Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

Optimal stochastic switching and the Dirichlet problem for the Bellman equation


Authors: Lawrence C. Evans and Avner Friedman
Journal: Trans. Amer. Math. Soc. 253 (1979), 365-389
MSC: Primary 93E20
MathSciNet review: 536953
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Let $ {L^i}$ be a sequence of second order elliptic operators in a bounded n-dimensional domain $ \Omega $, and let $ {f^i}$ be given functions. Consider the problem of finding a solution u to the Bellman equation $ {\sup _i}({L^i}u\, - \,{f^i})\, = \,0$ a.e. in $ \Omega $, subject to the Dirichlet boundary condition $ u\, = \,0$ on $ \partial \Omega $. It is proved that, provided the leading coefficients of the $ {L^i}$ are constants, there exists a unique solution u of this problem, belonging to $ {W^{1,\infty }}(\Omega )\, \cap \,W_{{\text{loc}}}^{2,\infty }(\Omega )$. The solution is obtained as a limit of solutions of certain weakly coupled systems of nonlinear elliptic equations; each component of the vector solution converges to u. Although the proof is entirely analytic, it is partially motivated by models of stochastic control. We solve also certain systems of variational inequalities corresponding to switching with cost.


References [Enhancements On Off] (What's this?)

  • [1] Viorel Barbu, Nonlinear semigroups and differential equations in Banach spaces, Editura Academiei Republicii Socialiste România, Bucharest; Noordhoff International Publishing, Leiden, 1976. Translated from the Romanian. MR 0390843
  • [2] Alain Bensoussan and Jacques-Louis Lions, Contrôle impulsionnel et systèmes d’inéquations quasi variationnelles, C. R. Acad. Sci. Paris Sér. A 278 (1974), 747–751 (French). MR 0341246
  • [3] A. Bensoussan and J.-L. Lions, Applications des inéquations variationnelles en contrôle stochastique, Dunod, Paris, 1978 (French). Méthodes Mathématiques de l’Informatique, No. 6. MR 0513618
  • [4] Jean-Michel Bony, Principe du maximum dans les espaces de Sobolev, C. R. Acad. Sci. Paris Sér. A-B 265 (1967), A333–A336 (French). MR 0223711
  • [5] H. Brézis and L. C. Evans, A variational inequality approach to the Bellman-Dirichlet equation for two elliptic operators, Arch. Rational Mech. Anal. 71 (1979), no. 1, 1–13. MR 522704, 10.1007/BF00250667
  • [6] Haïm Brézis and David Kinderlehrer, The smoothness of solutions to nonlinear variational inequalities, Indiana Univ. Math. J. 23 (1973/74), 831–844. MR 0361436
  • [7] Haïm Brézis and Walter A. Strauss, Semi-linear second-order elliptic equations in 𝐿¹, J. Math. Soc. Japan 25 (1973), 565–590. MR 0336050
  • [8] Maurizio Chicco, Principio di massimo per soluzioni di equazioni ellittiche del secondo ordine di tipo Cordes, Ann. Mat. Pura Appl. (4) 100 (1974), 239–258 (Italian, with English summary). MR 0377261
  • [9] Lawrence C. Evans, A convergence theorem for solutions of nonlinear second-order elliptic equations, Indiana Univ. Math. J. 27 (1978), no. 5, 875–887. MR 503721, 10.1512/iumj.1978.27.27059
  • [10] Lawrence C. Evans and Pierre-Louis Lions, Deux résultats de régularité pour le problème de Bellman-Dirichlet, C. R. Acad. Sci. Paris Sér. A-B 286 (1978), no. 13, A587–A589 (French, with English summary). MR 494517
  • [11] Avner Friedman, Stochastic differential equations and applications. Vol. 2, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1976. Probability and Mathematical Statistics, Vol. 28. MR 0494491
  • [12] N. V. Krylov, A certain estimate from the theory of stochastic integrals, Teor. Verojatnost. i Primenen. 16 (1971), 446–457 (Russian, with English summary). MR 0298792
  • [13] -, Control of a solution of a stochastic integral equation, Theor. Probability Appl. 17 (1972), 114-130.
  • [14] Makiko Nisio, Remarks on stochastic optimal controls, Japan. J. Math. (N.S.) 1 (1975/76), no. 1, 159–183. MR 0446697
  • [15] Stanley R. Pliska, A semigroup representation of the maximum expected reward vector in continuous parameter Markov decision theory, SIAM J. Control 13 (1975), no. 6, 1115–1129. MR 0395862
  • [16] Keniti Sato, On the generators of non-negative contraction semigroups in Banach lattices, J. Math. Soc. Japan 20 (1968), 423–436. MR 0231243
  • [17] Eugenio Sinestrari, Accretive differential operators, Boll. Un. Mat. Ital. B (5) 13 (1976), no. 1, 19–31 (English, with Italian summary). MR 0425682
  • [18] P. E. Sobolevskiĭ, On equations with operators forming an acute angle, Dokl. Akad. Nauk SSSR (N.S.) 116 (1957), 754–757. MR 0097727

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 93E20

Retrieve articles in all journals with MSC: 93E20


Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1979-0536953-4
Keywords: Bellman equation, elliptic operators, nonlinear elliptic equation, accretive operators, stochastic differential equations
Article copyright: © Copyright 1979 American Mathematical Society