Schedule for: 23w5139 - Geometry, Topology and Control System Design

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building.

Gather for a Meet and Greet at the BIRS Lounge on the 2nd floor of PDC.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

A brief introduction to BIRS with important logistical information, technology instruction, and opportunity for participants to ask questions.

Optimal mass transport (OMT) was posed as a problem in 1781 by Gaspar Monge. It provides a natural geometry for the space of probability distributions. As such it has been the cornerstone of many recent developments in physics, probability theory, image processing, and so on. The Schrödinger bridge problem (SBP) was posed by Erwin Schrödinger in 1931, in an attempt to provide a classical interpretation of quantum mechanics. It is rooted in statistical mechanics and large deviations theory, and provides an alternative model for flows of distributions (entropic interpolation). In Part I, we will explain the close relation between the two problems, their relevance in modeling and control, computational aspects, and their various extensions. In Part II, we will focus on the relevance of the topic in stochastic thermodynamics.

Optimal mass transport (OMT) was posed as a problem in 1781 by Gaspar Monge. It provides a natural geometry for the space of probability distributions. As such it has been the cornerstone of many recent developments in physics, probability theory, image processing, and so on. The Schrödinger bridge problem (SBP) was posed by Erwin Schrödinger in 1931, in an attempt to provide a classical interpretation of quantum mechanics. It is rooted in statistical mechanics and large deviations theory, and provides an alternative model for flows of distributions (entropic interpolation). In Part I, we will explain the close relation between the two problems, their relevance in modeling and control, computational aspects, and their various extensions. In Part II, we will focus on the relevance of the topic in stochastic thermodynamics.

Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

Meet in the PDC front desk for a guided tour of The Banff Centre campus.

Meet in foyer of TCPL to participate in the BIRS group photo. The photograph will be taken outdoors, so dress appropriately for the weather. Please don't be late, or you might not be in the official group photo!

The operation of pull-back of functions induces two sorts of operators, one where the mapping is fixed and the function varies (the composition operator) and another where the function is fixed and the mapping varies (the superposition operator). We overview a few classical results regarding these operators, indicate a new one, and show how these operators show up in chronological calculus and in classical Picard iteration. We then show how continuity of the superposition operator can be used to prove regularity results for a very general setting of time-varying, parameter-dependent ordinary differential equations.

For differential inclusions, hybrid systems, and other natural generalizations of classical dynamics, continuous dependence of solutions on initial conditions may be too much to ask for. Concepts of set convergence and (semi)continuity of set-valued mappings come to the rescue. The talk will present the foundations of these ideas; highlight how, under natural (semi)continuity assumptions on the data of a differential inclusion or a hybrid system, classical results on invariance, smooth Lyapunov functions, uniformity and robustness of asymptotic stability, etc., extend to these settings; and perhaps connect these extensions, in the hybrid setting, to a result from topological theory of dynamical systems called the Conley's decomposition. The talk will be low on technical details.

Consider a particle whose position q is governed by the dynamics dq = (aq + u)dt + dW, where W is Brownian motion, a is a real number, and u is a control variable. We would like to choose u to minimize a given cost function. If the parameter a is known, then it is a classical result that the optimal u is the linear-quadratic regulator. Suppose, however, that a is completely unknown; we refer to this as agnostic control. In this case it is not even clear what it means to choose u optimally. In this talk, I will introduce the notion of regret to characterize optimal strategies for agnostic control—-a strategy is optimal if it minimizes the regret. Given any ε>0, I will then exhibit an agnostic control strategy that minimizes the regret to within a factor of (1+ε). This is joint work with M. Eggl, C. Fefferman, C. Rowley.

