Second order Hamilton-Jacobi equations in Hilbert spaces and stochastic boundary control

SIAM Journal on Control and Optimization

Volumn 38, Issue 2, 2000, Pages 400-430

  Gozzi, Fausto ^a   Rouy, Elisabeth ^b,c   Świȩch, Andrzej ^d  

^a UNIVERSITY OF PISA   (Italy)

^b CNRS   (France)

^c UNIVERSITÉ DE TOURS   (France)

^d GEORGIA INSTITUTE OF TECHNOLOGY   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

CONTROL SYSTEM ANALYSIS; NONLINEAR EQUATIONS; OPTIMAL CONTROL SYSTEMS;

HILBERT SPACES; NONLINEAR SECOND ORDER HAMILTON-JACOBI-BELLMAN-ISAACS EQUATIONS; STOCHASTIC BOUNDARY CONTROL;

STOCHASTIC CONTROL SYSTEMS;

EID: 0033894023     PISSN: 03630129     EISSN: None     Source Type: Journal    
DOI: 10.1137/S0363012997324909     Document Type: Article

References (57)

1. S. ALBEVERIO AND M. RÖCKNER, Stochastic differential equations in infinite dimensions: solutions via Dirichlet forms, Probab. Theory Related Fields, 89 (1991), pp. 347-86.
2. V. BARBU AND G. DA PRATO, Hamilton-Jacobi Equations in Hubert Spaces, Res. Notes Math., Pitman, Boston, 1983.
3. A. BENSOUSSAN, G. DA PRATO, M. C. DELFOUR, AND S. K. MITTER, Representation and Control of Infinite Dimensional Systems, Birkhäuser, Boston, Cambridge, MA, 1992.
4. P. CANNARSA AND G. DA PRATO, Direct solution of a second order Hamilton-Jacobi equation in Hilbert spaces, in Stochastic Partial Differential Equations and Applications, G. Da Prato and L. Tubaro, eds., Pitman Res. Notes in Math. 268, Longman, Harlow, UK, 1992, pp. 72-85.
5. P. CANNARSA AND G. DA PRATO, On a functional analysis approach to parabolic equations in infinite dimensions, J. Funct. Anal., 118 (1993), pp. 22-42.
6. P. CANNARSA AND G. DA PRATO, Second order elliptic and parabolic equations with infinitely many variables, in Proceedings of the Amer. Math. Soc. Conference in honor of Lax and Nirenberg, Venice, 1996.
7. P. CANNARSA AND G. DA PRATO, Second-order Hamilton-Jacobi equations in infinite dimensions, SIAM J. Control Optim., 29 (1991), pp. 474-492.
8. P. CANNARSA, F. GOZZI, AND H. M. SONER, A dynamic programming approach to nonlinear boundary control problems of parabolic type, J. Funct. Anal., 117 (1992), pp. 25-61.
9. P. CANNARSA AND M. E. TESSITORE, Cauchy problem for the dynamic programming equation of boundary control, in Proceedings IFIP Workshop on Boundary Control and Boundary Variation, Marcel Dekker, New York, Basel, 1993.
10. P. CANNARSA AND M. E. TESSITORE, Infinite-dimensional Hamilton-Jacobi equations and Dirichlet boundary control problems of parabolic type, SIAM J. Control Optim., 34 (1996), pp. 1831-1847.
11. P. L. CHOW AND J. L. MENALDI, Infinite dimensional Hamilton-Jacobi-Bellman equations in Gauss-Sobolev spaces, Nonlinear Anal., 27 (1997), pp. 415-426.
12. 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), pp. 1-67.
13. M. G. CRANDALL, M. KOCAN, AND A. ŚWIȨCH, On partial sup-convolutions, a lemma of P.-L. Lions and viscosity solutions in Hilbert spaces, Adv. Math. Sci. Appl., 3 (1993/4), pp. 1-15.
14. M. G. CRANDALL AND P.-L. LIONS, Hamilton-Jacobi equations in infinite dimensions. Part I: Uniqueness of viscosity solutions, J. Funct. Anal., 62 (1985), pp. 379-396. Part II: Existence of viscosity solutions, J. Funct. Anal., 65 (1986), pp. 368-405. Part III, J. Funct. Anal., 68 (1986), pp. 214-247. Part IV: Hamiltonians with unbounded linear terms, J. Funct. Anal, 90 (1990), pp. 237-283. Part V: Unbounded linear terms and B-continuous solutions, J. Funct. Anal., 97 (1991), pp. 417-465. Part VI: Nonlinear A and Tataru's method refined, Evolution Equations, Control Theory, and Biomathematics (Han sur Lesse, 1991), Lecture Notes in Pure and Appl. Math. 155, Dekker, New York, 1994, pp. 51-89. Part VII: The HJB equation is not always satisfied, J. Funct. Anal., 125 (1994), pp. 111-148.
