Advertisement

The Comparison Principle: Nonlinearity and Nonconvexity

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

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.

Keywords

Nonlinearity Nonconvexity Comparison principle Unicycle  Boxes Zonotopes Ellipsoids 

References

  1. 3.
    Artstein, Z.: Yet another proof of the Lyapunov convexity theorem. Proc. Am. Math. Soc. 108(1), 89–91 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 16.
    Bardi, M., Capuzzo Dolcetta, I.: Optimal Control and Viscosity Solutions of Hamilton–Jacobi–Bellman Equations. SCFA. Birkhäuser, Boston (1997)CrossRefzbMATHGoogle Scholar
  3. 17.
    Barron, E.N., Jensen, R.: Semicontinuous viscosity solutions for Hamilton–Jacobi equations with convex Hamiltonians. Commun. Partial Differ. Equ. 15(12), 1713–1742 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 18.
    Başar, T., Bernhard, P.: H Optimal Control and Related Minimax Design Problems. SCFA, 2nd edn. Birkhäuser, Basel (1995)Google Scholar
  5. 19.
    Ba sar, T., Olsder, J.: Dynamic Noncooperative Game Theory. Academic, New York (1982)Google Scholar
  6. 21.
    Bellman, R., Dreyfus, S.: Applied Dynamic Programming. Princeton University Press, Princeton (1962)zbMATHGoogle Scholar
  7. 23.
    Bensoussan, A., Lions, J.-L.: Contrôle impulsionnel et inéquations quasi-variationnelles. Dunod, Paris (1982)zbMATHGoogle Scholar
  8. 24.
    Bertsekas, D.P.: Dynamic Programming and Optimal Control, vol. 1/2. Athena Scientific, Belmont (1996/2012)Google Scholar
  9. 48.
    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. 50.
    Crandall, M.G., Lions, P.-L.: Viscosity solutions of Hamilton–Jacobi equations. Trans. Am. Math. Soc. 277(1), 1–41 (1983)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 51.
    Courant, R., Hilbert, D.: Methods of Mathematical Physics, vol. 1 & 2. Wiley, New York (1989) (Wiley GmbH & Co., KGaA, 2008)Google Scholar
  12. 62.
    Demyanov, V.F., Rubinov, A.M.: Quasidifferential Calculus. Optimization Software, New York (1986)CrossRefzbMATHGoogle Scholar
  13. 72.
    Fan, K.: Existence theorems and extreme solutions for inequalities concerning convex functions or linear transformations. Mathematische Zeitschrift 68(1), 205–216 (1957)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 80.
    Fleming, W.H., Soner, H.M.: Controlled Markov Processes and Viscosity Solutions. Springer, New York (1993)zbMATHGoogle Scholar
  15. 95.
    Gurman, V.I.: The Extension Principle in Problems of Control. Nauka, Moscow (1997)Google Scholar
  16. 102.
    Isaacs, R.: Differential Games. Wiley, New York (1965)zbMATHGoogle Scholar
  17. 115.
    Kostousova, E.K.: Control synthesis via parallelotopes: optimization and parallel computations. Optim. Methods Softw. 14(4), 267–310 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 121.
    Krasovski, N.N.: Rendezvous Game Problems. National Technical Information Service, Springfield (1971)Google Scholar
  19. 123.
    Krasovskii, N.N., Subbotin, A.I.: Game-Theoretical Control Problems. Springer Series in Soviet Mathematics. Springer, New York (1988)CrossRefzbMATHGoogle Scholar
  20. 145.
    Kurzhanski, A.B.: Set-valued calculus and dynamic programming in problems of feedback control. Int. Ser. Numer. Math. 124, 163–174 (1998)MathSciNetGoogle Scholar
  21. 149.
    Kurzhanski, A.B.: Comparison principle for equations of the Hamilton–Jacobi type in control theory. Proc. Steklov Math. Inst. 253(1), 185–195 (2006)MathSciNetCrossRefGoogle Scholar
  22. 150.
    Kurzhanski, A.B.: “Selected Works of A.B. Kurzhanski”, 755 pp. Moscow State University Pub., Moscow (2009)Google Scholar
  23. 174.
    Kurzhanski, A.B., Vályi, I.: Ellipsoidal Calculus for Estimation and Control. SCFA. Birkhäuser, Boston (1997)CrossRefzbMATHGoogle Scholar
  24. 176.
    Kurzhanski, A.B., Varaiya, P.: On the reachability problem under persistent disturbances. Dokl. Math. 61(3), 3809–3814 (2000)MathSciNetGoogle Scholar
  25. 178.
    Kurzhanski, A.B., Varaiya, P.: Dynamic optimization for reachability problems. J. Optim. Theory Appl. 108(2), 227–251 (2001)MathSciNetCrossRefGoogle Scholar
  26. 180.
    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)MathSciNetzbMATHGoogle Scholar
  27. 183.
    Kurzhanski, A.B., Varaiya, P.: On reachability under uncertainty. SIAM J. Control Optim. 41(1), 181–216 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 186.
    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. 198.
    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. 202.
    Lyapunov, A.A: On fully additive vector functions. Izvestia USSR Acad. Sci., Ser. Math., 4(6),465–478 (1940)Google Scholar
  31. 221.
    Osher, S., Fedkiw, R.: Level Set Methods and Dynamic Implicit Surfaces. Applied Mathematical Sciences, vol. 153. Springer, New York (2003)Google Scholar
  32. 222.
    Osipov, Yu.S.: On the theory of differential games in distributed parameter systems. Sov. Math. Dokl. 223(6), 1314–1317 (1975)MathSciNetGoogle Scholar
  33. 225.
    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)MathSciNetzbMATHGoogle Scholar
  34. 226.
    Pontryagin, L.S., Boltyansky, V.G., Gamkrelidze, P.V., Mischenko, E.F.: The Mathematical Theory of Optimal Processes. Wiley Interscience, New York (1962)zbMATHGoogle Scholar
  35. 237.
    Rockafellar, R.T.: Convex Analysis, 2nd edn. Princeton University Press, Princeton (1999)Google Scholar
  36. 238.
    Rockafellar, R.T., Wets, R.J.: Variational Analysis. Springer, Berlin (2005)Google Scholar
  37. 244.
    Sethian, J.A.: Level Set Methods and Fast Marching Methods. Cambridge University Press, Cambridge (1999)zbMATHGoogle Scholar
  38. 247.
    Subbotin, A.I.: Generalized Solutions of First-Order PDE’s. The Dynamic Optimization Perspective. SCFA. Birkhäuser, Boston (1995)CrossRefGoogle Scholar
  39. 248.
    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. 255.
    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

Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  1. 1.Faculty of Computational Mathematics and CyberneticsMoscow State (Lomonosov) UniversityMoscowRussia
  2. 2.Electrical Engineering and Computer SciencesUniversity of California, BerkeleyBerkeleyUSA

Personalised recommendations