Figure skating is a beautiful sport combining elegance, precision, and athleticism. To understand some of the mechanics and complexity involved in this sport, we derive and analyze a model of a figure skater in continuous contact with the ice (i.e., no jumps). In the first part of the talk, we analyze the dynamics of a model figure skater and show that the behavior of the skater can either be regular (solvable) or chaotic, depending on the mass distribution. For regular dynamics, we show that there are constants of motion, one coming from symmetry (conservation of angular momentum) and one of a mysterious origin with no clear physical explanation. We then explain how a skater may control the trajectory on ice in a compulsory figures competition. For simplicity, we consider the model of a skater with no lean, with a controlled moving mass. We derive a control procedure by approximating the trajectories using circular arcs. We show that there is a control procedure of a 'lazy' (or 'efficient') figure skater, minimizing the 'relative kinetic energy' of the control mass, which leads to well-posed equations for the control masses. We demonstrate examples of our system tracing actual compulsory figure skating trajectories. We also discuss further extensions of the model and applications to real-life figure skating. Joint work with M. Rhodes and V. Gzenda

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

A broad class of models deal with spaces of directed paths avoiding some dynamic obstacles. (Examples include concurrency, in computer science, and collision avoidance in control theory.) The topology (say, the homotopy type) of these spaces is not yet well understood. It turns out, the study of these spaces is in many respects parallel to the theory of linear subspace arrangements. I will outline some recent results, and the key tools used to obtain them, - in particular, the apparatus of diagrams of spaces.

A broad class of models deal with spaces of directed paths avoiding some dynamic obstacles. (Examples include concurrency, in computer science, and collision avoidance in control theory.) The topology (say, the homotopy type) of these spaces is not yet well understood. It turns out, the study of these spaces is in many respects parallel to the theory of linear subspace arrangements. I will outline some recent results, and the key tools used to obtain them, - in particular, the apparatus of diagrams of spaces.

Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

Neural networks are increasingly deployed as controllers in real-world systems such as self-driving vehicles and mobile robots. Despite their impressive performance, neural networks are known to be highly sensitive to input disturbances, something which has caused serious concerns about their safety and reliability. In this talk, we develop a computationally efficient framework for estimating reachable sets of a nonlinear dynamical system with a neural network controller. The key idea is to embed the closed-loop dynamics into a larger system using an inclusion function of the neural network. By leveraging the monotone system theory, we compute hyper-rectangular over-approximations of the reachable sets using a single trajectory of the embedding system. Moreover, using non-Euclidean contraction theory, we show that if this embedding system is constructed in a certain way, the contraction rate of the embedding system is the same as the original closed-loop system. Thus, this embedding provides a scalable approach for reachability analysis of the neural control loop while preserving the nonlinear structure of the system. Finally, using numerical simulations, we compare the efficiency of our framework with the state-of-the-art approaches.

A retraction map is an essential tool in different research areas like optimization theory, numerical analysis, interpolation. We have recently extended the notion of retraction map to discretization map that focus on discretizing the manifold where the dynamics takes place in stead of discretizing the variational principle as in the foundational work by Marsden and West (2001). Indeed, the continuous dynamics is used to define the discretization rule. As a result, a systematic and geometric procedure is described to construct geometric integrators for mechanical system including constraints, controls, preserving symmetries, etc. This is a joint work with David Martín de Diego (ICMAT-CSIC, Spain).

In this talk I will discuss the problem of asymptotically stabilizing a compact subset of a manifold with a smooth, time-independent, feedback control law. I will explain that one obstruction to doing so arises from a coordinate-free generalization of Brockett's necessary condition from joint work with Daniel E. Koditschek (arXiv:2106.00215). I will then describe a stronger obstruction arising from the fact that any pair of vector fields asymptotically stabilizing the same compact set must be homotopic through nowhere-zero vector fields over a certain subset (arXiv:2205.07840). The latter obstruction leads to a generalization of the homological necessary conditions due to Coron and Mansouri. Finally, I will discuss joint work with Anthony M. Bloch on the case of periodic orbit stabilizability---which, curiously, cannot be detected by any of the topological obstructions from the literature---with applications to nonholonomic systems.

