SCIENTIFIC PROGRAMS AND ACTIVITIES
December 22, 2024

Seminar Series on Control Theory

Date: June 4, 1992

Topic: "Feedback Equivalence to Linear Prime Systems"

Speaker: Ricardo Marino, University of California, Santa Barbara

State space and output space change of coordinates and static state feedback are used to transform a multivariable nonlinear system with m-inputs and m-outputs into a linear prime system, i.e. m chain of integrators from the new outputs to the new inputs. Necessarky and sukfficient differential geometric conditions for the solvability of the problem are given. They generalize well-known results both on feedback linearization as well as on input output decoupling of nonlinear systems. From a computational point of view the output space change of coordinates is the crucial step, which is performed by constructing rectifying coordinates for a nested sequence of distributions on the output manifold.

Date: June 4, 1992

Topic: "Dynamic Output Feedback Linearization and Global Stabilization"

Speaker: Ricardo Marino, University of California, Santa Barbara

Differential geometric conditions are given to identify a class of single-input, single-output nonlinear systems which are globally transformable by dynamic output feedback control, "filtered' state space change of coordinates and output space change of coordinates into a linear observable minimum phase system. A global output tracking control can be designed for those systems. It is then shown that some conditions may be relaxed and a global output tracking course can still be designed without relying on the dynamic output feedback linearization property.

Date: May 15, 1992

Topic: "Elastic Curves and Related Variational Problems" - The Second of Two Talks

Speaker: Velimir Jurdjevic, Toronto and Fields Institute

Many classical variational problems with constraints, when viewed as optimal contol problems can be effectively analyzed through the Maximum Principle and its associated Hamiltonian formalism. In this lecture I will discuss the equilibrium configurations of a thin elastic, inextensible, rod in R^3, by treating it as a left-invariant variational problem on the group of motions of R^3. I will divide the equations of motion, show the connection with the equations of the rigid body (kinetic analogue of Kirchhoff), and discuss the integrability of the resulting systems.

Date: May 13, 1992

Topic: "Elastic Curves and Related Variational Problems" - The First of Two Talks

Speaker: Velimir Jurdjevic, Toronto and Fields Institute

Many classical variational problems with constraints, when viewed as optimal contol problems can be effectively analyzed through the Maximum Principle and its associated Hamiltonian formalism. In this lecture I will discuss the equilibrium configurations of a thin elastic, inextensible, rod in R^3, by treating it as a left-invariant variational problem on the group of motions of R^3. I will divide the equations of motion, show the connection with the equations of the rigid body (kinetic analogue of Kirchhoff), and discuss the integrability of the resulting systems.

Date: May 8, 1992

Topic: "Time Control of Tidal Generation"

Speaker: G.F.D. Duff, University of Toronto

Because of its almost periodic intermittent and time precessing nature, the availability of electric power from turbines in a tidal barrier requires time dependent controls. The nature of this problem will be described, together with some of the elaborations of design and operation that are possible. Systems with single, double or multiple basins are considered, with reference to the problem of construction cost.

Mathematical formulation by way of optimal control theory or by linear programming is described, and interaction with the hydrodynamical wave properties of the ocean system included. Numerical calculations, with references to stability and the frequent occurrence of focal points which act as limiting factors, will be described, and specific models based on the sites in Nova Scotia and New Brunswick discussed.

Date:May 1, 1992

Topic: "Integrable Foliations, the 2DFilament Equation, and the Modified KdV Equation"

Speaker: Ron Perline, Drexel University, Philadelphia

Date:April 30, 1992

Topic: "Elastic Curves, 3D Filaments and the Hasimoto Transformation "

Speaker: Ron Perline, Drexel University, Philadelphia

Date:April 22, 1992

Topic: "Sub-Riemannian Geometry and Heat Kernels on Lie Groups: Long Time Estimates"

Speaker: Nicholas Varpoulos, Pierre et Marie Paris IV

Date: April 10, 1992

Topic: "Global Properties of Compact Curves and Surfaces in Minkowski 3-space" - The Second of Two Talks

Speaker: Marek Kossowski, University of South Carolina

The classical global theorems for curves and surfaces in Euclidean 3-space have analogies in Minkowski 3-space if the curves and surfaces satisfy certain generic conditions. The talk will give an overview.

