Nonlinear Control and Geometry

John Baillieul (Boston University) Title: Topological aspects of optimal information acquisition in robotic exploration and multimodal sensor fusion. Abstract: The talk presents some recent work to construct mathematical models of the information content of spatially varying scalar fields defined on R,R, and R. The models are developed using analytical tools from both differential topology and information theory. The concept of topological persistence is described, and a refined notion that we refer to as topological information utility is presented. We describe applications in which the information content of measurements of continuously varying scalar fields can be used to guide robotic exploration or to support a new approach to multi-sensor data fusion with the aim of creating a composite image of a geometric locale that maximizes the information available to an observer. Because the model rests on information theory, it provides the foundation for treating search as a problem in optimal information acquisition. Because there is also a connection to topology and geometry, search problems may be thought of in terms of climbing information gradients. The talk presents some recent work to construct mathematical models of the information content of spatially varying scalar fields defined on R,R, and R. The models are developed using analytical tools from both differential topology and information theory. The concept of topological persistence is described, and a refined notion that we refer to as topological information utility is presented. We describe applications in which the information content of measurements of continuously varying scalar fields can be used to guide robotic exploration or to support a new approach to multi-sensor data fusion with the aim of creating a composite image of a geometric locale that maximizes the information available to an observer. Because the model rests on information theory, it provides the foundation for treating search as a problem in optimal information acquisition. Because there is also a connection to topology and geometry, search problems may be thought of in terms of climbing information gradients. Maria Barbero-Linan (Universidad Carlos III de Madrid-ICMAT) Title: Geometric description of controllability of hybrid control systems. Abstract: Hybrid systems with or without controls are those systems where there is an interaction between continuous dynamics and discrete events. Some examples of hybrid systems are given by a bouncing ball, an automobile with automatic or manual transmission, impacts, thermostats, etc. These systems attract the attention of engineers, computer scientists and mathematicians. Recently, mathematicians have focused on the geometrization of hybrid systems in order to bring more understanding to all the possible case studies [2,3,5]. A hybrid control system can be understood as a family of generalized dynamical systems with interactions among them. That information can be summarized in a directed graph. These interactions make necessary a casuistic approach to study hybrid control systems. The notion of controllability remains the same as in classical control theory, that is, a hybrid control system is controllable if for any two points there exists an admissible trajectory that joins them [1,4]. However, for hybrid systems, not only the involutive distribution of the control vector fields contributes to the reachable points, but also the set of points where jumps between systems take place. That is why single control systems could be no controllable, but interactions among them could make the hybrid control system controllable. In this talk we first define geometrically hybrid control systems, then focus on controllability tests for such systems. Some examples will be provided to elucidate the importance and necessary properties of the jump set to guarantee controllability. References: [1 ] Francesco Bullo and Andrew D. Lewis. Geometric control of mechanical systems, volume 49 of Texts in Applied Mathematics. Springer-Verlag, New York, 2005. Modeling, analysis, and design for simple mechanical control systems. [2 ] Rafal Goebel, Ricardo G. Sanfelice, and Andrew R. Teel. Hybrid dynamical systems. Princeton University Press, Princeton, NJ, 2012. Modeling, stability, and robustness. [3 ] Daniel Liberzon. Switching in systems and control. Systems & Control: Foundations & Applications. Birkhauser Boston Inc., Boston, MA, 2003. Hybrid systems with or without controls are those systems where there is an interaction between continuous dynamics and discrete events. Some examples of hybrid systems are given by a bouncing ball, an automobile with automatic or manual transmission, impacts, thermostats, etc. These systems attract the attention of engineers, computer scientists and mathematicians. Recently, mathematicians have focused on the geometrization of hybrid systems in order to bring more understanding to all the possible case studies [2,3,5]. A hybrid control system can be understood as a family of generalized dynamical systems with interactions among them. That information can be summarized in a directed graph. These interactions make necessary a casuistic approach to study hybrid control systems. The notion of controllability remains the same as in classical control theory, that is, a hybrid control system is controllable if for any two points there exists an admissible trajectory that joins them [1,4]. However, for hybrid systems, not only the involutive distribution of the control vector fields contributes to the reachable points, but also the set of points where jumps between systems take place. That is why single control systems could be no controllable, but interactions among them could make the hybrid control system controllable. In this talk we first define geometrically hybrid control systems, then focus on controllability tests for such systems. Some examples will be provided to elucidate the importance and necessary properties of the jump set to guarantee controllability. References: [1 ] Francesco Bullo and Andrew D. Lewis. Geometric control of mechanical systems, volume 49 of Texts in Applied Mathematics. Springer-Verlag, New York, 2005. Modeling, analysis, and design for simple mechanical control systems. [2 ] Rafal Goebel, Ricardo G. Sanfelice, and Andrew R. Teel. Hybrid dynamical systems. Princeton University Press, Princeton, NJ, 2012. Modeling, stability, and robustness. [3 ] Daniel Liberzon. Switching in systems and control. Systems & Control: Foundations & Applications. Birkhauser Boston Inc., Boston, MA, 2003.