Skip to main content
SpringerLink
Log in

The Comparison Principle: Nonlinearity and Nonconvexity

  • Chapter
  • First Online:
Book cover Dynamics and Control of Trajectory Tubes

Part of the book series: Systems & Control: Foundations & Applications ((SCFA,volume 85))

  • 1080 Accesses

Abstract

This chapter introduces generalizations and applications of the presented approach prescribed earlier to nonlinear systems, nonconvex reachability sets and systems subjected to non-ellipsoidal constraints. The key element for these issues lies in the Comparison Principle for HJB equations which indicates schemes of approximating their complicated solutions by arrays of simpler procedures. Given along these lines is a deductive derivation of ellipsoidal calculus in contrast with previous inductive derivation.

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

Access this chapter

Log in via an institution
eBook
USD 19.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 29.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 29.99
Price excludes VAT (USA)
  • Durable hardcover 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

Institutional subscriptions

Notes

  1. 1.

    Such considerations are also true for systems with uncertain (unknown but bounded) disturbances (the HJBI equation) [123, 176, 222], and for dynamic game-type problems. They involve more complicated equations of Hamilton–Jacobi–Bellman–Isaacs (HJBI) type, [18, 19, 102, 123]. [121, 150, 183]. These issues are mostly, except Chaps. 9 and 10, beyond the scope of the present volume.

  2. 2.

    In Sect. 5.1.2 we calculated the backward reach set \(\mathcal{W}[t]\) due to value function V (b)(t, x). Here, in a similar way, we calculate the forward reach set \(\mathcal{X}[t]\) due to value function V 0(t, x).

  3. 3.

    The illustrations for this example were done by Zakroischikov.

  4. 4.

    Do not confuse famous Lyapunov (1911–1973) with celebrated A.M. Lyapunov (1856–1918), founder of modern stability theory.

  5. 5.

    Here and below the conjugates of V are taken only in the second variables with τ fixed.

  6. 6.

    This example was worked out by O.L. Chucha.

  7. 7.

    This example was worked out by V.V.Sinyakov.

  8. 8.

    In the next subsection related to zonotopes we indicate a regularization procedure that ensures a numerical procedure that copes with such degeneracy. This is done by substituting cylinders for far-stretched ellipsoids along degenerate coordinates.

  9. 9.

    Illustrations to this example were worked out by M. Kirillin.

  10. 10.

    Examples illustrated in Figs. 5.16, 5.17, 5.18, and 5.19 of this section were worked out by M. Kirillin.

  11. 11.

    Note that designing external approximations of reachability sets \(\mathcal{X}[t]\) we have to apply them to nondegenerate \(\upvarepsilon\)-neighborhoods of ellipsoids \(\mathcal{E}_{ii}\) instead of exact \(\mathcal{E}_{ii}\), as in Sect. 5.4.2. This is because all ellipsoids \(\mathcal{E}_{ii}\) are degenerate, and their approximations may also turn out to be such. But since we need all externals to be nondegenerate, this will be guaranteed by applying our scheme to nondegenerate neighborhoods of \(\mathcal{E}_{ii}\).

References

  1. Artstein, Z.: Yet another proof of the Lyapunov convexity theorem. Proc. Am. Math. Soc. 108(1), 89–91 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  2. Bardi, M., Capuzzo Dolcetta, I.: Optimal Control and Viscosity Solutions of Hamilton–Jacobi–Bellman Equations. SCFA. Birkhäuser, Boston (1997)

    Book  MATH  Google Scholar 

  3. Barron, E.N., Jensen, R.: Semicontinuous viscosity solutions for Hamilton–Jacobi equations with convex Hamiltonians. Commun. Partial Differ. Equ. 15(12), 1713–1742 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  4. Başar, T., Bernhard, P.: H ∞ Optimal Control and Related Minimax Design Problems. SCFA, 2nd edn. Birkhäuser, Basel (1995)

