# Ivan IvanovSofia University "St. Kliment Ohridski" · Department of Statistics and Econometrics

Ivan Ivanov

Doctor Science

## About

98

Publications

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Citations since 2017

Introduction

## Publications

Publications (98)

In this paper, we solve a stochastic linear quadratic tracking problem. The controlled dynamical system is modeled by a system of linear Itô differential equations subject to jump Markov perturbations. We consider the case when there are two decision-makers and each of them wants to minimize the deviation of a preferential output of the controlled...

We investigate a linear quadratic tracking problem for impulsive controlled stochastic systems. The main tool in the derivation of the optimal control is the stabilizing solution for a class of backward jump matrix linear differential equation. We introduce the concept of mean square stability by impulses and consider the properties of stabilizing...

"We investigate a problem to solve the linear quadratic tracking problem for stochastic systems controlled by impulses. Two optimal control problems are investigated where the different objective functions are minimized. Explicit formulae for optimal controls are developed. The optimal controllers are computed based on the solution of the backward...

In this paper, we examine a sampled-data Nash equilibrium strategy for a stochastic linear quadratic (LQ) differential game, in which admissible strategies are assumed to be constant on the interval between consecutive measurements. Our solution first involves transforming the problem into a linear stochastic system with finite jumps. This allows u...

Hepatitis C can be a life threatening condition, whose severity depends on the stage of development. The stage of development of hepatitis C determines the chances of survival. In recent years machine learning algorithms have focused on predicting the stage of hepatitis development. The dataset called HCV is often used in machine learning experimen...

In this paper we consider the problem of minimization of the mean square value of the deviation of a random signal z(tf ) from a given target ζ. The random signal z(tf ) represents the value at instant time tf of an output of a controlled dynamical system described by an Itˆo differential equation. Both the case when the set of admissible controls...

In this paper we consider the problem of minimization of the mean square value of the deviation of a random signal z(tf ) from a given target ζ. The random signal z(tf ) represents the value at instant time tf of an output of a controlled dynamical system described by an Itˆo differential equation. Both the case when the set of admissible controls...

In classification problems, cross-validation chooses random samples from the dataset in order to improve the ability of the model to classify properly new observations in the respective class. Research articles from various fields show that when applied to regression problems, the bootstrap can improve either the prediction ability of the model or...

The purpose of this paper is to examine the causality between DUST, CO2 and temperature for the Vostok ice core data series [Vostok Data Series], dating from 420 000 years ago, and the EPICA C Dome data going back 800 000 years. In addition, the time-varying volatility and coefficient of variation in the CO2, dust and temperature is examined, as we...

The purpose of this paper is to examine the causality between DUST, CO2 and temperature for the Vostok ice core data series [Vostok Data Series], dating from 420 000 years ago, and the EPICA C Dome data going back 800 000 years. In addition, the time-varying volatility and coefficient of variation in the CO2, dust and temperature is examined, as we...

The problem of the existence of a Nash equilibrium strategy in a state feedback form is discussed for a class of stochastic linear quadratic two players differential games. It is assumed that only measurements at discrete-time instances of the state parameters are available. Both piecewise continuous admissible strategies as well as piecewise const...

Constrained Regulation Problem (CRP) for fractional-order nonlinear continuous-time systems is investigated in this paper. New existence condition of linear feedback control law for a class of fractional-order nonlinear continuous-time systems under constraints is proposed. Computation method for solving CRP of fractional-order nonlinear systems co...

Cross validation is often used to split input data into training and test set in Support vector machines. The two most commonly used cross validation versions are the tenfold and leave-one-out cross validation. Another commonly used resampling method is the random test/train split. The advantage of these methods is that they avoid overﬁtting in the...

The problem of the existence of a Nash equilibrium strategy in a state feedback form is discussed for a class of stochastic linear quadratic two players differential games. It is assumed that only measurements at discrete-time instances of the state parameters are available. Both piecewise continuous admissible strategies as well as piecewise const...

The problem of sampled-data Nash equilibrium strategy in a state feedback setting for a stochastic linear quadratic differential game is addressed. It is assumed that the admissible strategies are constant on the interval between two measurements. The original problem is converted into an equivalent one for a linear stochastic system with finite ju...

Feature selection is a powerful tool to identify the important characteristics of data for prediction. Feature selection, therefore, can be a tool for avoiding overfitting, improving prediction accuracy and reducing execution time. The applications of feature selection procedures are particularly important in Support vector machines, which is used...