Date:April 8, 1992

Topic: "Local Existence and Stability of Multivalued Solutions to Determined Nonlinear PDE on the Plane" - The First of Two Talks

Speaker: Marek Kossowski, University of South Carolina

Date:April 3, 1992

Topic: "Application of Singular Riemannian Foliations in Control Theory (Part II)"

Speaker: Miroslav Lovric, Mathematics and Statistics, McMaster University

Date: March 27, 1992

Topic: "Colombeau-Rosinger Theory of Generalized Function Algebras"

Speaker: Robert Hermann, ECS Department, Boston University

Date: March 27, 1992

Topic: "Input-Output Characterization of Hamiltonian Input-Output Systems"

Speaker: Peter Crouch, Arizona State

Date: March 27, 1992

Topic: "Reachable Sets and Singular Riemannian Foliations (Part I)"

Speaker: Miroslav Lovric, McMaster University

Date: March 27, 1992

Topic: "Stability and Instability via Geometric Methods "

Speaker: Tudor Ratiu, University of California, Santa Cruz

Date: March 26, 1992

Topic: "Geometric Phases in Mechanical Systems"

Speaker: Tudor Ratiu, University of California, Santa Cruz

Date: March 26, 1992

Topic: "Gradient Flows and Lie Theory"

Speaker: Robert Hermann, Boston

Date: March 25, 1992

Topic: "Variational Characterization of Hamiltonian Input Output Systems"

Speaker: Peter Crouch, Arizona State

Date: March 23, 1992

Topic: "Robust Stability and Performance in the Presence of Mixed Perturbations"

Speaker: Mohammed Dahleh, University of California, Santa Barbara

Date: March 20, 1992

Topic: "Smoke Rings and Springy Wires"

Speaker: Joel Langer, Case Western Reserve University

The vortex filament equation (VF) is a model for the evolution of a thin vortex filament in three dimensional hydrodyanmics. This model is known to be a completely integrable infinite dimensional Hamiltonian system, and it is also known that equilibrium configurations of the Kirchhoff elastic rod model yield soliton solutions of (VF). The latter model is also an integrable Hamiltonian system (finite dimensional). In the talk, relationships between the two integrable systems will be discussed.

Date: March 13, 1992

Topic: "Analytic Surgery for Mainfolds Degenerating along a Hypersurface"

Speaker: Parick McDonald, Ohio State University

Date: March 6, 1992.

Topic: "Local Invariants in Sub-Riemannian Geometry"

Speaker: George Wilkens, University of Hawaii

In sub-Riemannian geometry, one is given a smooth manifold, M, and a smooth, constant rank subbundle, D, of the tangent bundle of M. We assume that D satisfies the bracket generating condition, that is that at each point x in M, the tangent space to M at x (i.e. TxM) is spanned by the vectors in the Lie algebra generated by the vector fields of M which lie in D. Additionally, one is given a smooth function a: D -> R such that the restriction of a to each fiber is a positive definite quadratic form. With this structure, we say that a smooth curve in M is an integral curve of D if x'(t) lies in D for all t.We define an energy functional on integral curves by E(x) = 1/2 Integrate [a(x'(t)) dt]. Sub-Riemannian geometry is the geometry associated to the distribution D and the energy E.

I will take M to be a three dimensional manifold and D to be a rank two subbundle of TM. I will show that one can use geometrically nautral conditions to find all the local invariants of this structure. There are four fundamental invariants. These invariants all vanish if and only if the geometry is the one described by the Heisenberg fly wheel. Examples with three dimensional Lie groups of symmetries are easy to pick out, and the symmetry groups include semi-simple, solvable and nillpotent Lie groups. There is a related two dimensional manifold N with an invariantly defined volume form. This volume form describes part of the motion of integral curves. Special cases involving metrics on the surface are easily identified.

