11,960 research outputs found
Path-complete positivity of switching systems
The notion of path-complete positivity is introduced as a way to generalize the property of positivity from one LTI system to a family of switched LTI systems whose switching rule is constrained by a finite automaton. The generalization builds upon the analogy between stability and positivity, the former referring to the contraction of a norm, the latter referring to the contraction of a cone (or, equivalently, a projective norm). We motivate and investigate the potential of path-positivity and we propose an algorithm for the automatic verification of positivity.European Commission (670645
Fiber-optic realization of anisotropic depolarizing quantum channels
We employed an electrically-driven polarization controller to implement
anisotropic depolarizing quantum channels for the polarization state of single
photons. The channels were characterized by means of ancilla-assisted quantum
process tomography using polarization-entangled photons generated in the
process of spontaneous parametric down-conversion. The demonstrated
depolarization method offers good repeatability, low cost, and compatibility
with fiber-optic setups. It does not perturb the modal structure of single
photons, and therefore can be used to verify experimentally protocols for
managing decoherence effects based on multiphoton interference.Comment: 7 pages, 6 figure
Effective approach to the problem of time: general features and examples
The effective approach to quantum dynamics allows a reformulation of the
Dirac quantization procedure for constrained systems in terms of an
infinite-dimensional constrained system of classical type. For semiclassical
approximations, the quantum constrained system can be truncated to finite size
and solved by the reduced phase space or gauge-fixing methods. In particular,
the classical feasibility of local internal times is directly generalized to
quantum systems, overcoming the main difficulties associated with the general
problem of time in the semiclassical realm. The key features of local internal
times and the procedure of patching global solutions using overlapping
intervals of local internal times are described and illustrated by two quantum
mechanical examples. Relational evolution in a given choice of internal time is
most conveniently described and interpreted in a corresponding choice of gauge
at the effective level and changing the internal clock is, therefore,
essentially achieved by a gauge transformation. This article complements the
conceptual discussion in arXiv:1009.5953.Comment: 42 pages, 9 figures; v2: streamlined discussions, more compact
manuscrip
Path-Complete p-Dominant Switching Linear Systems
The notion of path-complete -dominance for switching linear systems (in
short, path-dominance) is introduced as a way to generalize the notion of
dominant/slow modes for LTI systems. Path-dominance is characterized by the
contraction property of a set of quadratic cones in the state space. We show
that path-dominant systems have a low-dimensional dominant behavior, and hence
allow for a simplified analysis of their dynamics. An algorithm for deciding
the path-dominance of a given system is presented
Joint Spectral Radius and Path-Complete Graph Lyapunov Functions
We introduce the framework of path-complete graph Lyapunov functions for
approximation of the joint spectral radius. The approach is based on the
analysis of the underlying switched system via inequalities imposed among
multiple Lyapunov functions associated to a labeled directed graph. Inspired by
concepts in automata theory and symbolic dynamics, we define a class of graphs
called path-complete graphs, and show that any such graph gives rise to a
method for proving stability of the switched system. This enables us to derive
several asymptotically tight hierarchies of semidefinite programming
relaxations that unify and generalize many existing techniques such as common
quadratic, common sum of squares, and maximum/minimum-of-quadratics Lyapunov
functions. We compare the quality of approximation obtained by certain classes
of path-complete graphs including a family of dual graphs and all path-complete
graphs with two nodes on an alphabet of two matrices. We provide approximation
guarantees for several families of path-complete graphs, such as the De Bruijn
graphs, establishing as a byproduct a constructive converse Lyapunov theorem
for maximum/minimum-of-quadratics Lyapunov functions.Comment: To appear in SIAM Journal on Control and Optimization. Version 2 has
gone through two major rounds of revision. In particular, a section on the
performance of our algorithm on application-motivated problems has been added
and a more comprehensive literature review is presente
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