The problem to classify big data is an actual one the subject. There are multiple ways to classify data but the k Nearest Neighbors (k-NN) has become a popular tool for the data scientist. In this paper we examine several modifications of the k Nearest Neighbors algorithm that achieve better efficiency in terms of accuracy and CPU time when classif...

A linear quadratic optimal control problem for a system described by Itô differential equations with state and control dependent white noise under the assumption that the set of admissible controls consists of a class of piecewise constant stochastic processes is considered. The considered LQ optimal control problem is converted into a LQ optimizat...

In this paper, based on Kou’s classical iterative methods with fifth-order of convergence, we propose new interval iterative methods for computing a real root of the nonlinear scalar equations. Some numerical experiments have executed with the program INTLAB in order to confirm the theoretical results. The computational results have described and c...

We consider a two players linear quadratic differential game associated to a positive theta-periodic dynamical system with jump Markov perturbations. We introduce the concept of stabilizing solution and strong stabilizing solution of the associated system of coupled matrix Riccati differential equations and we will point out the relation between th...

The infinite horizon linear quadratic differential games for positive linear systems with Markovian jumping is considered. The authors' goal is to propose a set of sufficient conditions that guarantee the existence of the stabilising solution of a system of game theoretic algebraic Riccati type equations associated to the considered differential ga...

We consider a two players linear quadratic differential game associated to a positive θ-periodic dynamical system with jump Markov perturbations. We introduce the concept of stabilizing solution and strong stabilizing solution of the associated system of coupled matrix Riccati differential equations and we will point out the relation between these...

We consider a generalized algebraic Riccati equation arising in stochastic control with an indefinite quadratic part. Three effective methods for computing a matrix sequence, which converges to the stabilizing solution of the considered type of Riccati equations with indefinite quadratic parts are explored. Convergence properties of these methods a...

We investigate the problem for solving a discrete-time periodic generalized Riccati equation with an indefinite sign of the quadratic term. A necessary condition for the existence of bounded and stabilizing solution of the discrete-time Riccati equation with an indefinite quadratic term is derived. The stabilizing solution is positive semidefinite...

We consider the linear quadratic differential games for positive linear
systems with the feedback information structure in the general case. Recently, several
iterative methods to obtain the stabilizing solution of a corresponding set of Riccati
equations are described in the literature - the Newton method and its accelerated
modi cation and the Ly...

A PhD class of lectures will be jointly developed on the topic of Linear Quadratic Differential Games and Applications. The class is intended to support the learning process of the PhD students as well as to increase their competencies through acquaintance with the latest advances in the field. The total study load is planned to amount to 30 lectur...

The paper studies N-player linear quadratic differential games on an infinite time horizon with deterministic feedback information structure. It introduces two iterative methods (the Newton method as well as its accelerated modification) in order to compute the stabilising solution of a set of generalised algebraic Riccati equations. The latter is...

The present paper develops an optimal linear quadratic boundary controller for 2×2 linear hyperbolic partial differential equations (PDEs) with actuation on only one end of the domain. First-order necessary conditions for optimality is derived via weak variations and an optimal controller in state-feedback form is presented. The linear quadratic re...

This paper addresses the problem of solving discrete-time H∞ control problems for periodic systems. The approach for solving such a type of equations is well known in the literature. However, the focus of our research is set on the numerical computation of the stabilizing solution. In particular, two effective methods for practical realization of t...

This study investigates the influence of the 2008 financial crisis on a number of European stock markets. The sample includes EU benchmark indices as well as European markets with slowed or hampered recovery over a period of ten years (2004-2014) thus allowing a comparison on their development before, during and after the crisis. We utilize a novel...

This paper addresses the problem of solving a generalized algebraic Riccati equation with an indefinite sign of its quadratic term. We extend the approach introduced by Lanzon, Feng, Anderson and Rotkowitz (2008) for solving similar Riccati equations. We numerically investigate two types of iterative methods for computing the stabilizing solution....

This paper demonstrates the utilization of wavelet-based tools for the analysis and prediction of financial time series exhibiting strong long-range dependence (LRD). Commonly emerging markets' stock returns are characterized by LRD. Therefore, we track the LRD evolvement for the return series of six Southeast European stock indices through the app...

We consider a discrete-time periodic generalized Riccati equation.
We investigate a few iterative methods for computing the stabilizing
solution. The first method is the Kleinman algorithm which is a special case of the classical Newton-Kantorovich procedure, the second one is a method of consistent iterations and two new Stein iterations. The prop...

