On a differential algebraic system structure theory

International audienceThe structure at infinity and the essential structure are two control theory notions which were first defined for linear, then for the so-called affine, systems. And they were shown to be useful tools for the study of the fundamental problem of noninteracting control. They also appeared as related to the solutions of other important control problems such as disturbance decoupling. Their definitions are however entirely in terms of an algorithm, namely the so-called structure algorithm. The present work proposes new definitions with some advantages: they extend the class of systems from linear and affine systems to systems which may be described by algebraic differential equations, they are not tied to specific algorithms, and finally they provide more information on system structure. Let a system be a set of differential equations in variables which are grouped as m inputs, p outputs and n latent variables. To each input component is attached a rational integer, which, for a single input single output system defined by a single differential equation, is the difference between the order in the output and the order in the input of the differential equation defining the system. The m-tuple of these rational integers is the new structure at infinity of the system. Associated to the structure at infinity is also defined a p-tuple of rational integers representing a new notion of essential structure. The old structure at infinity is shown to be recoverable from the new one. Computations of system structure based upon the suggested definitions are quite complex. The present paper focuses on proofs of algorithms which attempt to reduce the complexity of these computations

Ramanujan systems of Rankin-Cohen type and hyperbolic triangles

In the first part of the paper we characterize systems of first order nonlinear differential equations whose space of solutions is an $\mathfrak{sl}_2(\mathbb{C})$-module. We prove that such systems, called Ramanujan systems of Rankin-Cohen type, have a special shape and are precisely the ones whose solution space admits a Rankin-Cohen structure. In the second part of the paper we consider triangle groups $\Delta(n,m,\infty)$. By means of modular embeddings, we associate to every such group a number of systems of non linear ODEs whose solutions are algebraically independent twisted modular forms. In particular, all rational weight modular forms on $\Delta(n,m,\infty)$ are generated by the solutions of one such system (which is of Rankin-Cohen type). As a corollary we find new relations for the Gauss hypergeometric function evaluated at functions on the upper half-plane. To demonstrate the power of our approach in the non classical setting, we construct the space of integral weight twisted modular form on $\Delta(2,5,\infty)$ from solutions of systems of nonlinear ODEs.Comment: 23 page

Optimal control for polynomial systems using the sum of squares approach

The optimal control in linear systems is a widely known problem that leads to the solution of one or two equations of Ricatti. However, in non-linear systems is required to obtain the solution of the Hamilton-Jacobi-Bellman equation (HJB) or variations, which consist of quadratic first order and partial differential equations, that are really difficult to solve. On the other hand, many non-linear dynamical systems can be represented as polynomial functions, where thanks to abstract algebra there are several techniques that facilitate the analysis and work with polynomials. This is where the sum-of-squares approach can be used as a sufficient condition to determine the positivity of a polynomial, a tool that is used in the search for suboptimal solutions of the HJB equation for the synthesis of a controller. The main objective of this thesis is the analysis, improvement and/or extension of an optimal control algorithm for polynomial systems by using the sum of squares approach (SOS). To do this, I will explain the theory and advantages of the sum-of-squares approach and then present a controller, which will serve as the basis for our proposal. Next, improvements will be added in its performance criteria and the scope of the controller will be extended, so that rational systems can be controlled. Finally an alternative will be presented for its implementation, when it is not possible to measure or estimate the state-space variables of the system. Additionally, some examples that validated the results are also presented.Tesi

Methods in Mathematica for Solving Ordinary Differential Equations

An overview of the solution methods for ordinary differential equations in the Mathematica function DSolve is presented.Comment: 13 page

A multiple exp-function method for nonlinear differential equations and its application

A multiple exp-function method to exact multiple wave solutions of nonlinear partial differential equations is proposed. The method is oriented towards ease of use and capability of computer algebra systems, and provides a direct and systematical solution procedure which generalizes Hirota's perturbation scheme. With help of Maple, an application of the approach to the $3+1$ dimensional potential-Yu-Toda-Sasa-Fukuyama equation yields exact explicit 1-wave and 2-wave and 3-wave solutions, which include 1-soliton, 2-soliton and 3-soliton type solutions. Two cases with specific values of the involved parameters are plotted for each of 2-wave and 3-wave solutions.Comment: 12 pages, 16 figure

Non-Integrability of Some Higher-Order Painlev\'e Equations in the Sense of Liouville

In this paper we study the equation $$w^{(4)} = 5 w" (w^2 - w') + 5 w (w')^2 - w^5 + (\lambda z + \alpha)w + \gamma,$$ which is one of the higher-order Painlev\'e equations (i.e., equations in the polynomial class having the Painlev\'e property). Like the classical Painlev\'e equations, this equation admits a Hamiltonian formulation, B\"acklund transformations and families of rational and special functions. We prove that this equation considered as a Hamiltonian system with parameters $\gamma/\lambda = 3 k$, $\gamma/\lambda = 3 k - 1$, $k \in \mathbb{Z}$, is not integrable in Liouville sense by means of rational first integrals. To do that we use the Ziglin-Morales-Ruiz-Ramis approach. Then we study the integrability of the second and third members of the $\mathrm{P}_{\mathrm{II}}$-hierarchy. Again as in the previous case it turns out that the normal variational equations are particular cases of the generalized confluent hypergeometric equations whose differential Galois groups are non-commutative and hence, they are obstructions to integrability