15. M. G. CRANDALL AND P.-L. LIONS, Hamilton-Jacobi equations in infinite dimensions. Part I: Uniqueness of viscosity solutions, J. Funct. Anal., 62 (1985), pp. 379-396. Part II: Existence of viscosity solutions, J. Funct. Anal., 65 (1986), pp. 368-405. Part III, J. Funct. Anal., 68 (1986), pp. 214-247. Part IV: Hamiltonians with unbounded linear terms, J. Funct. Anal, 90 (1990), pp. 237-283. Part V: Unbounded linear terms and B-continuous solutions, J. Funct. Anal., 97 (1991), pp. 417-465. Part VI: Nonlinear A and Tataru's method refined, Evolution Equations, Control Theory, and Biomathematics (Han sur Lesse, 1991), Lecture Notes in Pure and Appl. Math. 155, Dekker, New York, 1994, pp. 51-89. Part VII: The HJB equation is not always satisfied, J. Funct. Anal., 125 (1994), pp. 111-148.
16. M. G. CRANDALL AND P.-L. LIONS, Hamilton-Jacobi equations in infinite dimensions. Part I: Uniqueness of viscosity solutions, J. Funct. Anal., 62 (1985), pp. 379-396. Part II: Existence of viscosity solutions, J. Funct. Anal., 65 (1986), pp. 368-405. Part III, J. Funct. Anal., 68 (1986), pp. 214-247. Part IV: Hamiltonians with unbounded linear terms, J. Funct. Anal, 90 (1990), pp. 237-283. Part V: Unbounded linear terms and B-continuous solutions, J. Funct. Anal., 97 (1991), pp. 417-465. Part VI: Nonlinear A and Tataru's method refined, Evolution Equations, Control Theory, and Biomathematics (Han sur Lesse, 1991), Lecture Notes in Pure and Appl. Math. 155, Dekker, New York, 1994, pp. 51-89. Part VII: The HJB equation is not always satisfied, J. Funct. Anal., 125 (1994), pp. 111-148.
17. M. G. CRANDALL AND P.-L. LIONS, Hamilton-Jacobi equations in infinite dimensions. Part I: Uniqueness of viscosity solutions, J. Funct. Anal., 62 (1985), pp. 379-396. Part II: Existence of viscosity solutions, J. Funct. Anal., 65 (1986), pp. 368-405. Part III, J. Funct. Anal., 68 (1986), pp. 214-247. Part IV: Hamiltonians with unbounded linear terms, J. Funct. Anal, 90 (1990), pp. 237-283. Part V: Unbounded linear terms and B-continuous solutions, J. Funct. Anal., 97 (1991), pp. 417-465. Part VI: Nonlinear A and Tataru's method refined, Evolution Equations, Control Theory, and Biomathematics (Han sur Lesse, 1991), Lecture Notes in Pure and Appl. Math. 155, Dekker, New York, 1994, pp. 51-89. Part VII: The HJB equation is not always satisfied, J. Funct. Anal., 125 (1994), pp. 111-148.
18. M. G. CRANDALL AND P.-L. LIONS, Hamilton-Jacobi equations in infinite dimensions. Part I: Uniqueness of viscosity solutions, J. Funct. Anal., 62 (1985), pp. 379-396. Part II: Existence of viscosity solutions, J. Funct. Anal., 65 (1986), pp. 368-405. Part III, J. Funct. Anal., 68 (1986), pp. 214-247. Part IV: Hamiltonians with unbounded linear terms, J. Funct. Anal, 90 (1990), pp. 237-283. Part V: Unbounded linear terms and B-continuous solutions, J. Funct. Anal., 97 (1991), pp. 417-465. Part VI: Nonlinear A and Tataru's method refined, Evolution Equations, Control Theory, and Biomathematics (Han sur Lesse, 1991), Lecture Notes in Pure and Appl. Math. 155, Dekker, New York, 1994, pp. 51-89. Part VII: The HJB equation is not always satisfied, J. Funct. Anal., 125 (1994), pp. 111-148.
19. M. G. CRANDALL AND P.-L. LIONS, Hamilton-Jacobi equations in infinite dimensions. Part I: Uniqueness of viscosity solutions, J. Funct. Anal., 62 (1985), pp. 379-396. Part II: Existence of viscosity solutions, J. Funct. Anal., 65 (1986), pp. 368-405. Part III, J. Funct. Anal., 68 (1986), pp. 214-247. Part IV: Hamiltonians with unbounded linear terms, J. Funct. Anal, 90 (1990), pp. 237-283. Part V: Unbounded linear terms and B-continuous solutions, J. Funct. Anal., 97 (1991), pp. 417-465. Part VI: Nonlinear A and Tataru's method refined, Evolution Equations, Control Theory, and Biomathematics (Han sur Lesse, 1991), Lecture Notes in Pure and Appl. Math. 155, Dekker, New York, 1994, pp. 51-89. Part VII: The HJB equation is not always satisfied, J. Funct. Anal., 125 (1994), pp. 111-148.