In the paper the infinite-horizon Linear Quadratic Regulator (LQR) problem of linear discrete time systems with non-negative state constraints is presented. Such kind of constraints on the system determine the class of positive systems. They have big application in many fields like economics, biology, ecology, ICT and others. The standard infinite...

This paper addresses the problem for solving a Continuous-time Riccati equation with an indefinite sign of the quadratic term. Such an equation is closely related to the so called full information H∞ control of linear time-invariant system with external disturbance. Recently, a simultaneous policy update algorithm (SPUA) for solving H∞ control prob...

Adaptive and relative market efficiency of seven Southeast European stock exchanges is investigated for a period of 11 years. A wavelet-based technique is utilized to the daily return series of the major stock indices in order to track the evolution of the LRD parameter, since its value is closely related to the degree of returns predictability. A...

In [1], the authors provided some comments on the paper [3]. In this note, we will address the points raised.

This paper is concerned with two kinds algebraic Riccati equations arising in the linear quadratic (LQ) control. The problem gives rise to the existence of a so-called stabilizing solution to a stochastic (generalized) algebraic Riccati equation, which is however
fundamentally different from the classical algebraic Riccati equation as a result fro...

SUMMARY This paper addresses the problem of solving a class of periodic discrete-time Riccati equation with an indefinite sign of its quadratic term. Such an equation is closely related to the so-called full-information H ∞ control of discrete-time periodic systems. A globally convergent iterative algorithm with a local quadratic convergence rate i...

We consider a set of discrete-time generalized Riccati equations that arise in quadratic optimal control of discrete-time stochastic systems subjected to both state-dependent noise and Markovian jumps. The iterative method to compute the maximal and stabilizing solution of wide class of discrete-time nonlinear equations is derived by Dragan et al....

This paper addresses the problem of solving a class of periodic discrete-time Riccati equation with an indefinite sign of it's quadratic term. More precisely, we focus on the computation of the stabilizing solutions of discrete-time game theoretic periodic Riccati equations. A convergent iterative algorithm is proposed for this purpose.

The optimal control problems for positive systems is an important and extensively studied topic in recent years. In this paper, the infinite horizon LQR problem of positive linear discrete time systems (PLDS) is studied. For solving the problem we consider the approach based on the admissibility of the solution of the standard infinite discrete LQR...

Stochastic linear systems subjected both to Markov jumps and to multiplicative white noise are considered. In order to stabilize such type of stochastic systems, the so-called set of generalized discrete-time algebraic Riccati equations has to be solved. The LMI approach for computing the stabilizing symmetric solution (which is in fact the equilib...

The authors investigate the numerical solution of a set of discrete-time generalised Riccati equations. The class of discrete-time non-linear equations involves in various control problems for discrete-time stochastic systems and it can be considered as an important tool for solving optimisation control for such type systems. A new procedure for co...

The presence of stock market efficiency is a distinctive characteristic of the effectively functioning market economy. Investigation of the market efficiency of seven emerging East-European stock exchanges is carried out as their major stock indices (BELEX15, BET, CROBEX, ISE100, PFTS, RTSI, SOFIX) are studied in respect of long-range dependence (L...

We consider a set of discrete-time coupled algebraic Riccati equations that arise in quadratic optimal control of Markovian jump linear systems. The LMI approach for computing the maximal symmetric solution of this system is studied. The special case of the Riccati equations with applications to financial modeling is commented. We construct two new...

The current paper investigates the major index of the Bulgarian Stock Exchange with respect to the presence of long-range dependence and principal predictability of the index. The wavelet transform is utilized in order to carry out the investigation since it is a well-suited tool for the analysis of fractal processes and sheds additional light to t...

This paper examines the problem of measuring sustainable governance in the European Union (EU-27) through the use of duality and the Slutsky equation. The proposed methodology is based on the application of a three-dimensional optimisation model, where the arguments of the objective (sustainable social welfare) function are economic goods that cont...

In this paper, the problem of the numerical computation of the stabilizing solution of the game theoretic algebraic Riccati
equation is investigated. The Riccati equation under consideration occurs in connection with the solution of the H
∞ control problem for a class of stochastic systems affected by state dependent and control dependent white n...

In this article, the problem of the numerical computation of the stabilising solution of the game theoretic algebraic Riccati equation is investigated. The Riccati equation under consideration occurs in connection with the solution of the H ∞ control problem for a class of stochastic systems affected by state-dependent and control-dependent white n...