    Google Scholar 

  5. Ba sar, T., Olsder, J.: Dynamic Noncooperative Game Theory. Academic, New York (1982)

    Google Scholar 

  6. Bellman, R., Dreyfus, S.: Applied Dynamic Programming. Princeton University Press, Princeton (1962)

    MATH  Google Scholar 

  7. Bensoussan, A., Lions, J.-L.: Contrôle impulsionnel et inéquations quasi-variationnelles. Dunod, Paris (1982)

    MATH  Google Scholar 

  8. Bertsekas, D.P.: Dynamic Programming and Optimal Control, vol. 1/2. Athena Scientific, Belmont (1996/2012)

    Google Scholar 

  9. Clarke, F.H., Ledyaev Yu, S., Stern, R.J., Wolenski, P.R.: Nonsmooth Analysis and Control Theory. Graduate Texts in Mathematics, vol. 178. Springer, New York (1998)

    Google Scholar 

  10. Crandall, M.G., Lions, P.-L.: Viscosity solutions of Hamilton–Jacobi equations. Trans. Am. Math. Soc. 277(1), 1–41 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  11. Courant, R., Hilbert, D.: Methods of Mathematical Physics, vol. 1 & 2. Wiley, New York (1989) (Wiley GmbH & Co., KGaA, 2008)

    Google Scholar 

  12. Demyanov, V.F., Rubinov, A.M.: Quasidifferential Calculus. Optimization Software, New York (1986)

    Book  MATH  Google Scholar 

  13. Fan, K.: Existence theorems and extreme solutions for inequalities concerning convex functions or linear transformations. Mathematische Zeitschrift 68(1), 205–216 (1957)

    Article  MathSciNet  MATH  Google Scholar 

  14. Fleming, W.H., Soner, H.M.: Controlled Markov Processes and Viscosity Solutions. Springer, New York (1993)

    MATH  Google Scholar 

  15. Gurman, V.I.: The Extension Principle in Problems of Control. Nauka, Moscow (1997)

    Google Scholar 

  16. Isaacs, R.: Differential Games. Wiley, New York (1965)

    MATH  Google Scholar 

  17. Kostousova, E.K.: Control synthesis via parallelotopes: optimization and parallel computations. Optim. Methods Softw. 14(4), 267–310 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  18. Krasovski, N.N.: Rendezvous Game Problems. National Technical Information Service, Springfield (1971)

    Google Scholar 

  19. Krasovskii, N.N., Subbotin, A.I.: Game-Theoretical Control Problems. Springer Series in Soviet Mathematics. Springer, New York (1988)

    Book  MATH  Google Scholar 

  20. Kurzhanski, A.B.: Set-valued calculus and dynamic programming in problems of feedback control. Int. Ser. Numer. Math. 124, 163–174 (1998)

    MathSciNet  Google Scholar 

  21. Kurzhanski, A.B.: Comparison principle for equations of the Hamilton–Jacobi type in control theory. Proc. Steklov Math. Inst. 253(1), 185–195 (2006)

    Article  MathSciNet  Google Scholar 

  22. Kurzhanski, A.B.: “Selected Works of A.B. Kurzhanski”, 755 pp. Moscow State University Pub., Moscow (2009)

    Google Scholar 

  23. Kurzhanski, A.B., Vályi, I.: Ellipsoidal Calculus for Estimation and Control. SCFA. Birkhäuser, Boston (1997)

    Book  MATH  Google Scholar 

  24. Kurzhanski, A.B., Varaiya, P.: On the reachability problem under persistent disturbances. Dokl. Math. 61(3), 3809–3814 (2000)

    MathSciNet  Google Scholar 

  25. Kurzhanski, A.B., Varaiya, P.: Dynamic optimization for reachability problems. J. Optim. Theory Appl. 108(2), 227–251 (2001)

    Article  MathSciNet  Google Scholar 

  26. Kurzhanski, A.B., Varaiya, P.: Reachability analysis for uncertain systems—the ellipsoidal technique. Dyn. Continuous Dis. Impulsive Syst. Ser. B 9(3), 347–367 (2002)

