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Adriano Da Silva

Publications and source records attributed to Adriano Da Silva.

At least 19 recordsLinked to original sources
arXiv2026

Maxwell Strata in the sub-Riemannian problem on solvable, nonnilpotent regular three-dimensional Lie groups

In this paper, we study the sub-Riemannian problem associated with contact structures on connected, simply connected, solvable, non-nilpotent, regular three-dimensional Lie groups. For these groups, the vertical component of the Hamiltonian system takes the form of a perturbed pendulum. A qualitative phase-space analysis allows us to prove that this vertical component exhibits nontrivial symmetries. In particular, we are able to fully characterize the Maxwell set corresponding to these symmetries, and show that its first Maxwell time coincides with the period of the pendulum for almost all geodesics. This result yields an explicit upper bound for the cut time in terms of the period of the pendulum.

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arXiv2025

Linear control systems on a 4D solvable Lie group used to model primary visual cortex $V1$

In this article, we study linear control systems on a 4-dimensional solvable Lie group. Our motivation stems from the model introduced in \cite{baspinar}, which presents a precise geometric framework in which the primary visual cortex $V1$ is interpreted as a fiber bundle over the retinal plane $M$ (identified with $\mathbb{R}^{2}$), with orientation $\theta \in S^{1}$, spatial frequency $\omega \in \mathbb{R}^{+}$, and phase $\phi \in S^{1}$ as intrinsic parameters. For each fixed frequency $\omega$, this model defines a Lie group $G(\omega) = \mathbb{R}^{2} \times S^{1} \times S^{1}$, which we adopt in this work as the state space group $G$ of our linear control system. We also present new results concerning controllability and characterize the control sets associated with this class of systems.

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arXiv2025

Dynamics of linear control systems and stabilization

In this paper, we study linear control systems with positive bounded orbits. We show that the existence of positive bounded orbits imposes strong algebraic and topological constraints on the state space. In fact, a linear control system has bounded positive orbits if and only if it can be decomposed as the product of the stable and central subgroups of the drift, with the central subgroup being compact. In particular, systems with bounded positive orbits admit a compact control set, and if the system is controllable, the entire state space is a compact group. As a byproduct, we obtain a complete characterization of the internal and BIBO stability of linear control systems.

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arXiv2025

Existence and uniqueness of control sets with a nonempty interior for linear control systems on solvable groups

In this paper, we obtain weak conditions for the existence of a control set with a nonempty interior for a linear control system on a solvable Lie group. We show that the Lie algebra rank condition together with the compactness of the nilpotent part of the generalized kernel of the drift are enough to assure the existence of such a control set. Moreover, this control set is unique and contains the whole generalized kernel in its closure.

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arXiv2025

Semigroups in semi-simple Lie groups: Flag type and estimation of cocycles

The flag type of a semigroup S of a noncompact semisimple Lie group is an algebraic tool related to the geometry of the invariant control set determined by S on the flag manifolds of G. In the present paper we show that it is possible to recover the flag type by studying the existence of lower bounds for cocycles on the maximal flag manifold

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arXiv2025

The chain recurrent set of flow of automorphisms on a decomposable Lie group

In this paper we show that the chain recurrent set of a flow of automorphisms on a connected Lie group coincides with the central subgroup of the flow, if the group is decomposable. Moreover, in the decomposable case, the flow satisfies the restriction property. Furthermore, the restriction of any flow of automorphisms to the connected component of the identity of its central subgroup is chain transitive.

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arXiv2024

Control sets of linear control systems on $\R^2$. The real case

In this paper, we study the dynamical behavior of a linear control system on $\R^2$ when the associated matrix has real eigenvalues. Different from the complex case, we show that the position of the control zero relative to the control range can have a strong interference in such dynamics if the matrix is not invertible. In the invertible case, we explicitly construct the unique control set with a nonempty interior.

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arXiv2024

One-input linear control systems on the homogeneous spaces of the Heisenberg group -- The singular case

The controllability issue of control-affine systems on smooth manifolds is one of the main problems in the theory, and it is recently known [Jouan P. Equivalence of control systems with linear systems on Lie groups and homogeneous spaces. ESAIM: Control Optim. Calc. Var. 2010, 16, 956-973] that it might be connected to that of a particular class of systems called linear control systems on (a homogeneous manifold of) a Lie group. Note that it may become very complicated to establish the controllability property of systems evolving on homogeneous spaces of Lie groups whose dynamics are induced by those of systems in the Lie group under consideration. In fact, even in low-dimensional certain homogeneous spaces, this is quite a challenging task, and for this reason, we have classified in [Da Silva, A., Kizil, E., Duman, O. Linear Control Systems on Homogeneous Spaces of the Heisenberg Group. J. Dyn. Control Syst. 2023, 29, 2065-2086] as a first goal all linear control systems on the homogeneous spaces of the 3-dimensional Heisenberg group $\mathbb{H}$ through its closed subgroups $L$ and, in particular, the controllability and the control sets have been studied for one of the homogeneous spaces $L\setminus \mathbb{H}$. In this paper, we study the controllability and control sets of the induced linear control systems in the homogeneous spaces left. In particular, we focus on the singularity of the induced drift vector fields that results in many cases and subcases to reveal control sets after quite a technical analysis. We give some nice illustrations to better understand what is going on geometrically.

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arXiv2023

Minimal-time trajectories of a linear control system on a homogeneous space of the 2D Lie group

Through the Pontryagin maximum principle, we solve a minimal-time problem for a linear control system on a cylinder, considered as a homogeneous space of the solvable Lie group of dimension two. The main result explicitly shows the existence of an optimal trajectory connecting every couple of arbitrary states on the manifold. It also gives a way to calculate the corresponding minimal time. Finally, the system admits points with two distinct minimal-time trajectories connecting them.

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arXiv2023

Control sets of one-input linear control systems on solvable, nonnilpotent 3D Lie groups

In this article, we completely describe the control sets of one-input linear control systems on solvable, nonnilpotent 3D Lie groups. We show that, if the restriction of the associate derivation to the nilradical is nontrivial, the Lie algebra rank condition is enough to assure the existence of a control set with a nonempty interior. Moreover, such a control set is unique and, up to conjugations, given as a cylinder of the state space. On the other hand, if such a restriction is trivial, one can obtain an infinite number of control sets with empty interiors or even controllability, depending on the group considered.

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arXiv2023

The chain control set of a linear control system

In this paper, we analyze the chain control sets of linear control systems on connected Lie groups. Our main result shows that the compactness of the central subgroup associated with the drift is a necessary and sufficient condition to assure the uniqueness and compactness of the chain control set.

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arXiv2022

Linear control systems on the homogeneous spaces of the Heisenberg group

Let H denote the 3-dimensional Heisenberg Lie group. The present paper classify all possible linear control systems on the homogeneous spaces of H through its closed subgroups and expose a detailed study on the control behavior (controllability property and control sets) of a particular dynamics evolving on a non simply connected homogeneous (state) space of dimension two.

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arXiv2022

Control sets of linear control systems on $\mathbb{R}^2$. The complex case

This paper explicitly computes the unique control set $D$ with non-empty interior of a linear control system on $\mathbb{R}^2$, when the associated matrix has complex eigenvalues. It turns out that the closure of $D$ coincides with the the region delimited by a computable periodic orbit $\mathcal{O}$ of the system.

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arXiv2022

Dynamics of LCSs on the group of proper motions

This article describes the control behavior of any linear control systems on the group of proper motions $SE(2)$. It characterizes the controllability property and the control sets of the system.

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