1999 年度実績報告書

粘性解理論の最適制御・微分ゲームへの応用

研究課題

研究課題/領域番号	09640242
研究機関	埼玉大学
研究代表者	小池茂昭埼玉大学, 理学部, 助教授 (90205295)
研究分担者	新居俊作埼玉大学, 理学部, 助手 (50282421) 福井敏純埼玉大学, 理学部, 助教授 (90218892) 桜井力埼玉大学, 理学部, 助教授 (40187084) 石井仁司東京都立大学, 大学院・理学研究科, 教授 (70102887)
キーワード	粘性解 / 退化楕円型方程式 / 一様楕円型方程式 / 最適制御 / 完全非線形方程式 / ハルナックの不等式 / 比較定理 / 最大値原理
研究概要	(1)状態拘束制御問題において、近似最適制御の構成を行った。従来、状態拘束制御問題では、対応する値関数(value functio)が満たすHamilton-Jacobi方程式と境界条件を決定し、粘性解としての一意性を研究が主に行われていた。ここでは、制御問題として、より重要なフィードバック(近似)最適制御を対応するHamilton-Jacobi方程式から直接的に導いた。方法は、領域近似による収束性・inf-軟化子による粘性解の近似・super-differentialの単調性公式等、粘性解理論の洗練された技巧を駆使して行われた。 (2)非発散型完全非線形二階一様楕円型偏微分方程式が1次以上の増大度を持つときの粘性解の(a)比較定理・(b)局所Holder連続性を示した。 (a)係数に関して連続性を仮定しないとき、一般には比較原理は成り立たないが、比較原理が成り立つような自然な十分条件を与えた。これは、Caffarelli,Crandall,Kocan Swiechの最近の結果の拡張である。方法は、二種類の近似とArea公式の巧妙な利用による。 (b)同じく、1次以上の増大度が比較的小さい場合に、粘性解の局所Holder連続性を示した。これは、Caffarelliの結果の拡張である。方法は、Alexandroff-Bakelman-Pucciの最大値原理の改良・古典優解の精密な構成・立方体分割法を改良して「爆発論法」が適用できるようにして行う。

研究成果

(26件)

すべてその他

すべて文献書誌 (26件)

