102 research outputs found
Equivalence of switching linear systems by bisimulation
A general notion of hybrid bisimulation is proposed for the class of switching linear systems. Connections between the notions of bisimulation-based equivalence, state-space equivalence, algebraic and input–output equivalence are investigated. An algebraic characterization of hybrid bisimulation and an algorithmic procedure converging in a finite number of steps to the maximal hybrid bisimulation are derived. Hybrid state space reduction is performed by hybrid bisimulation between the hybrid system and itself. By specializing the results obtained on bisimulation, also characterizations of simulation and abstraction are derived. Connections between observability, bisimulation-based reduction and simulation-based abstraction are studied.\ud
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Chern character, loop spaces and derived algebraic geometry.
International audienceIn this note we present a work in progress whose main purpose is to establish a categorified version of sheaf theory. We present a notion of derived categorical sheaves, which is a categorified version of the notion of complexes of sheaves of modules on schemes, as well as its quasi-coherent and perfect versions. We also explain how ideas from derived algebraic geometry and higher category theory can be used in order to construct a Chern character for these categorical sheaves, which is a categorified version of the Chern character for perfect complexes with values in cyclic homology. Our construction uses in an essential way the derived loop space of a scheme X, which is a derived scheme whose theory of functions is closely related to cyclic homology of X. This work can be seen as an attempt to define algebraic analogs of elliptic objects and characteristic classes for them. The present text is an overview of a work in progress and details will appear elsewhere
Simulation-based reachability analysis for nonlinear systems using componentwise contraction properties
A shortcoming of existing reachability approaches for nonlinear systems is
the poor scalability with the number of continuous state variables. To mitigate
this problem we present a simulation-based approach where we first sample a
number of trajectories of the system and next establish bounds on the
convergence or divergence between the samples and neighboring trajectories. We
compute these bounds using contraction theory and reduce the conservatism by
partitioning the state vector into several components and analyzing contraction
properties separately in each direction. Among other benefits this allows us to
analyze the effect of constant but uncertain parameters by treating them as
state variables and partitioning them into a separate direction. We next
present a numerical procedure to search for weighted norms that yield a
prescribed contraction rate, which can be incorporated in the reachability
algorithm to adjust the weights to minimize the growth of the reachable set
Lazy Abstraction-Based Controller Synthesis
We present lazy abstraction-based controller synthesis (ABCS) for
continuous-time nonlinear dynamical systems against reach-avoid and safety
specifications. State-of-the-art multi-layered ABCS pre-computes multiple
finite-state abstractions of varying granularity and applies reactive synthesis
to the coarsest abstraction whenever feasible, but adaptively considers finer
abstractions when necessary. Lazy ABCS improves this technique by constructing
abstractions on demand. Our insight is that the abstract transition relation
only needs to be locally computed for a small set of frontier states at the
precision currently required by the synthesis algorithm. We show that lazy ABCS
can significantly outperform previous multi-layered ABCS algorithms: on
standard benchmarks, lazy ABCS is more than 4 times faster
Noncommutative Geometry in the Framework of Differential Graded Categories
In this survey article we discuss a framework of noncommutative geometry with
differential graded categories as models for spaces. We outline a construction
of the category of noncommutative spaces and also include a discussion on
noncommutative motives. We propose a motivic measure with values in a motivic
ring. This enables us to introduce certain zeta functions of noncommutative
spaces.Comment: 19 pages. Minor corrections and one reference added; to appear in the
proceedings volume of AGAQ Istanbul, 200
Geometric Phantom Categories
In this paper we give a construction of phantom categories, i.e. admissible
triangulated subcategories in bounded derived categories of coherent sheaves on
smooth projective varieties that have trivial Hochschild homology and trivial
Grothendieck group. We also prove that these phantom categories are phantoms in
a stronger sense, namely, they have trivial K-motives and, hence, all their
higher K-groups are trivial too.Comment: LaTeX, 18 page
The homotopy theory of dg-categories and derived Morita theory
The main purpose of this work is the study of the homotopy theory of
dg-categories up to quasi-equivalences. Our main result provides a natural
description of the mapping spaces between two dg-categories and in
terms of the nerve of a certain category of -bimodules. We also prove
that the homotopy category is cartesian closed (i.e. possesses
internal Hom's relative to the tensor product). We use these two results in
order to prove a derived version of Morita theory, describing the morphisms
between dg-categories of modules over two dg-categories and as the
dg-category of -bi-modules. Finally, we give three applications of our
results. The first one expresses Hochschild cohomology as endomorphisms of the
identity functor, as well as higher homotopy groups of the \emph{classifying
space of dg-categories} (i.e. the nerve of the category of dg-categories and
quasi-equivalences between them). The second application is the existence of a
good theory of localization for dg-categories, defined in terms of a natural
universal property. Our last application states that the dg-category of
(continuous) morphisms between the dg-categories of quasi-coherent (resp.
