1,239 research outputs found
Stability and mass of point particles
In this paper we consider classical point particles in full interaction with
an arbitrary number of dynamical scalar and (abelian) vector fields. It is
shown that the requirement of stability ---vanishing self-force--- is
sufficient to remove the well-known inconsistencies of the classical theory:
the divergent self-energy, as well as the failure of Lorentz-covariance of the
energy-momentum when including the contributions of the fields. As a result, in
these models the mass of a point particle becomes finitely computable. It is
shown how these models are connected to quantum field theory via the
path-integral representation of the propagator.Comment: 23 page
Multi-scale modeling of follicular ovulation as a reachability problem
During each ovarian cycle, only a definite number of follicles ovulate, while
the others undergo a degeneration process called atresia. We have designed a
multi-scale mathematical model where ovulation and atresia result from a
hormonal controlled selection process. A 2D-conservation law describes the age
and maturity structuration of the follicular cell population. In this paper, we
focus on the operating mode of the control, through the study of the
characteristics of the conservation law. We describe in particular the set of
microscopic initial conditions leading to the macroscopic phenomenon of either
ovulation or atresia, in the framework of backwards reachable sets theory
The method of characteristics revisited. A viability approach
This mini-course provides a presentation of the method of characteristics to
initial/boundary-value problems for systems of first-order partial differential
equations and to Hamilton-Jacobi variational inequalities. In particular, these
results are indeed useful for the treatment of hybrid systems of control
theory.
We rely on tools forged by set-valued analysis and viability theory, which
happen to be both efficient and versatile to cover many problems. They find
here unexpected relevance
A tree structure algorithm for optimal control problems with state constraints
We present a tree structure algorithm for optimal control problems with state constraints. We prove a convergence result for a discrete time approximation of the value function based on a novel formulation in the case of convex constraints. Then the Dynamic Programming approach is developed by a discretization in time leading to a tree structure in space derived by the controlled dynamics, taking into account the state constraints to cut several branches of the tree. Moreover, an additional pruning allows for the reduction of the tree complexity as for the case without state constraints. Since the method does not use an a priori space grid, no interpolation is needed for the reconstruction of the value function and the accuracy essentially relies on the time step h. These features permit a reduction in CPU time and in memory allocations. The synthesis of optimal feedback controls is based on the values on the tree and an interpolation on the values obtained on the tree will be necessary if a different discretization in the control space is adopted, e.g. to improve the accuracy of the method in the reconstruction of the optimal trajectories. Several examples show how this algorithm can be applied to problems in low dimension and compare it to a classical DP method on a grid
Forward Euler Solutions and Weakly Invariant Time-Delayed Systems
This paper presents a necessary and sufficient condition for the weak invariance property of a time-delayed system parametrized by a differential inclusion. The aforementioned condition generalizes the well-known Hamilton-Jacobi inequality that characterizes weakly invariant systems in the nondelay setting. The forward Euler approximation scheme used in the theory of discontinuous differential equations is extended to the time-delayed context by incorporating the delay and tail functions featuring the dynamics. Accordingly, an existence theorem of weakly invariant trajectories is established under the extended forward Euler approach
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