PZ) ∆ r_{i} (t)‖}^{2} + {⟦PO ∆ h_{i} (t)⟧}^{2}]

\leq 2 β {‖∆ r_{i} (t)‖}^{2} + 2 {‖PO‖}^{2} {‖∆ h_{i} (t)‖}^{2} .

(16)

由此可得

G {‖∆ r_{i + 1}‖}_{μ}^{2} \leq 2 βG {‖∆ r_{i}‖}_{μ}^{2} + 2 {‖PO‖}^{2} G {‖∆ h_{i}‖}_{μ}^{2}

.(17)

把(15)代入(17)，可得

G {‖∆ r_{i + 1}‖}_{μ}^{2} \leq (2 β + \frac{6 c_{V}^{2} c^{2} a {‖PO‖}^{2} ({‖R‖}^{2} + c_{l})}{2 μ - 3 c_{V}^{2} c^{2} a c_{l}}) G {‖∆ r_{i}‖}_{μ}^{2} + \frac{12 μ c_{V}^{2} c^{2} a {(‖M‖ ‖PO‖)}^{2}}{2 μ - 3 c_{V}^{2} c^{2} a c_{l}} .

(18)

令

ζ = 2 β + \frac{6 c_{V}^{2} c^{2} a {‖PO‖}^{2} ({‖R‖}^{2} + c_{l})}{2 μ - 3 c_{V}^{2} c^{2} a c_{l}}

η = \frac{12 μ c_{V}^{2} c^{2} a {(‖M‖ ‖PO‖)}^{2}}{2 μ - 3 c_{V}^{2} c^{2} a c_{l}}

α_{i} = G {‖∆ r_{i}‖}_{μ}^{2}

则(18)可写为

α_{i + 1} \leq ζ α_{i} + η

.(19)

由于

2 β < 1

, 可选取充分大的

μ

使

0 < ζ < 1

. 由 (19)，有

i = 0, α_{1} \leq ζ α_{0} + η

i = 1, α_{2} \leq ζ α_{1} + η \leq ζ^{2} α_{0} + ζη + η,

i = 2, α_{3} \leq ζ α_{2} + η \leq ζ^{3} α_{0} + ζ^{2} η + ζη + η,

\dots \dots

i = n, α_{n + 1} \leq ζ α_{n} + η \leq ζ^{n + 1} α_{0} + η (ζ^{i} + \dots + ζ + 1) .

从而对于

i

，有

α_{i + 1} \leq ζ α_{i} + η \leq ζ^{i + 1} α_{0} + η (ζ^{i} + \dots + ζ + 1) = ζ^{i + 1} α_{0} + \frac{1 - ζ^{i + 1}}{1 - ζ} η .

(20)

式(20)表明

G {‖Δ r_{i + 1}‖}_{μ}^{2} \leq ζG {‖{Δr}_{i}‖}_{μ}^{2} + η \leq ζ^{i + 1} G {‖∆ r_{0}‖}_{μ}^{2} + \frac{1 - ζ^{i + 1}}{1 - ζ} η .

(21)

由于

G {‖∆ r_{i + 1} (t)‖}^{2} = e^{- 2 μt} G {‖∆ r_{i + 1} (t)‖}^{2} e^{2 μt} \leq G {‖∆ r_{i + 1}‖}_{μ}^{2} e^{2 μa}

.(22)

由(21)，有

e^{2 μa} G {‖∆ r_{i + 1} (t)‖}_{μ}^{2} \leq e^{2 μa} (ζ^{i + 1} G {‖∆ r_{0}‖}_{μ}^{2} + \frac{1 - ζ^{i + 1}}{1 - ζ} η)

.(23)

由(22)和(23)，可得

G {‖∆ r_{i + 1} (t)‖}^{2} \leq e^{2 μa} (ζ^{i + 1} G {‖Δ r_{0}‖}_{μ}^{2} + \frac{1 - ζ^{i + 1}}{1 - ζ} η),

而

0 < ζ < 1

，故

\lim_{i \to + \infty} G {‖∆ r_{i} (t)‖}^{2} \leq c_{1}, \forall \in [0, a],

(24)

