Hyperbolic Chain Control Sets on Flag Manifolds

2019

Isr. J. Math.

Section: Introductionsupporting

confidence: 60%

Section: Figure 1: Morse Spectramentioning

confidence: 98%

Section: Semisimple Theorymentioning

confidence: 99%

mentioning

confidence: 99%

See 2 more Smart Citations

Lyapunov exponents and partial hyperbolicity of chain control sets on flag manifolds

Silva

2019

Isr. J. Math.

“…Moreover, we introduce the following notion of hyperbolicity for controlled invariant sets of (1). A class of examples satisfying this definition is characterized in [4].…”

Section: The Entropy Formula For Hyperbolic Chain Control Setsmentioning

confidence: 99%

Minimal data rates and entropy in digitally networked systems

Proc Appl Math and Mech

2015

In digitally networked systems the assumption of classical control theory that information can be transmitted instantaneously, lossless and with arbitrary precision is violated. This raises the question about the smallest data rate above which a control task can be solved. For a single control loop and the problem to make a set Q of states invariant, the minimal data rate can be described by an entropy-like quantity, the so-called invariance entropy. Under some controllability and hyperbolicity assumptions, the invariance entropy can be expressed in terms of Lyapunov exponents. Furthermore, one can show that for making Q invariant with a data rate close to the smallest, no strategies more complicated than stabilization at periodic trajectories are necessary. For a network with n subsystems, which can all communicate with each other, there are different ways to formulate the question about the smallest data rate for the invariance problem, but also in this setting entropy-like quantities yield important information.

Invariance entropy for a class of partially hyperbolic sets

Math. Control Signals Syst.

Silva

2018

Invariance entropy is a measure for the smallest data rate in a noiseless digital channel above which a controller that only receives state information through this channel is able to render a given subset of the state space invariant. In this paper, we derive a lower bound on the invariance entropy for a class of partially hyperbolic sets. More precisely, we assume that Q is a compact controlled invariant set of a control-affine system whose extended tangent bundle decomposes into two invariant subbundles E + and E 0− with uniform expansion on E + and weak contraction on E 0− . Under the additional assumptions that Q is isolated and that the u-fibers of Q vary lower semicontinuously with the control u, we derive a lower bound on the invariance entropy of Q in terms of relative topological pressure with respect to the unstable determinant. Under the assumption that this bound is tight, our result provides a first quantitative explanation for the fact that the invariance entropy does not only depend on the dynamical complexity on the set of interest.the infimum taken over all invariant probability measures of the control flow that project to P and are supported on L(Q), we are able to recover the lower bound in (1) as a special case. Here we use that for a uniformly hyperbolic set without center bundle, the u-fibers of Q are finite (see [23] for a proof), implying that the measure-theoretic entropy of the bundle RDS ϕ |L(Q) vanishes.As in the uniformly hyperbolic case, examples to which our result can be applied, are provided by invariant systems on flag manifolds on semisimple Lie groups. In the paper at hand, we only discuss a special case, namely systems on projective space induced by bilinear systems on R d , while the more general case is studied in [15].The paper is organized as follows. In Section 2, we precisely formulate our main result, recalling all the concepts involved. The proof is carried out in Section 3 through a series of lemmas and propositions. Section 4 provides a class of examples and Section 5 discusses the contents of the result and relates it to other problems, e.g., submanifold stabilization. Finally, some technical results of independent interest, used in the proof, are collected in Section 6 (Appendix).Notation. We write N = {1, 2, 3, . . .} for the set of positive integers, Z for the set of all integers and R for the set of real numbers. We also write K + := {x ∈ K : x ≥ 0} for K ∈ {Z, R}. For x > 0, log x denotes the natural logarithm of x. If V, W are vector spaces, we write Hom(V, W ) and End(V ) for the spaces of homomorphisms from V to W and endomorphisms on V , respectively.If M is a smooth manifold, we write T x M for the tangent space to M at x ∈ M . If f : M → N is differentiable, Df (x) : T x M → T f (x) N denotes the derivative at x ∈ M . If (M, g) is a Riemannian manifold, we write | · | for the norm in each tangent space T x M , d(·, ·) for the geodesic distance and vol(·) for the Riemannian volume measure on M , respectively. Moreover, exp x denotes the Riemannian exponen...