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This article is the starting point of a series of works whose aim is the study of deterministic control problems where the dynamic and the running cost can be completely different in two (or more) complementary domains of the space ℝN. As a consequence, the dynamic and running cost present discontinuities at the boundary of these domains and this is the main difficulty of this type of problems. We address these questions by using a Bellman approach: our aim is to investigate how to define properly...
A deterministic affine-quadratic optimal control problem is considered. Due to the nature of the problem, optimal controls exist under some very mild conditions. Further, it is shown that under some assumptions, the optimal control is unique which leads to the differentiability of the value function. Therefore, the value function satisfies the corresponding Hamilton–Jacobi–Bellman equation in the classical sense, and the optimal control admits a state feedback representation. Under some additional...
This paper analyses the implementation of the generalized finite differences method for the HJB equation of stochastic control, introduced by two of the authors in [Bonnans and Zidani, SIAM J. Numer. Anal. 41 (2003) 1008–1021]. The computation of coefficients needs to solve at each point of the grid (and for each control) a linear programming problem. We show here that, for two dimensional problems, this linear programming problem can be solved in operations, where is the size of the stencil....
This paper analyses the implementation of the generalized
finite differences method for the HJB equation of
stochastic control, introduced by two of the authors in
[Bonnans and Zidani,
SIAM J. Numer. Anal.41 (2003) 1008–1021]. The computation of coefficients needs to
solve at each point of the grid (and for each control)
a linear programming problem.
We show here that, for two dimensional problems, this
linear programming problem can be solved in O(pmax)
operations, where pmax is the size of...
We provide a deterministic-control-based interpretation for a broad class of fully nonlinear parabolic and elliptic PDEs with continuous Neumann boundary conditions in a smooth domain. We construct families of two-person games depending on a small parameter ε which extend those proposed by Kohn and Serfaty [21]. These new games treat a Neumann boundary condition by introducing some specific rules near the boundary. We show that the value function converges, in the viscosity sense, to the solution...
The paper deals with deterministic optimal control problems with state constraints and non-linear dynamics. It is known for such problems that the value function is in general discontinuous and its characterization by means of a Hamilton-Jacobi equation requires some controllability assumptions involving the dynamics and the set of state constraints. Here, we first adopt the viability point of view and look at the value function as its epigraph. Then, we prove that this epigraph can always be described...
An approach for solving inverse problems involving obstacles is proposed.
The approach uses a level-set method which has been shown to be effective
in treating problems of moving boundaries, particularly those that involve
topological changes in the geometry.
We develop two computational methods based on this idea.
One method results in a nonlinear time-dependant partial differential
equation for the level-set function whose evolution minimizes the
residual in the data fit. The second method...
We prove that if f is a real valued lower semicontinuous function
on a Banach space X and if there exists a C^1, real valued Lipschitz continuous
function on X with bounded support and which is not identically equal to zero,
then f is Lipschitz continuous of constant K provided all lower subgradients of
f are bounded by K. As an application, we give a regularity result of viscosity
supersolutions (or subsolutions) of Hamilton-Jacobi equations in infinite dimensions
which satisfy a coercive condition....
We deal with the optimal portfolio problem in discrete-time setting. Employing the discrete Itô formula, which is developed by Fujita, we establish the discrete Hamilton–Jacobi–Bellman (d-HJB) equation for the value function. Simple examples of the d-HJB equation are also discussed.
We investigate the regularity of solutions of first order Hamilton-Jacobi equation with super linear growth in the gradient variable. We show that the solutions are locally Hölder continuous with Hölder exponent depending only on the growth of the hamiltonian. The proof relies on a reverse Hölder inequality.
We investigate the regularity of solutions of first order Hamilton-Jacobi equation with super linear growth in the gradient variable. We show that the solutions are locally Hölder continuous with Hölder exponent depending only on the growth of the Hamiltonian. The proof relies on a reverse Hölder inequality.
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