This paper shows how Grobner bases can be used to solve some common problems in nonlinear systems theory efficiently. These problems include finding critical levels of local Lyapunov functions and solving the equations that arise in the harmonic balancing method. The methods proposed are illustrated by some concrete examples in which the computer algebra system Maple is used for performing the necessary calculations. 1 Introduction. Grobner bases began to be studied seriously by mathematicians in the mid-70's. They have been used to solve difficult problems in commutative algebra constructively [14], and algorithms for constructing Grobner bases have been implemented on computers [6]. However, they seem to have found little use in control applications sofar. The problem of solving a system of nonlinear (algebraic) equations in several variables arises frequently in different applications. There is a general method for solving such systems in the case where the nonlinearities are of po...

In this paper, we explore the dynamical features of a neural network model which presents two types of adaptative parameters : the classical weights between the units and the time constants associated with each artificial neuron. The purpose of this study is to provide a strong theoretical basis for modeling and simulating dynamic recurrent neural networks. In order to achieve this, we study the effect of the statistical distribution of the weights and of the time constants on the network dynamics and we make a statistical analysis of the neural transformation. We examine the network power spectra (to draw some conclusions over the frequential behavior of the network) and we compute the stability regions to explore the stability of the model. We show that the network is sensitive to the variations of the mean values of the weights and the time constants (because of the temporal aspects of the learned tasks). Nevertheless, our results highlight the improvements in the network dynamics d...

We consider a time-invariant, continuous system x = f(x; u), where f is a rational function of the state x, linear in the input u. We introduce a Linear-Fractional Representation (LFR) for the system, which consists of viewing it as an LTI system, connected with a diagonal feedback operator linear in the state. Using this representation, we devise sufficient conditions for various properties to hold for the open-loop system. These include cheking whether a given polytope is stable, finding a lower bound on the decay rate on this polytope, etc. All these conditions are obtained by analyzing the properties of a related differential inclusion, and checked using convex optimization over Linear Matrix Inequalities (LMIs). The method extends to (static) state-feedback synthesis. 1. Introduction We consider a nonlinear, time-invariant, continuous-time system x = f(x; u); (1) where x : R+ ! R n , u : R+ ! R nu is the input, and y : R+ ! R ny is the output. We assume that f is a rati...

. Some connections between constructive real algebraic geometry and constrained optimization are exploited. We show how the problem of determining the projection of a real-algebraic variety on a certain axis is equivalent to a problem in nonlinear programming. As an application, Grobner bases are used to deal with an optimization problem arising in the theory of local Lyapunov functions. The problems addressed are: determining critical levels of local Lyapunov functions and investigating robustness using Lyapunov functions. Since the tools used come from commutative algebra and algebraic geometry the differential equations considered are of polynomial type and the Lyapunov functions used are polynomial. Keywords: constrained optimization, Lyapunov theory, stability, polynomial differential equations, robustness, Grobner bases, elimination theory, nonlinear equation solving, real algebraic geometry, quantifier elimination, commutative algebra 1 Introduction In this paper we discuss th...

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