# Bozenna Pasik-DuncanUniversity of Kansas | KU · Department of Mathematics

Bozenna Pasik-Duncan

## About

167

Publications

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Introduction

**Skills and Expertise**

## Publications

Publications (167)

This article examines mean-field-type game problems by means of a direct method.
We provide various solvable examples beyond the classical linear-quadratic game problems. These include quadratic-quadratic games and games with power, logarithmic, sine square, hyperbolic sine square payoffs. Non-linear state dynamics such as log-state, control-depend...

A stochastic calculus is given for processes described by stochastic integrals with respect to fractional Brownian motions and Rosenblatt processes somewhat analogous to the stochastic calculus for Itô processes. These processes for this stochastic calculus arise naturally from a stochastic chain rule for functionals of Rosenblatt processes; and so...

In this paper an infinite time horizon (ergodic) quadratic cost control problem for a linear two dimensional stochastic system with a two dimensional Rosenblatt noise process is solved by providing an explicit expression to determine the optimal feedback. The system has some symmetry properties that allow for an explicit determination of an optimal...

A stochastic linear-quadratic control problem is formulated and solved for some stochastic equations in an infinite dimensional Hilbert space for both finite and infinite time horizons. The equations are bilinear in the state and the noise process where the noise is a scalar Gauss-Volterra process. The Gauss-Volterra noise processes are obtained fr...

A stochastic calculus is given for processes described by stochastic integrals with respect to fractional Brownian motions and Rosenblatt processes somewhat analogous to the stochastic calculus for It\^{o} processes. These processes for this stochastic calculus arise naturally from a stochastic chain rule for functionals of Rosenblatt processes; an...

In this paper a two player zero sum stochastic differential game in a finite dimensional space having a linear stochastic equation with a state dependent Gauss-Volterra noise is formulated and solved with a quadratic payoff for the two players and a finite time horizon. The control strategies are continuous linear state feedbacks. A Nash equilibriu...

This article examines the solvability of mean-field-type game problems by means of a direct method. We provide various solvable examples beyond the classical LQ game problems. It includes quadratic-quadratic games, power, logarithmic, sine square, hyperbolic sine square payoffs. Non-linear state dynamics such as control-dependent regime switching,...

This paper focuses on innovative methods of teaching stochastic adaptive control with students who represent all science, technology, engineering and mathematics (STEM) disciplines. The Stochastic Adaptive Control course has been developed based on the author’s research area and it demonstrates the power, beauty and excitement of stochastic adaptiv...

This article examines the solvability of mean-field-type game problems by means of a direct method. We provide various solvable examples beyond the classical LQ game problems. It includes quadratic-quadratic games, power, logarithmic, sine square, hyperbolic sine square payoffs. Non-linear state dynamics such as control-dependent regime switching,...

Stochastic linear-quadratic control problems for bilinear equations with some stochastic coefficients and driven by fractional Brownian motions with the Hurst parameters H ϵ (½,1) are formulated and solved. The family of admissible controls is the collection of linear feedback controls. An optimal control in this family is not optimal in the admiss...

A two person, zero-sum, noncooperative stochastic differential game is formulated and solved for a multidimensional linear stochastic system that has a quadratic payoff and a linear state dependent scalar fractional Brownian motion noise process in the stochastic system. The strategies are restricted to be linear feedback. A Riccati equation is giv...

Some nonlinear stochastic differential games are formulated in the family of complex and quaternion projective spaces that are among the rank one compact symmetric spaces that consist of the spheres, the projective spaces over
R,C
, and
H
and one arising from an exceptional Lie algebra called the Cayley plane. The payoff functionals for the differ...

A direct approach is used to solve some linear-quadratic stochastic control problems for Brownian motion and other noise processes. This direct method does not require solving Hamilton-Jacobi-Bellman partial differential equations or backward stochastic differential equations with a stochastic maximum principle or the use of a dynamic programming p...

This paper focuses on innovative methods of teaching stochastic adaptive control with students who represent all science, technology, engineering and mathematics (STEM) disciplines. The Stochastic Adaptive Control course has been developed based on the authors’ research area and it demonstrates the power, beauty and excitement of stochastic adaptiv...

A stochastic control problem is formulated and a solution is described for a linear stochastic system that has a quadratic cost and a linear state dependent noise process appearing in the stochastic system. A stochastic Riccati equation is given that provides an optimal feedback control.