其中

c_{1} = e^{2 μa} \frac{η}{1 - ζ}

。

定理1表明，随机半线性广义分布参数系统在积分解意义下的输入误差的均方收敛性依赖于重复随机半线性广义分布参数系统的随机项的值。此外，有如下推论。

推论1 在随机半线性广义分布参数系统(1)中，如果

M = 0

，(H₁)-(H₄)成立，

{‖I - PZ‖}^{2} < \frac{1}{2}

，则

\lim_{i \to + \infty} {‖∆ r_{i}‖}_{μ}^{2} = 0

。

推论1表明，如果随机半线性广义分布参数系统(1)无随机项，则所得到的半线性广义分布参数系统在

{‖I - PZ‖}^{2} < \frac{1}{2}

的条件下，输入误差在

[0, a]

上一致收敛于零。

定理2 对于随机半线性广义分布参数系统(1)及P-型学习律(4)，设(H₁)-(H₄) 成立。如果学习增益算子

P

满足(9)，则存在常数

c_{2} > 0

，使

\lim_{t \to + \infty} G {‖∆ s_{i} (t)‖}^{2} \leq c_{2}, \forall t \in [0, a] .

(25)

证明由(5)和(11)，有

G {‖∆ s_{i} (t)‖}^{2} = G {‖O ∆ h_{i} (t) + Z ∆ r_{i} (t)‖}^{2} \leq 2 G ({‖O ∆ h_{i} (t)‖}^{2} + {‖Z ∆ r_{i} (t)‖}^{2})

\leq 6 {‖O‖}^{2} c_{V}^{2} c^{2} a [({‖R‖}^{2} + c_{l}) \int_{0}^{r} G {‖∆ r_{i} (s)‖}^{2} ds + c_{l} \int_{0}^{t} G {‖∆ h_{i} (s)‖}^{2} ds] + 6 {‖O‖}^{2} c_{V}^{2} c^{2} {‖M‖}^{2} a + 2 {‖Z‖}^{2} G {‖∆ r_{i}‖}^{2} .

(26)

由

μ -

范数的定义和 (26)，有

e^{- 2 μt} G {‖∆ s_{i} (t)‖}^{2} \leq 6 {‖O‖}^{2} c_{V}^{2} c^{2} a \times

[({‖R‖}^{2} + c_{l}) \int_{0}^{t} G {‖∆ r_{i} (s)‖}^{2} ds + c_{l} \int_{0}^{t} {G ‖∆ h_{i} (t)‖}^{2} ds] e^{- 2 μt} + (6 {‖O‖}^{2} c_{V}^{2} c^{2} {‖M‖}^{2} a + 2 {‖Z‖}^{2} G {‖∆ r_{i} (t)‖}^{2}) e^{- 2 μt} .

(27)

由(15)和(27)，有

G {‖∆ s_{i}‖}_{μ}^{2} \leq \frac{3}{μ} {‖O‖}^{2} c_{V}^{2} c^{2} a \times [({‖R‖}^{2} + c_{l}) G {‖∆ r_{i}‖}_{μ}^{2} + c_{l} G {‖∆ h_{i}‖}_{μ}^{2}] + 6 {‖O‖}^{2} c_{V}^{2} c^{2} a {‖M‖}^{2} + 2 {‖Z‖}^{2} G {‖∆ r_{i}‖}_{μ}^{2}

= ζ_{1} G {‖∆ r_{i}‖}_{μ}^{2} + η_{1},

(28)

其中

ζ_{1} = \frac{3 {‖O‖}^{2} c_{V}^{2} c^{2} a ({‖R‖}^{2} + c_{l})}{μ} (1 + \frac{3 c_{V}^{2} c^{2} a c_{l}}{2 μ - 3 c_{V}^{2} c^{2} a c_{l}}) + 2 {‖Z‖}^{2}

，

η_{1} = \frac{18 {‖O‖}^{2} c_{V}^{4} c^{4} a^{2} c_{l} {‖M‖}^{2}}{2 μ - 3 c_{V}^{2} c^{2} a c_{l}} + 6 {‖O‖}^{2} c_{V}^{2} c^{2} a {‖M‖}^{2}

。

根据(28)，利用(20)的推导过程，可得

G {‖∆ s_{i}‖}_{μ}^{2} \leq ζ_{1} ζ^{2} G {‖∆ r_{0}‖}_{μ}^{2} + \frac{ζ_{1} η}{1 - ζ} + η_{1} .

