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1999 Fiscal Year Final Research Report Summary

Applications to the optimal control and differential game via the viscosity solution theory

Research Project

Project/Area Number 09640242
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeSingle-year Grants
Section一般
Research Field General mathematics (including Probability theory/Statistical mathematics)
Research InstitutionSaitama University

Principal Investigator

KOIKE Shigeaki  Saitama Univ. Mathematics Associate Professor, 理学部, 助教授 (90205295)

Co-Investigator(Kenkyū-buntansha) NII Shunsaku  Saitama Univ. Mathematics Research Associate, 理学部, 助手 (50282421)
FUKUI Toshizumi  Saitama Univ. Mathematics Associate Professor, 理学部, 助教授 (90218892)
SAKURAI Tsutomu  Saitama Univ. Mathematics Associate Professor, 理学部, 助教授 (40187084)
TSUJIOKA Kunio  Saitama Univ. Mathematics Professor, 理学部, 教授 (30012412)
ISHII Hitoshi  Tokyo Metropolitan Univ. Mathematics Professor, 大学院・理学研究科, 教授 (70102887)
Project Period (FY) 1997 – 1999
Keywordsviscosity solutions / degenerate elliptic equation / optimal control / differential game / state constraint / dynamic programming / Hamilton-Jacobi-Bellman equation / Isaacs equation
Research Abstract

(1) We obtain the uniqueness and representation formula for lower semicontinuous (lsc for short) viscosity solutions of Hamilton-Jacobi (HJ for short) equations with degenerate coefficients.
This is a "lsc" version of a result of Ishii and Ramaswamy.
(2) We show the uniqueness of viscosity solutions of pursuit-evasion games of the state constraint (SC for short) problem and the convergence of time-discrete approximations to the value function. These results generalize those of Alziary.
(3) We show the uniqueness of lsc viscosity solutions for the minimum time problem. To show the equivalence between the Dirichlet type boundary condition and the SC type one, we prove in general that a function is a lsc viscosity solution of the associated PDE if and only if it satisfies the Dynamic Programming Principle. This extends the corresponding result by P.-L. Lions for continuous viscosity solutions.
(4) We construct ε-optimal feedback controls for the SC problem. To construct them, we use the information from the associated HJ equation. Furthermore, we need the inf-convolution approximations, the monotone formula for super-differentials and the convergence of value functions for SC problems of smaller domains.
(5) We obtain (a) the comparison principle and (b) interior Holder continuity of viscosity solutions of fully nonlinear, second-order, uniformly elliptic PDEs which involve superlinear growth terms for the gradient. The result (a) extends that of Caffarelli, Crandall, Kocan and Swiech while the result (b) generalizes that of Caffarelli.
To show (b), we need a generalized Alexandroff-Bakelman-Pucci maximum principle, a precise construction of classical supersolutions of the associated extremal PDEs, and a modified cube decomposition lemma.

  • Research Products

    (50 results)

All Other

All Publications (50 results)

  • [Publications] S.Koike: "Semicontinuous viscosity solutions for Hamilton-Jacobi eqautions with a degenerate coefficient"Differential and Integral Equations. 10巻3号. 455-472 (1997)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] 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)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] 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)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] 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)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] O.Alvarez,S.Koike & I.Nakayama: "Uniqueness of lower semicontinuous viscosity solutions for the minimum time problem"SIAM Jouranal on Control and Optimization. (掲載決定).

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] 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)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] H.Ishii & S.Koike: "Onε-optimal controls for state constraint problems"Annales de l'Institut Henri Poincare, Analyses Non Lineaire. (掲載決定).

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] S.Koike: "ε-optimal controls for state constraint problems"数理解析研究所講究録. (掲載決定).

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] S.Koike: "A new method of construction ofε-optimal feedback controls from Hamilton-Jacobi equations"Proceedings of the Ninth Tokyo Conference on Nonlinear PDE 1999. (掲載決定).

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] 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. (掲載決定).