Presents on the activities of the Control Education Technical Committee.

In this paper a direct approach is used to obtain explicit optimal controls for control problems for linear systems with general noise processes and quadratic cost functionals. An optimal control is determined by a sum of the well known linear feedback control from the associated deterministic linear-quadratic control problem and a prediction of th...

A control problem for a partially observed linear stochastic system with an exponential quadratic cost functional is formulated and explicitly solved. It is assumed given that the estimation of the state is described by the solution of the information filter which is known. This solution is a sufficient statistic for the unknown state based on the...

In this paper a two person zero sum stochastic differential game is formulated and explicitly solved where the state of the game evolves in a two dimensional sphere. The game is described by a stochastic equation that is the sum of the control strategies of the two players and a Brownian motion in the two-sphere. The problem formulation uses the pr...

In this paper a direct approach is given for the verification of the relation between the Riccati equation and the optimal cost for the solution of a linear exponential quadratic Gaussian control problem and the Riccati equation and the optimal payoff for the solution of a linear quadratic stochastic differential game. The cost functionals are expe...

In this technical note a control problem for a discrete time linear stochastic system with a general correlated noise process and a cost functional that is quadratic in the system state and the control is solved. An optimal control is given explicitly as the sum of the well known linear feedback control for the associated deterministic linear-quadr...

In this paper a control problem for a linear stochastic system driven by a noise process that is an arbitrary zero mean, square integrable stochastic process with continuous sample paths and a cost functional that is quadratic in the system state and the control is solved. An optimal control is given explicitly as the sum of the well-known linear f...

A control problem is formulated and solved for a stochastic system that evolves in the unit sphere in Euclidean three space. The cost functional depends on the system state only by its Riemannian distance from a point called the origin. An explicit optimal control is obtained by expressing the cost as a squared term in the control from which the op...

A control problem for a linear stochastic equation in a Hilbert space and an exponential quadratic cost functional of the state and the control is formulated and solved. The stochastic equation can model a variety of stochastic partial differential equations with the control restricted to the boundary or to discrete points in the domain. The soluti...

A linear-quadratic control problem with a finite time horizon for some infinite-dimensional controlled stochastic differential equations driven by a fractional Gaussian noise is formulated and solved. The feedback form of the optimal control and the optimal cost are given explicitly. The optimal control is the sum of the well-known linear feedback...

In this paper a method motivated by completion of squares is used to describe explicit optimal controls for some stochastic control problems that include the linear-quadratic control problem for systems with a general noise process, the linear exponential quadratic Gaussian control problem for systems with Brownian motion, and the control of Browni...

A control problem is formulated for a linear stochastic system with noisy, partial observations of the state and a cost that is a quadratic functional of the state and the control. Both the system noise and the observation noise can be fractional Brownian motions. An optimal control is explicitly described and this control is compared to the well k...

In this paper, stochastic differential equations in a Hilbert space with a standard, cylindrical fractional Brownian motion with the Hurst parameter in the interval (1/2,1) are investigated. Existence and uniqueness of mild solutions, continuity of the sample paths and state space regularity of the solutions, and the existence of limiting measures...

A control problem for a linear system in a Hilbert space with a fractional Brownian motion and a quadratic cost in the state and the control is solved. The feedback form of the optimal control and the optimal cost are given. The optimal control is the sum of the well known linear feedback control for the associated deterministic linear-quadratic co...

A continuous time long run growth optimal or optimal logarithmic utility portfolio with proportional transaction costs consisting
of a fixed proportional cost and a cost proportional to the volume of transaction is considered. The asset prices are modeled
as exponent of diffusion with jumps whose parameters depend on a finite state Markov process o...

In this paper a control problem for a linear stochastic system with a fractional Brownian motion and a cost functional that is quadratic in the state and the control is solved. An optimal control is given explicitly using fractional calculus and the control is shown to depend on a prediction of the future fractional Brownian motion and the well kno...

In this paper a control problem for a linear stochastic system driven by a fractional Brownian motion with a cost functional that is quadratic in the state and the control is considered. An optimal control is given explicitly using fractional calculus and the control is shown to depend on the prediction of the fractional Brownian motion as well as...

This presentation will discuss multiple challenges and opportunities that are presented to young investigators to prepare for careers in science and engineering. How can research and education be integrated? How is interdisciplinary research supported? How do graduate students gain value-added skills while obtaining their degrees? These questions w...