(29)

从而

e^{2 μa} G {‖∆ s_{i}‖}_{μ}^{2} \leq (ζ_{1} ζ^{i} G {‖∆ r_{i}‖}_{μ}^{2} + \frac{ζ_{1} η}{1 - ζ} + η_{1}) e^{2 μa}

.(30)

由

μ -

范数的定义，有

G {‖∆ s_{i} (t)‖}^{2} \leq e^{- 2 μt} G {‖∆ s_{i} (t)‖}^{2} e^{2 μt} \leq G {‖∆ s_{i}‖}_{μ}^{2} e^{2 μt}

.(31)

由(30)和(31)，可得

G {‖∆ s_{i} (t)‖}^{2} \leq e^{2 μa} (ζ_{1} ζ^{i} G {‖∆ r_{0}‖}_{μ}^{2} + \frac{ζ_{1} η}{1 - ζ} + η_{1})

.(32)

因此，由(32)，有

\lim_{i \to + \infty} G {‖∆ s_{i} (t)‖}^{2} \leq c_{2}, \forall t \in [0, a],

(33)

其中

c_{2} = e^{2 μa} (\frac{ζ_{1} η}{1 - ζ} + η_{1})

。

定理2表明，随机半线性广义分布参数系统在积分解意义下的输出跟踪误差的均方收敛性依赖于重复随机半线性广义分布参数系统的随机项的值。此外，有如下推论。

推论2 在随机半线性广义分布参数系统(1)中，如果

M = 0

，(H₁)-(H₄)成立，

{‖I - PZ‖}^{2} < \frac{1}{2}

，则

\lim_{i \to + \infty} {‖∆ s_{i}‖}_{μ}^{2} = 0

。

推论2表明，如果随机半线性广义分布参数系统(1)无随机项，则所得到的半线性广义分布参数系统在

{‖I - PZ‖}^{2} < \frac{1}{2}

时，输出跟踪误差在

[0, a]

上一致收敛于零。

例1 根据

[6]

, 经济学中的投入产出模型可以写成随机半线性广义分布参数系统(1)的形式。如果系统(1)中的算子取如下的形式：

H = [\begin{matrix} I_{1} & 0 \\ 0 & 0 \end{matrix}], J = [\begin{matrix} - 2 I_{1} & 0 \\ 0 & I_{2} \end{matrix}], R = [\begin{matrix} {3 I}_{1} & 0 \\ 0 & {2 I}_{2} \end{matrix}], M = [\begin{matrix} I_{1} & 0 \\ 0 & 3 I_{2} \end{matrix}], O = [\begin{matrix} 2 I_{1} & 0 \\ 0 & {3 I}_{2} \end{matrix}], Z = [\begin{matrix} I_{1} & 0 \\ 0 & 2 I_{2} \end{matrix}],

l (h_{i} (t), r_{i} (t), t) = [\begin{matrix} θ \\ l_{2} (h_{i} (t), r_{i} (t), t) \end{matrix}]

，

其中

I_{1}, I_{2}

分别是可析Hilbert空间

X_{1}, X_{2}

上的恒等算子;

X = X_{1} \oplus X_{2}

\oplus

表示直和;

X = Y = U = W;

h_{i} (t) = [\begin{matrix} h_{i}^{1} (t) \\ h_{i}^{2} (t) \end{matrix}], r_{i} (t) = [\begin{matrix} r_{i}^{1} (t) \\ r_{i}^{2} (t) \end{matrix}], w_{i} (t) = [\begin{matrix} w_{i}^{1} (t) \\ w_{i}^{2} (t) \end{matrix}]

s_{i} (t) = [\begin{matrix} s_{i}^{1} (t) \\ s_{i}^{2} (t) \end{matrix}]; P = [\begin{matrix} \frac{2}{3} I_{1} & 0 \\ 0 & \frac{1}{3} I_{2} \end{matrix}]

，

θ \in X_{1}

是常向量，

l

连续且满足(2)，则可得GE-半群

V (t)

为

V (t) = [\begin{matrix} e^{- 2 t} I_{1} & 0 \\ 0 & 0 \end{matrix}]

