## Optimal stochastic switching and the Dirichlet problem for the Bellman equation

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- by Lawrence C. Evans and Avner Friedman PDF
- Trans. Amer. Math. Soc.
**253**(1979), 365-389 Request permission

## Abstract:

Let ${L^i}$ be a sequence of second order elliptic operators in a bounded*n*-dimensional domain $\Omega$, and let ${f^i}$ be given functions. Consider the problem of finding a solution

*u*to the Bellman equation ${\sup _i}({L^i}u - {f^i}) = 0$ a.e. in $\Omega$, subject to the Dirichlet boundary condition $u = 0$ on $\partial \Omega$. It is proved that, provided the leading coefficients of the ${L^i}$ are constants, there exists a unique solution

*u*of this problem, belonging to ${W^{1,\infty }}(\Omega ) \cap W_{{\text {loc}}}^{2,\infty }(\Omega )$. The solution is obtained as a limit of solutions of certain weakly coupled systems of nonlinear elliptic equations; each component of the vector solution converges to

*u*. Although the proof is entirely analytic, it is partially motivated by models of stochastic control. We solve also certain systems of variational inequalities corresponding to switching with cost.

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## Additional Information

- © Copyright 1979 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**253**(1979), 365-389 - MSC: Primary 93E20
- DOI: https://doi.org/10.1090/S0002-9947-1979-0536953-4
- MathSciNet review: 536953