143,224 research outputs found
Polynomial Optimization with Applications to Stability Analysis and Control - Alternatives to Sum of Squares
In this paper, we explore the merits of various algorithms for polynomial
optimization problems, focusing on alternatives to sum of squares programming.
While we refer to advantages and disadvantages of Quantifier Elimination,
Reformulation Linear Techniques, Blossoming and Groebner basis methods, our
main focus is on algorithms defined by Polya's theorem, Bernstein's theorem and
Handelman's theorem. We first formulate polynomial optimization problems as
verifying the feasibility of semi-algebraic sets. Then, we discuss how Polya's
algorithm, Bernstein's algorithm and Handelman's algorithm reduce the
intractable problem of feasibility of semi-algebraic sets to linear and/or
semi-definite programming. We apply these algorithms to different problems in
robust stability analysis and stability of nonlinear dynamical systems. As one
contribution of this paper, we apply Polya's algorithm to the problem of
H_infinity control of systems with parametric uncertainty. Numerical examples
are provided to compare the accuracy of these algorithms with other polynomial
optimization algorithms in the literature.Comment: AIMS Journal of Discrete and Continuous Dynamical Systems - Series
Complexity Reduction for Parameter-Dependent Linear Systems
We present a complexity reduction algorithm for a family of
parameter-dependent linear systems when the system parameters belong to a
compact semi-algebraic set. This algorithm potentially describes the underlying
dynamical system with fewer parameters or state variables. To do so, it
minimizes the distance (i.e., H-infinity-norm of the difference) between the
original system and its reduced version. We present a sub-optimal solution to
this problem using sum-of-squares optimization methods. We present the results
for both continuous-time and discrete-time systems. Lastly, we illustrate the
applicability of our proposed algorithm on numerical examples
Survivability of Deterministic Dynamical Systems
The notion of a part of phase space containing desired (or allowed) states of
a dynamical system is important in a wide range of complex systems research. It
has been called the safe operating space, the viability kernel or the sunny
region. In this paper we define the notion of survivability: Given a random
initial condition, what is the likelihood that the transient behaviour of a
deterministic system does not leave a region of desirable states. We
demonstrate the utility of this novel stability measure by considering models
from climate science, neuronal networks and power grids. We also show that a
semi-analytic lower bound for the survivability of linear systems allows a
numerically very efficient survivability analysis in realistic models of power
grids. Our numerical and semi-analytic work underlines that the type of
stability measured by survivability is not captured by common asymptotic
stability measures.Comment: 21 pages, 6 figure
On the Structure of the Global Attractor for Infinite-Dimensional Non-Autonomous Dynamical Systems with Weak Convergence
The aim of this paper is to describe the structure of global attractors for infinite-dimensional non-autonomous dynamical systems with recurrent coefficients. We consider a special class of this type of systems (the so–called weak convergent systems). We study this problem in the framework of general
non-autonomous dynamical systems (cocycles). In particular, we apply the general results obtained in our previous paper to study the almost periodic (almost automorphic, recurrent, pseudo recurrent) and asymptotically almost periodic (asymptotically almost automorphic, asymptotically recurrent,
asymptotically pseudo recurrent) solutions of different classes of differential equations (functional-differential equations, evolution equation with monotone operator, semi-linear parabolic equations)
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