Abstract: In this talk, a second-order strongly elliptic (matrix) operator L will be discoursed in a bounded Lipschitz domain Q (n>1). We will study the problem Lu=f with homogeneous boundary conditions (Dirichlet, Neumann, or mixed). Our consideration allows to prove the Kato problem about the domain of the square root of the operator (1961) in a very simple way. It was solved in 2001 by A. McIntosh and others for bounded measurable coefficients, but our method allows to consider operator coefficients bounded in the Sobolev spaces with a small smoothness exponent as well.
The main approach is based on the Shneiberg theorem about extrapolation of the invertibility of a bounded operator in the interpolation scale.
The results can be used for the description of the space of the initial data for corresponding parabolic problems.

Abstract: We describe two unusual and apparently novel types of bifurcations leading to chaos in discrete dynamical systems and discuss applied models in which they occur. The first type occurs in 3-dimensional systems and is the result of the interaction of an attracting invariant torus with the stable and unstable manifolds of a nearby hyperbolic saddle point. These bifurcations occur, for example, in discrete dynamical systems models of walking droplet phenomena in elliptic corrals. Mathematical models of population dynamics is where the second variety of chaotic bifurcations can be observed. They are generated by horseshoe-like behavior in certain invariant sets that leads to the formation of chaotic strange attractors of the kind illustrated in the accompanying figure below.

Abstract: In stochastic systems, the effect of noise may cause the system to transition from one metastable state to another metastable state as a rare event. While there are many possible trajectories for this path to follow, there is a most probable path, which can be determined by minimizing the Friedlin-Wentzell rate functional. However, it is not clear how the most probable path may behave when the vector field is piecewise-smooth. In particular, we are motivated by models of Arctic energy balance, which typically have a piecewise-smooth geometry, and possible characterizations of tipping events as Arctic sea ice melts. With this application in mind, we will characterize the behavior of the most probable path between basins of attraction in a planar system of ODEs with additive noise, composed of two smooth vector fields separated by a switching manifold. We will explore mechanisms for the robust determination of most probable paths in stochastic piecewise-smooth systems of this type.

Abstract: Design and analysis of stable feedback controllers for systems with intermittent actuators and sensors are significantly challenging by the fact that the system could lose controllability and/or observability at specific time instants and/or over large time intervals due to the intermittence property of actuators and sensors. This talk will provide a broad overview of several control methods recently proposed by the speaker and his research team addressing control problems subject to intermittent sensing and actuation. Specifically, persistence of excitation condition is employed to describe the intermittence. A novel persistence filter construction is adopted to achieve the stabilization objectives for a large class of single-input linear systems with intermittent actuators. Next, a generalized result is presented for multi-input systems. In addition, an extended study on the rigid-body attitude control system shows that the proposed control scheme finds non-trivial applications to nonlinear systems including Euler-Lagrange class robotic systems. Furthermore, inspired by the proposed controller structure, a special persistence-filter-based state observer is designed for systems which are under the influence of sensor blackouts. A salient feature is that the states of the closed-loop system and also the estimation errors of the observer are shown satisfy exponential convergence properties. A relative proximity control problem of spacecraft, in which the on-off thruster schedules, and also the rise and fall time of thrusters are considered to highlight the effectiveness of the proposed approaches.

