40,391 research outputs found
Hamilton-Jacobi theory for optimal control problems on stratified domains
This thesis studies optimal control problems on stratified domains. We first establish a known proximal Hamilton-Jacobi characterization of the value function for problems with Lipschitz dynamics. This background gives the motivation for our results for systems over stratified domains, which is a system with non-Lipschitz dynamics that were introduced by Bressan and Hong. We provide an example that shows their attempt to derive a Hamilton-Jacobi characterization of the value function is incorrect, and discuss the nature of their error. A new construction of a multifunction is introduced that possesses properties similar to those of a Lipschitz multifunction, and is used to establish Hamiltonian criteria for weak and strong invariance. Finally, we use these characterizations to show that the minimal time function and the value function for a Mayer problem, both over stratified domains, satisfy and are the unique solutions to a proximal Hamilton-Jacobi equation
Approximation of the value function for optimal control problems on stratified domains
In optimal control problems defined on stratified domains, the dynamics and
the running cost may have discontinuities on a finite union of submanifolds of
RN. In [8, 5], the corresponding value function is characterized as the unique
viscosity solution of a discontinuous Hamilton-Jacobi equation satisfying
additional viscosity conditions on the submanifolds. In this paper, we consider
a semi-Lagrangian approximation scheme for the previous problem. Relying on a
classical stability argument in viscosity solution theory, we prove the
convergence of the scheme to the value function. We also present HJSD, a free
software we developed for the numerical solution of control problems on
stratified domains in two and three dimensions, showing, in various examples,
the particular phenomena that can arise with respect to the classical
continuous framework
(Almost) Everything You Always Wanted to Know About Deterministic Control Problems in Stratified Domains
We revisit the pioneering work of Bressan \& Hong on deterministic control
problems in stratified domains, i.e. control problems for which the dynamic and
the cost may have discontinuities on submanifolds of R N . By using slightly
different methods, involving more partial differential equations arguments, we
(i) slightly improve the assumptions on the dynamic and the cost; (ii) obtain a
comparison result for general semi-continuous sub and supersolutions (without
any continuity assumptions on the value function nor on the
sub/supersolutions); (iii) provide a general framework in which a stability
result holds
Junction conditions for finite horizon optimal control problems on multi-domains with continuous and discontinuous solutions
This paper deals with junction conditions for Hamilton-Jacobi-Bellman (HJB)
equations for finite horizon control problems on multi-domains. We consider two
different cases where the final cost is continuous or lower semi-continuous. In
the continuous case we extend the results of "Hamilton-Jacobi-Bellman equations
on multi-domains" by the second and third authors in a more general framework
with switching running costs and weaker controllability assumptions. The
comparison principle has been established to guarantee the uniqueness and the
stability results for the HJB system on such multi-domains. In the lower
semi-continuous case, we characterize the value function as the unique lower
semi-continuous viscosity solution of the HJB system, under a local
controllability assumption
Value function for regional control problems via dynamic programming and Pontryagin maximum principle
In this paper we focus on regional deterministic optimal control problems,
i.e., problems where the dynamics and the cost functional may be different in
several regions of the state space and present discontinuities at their
interface. Under the assumption that optimal trajectories have a locally finite
number of switchings (no Zeno phenomenon), we use the duplication technique to
show that the value function of the regional optimal control problem is the
minimum over all possible structures of trajectories of value functions
associated with classical optimal control problems settled over fixed
structures, each of them being the restriction to some submanifold of the value
function of a classical optimal control problem in higher dimension.The lifting
duplication technique is thus seen as a kind of desingularization of the value
function of the regional optimal control problem. In turn, we extend to
regional optimal control problems the classical sensitivity relations and we
prove that the regularity of this value function is the same (i.e., is not more
degenerate) than the one of the higher-dimensional classical optimal control
problem that lifts the problem
Flow Invariance on Stratified Domains
This paper studies conditions for invariance of dynamical systems on
stratified do- mains as originally introduced by Bressan and Hong. We establish
Hamiltonian conditions for both weak and strong invariance of trajectories on
systems with non-Lipschitz data. This is done via the identification of a new
multifunction, the essential velocity multifunction. Properties of this
multifunction are investigated and used to establish the relevant invariance
criteria
Stratified Transfer Learning for Cross-domain Activity Recognition
In activity recognition, it is often expensive and time-consuming to acquire
sufficient activity labels. To solve this problem, transfer learning leverages
the labeled samples from the source domain to annotate the target domain which
has few or none labels. Existing approaches typically consider learning a
global domain shift while ignoring the intra-affinity between classes, which
will hinder the performance of the algorithms. In this paper, we propose a
novel and general cross-domain learning framework that can exploit the
intra-affinity of classes to perform intra-class knowledge transfer. The
proposed framework, referred to as Stratified Transfer Learning (STL), can
dramatically improve the classification accuracy for cross-domain activity
recognition. Specifically, STL first obtains pseudo labels for the target
domain via majority voting technique. Then, it performs intra-class knowledge
transfer iteratively to transform both domains into the same subspaces.
Finally, the labels of target domain are obtained via the second annotation. To
evaluate the performance of STL, we conduct comprehensive experiments on three
large public activity recognition datasets~(i.e. OPPORTUNITY, PAMAP2, and UCI
DSADS), which demonstrates that STL significantly outperforms other
state-of-the-art methods w.r.t. classification accuracy (improvement of 7.68%).
Furthermore, we extensively investigate the performance of STL across different
degrees of similarities and activity levels between domains. And we also
discuss the potential of STL in other pervasive computing applications to
provide empirical experience for future research.Comment: 10 pages; accepted by IEEE PerCom 2018; full paper. (camera-ready
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