On the controllability of networks with nonidentical linear nodes

"The controllability of dynamical networks depends on both network structure and node dynamics. For networks of linearly coupled linear dynamical systems the controllability of the network can be determined using the well-known Kalman rank criterion. In the case of identical nodes the problem can be decomposed in local and structural contributions. However, for strictly different nodes an alternative approach is needed. We decomposed the controllability matrix into a structural component, which only depends on the networks structure and a dynamical component which includes the dynamical description of the nodes in the network. Using this approach we show that controllability of dynamical networks with strictly different linear nodes is dominated by the dynamical component. Therefore even a structurally uncontrollable network of different nn dimensional nodes becomes controllable if the dynamics of its nodes are properly chosen. Conversely, a structurally controllable network becomes uncontrollable for a given choice of the node’s dynamics. Furthermore, as nodes are not identical, we can have nodes that are uncontrollable in isolation, while the entire network is controllable, in this sense the node’s controllability is overwritten by the network even if the structure is uncontrollable. We illustrate our results using single-controller networks and extend our findings to conventional networks with large number of nodes.

El polinomio corchete en 4-trenzas

Tesis (Maestría en Control y Sistemas Dinámicos)"Un n-ovillo es una pareja (B3;T) donde B3 es la bola unitaria en R3 y T es un conjunto de n arcos propiamente encajados en B3. Un n-ovillo es llamado racional si, incluso moviendo B3, puede ser deformado en un n-ovillo que posee una proyección en donde sus cuerdas no tienen ningun cruce. Las n-trenzas son un subconjunto de los n-ovillos racionales. El polinomio corchete de Kauffman es un invariante bajo isotopia regular que ha sido usado para obtener una clasificacion de las 3-trenzas y de los 2-ovillos racionales, note que las 2-trenzas son un caso particular de los 2-ovillos racionales. En el caso de los 3-ovillos, el polinomio corchete de Kauffman es una funcion que, dado un 3-ovillo, le asigna cinco polinomios obtenidos de aplicar las relaciones del corchete al ovillo que corresponden a la descomposicion del algebra de los diagramas con una base de cinco 3-ovillos. Por otro lado, para el caso de los 4-ovillos, hay catorce polinomios, en lugar de cinco como en el caso de 3-ovillos, asociados a la base correspondiente de catorce 4-ovillos. En esta tesis se generaliza, de manera parcial, la clasificacion de 3-trenzas al caso de las 4-trenzas. Mas aun dada una 4-trenza se le asocia una matriz, que es un invariante de la 4-trenza, y esta asignacion tiene la propiedad de ser un homomorfismo entre las 4-trenzas y las matrices. Con este invariante se hizo la clasificacion de algunas familias de 4-trenzas.""An n-tangle is a pair (B3;T), where B3 is the 3-ball and T is a set of n disjoint properly embedded arcs in B3. An n-tangle is called rational if, by even moving B3 , it can be deformed into an n-tangle which possesses a projection with no crossings. n-Braids are a subset of rational n-tangles. The Kauffman bracket polynomial is an invariant under regular isotopy which has been used to obtain a classification of the 3-braids and rational 2-tangles, note that 2-braids are a particular case of rational 2-tangles. In the 3-tangle case, the Kauffman bracket polynomial is a function which, given a 3-tangle, assigns to it five polynomials obtained by applying the bracket relations to the tangle and corresponding to the algebra decomposition of the diagram with certain base of five 3-tangles. On the other hand, for the 4-tangle case there are fourteen polynomials, instead of five as in the 3-tangle case, associated to the corresponding base of fourteen 4-tangles. In this thesis we partially generalize the classification of 3-braids to 4-braids. Moreover, given a 4-braid a matrix, which is an invariant of the 4-braid, is associated to it and this assignation is a homomorphism between 4-braids and matrices. By using this invariant some 4-braids families are classified.