Speaker Bio: Maruthi R. Akella is a tenured full professor with the Department of Aerospace Engineering and Engineering Mechanics at The University of Texas at Austin (UT Austin) where he holds the E.P. Schoch Endowed Professorship in Engineering. He is the founding director for the Center for Autonomous Air Mobility (CAAM) and faculty lead for the Control, Autonomy, and Robotics area at UT Austin. He received his M.E. degree in Aerospace Engineering from the Indian Institute of Science (1994), and his Ph.D. in Aerospace Engineering in 1998 from Texas A&M University. Prior to joining the UT Austin faculty, he was a postdoctoral fellow with the Yale University Center for Systems Science. Dr. Akella’s research program encompasses control theoretic investigations of nonlinear and coordinated systems, vision-based sensing for state estimation, and development of integrated human and autonomous multivehicle systems. His research contributions have found several highly successful applications in astrodynamics, in the control of space systems, and vision-guided robotics. He published more than 150 research papers in peer-reviewed archival journals and professional conferences. For his far-reaching contributions to the field of attitude estimation and control, Dr. Akella received several prestigious awards including the Mechanics and Control of Flight Award from the American Institute of Aeronautics and Astronautics (AIAA) and the Judith A. Resnik Space Award by the IEEE Aerospace and Electronic Systems Society. He currently serves as Technical Editor for IEEE Transactions on Aerospace and Electronic Systems (TAES), and as Associate Editor for the Journal of Guidance, Control, and Dynamics (JGCD), and the Journal of the Astronautical Sciences (JAS). He is the Chair for the Space Flight Mechanics Technical Committee of the American Astronautical Society (AAS). Dr. Akella is a Fellow of the AAS, an Associate Fellow of the AIAA, and holds the title of IEEE Distinguished Lecturer.

Abstract: We propose a new epidemiological model, based on the classical SIR model, taking additionally into account a switching prevention strategy. The model has two distinct thresholds that determine the beginning and the end of an intervention and two different transmission rates. We study the global dynamics of the proposed two-dimensional model.

Description: We analyze the impact of public intervention (e.g. educating public, vaccination campaign, closing schools etc.) upon the dynamics of a SIR model. After a disease begins to spread the general public will begin to take action, which will reduce the transmission rate. However, as the threat of epidemics becomes less severe these preventative measures may be revoked by various reasons and so the transmission rate increases again. We propose a model for this phenomenon by introduces a switched system within the SIR model.

Abstract: In this talk, I will discuss about a recent paradigm shift in trajectory optimization problems for uncertain dynamical systems in which the boundary conditions are not prescribed states but probability distributions (distribution steering problems). In the case of Gaussian linear systems, the latter problem can be formulated as a finite-horizon stochastic optimal control problem with covariance assignment constraints. I will present a solution approach to the latter problem in a discrete-time framework for the case of partial and imperfect state information. The proposed solution approach utilizes separation-based control policies that correspond to sequences of control laws that are affine functions of either the complete history of the output estimation errors, that is, the differences between the actual output measurements and their corresponding estimated outputs produced by a discrete-time Kalman filter, or a truncation of the same history. With this particular feedback parametrization, one can associate the stochastic optimal control problem with a tractable semidefinite (convex) program which can be addressed by means of efficient and easily accessible computational tools. I will show that the proposed procedure for the reduction of the stochastic optimal control problem to a convex program has significant advantages in terms of improved scalability and tractability over the approaches proposed in the relevant literature.

Brief Bio: Dr. Bakolas joined the Department of Aerospace Engineering and Engineering Mechanics at The University of Texas at Austin as an Assistant Professor in fall 2012. He received his Diploma in Mechanical Engineering with highest honors from the National Technical University of Athens, Greece, in 2004 and his MS. and Ph.D. degrees in Aerospace Engineering from the Georgia Institute of Technology, Atlanta, in 2007 and 2011, respectively. His main research interests include optimal control theory (deterministic and stochastic), differential games and game theory, computational and algorithmic geometry, dynamic programming, nonlinear control, and their applications to problems of autonomous agents and networked multi-agent systems.

Abstract: We consider the type I and type II multiple orthogonal polynomials (MOPs), satisfying non-hermitian orthogonality with respect to the weight $\exp(-z^3)$ on two unbounded contours on the complex plane. Under the assumption that the orthogonality conditions are distributed with a fixed proportion $\alpha$, we ?nd the detailed (rescaled) asymptotics of these MOPs, and describe the phase transitions of this limit behavior as a function of $\alpha$. This description is given in terms of the vector critical measure, the saddle point of the energy functional comprising both attracting and repelling forces. These critical measures are characterized by a cubic equation (spectral curve), and their components live on trajectories of a canonical quadratic differential on the Riemann surface of this equation. The structure of these trajectories and their deformations as function of $\alpha$ plays the crucial role.

