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1995 | 115 | 3 | 291-310
Tytuł artykułu

Second order unbounded parabolic equations in separated form

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Języki publikacji
EN
Abstrakty
EN
We prove existence and uniqueness of viscosity solutions of Cauchy problems for fully nonlinear unbounded second order Hamilton-Jacobi-Bellman-Isaacs equations defined on the product of two infinite-dimensional Hilbert spaces H'× H'', where H'' is separable. The equations have a special "separated" form in the sense that the terms involving second derivatives are everywhere defined, continuous and depend only on derivatives with respect to x'' ∈ H'', while the unbounded terms are of first order and depend only on derivatives with respect to x' ∈ H'.
Słowa kluczowe
Kategorie tematyczne
Czasopismo
Rocznik
Tom
115
Numer
3
Strony
291-310
Opis fizyczny
Daty
wydano
1995
otrzymano
1994-12-05
Twórcy
autor
  • Department of Mathematics, University of California, Santa Barbara, California 93106, U.S.A.
  • School of Mathematics, Georgia Institute of Technology, Atlanta, Georgia 30332, U.S.A.
Bibliografia
  • [1] V. Barbu, Nonlinear Semigroups and Differential Equations in Banach Spaces, Noordhoff, Leyden, 1976.
  • [2] V. Barbu, The dynamic programming equation for the time optimal control problem in infinite dimensions, SIAM J. Control Optim. 29 (1991), 445-456.
  • [3] H. Brézis, Opérateurs Maximaux Monotones et Semi-groupes de Contractions dans les Espaces de Hilbert, North-Holland, Amsterdam, 1973.
  • [4] P. Cannarsa, F. Gozzi and H. M. Soner, A dynamic programming approach to nonlinear boundary control problems of parabolic type, J. Funct. Anal. 117 (1993), 25-61.
  • [5] M. G. Crandall, H. Ishii and P. L. Lions, User's guide to viscosity solutions of second order partial differential equations, Bull. Amer. Math. Soc. 27 (1992), 1-67.
  • [6] M. G. Crandall, M. Kocan and A. Święch, On partial sup-convolutions, a lemma of P. L. Lions and viscosity solutions in Hilbert spaces, Adv. Math. Sci. Appl. 3 (1993/4), 1-15.
  • [7] M. G. Crandall and P. L. Lions, Hamilton-Jacobi equations in infinite dimensions, Part IV. Unbounded linear terms, J. Funct. Anal. 90 (1990), 237-283.
  • [8] M. G. Crandall and P. L. Lions, Hamilton-Jacobi equations in infinite dimensions, Part V. B-continuous solutions, ibid. 97 (1991), 417-465.
  • [9] M. G. Crandall and P. L. Lions, Hamilton-Jacobi equations in infinite dimensions, Part VI. Nonlinear A and Tataru's method refined, in: Evolution Equations, Control Theory, and Biomathematics, Proc. Third International Workshop-Conference on Evolution Equations, Control Theory, and Biomathematics, Han-sur-Lesse, P. Clément and G. Lumer (eds.), Lecture Notes in Pure and Appl. Math. 155, Marcel Dekker, New York, 1994, 51-89.
  • [10] M. G. Crandall and P. L. Lions, Viscosity solutions of Hamilton-Jacobi equations in infinite dimensions, Part VII. The HJB equation is not always satisfied, J. Funct. Anal. 125 (1994), 111-148.
  • [11] I. Ekeland and G. Lebourg, Generic Fréchet differentiability and perturbed optimization problems in Banach spaces, Trans. Amer. Math. Soc. 224 (1976), 193-216.
  • [12] H. Ishii, Viscosity solutions for a class of Hamilton-Jacobi equations in Hilbert spaces, J. Funct. Anal. 105 (1992), 301-341.
  • [13] H. Ishii, Viscosity solutions of nonlinear second-order partial differential equations in Hilbert spaces, Comm. Partial Differential Equations 18 (1993), 601-651.
  • [14] M. Kocan, Some aspects of the theory of viscosity solutions of fully nonlinear partial differential equations in infinite dimensions, Thesis, UCSB, 1994.
  • [15] M. Kocan and A. Święch, Perturbed optimization on product spaces, Nonlinear Anal., to appear.
  • [16] P. L. Lions, Viscosity solutions of fully nonlinear second-order equations and optimal stochastic control in infinite dimensions. Part III. Uniqueness of viscosity solutions of general second order equations, J. Funct. Anal. 86 (1989), 1-18.
  • [17] P. L. Lions, Viscosity solutions of fully nonlinear second-order equations and optimal stochastic control in infinite dimensions. Part II. Optimal control of Zakai's equation, in: Stochastic Partial Differential Equations and Applications II, Proc. International Conference on Infinite Dimensional Stochastic Differential Equations, Trento, G. Da Prato and L. Tubaro (eds.), Lecture Notes in Math. 1390, Springer, Berlin, 1989, 147-170.
  • [18] H. M. Soner, On the Hamilton-Jacobi-Bellman equations in Banach spaces, J. Optim. Theory Appl. 57 (1988), 429-437.
  • [19] C. Stegall, Optimization of functions on certain subsets of Banach spaces, Math. Ann. 236 (1978), 171-176.
  • [20] A. Święch, Viscosity solutions of fully nonlinear partial differential equations with "unbounded" terms in infinite dimensions, Thesis, UCSB, 1993.
  • [21] A. Święch, Unbounded second order partial differential equations in infinite dimensional Hilbert spaces, Comm. Partial Differential Equations 19 (1994), 1999-2036.
  • [22] D. Tataru, Viscosity solutions for Hamilton-Jacobi equations with unbounded nonlinear terms, J. Math. Anal. Appl. 163 (1992), 345-392.
  • [23] D. Tataru, Viscosity solutions for the dynamic programming equations, Appl. Math. Optim. 25 (1992), 109-126.
  • [24] D. Tataru, Convergence results for Hamilton-Jacobi equations with unbounded nonlinear terms, in: Differential Equations and Control Theory, V. Barbu (ed.), Pitman Res. Notes in Math. 250, Longman, New York, 1991, 324-334.
  • [25] D. Tataru, Viscosity solutions for Hamilton-Jacobi equations with unbounded nonlinear term: a simplified approach, J. Differential Equations 111 (1994), 123-146.
  • [26] D. Tataru, On the equivalence between the dynamic programming principle and the dynamic programming equation, in: Estimation and Control of Distributed Parameter Systems, Proc. International Conference on Control and Estimation of Distributed Parameter Systems, Vorau, 1990, W. Desch, F. Kappel and K. Kunisch (eds.), Internat. Ser. Numer. Math. 100, Birkhäuser, Basel, 1991, 331-340.
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Bibliografia
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