;

γH {(γH - J)}^{- 1} \to [\begin{matrix} I_{1} & 0 \\ 0 & 0 \end{matrix}] (γ \to + \infty)

在强收敛的意义下成立。假定（H₄）成立，此时不难验证重复系统和期望系统分别有唯一的积分解，（H₁）-（H₃）成立，且

{‖I - PZ‖}^{2} = \frac{1}{9} < \frac{1}{2}

，

即(9)成立。由定理1和2，在以上条件下由随机半线性广义分布参数系统所描述的投入产出系统的输入和输出跟踪误差在均方意义下收敛到有界。由推论1和2，如果系统无随机项，则有广义分布参数系统所描述的投入产出系统的输入和输出跟踪误差在

[0, a]

上一致收敛于零。

此外，类似于文献

[24]

附录中例1，也可以给出由广义抛物型偏微分方程描述的随机半线性广义分布参数系统在本文意义下的迭代学习控制问题的收敛性来说明本文结果的有效性。对此不再做详细叙述。

4．结论

应用GE-半群理论在积分解的意义下讨论了可析Hilbert空间中随机半线性广义分布参数系统在P-型迭代学习控制下的收敛性问题。证明了输入和输出跟踪误差在一定的条件下可均方收敛到有界。表明了输入和输出跟踪误差的均方收敛依赖于重复随机半线性广义分布参数系统的随机项的值；当该系统中无随机项时，所得到的半线性广义分布参数系统的输入和输出跟踪误差一致收敛于零。

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Ge, Z. (2025). Convergence Analysis of Iterative Learning Control for Stocastic Semilinear Generalized Distributed Parameter Systems. Science Discovery, 13(5), 101-107. https://doi.org/10.11648/j.sd.20251305.14

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Ge, Z. Convergence Analysis of Iterative Learning Control for Stocastic Semilinear Generalized Distributed Parameter Systems. Sci. Discov. 2025, 13(5), 101-107. doi: 10.11648/j.sd.20251305.14

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Ge Z. Convergence Analysis of Iterative Learning Control for Stocastic Semilinear Generalized Distributed Parameter Systems. Sci Discov. 2025;13(5):101-107. doi: 10.11648/j.sd.20251305.14

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@article{10.11648/j.sd.20251305.14,
  author = {Zhaoqiang Ge},
  title = {Convergence Analysis of Iterative Learning Control for Stocastic Semilinear Generalized Distributed Parameter Systems
},
  journal = {Science Discovery},
  volume = {13},
  number = {5},
  pages = {101-107},
  doi = {10.11648/j.sd.20251305.14},
  url = {https://doi.org/10.11648/j.sd.20251305.14},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.sd.20251305.14},
  abstract = {The iterative learning control problem of generalized distributed parameter systems is one of the important issues in the research of generalized distributed parameter systems. In this paper convergence analysis of P-type iterative learning control for stochastic semilinear generalized distributed parameter systems is discussed in the sense of integral solution by GE-semigroup theory in separable Hilbert spaces. Firstly, the basic concepts of GE-semigroup and integral solution are presented, along with a description of the iterative learning control problem for stochastic semilinear generalized distributed parameter systems under P-type learning law. Secondly, some sufficient conditions are obtained, by which the input tracking and output tracking errors can converge to be bounded in the mean square sense. At the same time, we have also obtained sufficient conditions for the input tracking and output tracking errors of the semilinear generalized distributed parameter system to converge uniformly to zero.
},
 year = {2025}
}

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TY  - JOUR
T1  - Convergence Analysis of Iterative Learning Control for Stocastic Semilinear Generalized Distributed Parameter Systems

AU  - Zhaoqiang Ge
Y1  - 2025/10/29
PY  - 2025
N1  - https://doi.org/10.11648/j.sd.20251305.14
DO  - 10.11648/j.sd.20251305.14
T2  - Science Discovery
JF  - Science Discovery
JO  - Science Discovery
SP  - 101
EP  - 107
PB  - Science Publishing Group
SN  - 2331-0650
UR  - https://doi.org/10.11648/j.sd.20251305.14
AB  - The iterative learning control problem of generalized distributed parameter systems is one of the important issues in the research of generalized distributed parameter systems. In this paper convergence analysis of P-type iterative learning control for stochastic semilinear generalized distributed parameter systems is discussed in the sense of integral solution by GE-semigroup theory in separable Hilbert spaces. Firstly, the basic concepts of GE-semigroup and integral solution are presented, along with a description of the iterative learning control problem for stochastic semilinear generalized distributed parameter systems under P-type learning law. Secondly, some sufficient conditions are obtained, by which the input tracking and output tracking errors can converge to be bounded in the mean square sense. At the same time, we have also obtained sufficient conditions for the input tracking and output tracking errors of the semilinear generalized distributed parameter system to converge uniformly to zero.