20. M. G. CRANDALL AND P.-L. LIONS, Hamilton-Jacobi equations in infinite dimensions. Part I: Uniqueness of viscosity solutions, J. Funct. Anal., 62 (1985), pp. 379-396. Part II: Existence of viscosity solutions, J. Funct. Anal., 65 (1986), pp. 368-405. Part III, J. Funct. Anal., 68 (1986), pp. 214-247. Part IV: Hamiltonians with unbounded linear terms, J. Funct. Anal, 90 (1990), pp. 237-283. Part V: Unbounded linear terms and B-continuous solutions, J. Funct. Anal., 97 (1991), pp. 417-465. Part VI: Nonlinear A and Tataru's method refined, Evolution Equations, Control Theory, and Biomathematics (Han sur Lesse, 1991), Lecture Notes in Pure and Appl. Math. 155, Dekker, New York, 1994, pp. 51-89. Part VII: The HJB equation is not always satisfied, J. Funct. Anal., 125 (1994), pp. 111-148.
21. G. DA PRATO, Some results on Bellman equation in Hubert spaces, SIAM J. Control Optim., 23 (1985), pp. 61-71.
22. G. DA PRATO AND A. ICHIKAWA, Riccati equations with unbounded coefficient, Ann. Mat. Pura Appl., 140 (1985), pp. 209-221.
23. G. DA PRATO AND J. ZABCZYK, Stochastic Equations in Infinite Dimensions, Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, UK, 1992.
24. G. DA PRATO AND J. ZABCZYK, Ergodicity for Infinite Dimensional Systems, London Math. Soc. Lecture Note Ser. 229, Cambridge University Press, Cambridge, UK, 1996.
25. T. E. DUNCAN, B. MASLOWSKI, AND B. PASIK-DUNCAN, Adaptive boundary and point control of linear stochastic distributed parameter systems, SIAM J. Control Optim., 32 (1994), pp. 648-672.
26. T. E. DUNCAN, B. MASLOWSKI, AND B. PASIK-DUNCAN, Ergodic boundary/point control of stochastic semilinear systems, SIAM J. Control Optim., 36 (1998), pp. 1020-1047.
27. H. O. FATTORINI, Boundary control systems, SIAM J. Control Optim., 6 (1968), pp. 349-385.
28. F. FLANDOLI, Riccati equations arising in a boundary control problem with distributed parameters, SIAM J. Control Optim., 22 (1984), pp. 76-86.
29. F. FLANDOLI, A counterexample in the boundary control of parabolic systems, Appl. Math. Lett., 3 (1990), pp. 47-50.
30. W. H. FLEMING AND H. M. SONER, Controlled Markov Processes and Viscosity Solutions, Springer-Verlag, Berlin, New York, 1993.
31. F. GOZZI, Regularity of solutions of a second order Hamilton-Jacobi equation and application to a control problem, Comm. Partial Differential Equations, 20 (1995), pp. 775-826.
32. F. GOZZI, Global regular solutions of second order Hamilton-Jacobi equations in Hilbert spaces with locally Lipschitz nonlinearities, J. Math. Anal. Appl., 198 (1996), pp. 399-443.
33. F. GOZZI AND E. ROUY, Regular solutions of second order stationary Hamilton-Jacobi equations, J. Differential Equations 130 (1996), pp. 201-234.
34. T. HAVARNEANU, Existence for the dynamic programming equation of control diffusion processes in Hilbert space, Nonlinear Anal., 9 (1985), pp. 619-629.
35. D. HENRY, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Math., 840, Springer-Verlag, Berlin, 1981.
36. H. ISHII, On uniqueness and existence, of viscosity solutions of fully nonlinear second-order elliptic PDE's, Comm. Pure Appl. Math., 42 (1989), pp. 15-45.
37. H. ISHII, Viscosity solutions for a class of Hamilton-Jacobi equations in Hilbert spaces, J. Funct. Anal., 105 (1992), pp. 301-341.
38. H. ISHII, Viscosity solutions of nonlinear second-order partial differential equations in Hilbert spaces, Comm. Partial Differential Equations, 18 (1993), pp. 601-651.
39. G. JONA LASINIO AND S. K. MITTER, On the Stochastic Quantization of Field Theory, Pitman, Boston, 1982.
40. M. KOCAN AND A. ŚWIȨCH, Second order unbounded parabolic equations in separated form, Studia Math., 115 (1995), pp. 291-310.
41. N. V. KRYLOV, Controlled Diffusion Processes, Springer-Verlag, New York, 1980.
