Conjugate points and shocks in nonlinear optimal control

Caroff, N.; Frankowska, H.

doi:10.1090/S0002-9947-96-01577-2

Conjugate points and shocks in nonlinear optimal control
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Trans. Amer. Math. Soc. 348 (1996), 3133-3153 Request permission

Abstract:

We investigate characteristics of the Hamilton-Jacobi-Bellman equation arising in nonlinear optimal control and their relationship with weak and strong local minima. This leads to an extension of the Jacobi conjugate points theory to the Bolza control problem. Necessary and sufficient optimality conditions for weak and strong local minima are stated in terms of the existence of a solution to a corresponding matrix Riccati differential equation.

References

Jean-Pierre Aubin and Arrigo Cellina, Differential inclusions, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 264, Springer-Verlag, Berlin, 1984. Set-valued maps and viability theory. MR 755330, DOI 10.1007/978-3-642-69512-4
Jean-Pierre Aubin and Hélène Frankowska, Set-valued analysis, Systems & Control: Foundations & Applications, vol. 2, Birkhäuser Boston, Inc., Boston, MA, 1990. MR 1048347
Viorel Barbu and Giuseppe Da Prato, Hamilton-Jacobi equations and synthesis of nonlinear control processes in Hilbert spaces, J. Differential Equations 48 (1983), no. 3, 350–372. MR 702425, DOI 10.1016/0022-0396(83)90099-2
E. N. Barron and R. Jensen, Semicontinuous viscosity solutions for Hamilton-Jacobi equations with convex Hamiltonians, Comm. Partial Differential Equations 15 (1990), no. 12, 1713–1742. MR 1080619, DOI 10.1080/03605309908820745
Alain Bensoussan, Perturbation methods in optimal control, Wiley/Gauthier-Villars Series in Modern Applied Mathematics, John Wiley & Sons, Ltd., Chichester; Gauthier-Villars, Montrouge, 1988. Translated from the French by C. Tomson. MR 949208
Erich Rothe, Topological proofs of uniqueness theorems in the theory of differential and integral equations, Bull. Amer. Math. Soc. 45 (1939), 606–613. MR 93, DOI 10.1090/S0002-9904-1939-07048-1
Ch. Byrnes and H. Frankowska (submitted) Uniqueness of optimal trajectories and the nonexistence of shocks for Hamilton-Jacobi-Bellman and Riccati partial differential equations.
Piermarco Cannarsa and Halina Frankowska, Some characterizations of optimal trajectories in control theory, SIAM J. Control Optim. 29 (1991), no. 6, 1322–1347. MR 1132185, DOI 10.1137/0329068
N. Caroff (1994) Caractéristiques de l’équation d’Hamilton-Jacobi et conditions d’optimalité en contrôle optimal non linéaire, Thèse de Doctorat, Université Paris-Dauphine.
Nathalie Caroff and Hélène Frankowska, Optimality and characteristics of Hamilton-Jacobi-Bellman equations, Optimization, optimal control and partial differential equations (Iaşi, 1992) Internat. Ser. Numer. Math., vol. 107, Birkhäuser, Basel, 1992, pp. 169–180. MR 1223368
Lamberto Cesari, Optimization—theory and applications, Applications of Mathematics (New York), vol. 17, Springer-Verlag, New York, 1983. Problems with ordinary differential equations. MR 688142, DOI 10.1007/978-1-4613-8165-5
E. D. Conway and E. Hopf, Hamilton’s theory and generalized solutions of the Hamilton-Jacobi equation, J. Math. Mech. 13 (1964), 939–986. MR 0182761
Michael G. Crandall and Pierre-Louis Lions, Viscosity solutions of Hamilton-Jacobi equations, Trans. Amer. Math. Soc. 277 (1983), no. 1, 1–42. MR 690039, DOI 10.1090/S0002-9947-1983-0690039-8
C. M. Dafermos, Generalized characteristics and the structure of solutions of hyperbolic conservation laws, Indiana Univ. Math. J. 26 (1977), no. 6, 1097–1119. MR 457947, DOI 10.1512/iumj.1977.26.26088