This is a joint work with Gulherme L. Silva (University of Michigan, Ann Arbor).

Abstract: The safe operation of vehicles on roads depends, among other things, on a properly functioning brake system. Air brake systems are widely used in commercial vehicles such as trucks, tractor-trailers, and buses. In these brake systems, compressed air is used as the energy transmitting medium to actuate the foundation brakes mounted on the axles. In this paper, a model-based diagnostic system for air brakes is presented. This diagnostic system is based on a nonlinear model for predicting the pressure transients in the brake chamber that correlates the brake chamber pressure to the treadle valve (brake application valve) plunger displacement and the pressure of the air supplied to the brake system. Leaks and “out-of-adjustment” of push rods are two prominent defects that affect the performance of the air brake system. Diagnostic schemes that will monitor the brake system for these defects will be presented and corroborated with experimental data obtained from the brake testing facility at Texas A&M University.

Abstract: A PDE/ODE system can evolve in 3 time scales when the fast scales are associated with 2 small parameters that tend to zero at different rates. We investigate the limiting dynamics when the fast dynamics is generated by 2 skew-self-adjoint operators and the initial time derivative is uniformly bounded regardless of the small parameters. To find the subspace that the limiting dynamics resides, we rely on matrix perturbation theory.

Abstract: In this talk, I will first present a multi-cue multi-choice (MCMC) task model developed in our group for describing human decision making behavior based on two different approaches. The first approach is to leverage the existing multi-cue two-choice task model and the existing single-cue multi-choice task model to have an extended model by analogy. The second approach is to use a Markov chain technique to derive the discrete version of this MCMC model. Next, I will discuss asymptotic approximation of the first and second moment equations for this MCMC model. Specifically, we treat the first and second moment equations as switched linear nonautonomous systems. Then we use a combined numerical analysis and dynamical systems method to explore the asymptotic distribution of the first and second moment equations.

Abstract: Weak Galerkin (WG) finite element method is a numerical technique for PDEs where the differential operators in the variational form are reconstructed/approximated by using a framework that mimics the theory of distributions for piecewise polynomials. The usual regularity of the approximating functions is compensated by carefully-designed stabilizers. The fundamental difference between WG methods and other existing finite element methods is the use of weak derivatives and weak continuities in the design of numerical schemes based on conventional weak forms for the underlying PDE problems. Due to its great structural flexibility, WG methods are well suited to a wide class of PDEs by providing the needed stability and accuracy in approximations. The speaker will present a recent development of WG, called "Primal-Dual Weak Galerkin (PD-WG)", for problems for which the usual numerical methods are difficult to apply. The essential idea of PD-WG is to interpret the numerical solutions as a constrained minimization of some functionals with constraints that mimic the weak formulation of the PDEs by using weak derivatives. The resulting Euler-Lagrange equation offers a symmetric scheme involving both the primal variable and the dual variable (Lagrange multiplier). PD-WG method is applicable to several challenging problems for which existing methods may have difficulty in applying; these problems include the second order elliptic equations in nondivergence form, Fokker-Planck equation, and elliptic Cauchy problems. An abstract framework for PD-WG will be presented and discussed for its potential in other scientific applications.

Abstract: Multiple recurrent outbreak cycles have been commonly observed in infectious diseases such as measles and chicken pox. These complex disease outbreak behaviors are rarely captured by deterministic models. In this talk, we investigate a simple 2-dimensional SI model and propose that the coexistence of multiple attractors attributes to complex outbreak patterns. We first determine parameter conditions for the existence of an isolated center, then properly perturb the model to generate Hopf bifurcation and obtain limit cycles around the center. We further analytically prove that the maximum number of the coexisting limit cycles is three, and provide a corresponding parameter set for the existence of three limit cycles. Simulation results demonstrate three coexisting attractors, which contain one unstable endemic equilibrium and two stable limit cycles separated by one unstable limit cycle. Various disease dynamics can be predicted by a single nonlinear deterministic model based on different initial data.

