POLSYS GLP: A Parallel General Linear Product Homotopy Code for Solving Polynomial Systems of Equations

Globally convergent, probability-one homotopy methods have proven to be very effective for finding all the isolated solutions to polynomial systems of equations. After many years of development, homotopy path trackers based on probability-one homotopy methods are reliable and fast. Now, theoretical advances reducing the number of homotopy paths that must be tracked, and in the handling of singular solutions, have made probability-one homotopy methods even more practical. POLSYS GLP consists of Fortran 95 modules for nding all isolated solutions of a complex coefficient polynomial system of equations. The package is intended to be used on a distributed memory multiprocessor in conjunction with HOMPACK90 (Algorithm 777), and makes extensive use of Fortran 95 derived data types and MPI to support a general linear product (GLP) polynomial system structure. GLP structure is intermediate between the partitioned linear product structure used by POLSYS PLP (Algorithm 801) and the BKK-based structure used by PHCPACK. The code requires a GLP structure as input, and although nding the optimal GLP structure is a dicult combinatorial problem, generally physical or engineering intuition about a problem yields a very good GLP structure. POLSYS GLP employs a sophisticated power series end game for handling singular solutions, and provides support for problem denition both at a high level and via hand-crafted code. Dierent GLP structures and their corresponding Bezout numbers can be systematically explored before committing to root finding

Control Theory: A Mathematical Perspective on Cyber-Physical Systems

Control theory is an interdisciplinary field that is located at the crossroads of pure and applied mathematics with systems engineering and the sciences. Recently the control field is facing new challenges motivated by application domains that involve networks of systems. Examples are interacting robots, networks of autonomous cars or the smart grid. In order to address the new challenges posed by these application disciplines, the special focus of this workshop has been on the currently very active field of Cyber-Physical Systems, which forms the underlying basis for many network control applications. A series of lectures in this workshop was devoted to give an overview on current theoretical developments in Cyber-Physical Systems, emphasizing in particular the mathematical aspects of the field. Special focus was on the dynamics and control of networks of systems, distributed optimization and formation control, fundamentals of nonlinear interconnected systems, as well as open problems in control