[文献書誌] S. KOIKE: "Semicontinuous viscosity solutions for Hamilton-Jacobi equations with a degenerate coefficient"Differential and Integral Equations. 10巻3号. 455-472 (1997)
[文献書誌] S. KOIKE & I. NAKAYAMA: "Lower semicontinuous viscosity solutions for the minimum time problem"Proceedings of the Sixth Tokyo Conference on Nonlinear PDE 1997. 6巻. 1-8 (1997)
[文献書誌] S. KOIKE: "A comparison result for the state constraint problem of differential games"Proceedings of Korea-Japan Partial Differential Equations Conference. 39巻. 87-94 (1997)
[文献書誌] S. KOIKE: "Introduction to the viscosity solution theory of first order PDEs"数理解析研究所講究録. 1111巻. 107-116 (1999)
[文献書誌] O. ALVAREZ, S. KOIKE & I. .NAKAYAMA: "Uniqueness of lower semicontinuous viscosity solutions for the minimum time problem"SIAM Jouranal on Control and Optimization.
[文献書誌] M. BARDI, S. KOIKE & P. SORAVIA: "Pursuit-evasion games with state constraints: dymanic programming and discrete-time approximations"Discrete and Continuous Dynamical Systems. 6巻. 361-380 (2000)
[文献書誌] H. ISHII & S. KOIKE: "On ε-optimal controls for state constraint problems"Annales de l'Institut Henri Poincare, Analyses Non Lineaire.
[文献書誌] S. KOIKE: "ε-optimal controls for stste constraint problems"数理解析研究所講究緑.
[文献書誌] S. KOIKE: "A new method of construction of ε-optimal feedback controls from Hamilton-Jacobi equations"Proceedings of the Ninth Tokyo Conference on Nonlinear PDE 1999.
[文献書誌] S. KOIKE: "Tiny results on L^p-viscosity solutions of fully nonlinear uniformly elliptic PDEs"Proceedings of the Tenth Tokyo Conference on Nonlinear PDE 2000.
[文献書誌] K. TSUJIOKA: "Remarks on controllability of Euler-Bernoulli equations with singular bound-ary condition"Saitama Mathematical Journal. 15巻. 73-83 (1997)
[文献書誌] K. TSUJIOKA: "Euler-Bernoulli equation in a higher dimensional space with a singular bound-ary condition"Proceedings of the Fifth International Colloquium on Complex Analysis. 317-321 (1997)
[文献書誌] T. FUKUI: "An algebraic formula for a topological invariant of bifurcation of 1-parameter family of function germ"Stratifications, singularities and differential equations II Stratifications and topology of singular spaces, Travaux en cours. 55巻. 45-54 (1997)
[文献書誌] T. FUKUI: "Seeking invariants for blow-analytic equivalence"Compositio Mathematica. 105巻. 95-108 (1997)
[文献書誌] T. FUKUI, S. KOIKE & M. SHIOTA: "Modified Nash triviality of a family of zero-sets of real polynomial mappings"Annales de l'Institut Fourier, Grenoble. 48巻5号. 1395-1440 (1998)
[文献書誌] T. FUKUI, S. KOIKE & T. C. KUO: "Blow-analytic equisingularities, properties, problems and progress"Pitman Research Notes in Mathematics Series. 381巻. 8-29 (1998)
[文献書誌] T. FUKUI, J. J. NUNO BALLESTERCES & M. J. SAIA: "On the number of singularities in generic deformations of map germs"Journal London Mathematical Society. 58巻2号. 141-152 (1998)
[文献書誌] T. FUKUI: "Newton polygons and topology of real zero loci of real polynomials"Journal London Mathematical Society. 58巻3号. 545-563 (1998)
[文献書誌] S. NII: "A topological proof of stability of N-front solutions of the FitzHugh-Nagumo equations"Journal of Dynamical Differential Equations. 11巻. 525-555 (1999)
[文献書誌] D. ADALSTEINSSON, L. C. EVANS & H. ISHII: "The level set method for etching and deposition"Mathematical Models, Methods and Applications in Scieneces. 7巻. 1153-1186 (1997)
[文献書誌] H. ISHII: "Comparison results for Hamilton-Jacobi equations without growth condition on solutions from above"Applicable Analysis. 67巻. 357-372 (1997)
[文献書誌] M. ARISAWA, H. ISHII & P. L. LIONS: "Un contre-exemple pour la theorie du controle ergodique en dimension infinie"Conptes Rendus de l'Academie des Sciences, I, Mathematique. 325巻. 37-41 (1997)
[文献書誌] M. ARISAWA & H. ISHII: "Some properties of ergodic attractors for controlled dynamical systems"Discrete and Continuous Dynamical Systems. 4巻. 43-54 (1998)
[文献書誌] K. HORIE & S. KOIKE: "Homogenization of Hamilton-Jacobi equations on domains with small scale periodic structure"Indiana University Mathematics Journal. 47巻. 1011-1058 (1998)
[文献書誌] H .ISHII: "Homogenization of the Cauchy problem for Hamilton-Jacobi equations"Stochastic Analysis, Control, Optimization and Applications. 305-324 (1999)
[文献書誌] H. ISHII, G. E. PIRES & P. E. SOUGANIDIS: "Threshhold dynamics type apporoximations schemes for propagating fronts"Journal of Mathematical Society of Japan. 51巻2号. 267-308 (1999)

1999 年度 実績報告書

粘性解理論の最適制御・微分ゲームへの応用

研究代表者

小池 茂昭 埼玉大学, 理学部, 助教授 (90205295)

研究成果

[文献書誌] S. KOIKE: "Semicontinuous viscosity solutions for Hamilton-Jacobi equations with a degenerate coefficient"Differential and Integral Equations. 10巻3号. 455-472 (1997)

[文献書誌] S. KOIKE & I. NAKAYAMA: "Lower semicontinuous viscosity solutions for the minimum time problem"Proceedings of the Sixth Tokyo Conference on Nonlinear PDE 1997. 6巻. 1-8 (1997)

[文献書誌] S. KOIKE: "A comparison result for the state constraint problem of differential games"Proceedings of Korea-Japan Partial Differential Equations Conference. 39巻. 87-94 (1997)

[文献書誌] S. KOIKE: "Introduction to the viscosity solution theory of first order PDEs"数理解析研究所講究録. 1111巻. 107-116 (1999)

[文献書誌] O. ALVAREZ, S. KOIKE & I. .NAKAYAMA: "Uniqueness of lower semicontinuous viscosity solutions for the minimum time problem"SIAM Jouranal on Control and Optimization.