VL  - 13
IS  - 5
ER  -

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Zhaoqiang Ge

School of Mathematics and Statistics, Xi’an Jiaotong University, Xi’an, China

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http://orcid.org/0000-0002-1550-096X

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Ge, Z. (2025). Convergence Analysis of Iterative Learning Control for Stocastic Semilinear Generalized Distributed Parameter Systems. Science Discovery, 13(5), 101-107. https://doi.org/10.11648/j.sd.20251305.14

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Ge, Z. Convergence Analysis of Iterative Learning Control for Stocastic Semilinear Generalized Distributed Parameter Systems. Sci. Discov. 2025, 13(5), 101-107. doi: 10.11648/j.sd.20251305.14

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Ge Z. Convergence Analysis of Iterative Learning Control for Stocastic Semilinear Generalized Distributed Parameter Systems. Sci Discov. 2025;13(5):101-107. doi: 10.11648/j.sd.20251305.14

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@article{10.11648/j.sd.20251305.14,
  author = {Zhaoqiang Ge},
  title = {Convergence Analysis of Iterative Learning Control for Stocastic Semilinear Generalized Distributed Parameter Systems
},
  journal = {Science Discovery},
  volume = {13},
  number = {5},
  pages = {101-107},
  doi = {10.11648/j.sd.20251305.14},
  url = {https://doi.org/10.11648/j.sd.20251305.14},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.sd.20251305.14},
  abstract = {The iterative learning control problem of generalized distributed parameter systems is one of the important issues in the research of generalized distributed parameter systems. In this paper convergence analysis of P-type iterative learning control for stochastic semilinear generalized distributed parameter systems is discussed in the sense of integral solution by GE-semigroup theory in separable Hilbert spaces. Firstly, the basic concepts of GE-semigroup and integral solution are presented, along with a description of the iterative learning control problem for stochastic semilinear generalized distributed parameter systems under P-type learning law. Secondly, some sufficient conditions are obtained, by which the input tracking and output tracking errors can converge to be bounded in the mean square sense. At the same time, we have also obtained sufficient conditions for the input tracking and output tracking errors of the semilinear generalized distributed parameter system to converge uniformly to zero.
},
 year = {2025}
}

Copy | Download

TY  - JOUR
T1  - Convergence Analysis of Iterative Learning Control for Stocastic Semilinear Generalized Distributed Parameter Systems

AU  - Zhaoqiang Ge
Y1  - 2025/10/29
PY  - 2025
N1  - https://doi.org/10.11648/j.sd.20251305.14
DO  - 10.11648/j.sd.20251305.14
T2  - Science Discovery
JF  - Science Discovery
JO  - Science Discovery
SP  - 101
EP  - 107
PB  - Science Publishing Group
SN  - 2331-0650
UR  - https://doi.org/10.11648/j.sd.20251305.14
AB  - The iterative learning control problem of generalized distributed parameter systems is one of the important issues in the research of generalized distributed parameter systems. In this paper convergence analysis of P-type iterative learning control for stochastic semilinear generalized distributed parameter systems is discussed in the sense of integral solution by GE-semigroup theory in separable Hilbert spaces. Firstly, the basic concepts of GE-semigroup and integral solution are presented, along with a description of the iterative learning control problem for stochastic semilinear generalized distributed parameter systems under P-type learning law. Secondly, some sufficient conditions are obtained, by which the input tracking and output tracking errors can converge to be bounded in the mean square sense. At the same time, we have also obtained sufficient conditions for the input tracking and output tracking errors of the semilinear generalized distributed parameter system to converge uniformly to zero.

VL  - 13
IS  - 5
ER  -

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