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 Mat. Sb. (N.S.), 1978, Volume 106(148), Number 2(6), Pages 214–233 (Mi msb2567)

On passing to the limit in degenerate Bellman equations. I

N. V. Krylov

Abstract: In this paper the author proves theorems on passage to the limit in nonlinear parabolic equations of the form $Fu=0$, arising in the theory of optimal control of random processes of diffusion type. Under the assumptions that i) the functions $u_n$ and $u$ have bounded Sobolev derivatives in $t$, ii) the $u_n$ and $u$ are convex downwards in $x$, iii) the $u_n$ are uniformly bounded in some domain $Q$, iv) $u_n\to u$ a.e. in $Q$, v) the coefficients of linear combinations of $F$ satisfy certain smoothness conditions, it is proved that $Fu_n=0$ on $Q$ for all $n$ implies $Fu=0$ on $Q$. The second derivatives of the $u_n$ and $u$ with respect to $x$ are understood in the generalized sense (as measures), and the equations $Fu_n=0$ and $Fu=0$ are considered in the lattice of measures.
Bibliography: 10 titles.

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English version:
Mathematics of the USSR-Sbornik, 1978, 34:6, 765–783

Bibliographic databases:

UDC: 519.2+517.9
MSC: Primary 60J60; Secondary 93E20

Citation: N. V. Krylov, “On passing to the limit in degenerate Bellman equations. I”, Mat. Sb. (N.S.), 106(148):2(6) (1978), 214–233; Math. USSR-Sb., 34:6 (1978), 765–783

Citation in format AMSBIB
\Bibitem{Kry78} \by N.~V.~Krylov \paper On passing to the limit in degenerate Bellman equations.~I \jour Mat. Sb. (N.S.) \yr 1978 \vol 106(148) \issue 2(6) \pages 214--233 \mathnet{http://mi.mathnet.ru/msb2567} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=503593} \zmath{https://zbmath.org/?q=an:0403.35054|0445.35067} \transl \jour Math. USSR-Sb. \yr 1978 \vol 34 \issue 6 \pages 765--783 \crossref{https://doi.org/10.1070/SM1978v034n06ABEH001356} 

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This publication is cited in the following articles:
1. N. V. Krylov, “On passing to the limit in degenerate Bellman equations. II”, Math. USSR-Sb., 35:3 (1979), 351–362
2. N. V. Krylov, “Some new results in the theory of controlled diffusion processes”, Math. USSR-Sb., 37:1 (1980), 133–149
3. N. V. Krylov, “On controlled diffusion processes with unbounded coefficients”, Math. USSR-Izv., 19:1 (1982), 41–64
4. Subbotin A. Subbotina N., “Optimum Result Function in the Problem of Control”, 266, no. 2, 1982, 294–299
5. P. L. Lions, “Optimal control of diffusion processes and Hamilton–Jacobi–Bellman equations part 2 : viscosity solutions and uniqueness”, Communications in Partial Differential Equations, 8:11 (1983), 1229
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