Skip to main content
SpringerLink
Log in
Book cover

Advances in Nonlinear Dynamics and Control: A Report from Russia pp 189–229Cite as

Birkhäuser

A Theory of Generalized Solutions to First-Order PDEs with an Emphasis on Differential Games

  • Chapter

Part of the book series: Progress in Systems and Control Theory ((PSCT,volume 17))

Abstract

It is well known that in many applications connected with boundary value problems and Cauchy problems for Hamilton-Jacobi equations and other types of partial differential equations (PDE) of the first order there exist no classical nonlocal solutions. On the other hand, there exist nonsmooth functions that are crucial for the considered problems (e.g., optimal value functions in control problems, alsoknown as Bellman functions). These functions satisfy the considered equations at each point of differentiability. Thus, there arises a necessity to introduce a notion of generalized solution and to develop the theory and methods for constructing these solutions. Investigations of various approaches to the definition of generalized solutions are dealt with in many papers (see, e.g., the articles [13,28] and the book [32], which contain reviews and bibliography of investigations of 1950s–70s among which we should name the results of S.N.Kruzhkov for the Hamilton-Jacobi equation whose Hamiltonian is convex with respect to the impulse variable).

This is a preview of subscription content, log in via an institution.

Chapter
USD   29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   39.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Aubin J.-P., A survey of viability theory, SIAM J. on Control and Optimization 28 (1990), 749–788.

    Article  Google Scholar 

  2. Aubin J.-P. and Ekeland I., Applied nonlinear analysis, John Wiley & Sons, Inc., New York, 1984.

    Google Scholar 

  3. Bardi M. and Soravia P., Hamilton-Jacobi equations with singular boundary conditions on a free boundary and applications to differential games, Trans. Amer. Math. Soc. 325 (1991), 205–229.

    Article  Google Scholar 

  4. Barron E.N., Evans L.C., and Jensen R., Viscosity solutions of Isaacs’ equations and differential games with Lipschitz controls, J. Different Equat 53 (1984), 213–233.

    Article  Google Scholar 

  5. Chentsov A.G., On a game problem of converging at a given instant of time, Math. of the USSR Sbornik 28 (1976), 353–376.

    Article  Google Scholar 

  6. Chistyakov S.V., On solutions of game problems of pursuit, Prikl. Matemat. i Mehanika 41 (1977), No. 5 (in Russian).

    Google Scholar 

  7. Clarke F.H., Generalized gradients and applications, Trans. Amer. Math. Soc. 205 (1975), 247–262.

    Article  Google Scholar 

  8. Clarke F.H., Optimization and nonsmooth analysis, John Wiley & Sons, Inc., New York, 1983.

    Google Scholar 

  9. Courant R. and Hilbert D., Methods of mathematical physics, Vol. I and II, John Wiley & Sons, New-York, 1953, 1962.

    Google Scholar 

  10. Crandall M.G., A generalization of Peano’s existence theorem and flow invariance, Proc. Amer. Math. Soc. 36 (1972), 151–155.

    Google Scholar 

  11. Crandall M.G., Ishii H., and Lions P.-L., Uniqueness of viscosity solutions revisited, J. Math. Soc. Japan 39 (1987), 581–596.

    Article  Google Scholar 

  12. Crandall M.G. and Lions P.-L., Condition d’unicité pour les solutions généralisées des équations Hamilton-Jacobi du 1er ordre, C. R. Acad. Sci. Paris Sér. A — B 292 (1981), 183–186.

    Google Scholar 

  13. Crandall M.G. and Lions P.-L., Viscosity solutions of Hamilton-Jacobi equations, Trans. Amer. Math. Soc. 277 (1983), 1–42.

    Article  Google Scholar 

  14. Crandall M.G. and Lions P.-L., Hamilton-Jacobi equations in infinite dimensions, Part II. Existence of viscosity solutions, J. Func. Anal. 65 (1986), 368–405.

    Article  Google Scholar 

  15. Davy J.L., Properties of the solution set of generalized differential equation, Bull. Austral. Math. Soc. 6 (1972), 379–398.

    Article  Google Scholar 

  16. Demyanov V.F. and Rubinov A.M., Foundations of nonsmooth analysis and quasidifferential calculus, Nauka, Moscow, 1990 (in Russian).

    Google Scholar 

  17. Evans L.C. and Souganidis P.E., Differential games and representation formulas for solutions of Hamilton-Jacobi-Isaacs equations, Indiana Univ. Math. J. 33 (1984), 773–797.

    Article  Google Scholar 

  18. Fleming W.H., A note on differential games of prescribed durations, Contributions to the theory of games, Ann. Math. Stud. 3 (1957), 407–412.

    Google Scholar 

  19. Fleming W.H., The Cauchy problem for degenerate parabolic equations, J. Math. Mech. 13 (1964), 987–1008.

    Google Scholar 

  20. Guseinov H.G., Subbotin A.I., and Ushakov V.N., Derivatives for multivalued mappings with applications to game theoretical problems of control, Problems of Contr. Inform. Theory 14 (1985), 155–167.

    Google Scholar 

  21. Haddad G., Monotone trajectories of differential inclusions and functional — differential inclusions with memory, Izrael J. of Math. 39 (1981), 83–100.

