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A fundamental problem from invariant theory is to describe the endomorphism algebra of multilinear functions on a representation V invariant under the action of a group G. According to Weyl's classic, a first main (later: fundamental)…
The purpose of this note is to survey a methodology to solve systems of polynomial equations and inequalities. The techniques we discuss use the algebra of multivariate polynomials with coefficients over a field to create large-scale linear…
The purpose of this paper is to use the framework of Lie algebroids to study optimal control problems for affine connection control systems on Lie groups. In this context, the equations for critical trajectories of the problem are…
Reparable systems are systems that are characterized by their ability to undergo maintenance actions when failures occur. These systems are often described by transport equations, all coupled through an integro-differential equation. In…
Quantum gauge theory in the connection representation uses functions of holonomies as configuration observables. Physical observables (gauge and diffeomorphism invariant) are represented in the Hilbert space of physical states; physical…
This paper studies a class of complex-valued linear systems whose state evolution dependents on both the state vector and its conjugate. The complex-valued linear system comes from linear dynamical quantum control theory and is also…
In this paper, we tackle the long-standing challenges of ensemble control analysis and design using a convex-geometric approach in a Hilbert space setting. Specifically, we formulate the control of linear ensemble systems as a convex…
Many relevant applications of group theoretical methods to physical problems are related, in some manner, to classification schemes by means of symmetry groups. In these schemes, irreducible representations of a Lie group have to be…
This paper analyzes the optimal control problem of cubic polynomials on compact Lie groups from a Hamiltonian point of view and its symmetries. The dynamics of the problem is described by a presymplectic formalism associated with the…
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.
For linear control systems with bounded control range, chain controllability properties are analyzed. It is shown that there exists a unique chain control set and that it equals the sum of the control set around the origin and the center…
In this paper, we present a state-feedback controller design method for bilinear systems. To this end, we write the bilinear system as a linear fractional representation by interpreting the state in the bilinearity as a structured…
Graphs are a natural representation for systems based on relations between connected entities. Combinatorial optimization problems, which arise when considering an objective function related to a process of interest on discrete structures,…
This paper is, essentially, a survey related to the problem of understanding the combinatorics of the action of the monoidal category of finite dimensional modules over a simple finite dimensional Lie algebra on various categories of Lie…
In this paper we investigate the controllability and observability properties of a family of linear dynamical systems, whose structure is induced by the Laplacian of a grid graph. This analysis is motivated by several applications in…
Framed combinatorial topology is a novel theory describing combinatorial phenomena arising at the intersection of stratified topology, singularity theory, and higher algebra. The theory synthesizes elements of classical combinatorial…
In this work, we will investigate the question of optimal control for bilinear systems with constrained endpoint. The optimal control will be characterized through a set of unconstrained minimization problems that approximate the former.…
In this paper, we show how to use the analysis of the Lie algebra associated with a quantum mechanical system to study its dynamics and facilitate the design of controls. We give algorithms to decompose the dynamics and describe their…
Graph-based signal processing techniques have become essential for handling data in non-Euclidean spaces. However, there is a growing awareness that these graph models might need to be expanded into `higher-order' domains to effectively…
This is a book on higher-categorical diagrams, including pasting diagrams. It aims to provide a thorough and modern reference on the subject, collecting, revisiting and expanding results scattered across the literature, informed by recent…