42. I. LASIECKA AND R. TRIGGIANI, Differential and Algebraic Riccati Equations with Application to Boundary/Point Control Problems: Continuous Theory and Approximation Theory, Lecture Notes in Control and Inform. Sci., 164, Springer Verlag, Berlin, 1991.
43. X. LI, J. YONG, AND X. Y. ZHOU, Stochastic verification theorems within the framework of viscosity solutions, SIAM J. Control Optim., 35 (1997), pp. 243-253.
44. P.-L. LIONS, Optimal control of diffusion processes and Hamilton-Jacobi-Bellman equations. Part 1: The dynamic programming principle and applications. Part 2: Viscosity solutions and uniqueness, Comm. Partial Differential Equations, 8 (1983), pp. 1101-1174 and pp. 1229-1276.
45. P.-L. LIONS, Viscosity solutions of fully nonlinear second-order equations and optimal stochastic control in infinite dimensions. Part I: The case of bounded stochastic evolution, Acta Math., 161 (1988), pp. 243-278. Part II: Optimal control of Zakai's equation, in Stochastic Partial Differential Equations and Applications, Lecture Notes in Math. 1390, G. Da Prato and L. Tubaro, eds. Springer-Verlag, Berlin, 1989, pp. 147-170. Part III: Uniqueness of viscosity solutions for general second order equations, J. Funct. Anal., 86 (1989), pp. 1-18.
46. P.-L. LIONS, Viscosity solutions of fully nonlinear second-order equations and optimal stochastic control in infinite dimensions. Part I: The case of bounded stochastic evolution, Acta Math., 161 (1988), pp. 243-278. Part II: Optimal control of Zakai's equation, in Stochastic Partial Differential Equations and Applications, Lecture Notes in Math. 1390, G. Da Prato and L. Tubaro, eds. Springer-Verlag, Berlin, 1989, pp. 147-170. Part III: Uniqueness of viscosity solutions for general second order equations, J. Funct. Anal., 86 (1989), pp. 1-18.
47. P.-L. LIONS, Viscosity solutions of fully nonlinear second-order equations and optimal stochastic control in infinite dimensions. Part I: The case of bounded stochastic evolution, Acta Math., 161 (1988), pp. 243-278. Part II: Optimal control of Zakai's equation, in Stochastic Partial Differential Equations and Applications, Lecture Notes in Math. 1390, G. Da Prato and L. Tubaro, eds. Springer-Verlag, Berlin, 1989, pp. 147-170. Part III: Uniqueness of viscosity solutions for general second order equations, J. Funct. Anal., 86 (1989), pp. 1-18.
48. J.-L. LIONS AND E. MAGENES, Problèmes aux limites non homogènes et applications II, Dunod, Paris, 1968.
49. A. PAZY, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, Berlin, New York, 1983.
50. H. M. SONER, On the Hamilton-Jacobi-Bellman equations in Banach spaces, J. Optim. Theory Appl., 57 (1988), pp. 429-437.
51. A. ŚWIȨCH, Unbounded second order partial differential equations in infinite dimensional Hilbert spaces, Comm. Partial Differential Equations, 19 (1994), pp. 1999-2036.
52. D. TATARU, Viscosity solutions for Hamilton-Jacobi equations with unbounded nonlinear terms, J. Math. Anal. Appl., 163 (1992), pp. 345-392.
53. D. TATARU, Viscosity solutions for Hamilton-Jacobi equations with unbounded nonlinear term: a simplified approach, J. Differential Equations, 111 (1994), pp. 123-146.
54. D. TATARU, Viscosity solutions for the dynamic programming equations, Appl. Math. Optim., 25 (1992), pp. 109-126.
55. G. TESSITORE, Linear quadratic optical control for a stochastic system with control on the boundary and hyperbolic dynamics, J. Math. Systems Estim. Control, 2 (1992), pp. 717-744.
56. G. TESSITORE, Linear quadratic boundary control for an age equation perturbed by noise, in Proceedings Trento, Dekker, New York, 1994, pp. 197-213.
57. J. ZABCZYK, Stabilization of boundary control systems, in International Symposium on Systems Optimization and Analysis, Lecture Notes in Control and Inform. Sci. 14, Springer-Verlag, Berlin, New York, 1978, pp. 321-332.