This paper discusses the numerical solution of the coupled algebraic Riccati equations associated with the linear quadratic differential games. The Lyapunov iteration for solving the considered coupled equations is discussed by Li and Gajic (1994). We modify this iteration and derive the new algorithm with typically convergence properties for metho...

In this paper, the stochastic Nash games for weakly coupled large-scale systems with state-dependent noise are considered. The considered stochastic algebraic Riccati equations is quite different from the existing results in the sense that the equations have the additional linear term. The numerical algorithm based on Lyapunov iterations for solvin...

We consider a set of discrete-time coupled algebraic Riccati equations that arise in quadratic optimal control of Markovian jump linear systems. Two iterations for computing a symmetric (maximal) solution of this system are investigated. We construct sequences of the solutions of the decoupled Stein equations and show that these sequences converge...

We consider a new type nonlinear matrix equation. We investigate the existence a positive definite solution and two iterative methods for computing this solution. The first method is the classical Newton procedure and the second is a new Stein iteration. In this paper it is proved that a new Stein iteration has convergence properties to those of th...

We consider a set of discrete-time coupled algebraic Riccati equations that arise in quadratic optimal control. Two iterations for computing a symmetric solution of this system are investigated. New iterations are based on the properties of a Stein equation. It is necessary to solve a Stein equation at each step of proposed algorithms. We adapt the...

We consider different iterative methods for computing Hermitian solutions of the coupled Riccati equations of the optimal control problem for jump linear systems. We have constructed a sequence of perturbed Lyapunov algebraic equations whose solutions define matrix sequences with special properties proved under proper initial conditions. Several nu...

We consider the numerical solution of a set of discrete-time coupled algebraic Riccati equations that arise in quadratic optimal
control. Several iterations for computing a symmetric solution of this system are investigated and compared. New iterations
are based on the properties of a Stein equation. It is necessary to solve a Stein equation at eac...

We consider different iterative methods for computing a Hermitian or maximal Hermitian solution of two types of rational Riccati equations arising in stochastic control. The classical Newton procedure and its modification applied to equations are very expensive. New less expensive iterations for these equations are introduced and some convergence p...

We analyze the solution of the system of coupled algebraic Riccati equations of the optimal control problem of jump linear
system. We prove that the Lyapunov iterations converge to a positive semidefinite stabilizing solution under mild conditions.

We consider different iterative methods for computing a Hermitian or maximal Hermitian solution of two types rational Riccati equations arising in stochastic control. The classical Newton procedure and its modification applied to equations are very expensive. New less expensive iterations for these equations are introduced and some convergence prop...

The positive semidefinite solutions of the nonlinear matrix equation X + S* X†S = Q are investigated. We consider an iterative method converges to a positive semidefinite solution of this equation under the condition Ker X ⊆ Ker S*. The new results are illustrated by numerical examples.

The nonlinear matrix equation X+A*X-nA=Q and properties of its positive definite solutions are studied. Sufficient conditions for existence the minimal XS and special Xl positive definite solutions are derived and iterative procedures for computing these solution are discussed.

Two perturbation estimates for maximal positive definite solutions of equations X+A*X-1 A = Q and X - A*X(-1)A = Q are considered. These estimates are proved in [Hasanov et al., Improved perturbation Estimates for the Matrix Equations X A*X(-1)A = Q,LinearAlgebra Appl. 379 (2004) 113-135]. We derive new perturbation estimates under weaker restricti...

The nonlinear matrix equation X−A∗X−nA=I is discussed (n is a positive integer). Convergence of the iterative methods proposed by El-Sayed [Comput. Math. Appl. 41 (2001) 579–588] is proved under weaker restrictions for the matrix A and an arbitrary integer n.

The nonlinear matrix equations X ± A*X−nA = Q are investigated. New perturbation estimates for positive definite solutions of these equations are derived. The new results are illustrated by numerical examples.

The two matrix iterations X k+1 = I A * X −1 k A are known to converge linearly to a positive definite solution of the matrix equations X ± A * X −1 A = I, respectively, for known choices of X 0 and under certain restrictions on A. The convergence for previously suggested starting matrices X 0 is generally very slow. This paper explores different i...

The two matrix iterations Xk+1 = I AX 1 k A are known to converge linearly to a positive denite solution of the matrix equations X AX 1A = I, respectively, for known choices of X0 and under certain re- strictions on A. The convergence for previously suggested starting matrices X0 is generally very slow. This paper explores dierent initial choices o...

In this paper we investigate the equations X±A∗X−nA=Q for the existence of positive definite solutions and perturbation bounds for these solutions are derived. A sufficient condition for uniqueness of a unique positive definite solution of the equation X−A∗X−nA=Q is given. The results are illustrated by using numerical examples.