In this paper, some linear and semilinear distributed parameter equations (equations in a Hilbert space) with a (cylindrical) fractional Brownian motion are considered. Solutions and sample path properties of these solutions are given for the stochastic distributed parameter equations. The fractional Brownian motions are indexed by the Hurst parame...

A continuous time long run growth optimal portfolio with proportional cost consisting of the sum of a fixed proportional cost and a cost proportional to the volume of transactions is considered. An obligatory portfolio diversification is introduced according to which it is required to invest at least a fixed small portion of the wealth in each asse...

Some results are given for a continuous time long run growth optimal portfolio that has proportional costs consisting of the sum of a fixed proportional cost and a cost that is proportional to the volume of each transaction. An obligatory portfolio diversification is given that requires at least a small portion of the wealth be invested in each ass...

The solutions of a family of semilinear stochastic equations in a Hilbert space with a fractional Brownian motion are investigated. The nonlinear term in these equations has primarily only a growth condition assumption. An arbitrary member of the family of fractional Brownian motions can be used in these equations. Existence and uniqueness for both...

Engineering education has seen an explosion of interest in recent years,
fueled simultaneously by reports from both industry and academia. Automatic control education has recently become a core issue for the international control community. This has occurred in tandem with the explosion of interest in engineering education as a whole. The applica...

The purpose of this workshop is to increase the general awareness of the importance of systems and control technology and its cross-disciplinary nature among high school teachers and students. The workshop activities include presentations by control scholars and remarkable speakers, informal discussions, and the opportunity for teachers to meet pas...

A discrete time, linear, stochastic control system is constructed to model the risk reserves for an insurance company. The model has the autoregressive form. A control is used to regulate the risk reserve. The sequence of controls is determined by two approximations, the normal power approximation of order two and a log normal approximation. These...

In this paper, an adaptive control problem is formulated and solved for a scalar linear stochastic system perturbed by a fractional Brownian motion and an ergodic (or average cost per unit time) quadratic cost functional. The Hurst parameter for the fractional Brownian motion may take any value in (1/2, 1).

An additive Gaussian channel is considered where there are a stochastic signal and a fractional Gaussian noise. The fractional Gaussian noise is the formal derivative of a fractional Brownian motion with the Hurst parameter in the interval (1/2, 1). These processes have a long range dependence and seem to be meaningful models for many physical phen...

Stochastic equations in a Hilbert space with a fractional Brownian motion are used to model stochastic partial differential equations with a space-time noise. Some semilinear stochastic equations are shown to possess one and only one weak solution. These weak solutions are constructed from the solutions of the corresponding linear equations by an a...

A solution is obtained for a linear stochastic equation in a Hilbert space with a fractional Brownian motion. The Hurst parameter
for the fractional Brownian motion is not restricted. Sample path properties of the solution are obtained that depend on the
Hurst parameter. An example of a stochastic partial differential equation is given.

The solution of adaptive control of continuous time linear stochastic systems without and with delays is presented. The asymptotic distribution of the cost in the adaptive control of a continuous time linear stochastic system without and with delays is given. The consistency of a least squares identification procedure for continuous time linear sto...

Some techniques are described for the solution of a family of related problems in stochastic adaptive control, the models being assumed to evolve in continous time rather than discrete time. The families of linear systems that are considered include finite dimensional systems, delay time systems, and infinite dimensional systems. The general approa...

A scalar input-scalar output linear second order system has a fractional Gaussian noise input. The fractional Gaussian noise is the formal derivative of a fractional Brownian motion with the Hurst parameter in the interval (1/2,1). A family of estimators for a linear system with a fractional Gaussian noise and some unknown parameters that is obtain...

The study of adaptive control problems with multiplicative cost functionals, which appear in particular in risk sensitive control problems requires a different approach than that for risk neutral (average cost per unit time) problems. Assuming that the unknown parameter is selected from a known compact set, it is proved that there is a finite class...

In this paper, some explicit solutions are given for stochastic differential equations in a Hilbert space with a multiplicative fractional Gaussian noise. This noise is the formal derivative of a fractional Brownian motion with the Hurst parameter in the interval (1/2,1). These solutions can be weak, strong or mild depending on the specific assumpt...

This article focuses on educational aspects of identification and stochastic adaptive control with applications to finance. Identification and adaptive control problem is illustrated using the Merton Portfolio model. It will be shown that methods used in mathematics of finance have been suc-cessfully used in other fields as well as mathematics of f...

