Comparison principle for equations of the Hamilton–Jacobi type in control theory

A. B. Kurzhanskii

Trudy Instituta matematiki i mehaniki, Dynamical systems: modeling, optimization, and control, Tome 12 (2006) no. 1, pp. 173-183

Résumé

This paper deals with the comparison principle for the first-order ODEs of the Hamilton–Jacobi–Bellman and Hamilton–Jacobi–Bellman–Isaacs type which describe solutions to the problems of reachability and control synthesis under complete as well as under limited information on the system disturbances. Since the exact solutions require fairly complicated calculation, this paper presents the upper and lower bounds to these solutions, which in some cases may suffice for solving such problems as the investigation of safety zones in motion planning, verification of control strategies or of conditions for the nonintersection of reachability tubes, etc. For systems with original linear structure it is indicated that present among the suggested estimates are those of ellipsoidal type, which ensure tight approximations of the convex reachability sets as well as of the solvability sets for the problem of control synthesis.

Bibliographie

[1] Pontryagin L. S., Boltyanskii V. G., Gamkrelidze R. V., Mischenko E. F., Matematicheskaya teoriya optimalnykh protsessov, Nauka, M., 1961 | Zbl

[2] Bellman R., Kalaba R., Dinamicheskoe programmirovanie i sovremennye problemy upravleniya, Nauka, M., 1968

[3] Krasovskii N. N., Subbotin A. I., Game-Theoretical Control Problems, Springer-Verlag, New York, 1988 | MR | Zbl

[4] Osipov Yu. S., Kryazhimskii A. V., Inverse problem of ordinary differential equations: dynamic solutions, Gordon and Breach, 1995 | MR | Zbl

[5] Bertsekas D. P., Dynamic Programming and Optimal Control, V. I, II, Athena Scientific, Belmont, 1995

[6] Kurzhanski A. B., Varaiya P., “Dynamic optimization for reachability problems”, J. Optim. Theory Appl., 108:2 (2001), 227–251 | DOI | MR | Zbl

[7] Aizeks R., Differentsialnye igry, Mir, M., 1967 | MR

[8] Basar T., Bernhard P., $H^\infty$-optimal control and related minimax design problems, 2nd ed., Birkhäuser, 1995 | MR

[9] Basar T., Olsder J., Dynamic noncooperative game theory, Acad. Press, New York, 1982 | MR

[10] Leitmann G., “Optimality and reachability via feedback control”, Dynamic Systems and Microphysics, eds. Blaquière A., Leitmann G., Academic Press, New York–London, 1982, 119–141 | MR

[11] Krasovskii N. N., Igrovye zadachi o vstreche dvizhenii, Nauka, M., 1970 | MR

[12] Lee E. B., Marcus L., Foundations of Optimal Control Theory, Wiley, New York, 1967 | MR | Zbl

[13] Kurzhanskii A. B., Upravlenie i nablyudenie v usloviyakh neopredelennosti, Nauka, M., 1977 | MR | Zbl

[14] Osipov Yu. S., “K teorii differentsialnykh igr s raspredelennymi parametrami”, Dokl. AN SSSR, 223:6 (1975), 1314–1317 | MR | Zbl

[15] Kurzhanski A. B., Varaiya P., “On reachability under uncertainty”, SIAM Journal on Control and Optimization, 41:1 (2002), 181–216 | DOI | MR | Zbl

[16] Osher S., Fedkiw R., Level set methods and dynamic implicit surfaces, Applied Mathematical Sciences, 153, Springer-Verlag, New York, 2003 | MR | Zbl

[17] Sethian J. A., Level Set Methods and Fast Marching Methods, Second ed., Cambridge Univ. Press, 1999 | MR | Zbl

[18] Bensussan A., Lions Zh.-L., Impulsnoe upravlenie i kvazivariatsionnye neravenstva, Nauka, M., 1987

[19] Clarke F. H., Ledyaev Yu. S., Stern R. J., Wolenski P. R., Nonsmooth Analysis and Control Theory, Graduate Texts in Mathematics, 178, Springer-Verlag, 1998 | MR | Zbl

[20] Gurman V. I., Printsip rasshireniya v zadachakh upravleniya, Fizmatlit, M., 1997 | MR | Zbl

[21] Patsko V. S., Pyatko S. G., Fedotov A. A., “Three-dimensional reachability set for a nonlinear control system”, J. Computer and Syst. Sci. Intl., 42:3 (2003), 320–328 | MR | Zbl

[22] Guseinov Kh. G., Moiseev A. N., Ushakov V. N., “Ob approksimatsii oblastei dostizhimosti sistem upravleniya”, Prikl. matematika i mekhanika, 1998, no. 2, 179–186 | MR

[23] Crandall M. G., Evans L. C., Lions P.-L., “Some properties of viscosity solutions of Hamilton–Jacobi equations”, Trans. Amer. Math. Soc., 282:2 (1984), 487–502 | DOI | MR | Zbl

[24] Fleming W. H., Soner H. M., Controlled Markov processes and viscosity solutions, Springer-Verlag, New York, 1993 | MR | Zbl

[25] Bardi M., Capuzzo-Dolcetta I., Optimal control and viscosity solutions of Hamilton–Jacobi–Bellman Equations, Systems Control: Foundations Applications, Birkhäuser, Boston, 1997 | MR | Zbl

[26] Subbotin A. I., Generalized solutions of first-order PDE's. The dynamical optimization perspective, Systems Control: Foundations Applications, Birkhäuser, Boston, 1995 | MR

[27] Filippov A. F., Differential Equations with Discontinuous Righthand Sides, Kluwer, Dordrecht, 1988 | MR | Zbl

[28] Kurzhanski A. B., Vályi I., Ellipsoidal Calculus for Estimation and Control, Progr. Systems Control Theory, 22, Birkhäuser, Boston, 1997 | MR | Zbl

[29] Varaiya P., Kurzhanskii A. B., “Ellipsoidalnye metody dlya zadach dinamiki i upravleniya. Ch. 1”, Sovremennaya matematika i ee prilozheniya, 23, In-t kibernetiki AN Gruzii, Tbilisi, 2005, 34–72 | MR

[30] Kurzhanski A. B., Varaiya P., “Reachability analysis for uncertain systems-the ellipsoidal technique”, Dynamics Contin., Discrete, Impuls. Systems. Ser. B (Appl. Algorithms), 9:3 (2002), 347–367 | MR | Zbl