Wendell H. Fleming and Raymond W. Rishel, Deterministic and stochastic optimal control, Applications of Mathematics, No. 1, Springer-Verlag, Berlin-New York, 1975. MR 0454768, DOI 10.1007/978-1-4612-6380-7
Halina Frankowska, Contingent cones to reachable sets of control systems, SIAM J. Control Optim. 27 (1989), no. 1, 170–198. MR 980229, DOI 10.1137/0327010
H. Frankowska (1991) Lower semicontinuous solutions of Hamilton-Jacobi-Bellman equation, Proceedings of 30th IEEE Conference on Decision and Control, Brighton, GB, December 11-13.
Hélène Frankowska, Lower semicontinuous solutions of Hamilton-Jacobi-Bellman equations, SIAM J. Control Optim. 31 (1993), no. 1, 257–272. MR 1200233, DOI 10.1137/0331016
Magnus R. Hestenes, Calculus of variations and optimal control theory, John Wiley & Sons, Inc., New York-London-Sydney, 1966. MR 0203540
A. D. Ioffe and V. M. Tihomirov, Theorie der Extremalaufgaben, VEB Deutscher Verlag der Wissenschaften, Berlin, 1979 (German). Translated from the Russian by Bernd Luderer. MR 527119
E. B. Lee and L. Markus, Foundations of optimal control theory, John Wiley & Sons, Inc., New York-London-Sydney, 1967. MR 0220537
D. Q. Mayne, Sufficient conditions for a control to be a strong minimum, J. Optim. Theory Appl. 21 (1977), no. 3, 339–351. MR 458283, DOI 10.1007/BF00933535
D. Orrell and V. Zeidan, Another Jacobi sufficiency criterion for optimal control with smooth constraints, J. Optim. Theory Appl. 58 (1988), no. 2, 283–300. MR 953658, DOI 10.1007/BF00939686
Count J. F. Riccati (1724) Animadversationes in aequationes differentiales secundi gradus, Actorum Eruditorum quae Lipsiae Publicantur, Supplementa 8, 66-73.
William T. Reid, Riccati differential equations, Mathematics in Science and Engineering, Vol. 86, Academic Press, New York-London, 1972. MR 0357936
Joel Smoller, Shock waves and reaction-diffusion equations, 2nd ed., Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 258, Springer-Verlag, New York, 1994. MR 1301779, DOI 10.1007/978-1-4612-0873-0
Michael Spivak, A comprehensive introduction to differential geometry. Vol. IV, Publish or Perish, Inc., Boston, Mass., 1975. MR 0394452
Vera Zeidan, Sufficient conditions for the generalized problem of Bolza, Trans. Amer. Math. Soc. 275 (1983), no. 2, 561–586. MR 682718, DOI 10.1090/S0002-9947-1983-0682718-3
Vera Zeidan, A modified Hamilton-Jacobi approach in the generalized problem of Bolza, Appl. Math. Optim. 11 (1984), no. 2, 97–109. MR 743921, DOI 10.1007/BF01442172
Vera Zeidan, First and second order sufficient conditions for optimal control and the calculus of variations, Appl. Math. Optim. 11 (1984), no. 3, 209–226. MR 748180, DOI 10.1007/BF01442179
Vera Zeidan, Extended Jacobi sufficiency criterion for optimal control, SIAM J. Control Optim. 22 (1984), no. 2, 294–301. MR 732429, DOI 10.1137/0322020
Vera Zeidan, Sufficiency conditions with minimal regularity assumptions, Appl. Math. Optim. 20 (1989), no. 1, 19–31. MR 989428, DOI 10.1007/BF01447643
V. Zeidan and P. Zezza, Necessary conditions for optimal control problems: conjugate points, SIAM J. Control Optim. 26 (1988), no. 3, 592–608. MR 937674, DOI 10.1137/0326035
V. Zeidan and P. Zezza, The conjugate point condition for smooth control sets, J. Math. Anal. Appl. 132 (1988), no. 2, 572–589. MR 943530, DOI 10.1016/0022-247X(88)90085-6
Vera Zeidan and Pier Luigi Zezza, Coupled points in the calculus of variations and applications to periodic problems, Trans. Amer. Math. Soc. 315 (1989), no. 1, 323–335. MR 961599, DOI 10.1090/S0002-9947-1989-0961599-X
Vera Zeidan and Pier Luigi Zezza, Coupled points in optimal control theory, IEEE Trans. Automat. Control 36 (1991), no. 11, 1276–1281. MR 1130498, DOI 10.1109/9.100937