Theory of abstract partial differential equations -- as a method of analyzing population models

• Project/Area Number: 17K05273
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Basic analysis
• Research Institution: Kanazawa University
• Principal Investigator: Kato Nobuyuki 金沢大学, 電子情報通信学系, 教授 (40177423)
• Project Period (FY): 2017-04-01 – 2022-03-31
• Project Status: Completed (Fiscal Year 2021)
• Budget Amount: ¥4,160,000 (Direct Cost: ¥3,200,000、Indirect Cost: ¥960,000); Fiscal Year 2021: ¥780,000 (Direct Cost: ¥600,000、Indirect Cost: ¥180,000); Fiscal Year 2020: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000); Fiscal Year 2019: ¥780,000 (Direct Cost: ¥600,000、Indirect Cost: ¥180,000); Fiscal Year 2018: ¥780,000 (Direct Cost: ¥600,000、Indirect Cost: ¥180,000); Fiscal Year 2017: ¥780,000 (Direct Cost: ¥600,000、Indirect Cost: ¥180,000)
• Keywords: 抽象偏微分方程式 / 個体数変動モデル / 個体数変動 / 測度値最適制御 / 最適制御 / 測度値制御 / 線形化安定性 / 最適制御問題 / 解析学

Outline of Final Research Achievements

For abstract partial differential equations with time delay in Banach spaces, the linearized stability of steady states is shown, which is applicable to age- or size-structured population models with spatial diffusion and time delay.
For optimal harvesting problems for size-structured population models with spatial diffusion, the existence of an optimal control and an optimality condition are shown. In addition, the existence of a measure-valued optimal control is shown for age-structured harvesting models.

Academic Significance and Societal Importance of the Research Achievements

研究成果の学術的意義の一つは、抽象的偏微分方程式の理論を展開することにより，時間，場所，サイズという３種類の変数を持つ数理モデルの解析が可能となったことである。従来の抽象的発展方程式の理論は，時間と場所，または時間とサイズといった２種類の変数を持つ数理モデルに対し有効であったことと比較して，より応用の範囲が広がったと言える。
もう一つは，最適収穫問題について得られた結果は，農業や水産業における収穫量や利益を最大にするため制御モデルからくる問題で，最適解であるための必要条件や解の存在条件を与えることができた。

Research Products

• [Int'l Joint Research] Prairie View A&M University/Dunham College of Business(米国)
• [Int'l Joint Research] Prairie View A&M University/Houston Baptist University(米国)
• [Int'l Joint Research] Prairie View A&M University/Houston Baptist University(米国)
• [Journal Article] Existence of measure-valued solutions in optimal control age-structured populations (2021)
  • Author(s): N. Hritonenko, N. Kato, and Y. Yatsenko
  • Journal Title: Applicable Analysis Volume: - Issue: 5 Pages: 1281-1293
  • DOI: 10.1080/00036811.2021.1981876
  • Peer Reviewed
• [Journal Article] Dirac Delta-Function in Optimal Control of Age-Structured Populations (2020)
  • Author(s): Natali Hritonenko, Nobuyuki Kato, Yuri Yatsenko
  • Journal Title: Proceeding of 12th International ISAAC Congress Volume: -
  • Peer Reviewed / Int'l Joint Research
• [Journal Article] Optimal harvesting for linear size-structured population models with diffusion (2017)
  • Author(s): Nobuyuki Kato
  • Journal Title: Journal of Nonlinear and Convex Analysis Volume: 18 Pages: 1335-1348
  • Peer Reviewed
• [Presentation] Linearized stability for abstract size-structured equations with delay (2019)
  • Author(s): Nobuyuki Kato
  • Organizer: Equadiff2019
  • Int'l Joint Research
• [Presentation] Linearized stability for abstract age-structured population equations with delay in Banach spaces (2018)
  • Author(s): Nobuyuki Kato
  • Organizer: The 6th Asian Conference on Nao-Asia 2018
  • Int'l Joint Research / Invited
• [Presentation] Optimal harvesting for size-structured population models with spatial diffusion (2017)
  • Author(s): Nobuyuki Kato
  • Organizer: Equadiff 2017
  • Int'l Joint Research