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

Roger Brockett’s 1973 paper – Lie Theory and Control Systems Defined on Spheres, has illuminated the field through the many questions and threads of inquiry it initiated. In this talk, we consider one aspect of optimal control and Brockett’s approach to it. In joint work with Eric Justh, we had derived a synchronization result for interacting agents subject to a quadratic cost functional. We used methods of Poisson reduction and illustrated the same in the setting of the rigid motion group SE(2). On the other hand, Brockett’s approach in his 1973 paper suggests a convenient alternative, best described as enlargement. We will discuss these results and questions evolving from them.

We consider Linear Quadratic Regulation (LQR) for the boundary control of the one dimensional, undamped, linear wave equation under Neumann actuation. We present a Riccati partial differential equation, the derivation of is by the simple and explicit techniques of integration by parts and completing the square. The Fourier expansion of the solution of the Riccati PDE leads to an infinite dimensional algebraic Riccati equation that can be solved by policy iteration. Since the system is undamped, all the open loop eigenvalues lie on the imaginary axis. Under suitable assumptions a Neumann LQR feedback moves all these eigenvalues into the open left half plane. The closed loop eigenvalues converge to the open loop eigenvalues as the wave number increases so the closed loop system is asymptotically stable but not exponentially stable. An interesting fact is that the closed loop modal shapes are complex sinusoids and they appear to converge to real sinusoids as the wave number increases.

Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

In recent years the task of controlling a large, potentially infinite, number of states or systems at once using only a single open-loop input or a single feedback controller has posed a challenge in mathematical systems and control theory. More specifically, we consider families of systems which are defined by parameter-dependent linear systems and treat the question how to (approximately) control the entire family using open-loop inputs or feedback controllers which are independent of the parameters of the systems. Loosely speaking, a family of parameter-dependent linear systems is called ensemble reachable if for every family of terminal states and for each neighborhood of it there are parameter-independent inputs such that the origin can be steered into the neighborhood of the terminal states in some finite time. In this talk we treat continuous-time and discrete-time systems and present criteria for the existence of such open-loop inputs. Also we discuss methods to design suitable inputs. Though parameter-dependent systems are located in infinite dimensional systems theory they are at the crossroads to finite-dimensional control theory as well as to other mathematical areas, such as approximation theory, complex analysis, inverse problems and operator theory. In addition, we also present some recent results on feedback methods for parameter-dependent linear systems.

This talk is concerned with the universal approximation capabilities of deep residual neural networks with respect to the uniform norm. We relate the universal approximation problem to controllability of an ensemble of control systems corresponding to sample points on the function to be approximated. We leverage classical Lie algebraic techniques to identify a class of activation functions, encompassing many known ones, that ensure controllability. We then identify monotonicity as the bridge between controllability of finite ensembles and uniform approximability on compact sets.

It is often handy in trajectory planning problems to have a linear control system or linearizable control system. In the case that a control system is ``intrinsically nonlinear" the next best thing is flat outputs or a dynamic feedback linearization. It is also often the case that control systems inspired by nature/engineering have symmetries that may provide insight into many questions about a given control system. It turns out we can use symmetries to probe the existence of dynamic feedback linearizations and even construct them explicitly! I'll present this procedure via explicit examples. This approach is known as cascade feedback linearizability and is joint work with Peter J. Vassiliou and Jeanne N. Clelland.

We describe a new algorithm for computing Whitney stratifications of both real and complex algebraic varieties. The main ingredients are (a) an algebraic criterion, due to Lê and Teissier, which reformulates Whitney regularity in terms of conormal spaces and maps, and (b) a new interpretation of this conormal criterion via primary decomposition, which can be practically implemented on a computer. Using this we also give an algorithm to stratify algebraic maps between varieties. Potential applications of this stratification to the study of singularities of kinematic maps and ongoing work in this setting will be discussed. This is joint work with Vidit Nanda (University of Oxford).

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

5-day workshop participants are welcome to use BIRS facilities (TCPL ) until 3 pm on Friday, although participants are still required to checkout of the guest rooms by 11AM.