1,721,086 research outputs found
A stability property for the generalized mean curvature flow equation
No full textIn this paper we will study stability properties for
viscosity solutions of geometric equations. We will prove that, if
the interface is regular (i.e. it is the boundary of an open set and it
is not fat), the signed distance function from the front is stable for
geometric perturbations of the equation. This result is based on
representation formulas for viscosity solutions in terms of distance
functions from the level sets. An application of the previous result
to stability of approximation schemes is also presente
A stability property for the generalized mean curvature flow equation
No full textIn this paper we will study stability properties for
viscosity solutions of geometric equations. We will prove that, if
the interface is regular (i.e. it is the boundary of an open set and it
is not fat), the signed distance function from the front is stable for
geometric perturbations of the equation. This result is based on
representation formulas for viscosity solutions in terms of distance
functions from the level sets. An application of the previous result
to stability of approximation schemes is also presente
Approximation of integro-differential equations associated with piecewise deterministic process
No full textAim of this paper is to present an approximation scheme for optimal control problems of piecewise deterministic processes and corresponding integro-differential Hamilton-Jacobi-Bellman equations.
The method is based on a discrete dynamic programming approach. We discretize the continuous process and the cost functional obtaining a discrete time optimal control problem. The corresponding dynamic programming equation gives an approximation of the integro-differential equation.
The main feature of the method is the uniform convergence to the value function of the continuous control problem, which can be characterized as the unique weal solution (in viscosity sense) of the dynamic programming equation. Moreover, under appropriate assumptions, an error estimate on the truncation error is derived.
It is worth noting that the method provides approximate feedback controls at any point of the grid without extra computations.
An application of the approximation scheme to the numerical solution of an optimal control problem for a storage process is also detailed
A characterization of the value function for a class of degenerate control problems
No full textWe review the results contained in [1] and [3] where approximation schemes for
the effective Hamiltonian and for the limit equation arising in homogenization of periodic
Hamilton-Jacobi equations are discussed. The main feature of these schemes is a global
error estimates in all the parameters involved in the approximation
A note on convergence of level sets
No full textGiven a sequence of functions fn converging in some topology to a function f, in
general the 0-level set of fn does not give a good approximation of the one of f. In this paper
we show that, if we consider an appropriate perturbation of the 0-level set of fn, we get a
sequence of sets converging to the 0-level set of f, where the type of set convergence depends
on the type of convergence of fn to f
Computation of the H∞ norm for nonlinear systems: a convergence result
No full textIn this paper we show that H-infinity, norm for a class of nonlinear continuous-time systems is the limit of the H-infinity norm for approximating discrete-time systems. The convergence result is proved by means of a characterization of the H-infinity, norm in terms of the value for an ergodic control problem
An Hopf-Lax formula for a class of measurable Hamilton-Jacobi equations
No full textWe consider the Cauchy problem
u(t) + H(x, Du) = 0, (x, t) epsilon R-N x (0, infinity),
u(x, 0) = u(0)(x), x epsilon R-N,
where H is measurable in x, continuous, convex and positive homogeneous in p. We adapt the definition of viscosity solution to the measurable framework and we prove that the unique viscosity solution is given by a representation formula of Hopf-Lax type
Discontinuous solutions of an Hamilton-Jacobi equation with infinite speed of propagation
No full textAn approximation scheme for the optimal control of diffusion processes
Full text linkWe present a numerical approximation scheme for the infinite horizon problem related to diffusion processes. The scheme is based on a discrete version of the dynamic programming principle and converges to the viscosity solution of the second order Hamilton-Jacobi-Bellman equation. The diffusion can be degenerate. The problem R(n) is solved in a bounded domain Omega using a truncation technique and without imposing invariance conditions on Omega. We prove explicit estimates of the error due to the truncation technique
Large-time stability of travelling waves for a class of fully nonlinear parabolic equations
No full textIn this paper we prove existence and L-1 stability of travelling waves for a, class of second-order nonlinear parabolic equations in divergence form. As a consequence of the previous result, we get stability in the L-infinity norm of travelling waves for a class of fully nonlinear second-order equations
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