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] K.Tsujioka: "Remarks on controllability of Euler-Bernoulli equations with singular boundary condition"Saitama Mathematical Journal. 15巻. 73-83 (1997)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] K.Tsujioka: "Euler-Bernoulli equation in a higher dimensional space with a singular boundary condition"Proceedings of the Fifth International Colloquium on Complex Analysis. 317-321 (1997)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] T.Fikui: "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)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] T.Fukui: "Seeking invariants for blow-analytic equivalence"Compositio Mathematica. 105巻. 95-108 (1997)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] 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)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] 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)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] 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)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] T.Fukui: "Newton polygons and topology of real zero loci of real polynomials"Journal London Mathematical Society. 58巻3号. 545-563 (1998)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] 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)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] D.Adalsteinsson,L.C.Evans & H.Ishii: "The level set method for etching and deposition"Mathematical Models, Methods and Applications in Sciences. 7巻. 1153-1186 (1997)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] H.Ishii: "Comparison results for Hamilton-Jacobi equations without growth condition on solutions from above"Applicable Analysis. 67巻. 357-372 (1997)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] 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)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] M.Arisawa & H.Ishii: "Some Properties of ergodic attractors for controlled dynamical systems"Discrete and Continuous Dynamical Systems. 4巻. 43-54 (1998)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] 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)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] H.Ishii: "Homogenization of the Cauchy problem for Hamilton-Jacobi equations"Stochastic Analysis Control, Optimization and Applications. 305-324 (1999)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] H.Ishii: "G.E.Pires & P.E.Souganidis,Threshhold dynamics type approximations schemes for propagating fronts"Journal of Mathematical Society of Japan. 51巻2号. 267-308 (1999)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] S. KOIKE: "Semicontinuous viscosity solutions for Hamilton-Jacobi equations with a degenerate coefficient"Differential and Integral Equations. 10(3). 455-472 (1997)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] S. KOIKE & I. NAKAYAMA: "Lower semicontinuous viscosity solutions for the minimum time problem"Proceedings of the Sixth Tokyo Conference on Nonlinear PDE 1997. 6. 1-8 (1997)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] S. KOIKE: "A comparison result for the state constraint problem of differential games"Proceedings of Korea-Japan Partial Differential Equations Conference. 39. 87-94 (1997)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] S. KOIKE: "Introduction to the viscosity solution theory of first order PDEs"Surikaisekikenkyusho Kokyuroku. 1111. 107-116 (1999)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] O. ALVAREZ, S. KOIKE & I. NAKAYAMA: "Uniqueness of lower semicontinuous viscosity solutions for the minimum time problem"SIAM Journal on Control and Optimization.. (to appear).

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] M. BARDI, S. KOIKE & P. SORAVIA: "Pursuit-evasion games with state constraints : dynamic programming and discrete-time approximations"Discrete and Continuous Dynamical Systems. 6. 361-380 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] H. ISHII & S. KOIKE: "On ε-optimal controls for state constraint problems"Annales de l'Institut Henri Poincare, Analyses Non Lineaire. (to appear).

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] S. KOIKE: "ε-optimal controls for state constraint problems"Surikaisekikenkyusho Kokyuroku. (to appear).

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] K. TSUJIOKA: "Remarks on controllability of Euler-Bernoulli equations with singular boundary condition"Saitama Mathematical Journal. 15. 73-83 (1997)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] K. TSUJIOKA: "Euler-Bernoulli equation in a higher dimensional space with a singular boundary condition"Proceedings of the Fifth International Colloquium on Complex Analysis. 317-321 (1997)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] 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. 45-54 (1997)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] T. FUKUI: "Seeking invariants for blow-analytic equivalence"Compositio Mathematica. 105. 95-108 (1997)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] 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(5). 1395-1440 (1998)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] T. FUKUI, S. KOIKE & T.C. KUO: "Blow-analytic equisingularities, properties, problems and progress"Pitman Research Motes in Mathematics Series. 381. 8-29 (1998)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] 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(2). 141-152 (1998)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] T. FUKUI: "Newton polygons and topology of real zero loci of real polynomials"Journal London Mathematical Society. 58(3). 545-563 (1998)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] D.ADALSTEINSSON, L.C.EVANS & H.ISHII: "The level set method for etching and deposition, Mathematical Models"Methods and Applications in Sciences. 7. 1153-1186 (1997)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] H.ISHII: "Comparison results for Hamilton-Jacobi equations without growth condition on solutions from above"Applicable Analysis. 67. 357-372 (1997)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] 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. 37-41 (1997)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] M.ARISAWA & H.ISHII: "Some properties of ergodic attractors for controlled dynamical systems"Discrete and continuous Dynamical Systems. 4. 43-54 (1998)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] K. HORIE & S.KOIKE: "Homogenization of Hamilton-Jacobi equations on domains with small scale periodic structure"Indiana University Mathematics Journal. 47. 1011-1058 (1998)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] H.ISHII: "Homogenization of the Cauchy problem for Hamilton-Jacobi equations"Stochastic Analysis, Control, Optimization and Applications. 305-324 (1999)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] H.ISHII, G.E.PIRES & P.E.SOUGANIDIS: "Threshold dynamics type approximations schemes for propagating fronts"Journal of Mathematical Society of Japan. 51(2). 267-308 (1999)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] S.NII: "A topological proof of stability of N-front solutions of the FitzHugh-Nagumo equations"Journal of Dynamical Differential Equations. 11. 525-555 (1999)

    • Description
      「研究成果報告書概要(欧文)」より

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Published: 2001-10-23  

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