The removal of ocular artifact from scalp electroencephalograms (EEGs) is of con-siderable importance for both the automated and visual analysis of underlying brainwave activity. Traditionally, subtraction techniques use linear regression to estimate the influence of eye movements on the electrodes of interest. These methods are based on the assump...

The dynamics of a discrete time, state process are assumed to depend on the current value of the state of a possibly unobserved hidden Markov model. Both the state and the hidden process are controlled with a control that depends on the available observations. An ergodic or average cost per unit time control problem is solved making some regularity...

This presentation will focus on vertical and horizontal research on stochastic theory and control. In ten poster presentations, the importance of the vertical and horizontal teaching and research will be shown. Through those ten posters prepared by undergraduate, graduate, K-12 students and established researchers the development of complexity of c...

This paper presents the version of the robust maximum principle in the context of multi-model control formulated as the minimax Bolza problem. The cost function contains a terminal term as well as an integral one. A fixed horizon and terminal set are considered. The necessary conditions of the optimality are derived for the class of uncertain syste...

Backlash is one of the most important non-linearities that limit the performance of speed and position control in industrial, robotics, automotive, automation and other applications. The control of systems with backlash has been the subject of study ...

This paper develops a version of the robust maximum principle applied to the minimax Mayer problem formulated for stochastic differential equations with a control-dependent diffusion term. The parametric families of first and second order adjoint stochastic processes are introduced to construct the corresponding Hamiltonian formalism. The Hamiltoni...

The robust maximum principle applied to the minimax linear quadratic problem is derived for stochastic differential equations containing a control-dependent diffusion term. The parametric families of the first and second order adjoint stochastic processes are obtained to construct the corresponding Hamiltonian formalism. The Hamiltonian function us...

A Hilb alued stochastic integration is defined for an integrator that is acylindrical fractional Brownian motion in a Hilbert space. Since the integrator is not a semimartingale for the fractional Brownian motions considered, a di#erent definition of integration is required. Both deterministic and stochastic operatoralued integrands are used. The a...

Some problems and solutions of adaptive control problems for continuous time stochastic systems are described. A solution is given to the adaptive linear quadratic Gaussian control problem under the natural assumptions of controllability and observability. The effects of sampling and numerical differentiation on a least-squares estimation algorithm...

This article focuses on six women who made important contributions
to the control systems field. For each, biographical data is provided,
as well as a sketch of their research interests and accomplishments. All
of these women had to make their way in a field dominated by men. They
are: Irmgard Flugge-Lotz (1903-1974); Violet Haas (1928-1986); Faina...

This paper focuses on outreach programs for grades K through 12. The KU outreach program, which has functioned for ten years, has reached thousands of K-12 students through the past ten years of its existence. The wellknown semi-annual KU activity, "Workshop + Students = Fun + Success," involves hundreds of undergraduate students in thoughtfully pr...

A few undergraduate success stories will be provided. It will be shown that the REU experience, plus a strong stochastic theory and control systems program, equals success. The examples provided include different types of partnerships in undergraduate education; mathematics and engineering; mathematics, physics, and astronomy; mathematics and philo...

An optimal production planning for a stochastic manufacturing system is considered. The system consists of a single, failure-prone
machine that produces a finite number of different products. The objective is to determine a rate of production that minimizes
an average cost per unit time criterion where the demand is random. The results given in thi...

An important class of controlled linear stochastic distributed parameter systems is that with boundary or point control. A survey of some existing adaptive control problems with their solutions for the boundary or the point control of a partially known linear stochastic distributed parameter systems is presented. The distributed parameter system is...

This paper focuses on the interdisciplinary character of the problems of stochastic adaptive Control Of continuous-time systems. It shows the complexity of these problems and how their solutions can be provided by a team work. The paper sends a strong message that there is a room for every stochastic person in stochastic adaptive control theory and...

In this paper, a family of estimators for an unknown parameter in the drift term of a scalar linear stochastic differential equation is given. The linear stochastic differential equation has a Brownian motion replaced by a fractional Brownian motion with the Hurst parameter in the interval (1/2,1). Such a fractional Brownian motion is not a semimar...

A fractional Brownian motion with Hurst parameter in the interval
(½, 1) is used for the Gaussian noise process in a linear
stochastic distributed system or a linear stochastic partial
differential equation. These noise processes have properties that have
been important for finite dimensional systems. The notion of a mild
solution is given a...