    MathSciNet  MATH  Google Scholar 

  27. Kurzhanski, A.B., Varaiya, P.: On reachability under uncertainty. SIAM J. Control Optim. 41(1), 181–216 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  28. Kurzhanski, A.B., Varaiya, P.: The Hamilton–Jacobi equations for nonlinear target control and their approximation. In: Analysis and Design of Nonlinear Control Systems (in Honor of Alberto Isidori), pp. 77–90. Springer, Berlin (2007)

    Google Scholar 

  29. Leitmann, G.: Optimality and reachability with feedback controls. In: Blaquière, A., Leitmann, G. (eds.) Dynamical Systems and Microphysics: Control Theory and Mechanics. Academic, Orlando (1982)

    Google Scholar 

  30. Lyapunov, A.A: On fully additive vector functions. Izvestia USSR Acad. Sci., Ser. Math., 4(6),465–478 (1940)

    Google Scholar 

  31. Osher, S., Fedkiw, R.: Level Set Methods and Dynamic Implicit Surfaces. Applied Mathematical Sciences, vol. 153. Springer, New York (2003)

    Google Scholar 

  32. Osipov, Yu.S.: On the theory of differential games in distributed parameter systems. Sov. Math. Dokl. 223(6), 1314–1317 (1975)

    MathSciNet  Google Scholar 

  33. Patsko, V.S., Pyatko, S.G., Fedotov, A.A.: Three-dimensional reachability set for a nonlinear control system. J. Comput. Syst. Sci. Int. 42(3), 320–328 (2003)

    MathSciNet  MATH  Google Scholar 

  34. Pontryagin, L.S., Boltyansky, V.G., Gamkrelidze, P.V., Mischenko, E.F.: The Mathematical Theory of Optimal Processes. Wiley Interscience, New York (1962)

    MATH  Google Scholar 

  35. Rockafellar, R.T.: Convex Analysis, 2nd edn. Princeton University Press, Princeton (1999)

    Google Scholar 

  36. Rockafellar, R.T., Wets, R.J.: Variational Analysis. Springer, Berlin (2005)

    Google Scholar 

  37. Sethian, J.A.: Level Set Methods and Fast Marching Methods. Cambridge University Press, Cambridge (1999)

    MATH  Google Scholar 

  38. Subbotin, A.I.: Generalized Solutions of First-Order PDE’s. The Dynamic Optimization Perspective. SCFA. Birkhäuser, Boston (1995)

    Book  Google Scholar 

  39. Subbotina, N.N., Kolpakova, E.A., Tokmantsev, T.B., Shagalova, L.G.: Method of Characteristics for the Hamilton-Jacobi-Bellman Equation. Ural Scientific Center/Russian Academy of Sciences, Yekaterinburg (2013)

    Google Scholar 

  40. Ushakov, V.N.: Construction of solutions in differential games of pursuit-evasion. Lecture Notes in Nonlinear Analysis, vol. 2(1), pp. 269–281 Polish Academic Publishers, Warsaw (1997)

    Google Scholar 

Download references

Author information

Authors and Affiliations

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

    Alexander B. Kurzhanski

  2. Electrical Engineering and Computer Sciences, University of California, Berkeley, Berkeley, CA, USA

    Alexander B. Kurzhanski & Pravin Varaiya

Authors
  1. Alexander B. Kurzhanski
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Pravin Varaiya
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

Copyright information

© 2014 Springer International Publishing Switzerland

About this chapter

Cite this chapter

Kurzhanski, A.B., Varaiya, P. (2014). The Comparison Principle: Nonlinearity and Nonconvexity. In: Dynamics and Control of Trajectory Tubes. Systems & Control: Foundations & Applications, vol 85. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-10277-1_5

Download citation

Publish with us

Policies and ethics

Search

Navigation