perfect) complexes on two schemes (resp. smooth and proper schemes) is
quasi-equivalent to the dg-category of quasi-coherent complexes (resp. perfect)
on their product.Comment: 50 pages. Few mistakes corrected, and some references added. Thm.
8.15 is new. Minor corrections. Final version, to appear in Inventione
Discrete-State Abstractions of Nonlinear Systems Using Multi-resolution Quantizer
Abstract. This paper proposes a design method for discrete abstrac-tions of nonlinear systems using multi-resolution quantizer, which is ca-pable of handling state dependent approximation precision requirements. To this aim, we extend the notion of quantizer embedding, which has been proposed by the authors ’ previous works as a transformation from continuous-state systems to discrete-state systems, to a multi-resolution setting. Then, we propose a computational method that analyzes how a locally generated quantization error is propagated through the state space. Based on this method, we present an algorithm that generates a multi-resolution quantizer with a specified error precision by finite refine-ments. Discrete abstractions produced by the proposed method exhibit non-uniform distribution of discrete states and inputs.
On Abstraction-Based Controller Design With Output Feedback
We consider abstraction-based design of output-feedback controllers for
dynamical systems with a finite set of inputs and outputs against
specifications in linear-time temporal logic. The usual procedure for
abstraction-based controller design (ABCD) first constructs a finite-state
abstraction of the underlying dynamical system, and second, uses reactive
synthesis techniques to compute an abstract state-feedback controller on the
abstraction. In this context, our contribution is two-fold: (I) we define a
suitable relation between the original system and its abstraction which
characterizes the soundness and completeness conditions for an abstract
state-feedback controller to be refined to a concrete output-feedback
controller for the original system, and (II) we provide an algorithm to compute
a sound finite-state abstraction fulfilling this relation.
Our relation generalizes feedback-refinement relations from ABCD with
state-feedback. Our algorithm for constructing sound finite-state abstractions
is inspired by the simultaneous reachability and bisimulation minimization
algorithm of Lee and Yannakakis. We lift their idea to the computation of an
observation-equivalent system and show how sound abstractions can be obtained
by stopping this algorithm at any point. Additionally, our new algorithm
produces a realization of the topological closure of the input/output behavior
of the original system if it is finite-state realizable
Formalising the Continuous/Discrete Modeling Step
Formally capturing the transition from a continuous model to a discrete model
is investigated using model based refinement techniques. A very simple model
for stopping (eg. of a train) is developed in both the continuous and discrete
domains. The difference between the two is quantified using generic results
from ODE theory, and these estimates can be compared with the exact solutions.
Such results do not fit well into a conventional model based refinement
framework; however they can be accommodated into a model based retrenchment.
The retrenchment is described, and the way it can interface to refinement
development on both the continuous and discrete sides is outlined. The approach
is compared to what can be achieved using hybrid systems techniques.Comment: In Proceedings Refine 2011, arXiv:1106.348
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