arXiv subjects
Publications and source records attributed to Adriano Da Silva.
The present paper shows that the bounded control set of a linear system on a connected Lie group $G$ contains all the bounded orbits of the system. As a consequence, its closure is the continuous image of the cartesian product of the set of control functions by the central subgroup associated with the drift of the system.
In this paper, we introduce the concept of central periodic points of a linear system as points which lies on orbits starting and ending at the central subgroup of the system. We show that this set is bounded if and only if the central subgroup is compact. Moreover, if the system admits a control set containing the identity element of $G$ then, the set of central periodic points, coincides with its interior.
In this paper we show that any linear vector field $\mathcal{X}$ on a connected Lie group $G$ admits a Jordan decomposition and the recurrent set of the associated ow of automorphisms is given as the intersection of the fixed points of the hyperbolic and nilpotent components of its Jordan decomposition.
In this paper we introduce a weak version of level and epigraph convergence for level functions on topological spaces. In the particular case of topological groups we are able to define convolutions in the set of level functions and show that any such function is the limit in level and epigraph of robust functions.
In this paper we show that a complete characterization of the controllability property for linear control system on three-dimensional solvable nonnilpotent Lie groups is possible by the LARC and the knowledge of the eigenvalues of the derivation associated with the drift of the system.
In this paper we study the main properties of control sets with nonempty interior of linear control systems on semisimple Lie groups. We show that, unlike the solvable case, linear control systems on semisimple Lie groups may have more than one control set with nonempty interior and that they are contained in right translations of the one around the identity.
In this paper we explicitly calculate the control sets associated with a linear control system on the two dimensional solvable Lie group. We show that a linear control system of such kind admits exactly one control set or infinite control sets depending on some algebraic conditions.
Given a compact metric space (X; \varrho) and a continuous function f:X\rightarrow X, we study the dynamics of the induced map \bar{f} on the hyperspace of the compact subsets of X. We show how the chain recurrent set of f and its components are related with the one of the induced map. The main result of the paper proves that, under mild conditions, the numbers of chain components of \bar{f} is greater than the ones of f. Showing the richness in the dynamics of \bar{f} which cannot be perceived by f.
For a right-invariant control system on a flag manifold $\mathbb{F}_Θ$ of a real semisimple Lie group, we relate the $\mathfrak{a}$-Lyapunov exponents to the Lyapunov exponents of the system over regular points. Moreover, we adapt the concept of partial hyperbolicity from the theory of smooth dynamical systems to control-affine systems, and we completely characterize the partially hyperbolic chain control sets on $\mathbb{F}_Θ$.
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.
In this paper we study affine and bilinear systems on Lie groups. We show that there is an intrinsic connection between the solutions of both systems. Such relation allows us to obtain some preliminary controllability results of affne systems on compact and solvable Lie groups. We also show that the controllability property of bilinear systems is very restricted and may only be achieved if the state space G is an Euclidean space.
Let ϕ be an automorphism on a connected Lie group G. Through several G-subgroups associated to the dynamics of ϕ we analyze their topological entropy. Assume that G belongs to the class of finite semisimple center Lie groups which admits a ϕ invariant Levi subgroup. Then we prove that the topological entropy information of ϕ is contained in the toral component of the unstable subgroup of ϕ in the radical of G. We specialize the main result in a couple of interesting situations.
In this paper we address the question of robustness of critical bit rates for the stabilization of networked control systems over digital communication channels. For a deterministic nonlinear system, the smallest bit rate above which practical stabilization (in the sense of set-invariance) can be achieved is measured by the invariance entropy of the system. Under the assumptions of chain controllability and a uniformly hyperbolic structure on the set of interest, we prove that the invariance entropy varies continuously with respect to system parameters. Hence, in this case the critical bit rate is robust with respect to small perturbations.
In this paper we give a direct proof that for a restricted affine control system on a connected manilfold M, the associated reachable sets up to time t varies continuously with the Haudorff metric.
In this paper we shown that the chain control sets of induced systems on flag manifolds coincides with their analogous one defined via semigroup actions. Consequently, any chain control set of the system contains a control set with nonempty interior and, if the number of the control sets with nonempty interior coincides with the number of the chain control sets, then the closure of any control set with nonempty interior is a chain control set.
Linear systems on Lie groups are a natural generalization of linear system on Euclidian spaces. For such systems, this paper studies the properties of the maximal sets of approximate controllability.
Linear systems on Lie groups are a natural generalization of linear system on Euclidian spaces. For such systems, this paper studies controllability by taking in consideration the eigenvalues of an associated derivation D. When the state space is a solvable connected Lie group, controllability of the systems is guaranteed if the reachable set of the neutral element is open and the derivation D has only pure imaginary eigenvalues. For bounded systems on nilpotent Lie groups such conditions are also necessary.
In this paper we extend the results on controllability of linear systems obtained in "Controllability of linear systems on solvable Lie groups", from solvable Lie groups to Lie groups with finite semisimple center.