    Article  Google Scholar 

  22. Ishii H., Perron’s method for Hamilton-Jacobi equations, Duke math. J. 55 (1987), 369–384.

    Article  Google Scholar 

  23. Krasovskiĭ N.N., Game — theoretical problems on the encounter of motions, Nauka, Moscow, 1970 (in Russian).

    Google Scholar 

  24. Krasovskiĭ N.N., Control of dynamical system, Nauka, Moscow, 1985 (in Russian).

    Google Scholar 

  25. Krasovskiĭ N.N. and Subbotin A.I., Positional differential games, Nauka, Moscow, 1974 (in Russian).

    Google Scholar 

  26. Krasovskiĭ N. N. and Subbotin A. I., Game — Theoretical Control Problems, Springer, New-York, 1988.

    Book  Google Scholar 

  27. Kruzhkov S.N., Generalized solutions of nonlinear first order equations with several variables, I, Matemat. Sbornik (N.S.) 70(112) (1966), 394–415 (in Russian).

    Google Scholar 

  28. Kruzhkov S.N., Generalized solutions of the Hamilton-Jacobi equations of Eikonal type, I, Math. USSR-Sb. 27 (1975), 406–446.

    Article  Google Scholar 

  29. Kurzhanski A.B., On the analytical description of the set of viable trajectories to a differential system, Uspehi mat. nauk, 40 (1985), 183–194 (in Russian).

    Google Scholar 

  30. Kurzhanski A.B. and Filippova T.F., On the set-valued calculus in problems of viablility and control for dynamic processes: the evolution equation, Les Annales de l’Institut Henri Poincare, Analyse non-linear (1989), 339–363.

    Google Scholar 

  31. Ledyaev Yu.S., Criteria for viability of trajectories of nonautonomous differential inclusion and their applications, Preprint CMR-1583, Centre de Recherches Math., Univ. de Montreal (1988), 1–22.

    Google Scholar 

  32. Lions P.-L., Generalized solutions of Hamilton-Jacobi equations, Pitman, Boston, 1982.

    Google Scholar 

  33. Lions P.-L. and Souganidis P.E., Differential games, optimal control and directional derivatives of viscosity solutions of Bellman’s and Isaacs equations, SIAM J. on Control and Optimization 23 (1985), 556–583.

    Article  Google Scholar 

  34. Nagumo M., Uber die Laga der integralkurven gewöhnlicher differential Gleichungen, Proc. Phys. Math. Japan 24 (1942), 399–414.

    Google Scholar 

  35. Osipov Yu.S., On the theory of differential games to systems with distributed parameters, Dokl. Akad. Nauk SSSR 223 (1975), 1314–1317 (in Russian).

    Google Scholar 

  36. Petrovskiĭ LG., Lectures on the theory of ordinary differential equations, Nauka, Moscow, 1964 (in Russian).

    Google Scholar 

  37. Pontryagin L.S., On the theory of differential games, Uspehi mat. nauk, 21 (1966), 219–274 (in Russian).

    Google Scholar 

  38. Rockafellar R.T., Existence and duality theorems for convex problems of Bolza, Trans. Amer. Math. Soc. 159 (1971), 1–40.

    Article  Google Scholar 

  39. Sonnevend G., Existence and numerical computation of extremal invariant sets in linear differential games, Lect. Notes in Control Inf. Sci. 22 (1981), 251–260.

    Article  Google Scholar 

  40. Subbotin A.I., A generalization of the basic equation of the theory of differential games, Soviet Math. Dokl. 22 (1980), 358–362.

    Google Scholar 

  41. Subbotin A.I., Generalization of the main equation of differential game theory, J. Optimiz. Theory and Appl. 43 (1984), 103–133.

    Article  Google Scholar 

  42. Subbotin A.I., Minimax inequalities and Hamilton — Jacobi equations, Nauka, Moscow, 1990 (in Russian).

    Google Scholar 

  43. Subbotin A.I., Existence and uniqueness results for Hamilton-Jacobi equations, Nonlinear Analysis 16 (1991), 683–689.

    Article  Google Scholar 

  44. Subbotin A.I., On a property of subdifferential, Matemat. Sbornik 182 (1991), 1315–1330 (in Russian).

    Google Scholar 

  45. Subbotin A.I. and Subbotina N.N., Necessary and sufficient conditions for a piecewise smooth value of a differential game, Soviet Math. Dokl. 19 (1978), 1447–1451.

    Google Scholar 

  46. Subbotin A.I. and Tarasyev A.M., Stability properties of the value function of a differential game and viscosity solutions of Hamilton-Jacobi equations, Problems Contr. Inform. Theory 15 (1986), 451–463.

    Google Scholar 

  47. Subbotina N.N., The method of Cauchy characteristics and generalized solutions of Hamilton-Jacobi-Bellman equations, Dokl. Akad. Nauk SSSR 320 (1991), 556–561 (in Russian).

    Google Scholar 

  48. Young L.C., Lectures on the calculus of variations and optimal control theory, Sanders, Philadelphia, 1969.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institute of Mathematics and Mechanics, Ural Branch, Russian Acad. Sci., S.Kovalevsky st. 16, Ekaterinburg, 620219, Russia

    A. I. Subbotin

Authors
  1. A. I. Subbotin
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Faculty of Computational Mathematics and Cybernetics (VMK), Moscow State University, 119899, Moscow, Russia

    Alexander B. Kurzhanski

Rights and permissions

Reprints and permissions

Copyright information

© 1993 Springer Science+Business Media New York

About this chapter

Cite this chapter

Subbotin, A.I. (1993). A Theory of Generalized Solutions to First-Order PDEs with an Emphasis on Differential Games. In: Kurzhanski, A.B. (eds) Advances in Nonlinear Dynamics and Control: A Report from Russia . Progress in Systems and Control Theory, vol 17. Birkhäuser, Boston, MA. https://doi.org/10.1007/978-1-4612-0349-0_5

Download citation

  • DOI: https://doi.org/10.1007/978-1-4612-0349-0_5

  • Publisher Name: Birkhäuser, Boston, MA

  • Print ISBN: 978-1-4612-6719-5

  • Online ISBN: 978-1-4612-0349-0

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation