823 research outputs found
Weak Dynamic Programming for Generalized State Constraints
We provide a dynamic programming principle for stochastic optimal control
problems with expectation constraints. A weak formulation, using test functions
and a probabilistic relaxation of the constraint, avoids restrictions related
to a measurable selection but still implies the Hamilton-Jacobi-Bellman
equation in the viscosity sense. We treat open state constraints as a special
case of expectation constraints and prove a comparison theorem to obtain the
equation for closed state constraints.Comment: 36 pages;forthcoming in 'SIAM Journal on Control and Optimization
Some results on second order controllability conditions
For a symmetric system, we want to study the problem of crossing an
hypersurface in the neighborhood of a given point, when we suppose that all of
the available vector fields are tangent to the hypersurface at the point.
Classically one requires transversality of at least one Lie bracket generated
by two available vector fields. However such condition does not take into
account neither the geometry of the hypersurface nor the practical fact that in
order to realize the direction of a Lie bracket one needs three switches among
the vector fields in a short time. We find a new sufficient condition that
requires a symmetric matrix to have a negative eigenvalue. This sufficient
condition, which contains either the case of a transversal Lie bracket and the
case of a favorable geometry of the hypersurface, is thus weaker than the
classical one and easy to check. Moreover it is constructive since it provides
the controls for the vector fields to be used and produces a trajectory with at
most one switch to reach the goal
Singular limits of reaction diffusion equations and geometric flows with discontinuous velocity
We consider the singular limit of a bistable reaction diffusion equation in
the case when the velocity of the traveling wave solution depends on the space
variable and converges to a discontinuous function. We show that the family of
solutions converges to the stable equilibria off a front propagating with a
discontinuous velocity. The convergence is global in time by applying the weak
geometric flow uniquely defined through the theory of viscosity solutions and
the level-set equation
Viscosity solutions of the bellman equation for exit time optimal control problems with non-lipschitz dynamics
We study the Bellman equation for undiscounted exit time optimal control problems with fully nonlinear Lagrangians and fully nonlinear dynamics using the dynamic programming approach. We allow problems whose non-Lipschitz dynamics admit more than one solution trajectory for some choices of open loop controls and initial positions. We prove a uniqueness theorem which characterizes the value functions of these problems as the unique viscosity solutions of the corresponding Bellman equations that satisfy appropriate boundary conditions. We deduce that the value function for Sussmann’s Reflected Brachystochrone Problem for an arbitrary singleton target is the unique viscosity solution of the corresponding Bellman equation in the class of functions which are continuous in the plane, null at the target, and bounded below. Our results also apply to degenerate eikonal equations, and to problems whose targets can be unbounded and whose Lagrangians vanish for some points in the state space which are outside the target, including Fuller’s Example. © 2001 EDP Sciences, SMAI
The Aronsson equation, Lyapunov functions and local Lipschitz regularity of the minimum time function
We define and study solutions of the Aronsson equation (AE), a second order quasi linear equation. We show that such super/subsolutions make the Hamiltonian monotone on the trajectories of the closed loop Hamiltonian dynamics. We give a short, general proof that solutions are absolutely minimizing functions. We discuss how supersolutions of (AE) become special Lyapunov functions of symmetric control systems, and allow to find continuous feedbacks driving the system to a target in finite time, except on a singular manifold. A consequence is a simple proof that the corresponding minimum time function is locally Lipschitz continuous away from the singular manifold, despite classical results show that it should only be H"older continuous unless appropriate conditions hold. We provide two examples for H"ormander and Grushin families of vector fields where we construct solutions (even classical) explicitly
A Hamiltonian approach to small time local attainability of manifolds for nonlinear control systems
This paper develops a new approach to small time local attainability of
smooth manifolds of any dimension, possibly with boundary and to prove H\"older
continuity of the minimum time function. We give explicit pointwise conditions
of any order by using higher order hamiltonians which combine derivatives of
the controlled vector field and the functions that locally define the target.
For the controllability of a point our sufficient conditions extend some
classically known results for symmetric or control affine systems, using the
Lie algebra instead, but for targets of higher dimension our approach and
results are new. We find our sufficient higher order conditions explicit and
easy to use for targets with curvature and general control systems
The value function of an asymptotic exit-time optimal control problem
We consider a class of exit--time control problems for nonlinear systems with
a nonnegative vanishing Lagrangian. In general, the associated PDE may have
multiple solutions, and known regularity and stability properties do not hold.
In this paper we obtain such properties and a uniqueness result under some
explicit sufficient conditions. We briefly investigate also the infinite
horizon problem
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