[文献書誌] M. BARDI, S. KOIKE & P. SORAVIA: "Pursuit-evasion games with state constraints: dymanic programming and discrete-time approximations"Discrete and Continuous Dynamical Systems. 6巻. 361-380 (2000)

[文献書誌] H. ISHII & S. KOIKE: "On ε-optimal controls for state constraint problems"Annales de l'Institut Henri Poincare, Analyses Non Lineaire.

[文献書誌] S. KOIKE: "ε-optimal controls for stste constraint problems"数理解析研究所講究緑.

[文献書誌] S. KOIKE: "A new method of construction of ε-optimal feedback controls from Hamilton-Jacobi equations"Proceedings of the Ninth Tokyo Conference on Nonlinear PDE 1999.

[文献書誌] S. KOIKE: "Tiny results on L^p-viscosity solutions of fully nonlinear uniformly elliptic PDEs"Proceedings of the Tenth Tokyo Conference on Nonlinear PDE 2000.

[文献書誌] K. TSUJIOKA: "Remarks on controllability of Euler-Bernoulli equations with singular bound-ary condition"Saitama Mathematical Journal. 15巻. 73-83 (1997)

[文献書誌] K. TSUJIOKA: "Euler-Bernoulli equation in a higher dimensional space with a singular bound-ary condition"Proceedings of the Fifth International Colloquium on Complex Analysis. 317-321 (1997)

[文献書誌] T. FUKUI: "An algebraic formula for a topological invariant of bifurcation of 1-parameter family of function germ"Stratifications, singularities and differential equations II Stratifications and topology of singular spaces, Travaux en cours. 55巻. 45-54 (1997)

[文献書誌] T. FUKUI: "Seeking invariants for blow-analytic equivalence"Compositio Mathematica. 105巻. 95-108 (1997)

[文献書誌] T. FUKUI, S. KOIKE & M. SHIOTA: "Modified Nash triviality of a family of zero-sets of real polynomial mappings"Annales de l'Institut Fourier, Grenoble. 48巻5号. 1395-1440 (1998)

[文献書誌] T. FUKUI, S. KOIKE & T. C. KUO: "Blow-analytic equisingularities, properties, problems and progress"Pitman Research Notes in Mathematics Series. 381巻. 8-29 (1998)

[文献書誌] T. FUKUI, J. J. NUNO BALLESTERCES & M. J. SAIA: "On the number of singularities in generic deformations of map germs"Journal London Mathematical Society. 58巻2号. 141-152 (1998)

[文献書誌] T. FUKUI: "Newton polygons and topology of real zero loci of real polynomials"Journal London Mathematical Society. 58巻3号. 545-563 (1998)

[文献書誌] S. NII: "A topological proof of stability of N-front solutions of the FitzHugh-Nagumo equations"Journal of Dynamical Differential Equations. 11巻. 525-555 (1999)

[文献書誌] D. ADALSTEINSSON, L. C. EVANS & H. ISHII: "The level set method for etching and deposition"Mathematical Models, Methods and Applications in Scieneces. 7巻. 1153-1186 (1997)

[文献書誌] H. ISHII: "Comparison results for Hamilton-Jacobi equations without growth condition on solutions from above"Applicable Analysis. 67巻. 357-372 (1997)

[文献書誌] M. ARISAWA, H. ISHII & P. L. LIONS: "Un contre-exemple pour la theorie du controle ergodique en dimension infinie"Conptes Rendus de l'Academie des Sciences, I, Mathematique. 325巻. 37-41 (1997)

[文献書誌] M. ARISAWA & H. ISHII: "Some properties of ergodic attractors for controlled dynamical systems"Discrete and Continuous Dynamical Systems. 4巻. 43-54 (1998)

[文献書誌] K. HORIE & S. KOIKE: "Homogenization of Hamilton-Jacobi equations on domains with small scale periodic structure"Indiana University Mathematics Journal. 47巻. 1011-1058 (1998)

[文献書誌] H .ISHII: "Homogenization of the Cauchy problem for Hamilton-Jacobi equations"Stochastic Analysis, Control, Optimization and Applications. 305-324 (1999)

[文献書誌] H. ISHII, G. E. PIRES & P. E. SOUGANIDIS: "Threshhold dynamics type apporoximations schemes for propagating fronts"Journal of Mathematical Society of Japan. 51巻2号. 267-308 (1999)

1999 年度実績報告書

小池茂昭埼玉大学, 理学部, 助教授 (90205295)