Abstract: Navier-Stokes system is the system of PDEs thought to describe the dynamics of viscous incompressible fluid. Using dimensional arguments applied to this system, Kolmogorov in 1941 presented the theory of turbulence of the flow as a power law cascade of the energy function. I will try to overview known results in the theory of 2d Navier-Stokes system and present one particular result regarding the spectral properties of energy function. The talk is based on joint work with W. Craig, passed away this January.

Abstract: In this paper, planar Filippov systems with a single discontinuity boundary are considered. We contribute to a codimension-two critical crossing cycle bifurcation by assuming that a crossing limit cycle is tangent to the discontinuity boundary at two distinct points. Analyzing sliding dynamics and an appropriate Pioncare map, we prove that one limit cycle preserving the stability character persists as parameters are perturbed. In particular, the bifurcation diagram is provided, as well as the asymptotics of all codimension-one bifurcation curves. Finally, our theoretical result is applied to linear Filippov systems and the Morris-Lecar model in neural activities.

Abstract: We revisit the classical real option problem, with additional features. The classical situation concerns choosing the right timing for an investment. We consider a stochastic optimization problem which involves an option of one time investment and an option of closing the activities. The timing and amount of an investment and the timing of the closure are parameters to be optimized in order to maximize the expected value of the profit. We reduce the stochastic optimization problem to a deterministic variational inequality using the dynamic programming technique and discuss the properties of the solutions to the variational inequality.

Abstract: We study the equations of motion of n vortices of equal circulation in the plane, in a disk and on a sphere. The n vortices form a polygonal equilibrium in a rotating frame of reference. We investigate the bifurcation and numerical continuation of Lyapunov families of periodic orbits that arise from the polygonal relative equilibrium. When the frequency of a Lyapunov orbit and the frequency of the rotating frame have a rational relationship then the orbit is also periodic in the inertial frame. We prove that a dense set of Lyapunov orbits, with frequencies satisfying a diophantine equation, correspond to choreographies of the n vortices. We include numerical results for all cases, and for various values of n.

Abstract: I will talk about a geometric approach to controlling a spherical rolling robot like the Sphero (a programmable spherical toy robot that can be controlled by a cell-phone app). Specifically, we formulate the rolling sphere controlled by internal wheels as a control system on a principal bundle. We analyze its controllability using the connection naturally defined by the system as well as its curvature. We will also address an optimal control problem of the system and its relationship with an integrable Hamiltonian system.

Abstract: Cluster algebras provide a unified framework to explain certain phenomena arising in discrete dynamical systems---for example, both the integrality of the Somos-4 sequence and the periodicity of the pentagonal recurrence can be explained by appropriate cluster algebras. I will present a few recently-described related periodicity phenomena arising from root systems, and indicate some directions of generalization.

Abstract: In the theory of mathematical billiards, caustics correspond to the invariant curves splitting the phase space of the billiard transformation. The correspondence between the billiard table and its caustics resembles the correspondence between wave-fronts during the evolution. String construction helps to construct a table having particular caustic. I will discuss some properties of this construction. The talk is based on joint work with M. Bialy.

Abstract: A Fully Convex Control (FCC) problem has the appearance of the classical calculus of variations Bolza problem $$\min \int_0^T L(x(t),\dot x(t))dt+l(x(0),x(T)),$$ where the minimization is over $x(\cdot)$ belonging to some class of arcs. The distinguishing features of FCC are that the data $L(\cdot, \cdot)$ and $l(\cdot,\cdot)$ (i) may take on the value $+\infty$ and (ii) are convex functions. Allowance of (i) provides great flexibility incorporating constraints so that most standard control problems come under its purlieu. However, broad generality is restrained by (ii), which although quite special, nonetheless includes the classical linear quadratic regulator and many of its generalizations. Moreover, the speciality of (ii) opens up the possibility of using convex dual formulations.

Rockafellar developed FCC in the 1960-70's that evolved into more gen- eral nonsmooth control and variational analysis. We shall first survey these early results. Then we shall present new developments with state constraints and involve impulsive arcs.

Abstract: Since its introduction to mathematics, about 110 years ago, the topological degree was widely used as a tool to show the existence and multiplicity solutions in various types of differential equations. In my talk I would introduce the equivariant version of the Brouwer degree, provide its connection to the original Brouwer degree and list its most important properties and computational algorithms. In addition, we will discuss various problems related to the existence of periodic solutions in second order systems (discrete and continuous) with only minimal symmetry properties.

Abstract: We study nonlinear vibrational modes of oscillations for a tetrahedral configuration of particles interacting by general force fields. In the case of Tetraphosphorus, the interaction of atoms is given by bond stretching, van der Waals and Coulomb forces. Using gradient equivariant degree, we present a full topological classification of the periodic solutions with spatio-temporal symmetries, i.e. we give all the symmetries of the vibrations that exists for any cubic or higher order perturbation of force fields.

Abstract: Hybrid zero dynamics-based control is a promising framework for controlling underactuated biped robots and powered prosthetic legs. In this control paradigm, stable walking gaits are implicitly encoded in polynomial output functions of the robot configuration variables, which are to be stabilized via feedback. The biped polynomial output functions are parameterized by a suitable mechanical phasing variable whose evolution determines the biped gait progression during each step. Determining a proper phase variable, however, might not always be a trivial task. In this paper, we present an elimination method for generating output functions from given parametric walking gaits without any explicit knowledge of the phase variable. Our elimination method is based on computing the resultant of polynomials, a fundamental tool in computer algebra.

Abstract:We illustrate differential equations with threshold-type state-dependent delays and their applications in modeling regulatory dynamics. We model the diffusion processes in regulatory dynamics with differential equations with threshold-type state-dependent delay and investigate the effect of the state-dependent diffusion time on intracellular regulatory dynamics. A general model which is an extension of the classic differential equation models with constant or zero time delays is developed to study the stability of steady state, the occurrence and stability of periodic oscillations in regulatory dynamics. Using the method of multiple time scales, we compute the normal form of the general model and show that the state-dependent diffusion time may give arise to both supercritical and subcritical Hopf bifurcations. Numerical simulations of the prototype model of genetic regulatory dynamics are also given to illustrate the general results.

Abstract:Switching dynamical systems have be used to generate multiscroll attractors by different approaches as series of step functions, saturation, or hysteresis. This kind of switching systems are based on piecewise linear (PWL) system whose each subsystem is a dissipative system with unstable dynamics (the latter is called an unstable dissipative system, UDS). Generalized multi-stability appears when two PWL systems are coupled as master-slave, or the equilibria of a PWL system are separated in pairs generating at least two double scroll attractors. An application of multiscroll attractors is also presented.

Authors: Eyram Kwame and Dmitrii Rachinskii
Abstract:We consider dynamics of discrete time piecewise linear systems, which can be also viewed as a system with the so-called play operator. This operator is used as a building block for several models of hysteresis phenomena observed in plasticity, dry friction, applications of smart materials, and fatigue/damage estimation. We will argue that the play operator may also be potentially useful for modeling “boundedly rational” behavior of economic agents leading to path dependence, multiplicity of equilibrium states and stickiness of information/prices in economy. As a test, we will consider a simple DSGE type economic model with the play operator. Some conditions for global stability will be presented. Then we will discuss various types of dynamics. For a small prototype model, global dynamics and bifurcations will be described depending on two parameters.

Abstract: We will discuss shortly the ideas behind the equivariant gradient degree and how it is used to study the existence and multiplicity of solutions in variational problems.

Abstract: We study the dynamics of a two-degree-of-freedom (pitch and plunge) aeroelastic system where the aerodynamic forces are modeled as a piecewise linear function of the effective angle of attack. Stability and bifurcations of equilibria are analyzed. We dind border collision and rapid bifurcations. Bifurcation diagrams of the system were calculated utilizing MATCONT and Mathematica. Chaotic behavior with intermittent switches about the two nontrivial equilibria was also observed.

Abstract: I will discuss the dynamics of cyclic evasion and provide some partial results on the stability of the stationary shapes.