Date: February 28, 1992

Topic: "Robust Control on Metric Spaces of Systems"

Speaker: Li Qiu, Postdoctoral Fellow, Fields Institute

Robust control deals with the control of uncertain systems, or equivalently sets of systems. An uncertain system is usually described as a neighborhood of a fixed nominal system. Such a neighborhood can therefore be defined through a metric in the space of all systems. In this talk, several such metrics in the space of linear time-invariant systems will be introduced and an overview of a robust control theory based on these metrics will be given.

Date: February 21, 1992

Topic: "New Integrable Problem of Classical Mechanics"

Speaker: Oleg Bogoyavlenskij, Steklov Mathematical Institute, Moscow

The talk will be devoted to the proof that the dynamics of an arbitrary rigid body in a gravitational field with an abitrary quadratic potential is completely integrable in Liouville's sense. The dynamics of the centre of mass of the rigid body is integrable in elementary functions; rotation of the rigid body around its centre of mass is integrable in theta-functions on Riemannian surface.

Date: February 14, 1992

Topic: "Centro-Affine Plane Curves and Feedback Control "

Speaker: George Wilkens, University of Hawaii

Date: February 14, 1992

Topic: "Geometric Phases and the Control of Super-Articulated Mechanical Systems"

Speaker: John Ballieul, Aerospace/Mechanical Engineering, Boston University

Super-articulated mechanical systems have more degrees of freedom than actuators. For these systems, the relationship between actuator inputs and configuration space trajectories is nonholonomic in the sense that when the input variables return to their intial values it will not generally be the case that configuration variables return to their initial values. In this talk, the long-term effects produced by periodic forcing of super-articulated mechanical systems are studied. For Lagrangian control systems with symmetry, it is shown that the stability of equilibrium motions may be assessed in terms of a quantity which we call the average potential. Although the definition of this may be motivated by classical averaging theory, we shall argue that it is essentially a geometric invariant.

Date: February 13, 1992

Topic: "Resolution of Kinematic Redundancy and Non-holonomic Motion Planning for Robots with Elastic Components"

Speaker: John Ballieul, Aerospace/Mechanical Engineering, Boston University

Date: February 7, 1992

Topic: "Control Theory and Sub-Riemannian Geometry (Part II)"

Speaker: Zhong Ge, Fields Institute

Date: February 7, 1992

Topic: "Recovery of Parameters in Age-structured Populations "

Speaker: William Rundell, Texas A & M

Date: February 5, 1992

Topic: "Centro-affine Plane Curves and Feedback Control "

Speaker: George Wilkens, Mathematics, Hawaii

Date: January 31, 1992

Topic: "Control Theory and Sub-Riemannian Geometry (Part I)"

Speaker: Zhong Ge, Fields Institute

Date: December 5, 1991

Topic: "Optimal Control of Uncertain Quantum Systems"

Speaker: Anthony Peirce, Mathematics and Statistics, McMaster University

Date: Novemeber 20, 1991

Topic: "Input-Output Stability for Accelerometer Control Systems"

Speaker: Kirsten Morris, Applied Mathematics, University of Waterloo

Date: Novemeber 6, 1991

Topic: As below, Part IV

Speaker: Michael Enos, Fields Institute

Date: October 30, 1991

Topic: As below, Part III

Speaker: Michael Enos, Fields Institute

Date: October 23, 1991

Topic: As below, Part II

Speaker: Michael Enos, Fields Institute

Date: October 16, 1991

Topic: "The Time-Optimal Problem for a Force-Free Systems of Two Symmetric Rigid Bodies in Three Space (Part I)"

Speaker: Michael Enos, Fields Institute