An adaptive, ergodic cost stochastic control problem for a partially known, semilinear, stochastic system in an infinite dimensional space is formulated and solved. The solutions of the Hamilton--Jacobi--Bellman equations for the discounted cost and the ergodic cost stochastic control problems require some special interpretations because they do no...

In this paper a stochastic calculus is given for the fractional Brownian motions that have the Hurst parameter in (1/2, 1). A stochastic integral of Ito type is defined for a family of integrands so that the integral has zero mean and an explicit expression for the second moment. This integral uses the Wick product and a derivative in the path spac...

In this paper a stochastic calculus is given for the fractional Brownian motions that have the Hurst parameter in (1/2, 1). A stochastic integral of Ito type is defined for a family of integrands so that the integral has zero mean and an explicit expression for the second moment. This integral uses the Wick product and a derivative in the path spac...

Numerical differentiation formulas that yield consistent least
squares parameter estimates from sampled observations of linear, time
invariant higher order systems have been introduced previously by Duncan
et al. (1994). The formulas given by Duncan et al. have the same
limiting system of equations as in the continuous time case. The formula
presen...

©1999 IEEE. Personal use of this material is permitted. However, permission to reprint/republish this material for advertising or promotional purposes or for creating new collective works for resale or redistribution to servers or lists, or to reuse any copyrighted component of this work in other works must be obtained from the IEEE. The adaptive l...

Adaptive control of discrete time Markov processes with an infinite time horizon risk sensitive cost functional is investigated. The transition probability for the Markov process depends on an unknown parameter. It is shown that the optimal risk sensitive cost is a continuous function of the parameter. Two almost optimal adaptive procedures that ar...

A Kiefer-Wolfowitz or simultaneous perturbation algorithm that
uses either one-sided or two-sided randomized differences and
truncations at randomly varying bounds is given in this paper. At each
iteration of the algorithm only two observations are required in
contrast to 2l observations, where l is the dimension, in the classical
algorithm. The al...

©1999 IEEE. Personal use of this material is permitted. However, permission to reprint/republish this material for advertising or promotional purposes or for creating new collective works for resale or redistribution to servers or lists, or to reuse any copyrighted component of this work in other works must be obtained from the IEEE. The adaptive p...

This paper describes a study of asynchronous transfer mode (ATM)
cell data, considering both the analysis and the modeling of the data.
For the data analysis portion of the work, cell counts per unit time are
done, and interarrival times between cells are determined. Source
modeling is done for the two large variable bit rate users and a
constant b...

Some results for stochastic calculus for a fractional Brownian
motion are described and an application to identification is given. A
stochastic integral is defined that has mean zero and an explicit
expression is given for the second moment. Another stochastic integral
is defined and the two stochastic integrals are explicitly related. An
Ito formu...

A controlled Markov process in a Hilbert space and an ergodic cost
functional are given for a control problem that is solved where the
process is a solution of a parameter dependent semilinear stochastic
differential equation and the control can occur only on the boundary or
at discrete points in the domain. The linear term of the semilinear
differ...

An adaptive, ergodic cost stochastic control problem for a
partially known, semilinear, stochastic system in an infinite
dimensional space is formulated and solved. Since the results for
discounted cost and ergodic cost stochastic control problems for known
semilinear stochastic systems are only relatively recent, it seems that
this is the first wo...

Some conditions are given for consistency of a family of least squares estimates of some unknown parameters for a linear stochastic distributed parameter system. The distributed parameter system is described by an analytic semigroup with cylindrical white noise and control that occurs only on the boundary or at discrete points. The consistency of t...

An adaptive control problem of a discrete time Markov process that is completely observed in a fixed recurrent domain and
is partially observed elsewhere is formulated and a solution is given by constructing an approximately self-optimal strategy.
The state space of the Markov process is either a closed subset of Euclidean space or a countable set....

In the control or adaptive control of linear stochastic evolution systems with complete observations of the state it is important to know the asymptotic distribution of the quadratic cost or the asymptotic bounds for the fluctuation of the average cost around the optimal average cost. In this paper stochastic evolution systems are considered. These...

Three distinct controlled ergodic Markov models are considered here. The models are a discrete time controlled Markov process with complete observations, a controlled diffusion process with complete observations, and a discrete time controlled Markov process with partial observations. The partial observations for the third model have the special fo...