The equation of dynamic programming for a time-optimal problem with phase constraints

V. A. Komarov

Sbornik. Mathematics, Tome 63 (1989) no. 1, pp. 47-58

Résumé

The time-optimal problem with a phase constraint given by a compact set $K$ is considered for a differential inclusion $\dot x\in F(x)$ with right-hand side that is upper semicontinuous, convex, and compact for all $x\in F^n$. It is shown that a nonnegative lower semicontinuous function $\tau(x)$ vanishing only on the terminal set $M$ and continuous on the solutions of the differential inclusion $\dot x\in-F(x)$ is the optimal time in this problem if it satisfies the relation $$ \min_{f\in F_K(x)}D^+\tau(x;f)=-1. $$ for all $x$ with $\tau(x)<\infty$. Here $D^+\tau(x;f)$ is the upper contingent derivative of $\tau$ in the direction of $f$, $F_K(x)=T_K(x)\cap F(x)$, and $T_K(x)$ is the lower contingent tangent cone to $K$ at the point $x$. It is also shown that if $F$ is continuous and $\tau$ satisfies a one-sided Lipschitz condition, then the conditions given are necessary. Figures: 1. Bibliography: 14 titles.

Bibliographie

[1] Pontryagin L. S., Boltyanskii V. G., Gamkrelidze R. V., Mischenko E. F., Matematicheskie metody optimalnykh protsessov, Nauka, M., 1976 | Zbl

[2] Rozhdestvenskii B. L., Yanenko N. N., Sistemy kvazilineinykh uravnenii, Nauka, M., 1977 | Zbl

[3] Lions P. L., Generalized solutions of Hamilton–Jacobi equations, Pitman, Boston, 1982 | MR

[4] Blagodatskikh V. I., Filippov A. F., “Differentsialnye vklyucheniya i optimalnoe upravlenie”, MIAN, 169 (1985), 194–252 | MR | Zbl

[5] Clarke F. H., “Generalized gradients and applications”, Trans. Amer. Math. Soc., 205 (1975), 247–262 | DOI | MR | Zbl

[6] Rockafellar R. T., “Generalized directional derivatives and subgradients of nonconvex functions”, Canad. J. Math., 32:2 (1980), 257–280 | MR | Zbl

[7] Pshenichnyi B. N., Vypuklyi analiz i ekstremalnye zadachi, Nauka, M., 1980 | MR

[8] Polovinkin E. S., Teoriya mnogoznachnykh otobrazhenii, Izd-vo MFTI, M., 1983

[9] Blagodatskikh V. I., “Sufficient conditions for optimality in problems with state constraints”, Appl. Math. Optim., 7:2 (1981), 149–157 | DOI | MR | Zbl

[10] Boltyanskii V. G., Matematicheskie metody optimalnogo upravleniya, Nauka, M., 1969 | MR

[11] Aubin J.-P., Cellina A., Differential inclusions, Springer, Berlin, 1984 | MR

[12] Chugunov P. I., “O pravilnykh resheniyakh differentsialnykh vklyuchenii”, Differents. uravneniya, 17:4 (1981), 660–668 | MR | Zbl

[13] Filippov A. F., Differentsialnye uravneniya s razryvnoi pravoi chastyu, Nauka, M., 1985 | MR

[14] Kozlov R. I., “K teorii differentsialnykh uravnenii s razryvnymi pravymi chastyami”, Differents. uravneniya, 10:7 (1974), 1264–1275 | Zbl