We give new and improved perturbation estimates for the solution of the matrix quadratic
equations $X \pm A^∗X^{−1}A = Q$. Some of the estimates depend and some do not depend on
knowledge of the exact solution X. These bounds are compared numerically against other
known bounds from the literature.

A new effective method and its two modifications for solving Hermitian pentadiagonal block circulant systems of linear equations are proposed. New algorithms based on the proposed method are constructed. Our algorithms are then compared with some classical techniques as far as implementation time is concerned, number of operations and storage. Nume...

A special nonlinear matrix equation is considered. Theorems for the existence of a special positive definite solution $X_l$ and a minimal positive definite solution $X_S$ are proved. Some inequalities of these solutions are derived.

Perturbation theory of a special nonlinear matrix equation is discussed. New perturbation bounds for a special solution $X_l$ of this equation are derived. The results are illustrated by using numerical examples.

The solution of Hermitian block circulant tridiagonal linear system is investigated. This special kind of system appears in many applications. We propose new approach of El-sayed's method and develop two new algorithms for solving such kind of systems. Numerical experiments with our algorithms and some classical algorithms are executed. Numerical e...

A new iterative modification of Newton's method for solving nonlinear scalar equations are proposed. Weerakoon and Fernando have been propose a variant of Newton's method in which they approximate the indefinite integral by a trapezoid instead of a rectangle. This modification has third-order convergence. We approximate the indefinite integral by a...

The two matrix equations X+A*X−2A=I and X−A*X−2A=I are studied. We construct iterative methods for obtaining positive definite solutions of these equations. Sufficient conditions for the existence of two different solutions of the equation X+A*X−2A=I are derived. Sufficient conditions for the existence of positive definite solutions of the equation...

The general nonlinear matrix equation X + A*X
-nA I is discussed (n is a positive integer). Some necessary and sufficient conditions for existence a solution are given. Two methods for iterative computing a positive definite solution are investigated. Numerical experiments to illustrate the performance of the methods are reported.

Iterative methods for computing the square matrix root of positive definite matrices based on Newton's method are investigated. The methods consist two matrix sequences with scaling param-eter are proposed. Sufficiently conditions for convergence of these methods are given. Numerical experiments are included.

Iterative methods for computing the square matrix root of positive definite matrices based on Newton's method are investigated. The methods consist two matrix sequences with scaling param-eter are proposed. Sufficiently conditions for convergence of these methods are given. Numerical experiments are included.

The general nonlinear matrix equation $X + A^∗X^{−n}A = I$ is
discussed (n is a positive integer). Some necessary and sufficient conditions
for existence a solution are given. Two methods for iterative
computing a positive definite solution are investigated. Numerical experiments
to illustrate the performance of the methods are reported.

Iterative methods for computing the square matrix root of positive definite matrices based on Newton's method are investigated. The methods consist two matrix sequences with scaling param-eter are proposed. Sufficiently conditions for convergence of these methods are given. Numerical experiments are included.

A new effective modification of the method which is described by S. M. El-Sayed, I. G. Ivanov and M. G. Petkov [A new modification of the Rojo method for solving symmetric circulant five-diagonal systems of linear equations. Computers Math. Appl. 35, 35-44 (1998)] for solving of real symmetric circulant pentadiagonal systems of linear equations is...

In this paper, the stochastic Nash games for weakly coupled large-scale systems with state-dependent noise are con-sidered. The considered stochastic algebraic Riccati equations is quite different from the existing results in the sense that the equations have the additional linear term. The numerical methods based on the solution of linear matrix e...

The matrix equation $X-A^*\sqrt{X^{-1}}A = I$ in this paper is studied. There is an iterative method for obtaining of a positive definite solution of this equation. Sucient conditions for existence of positive definite solutions are proved. Results of numerical expiriments are given.

In this paper we discuss some properties of a positive definite solution of the matrix equation X + A∗X−2 A = I. Two effective iterative methods for computing a positive definite solution of this equation are proposed. Necessary and sufficient conditions for existence of a positive definite solution are derived. Numerical experiments are executed w...

A new, effective, and stable modification of the Rojo method [1] for solving of real symmetric circulant five-diagonal systems of linear equations is proposed. This special kind of system appears in many applications: spline approximation, difference solution of partial differential equations, etc. The method presented in this paper a very efficien...

Abstract The two matrix equations X + A,2A = I are given. Numerical experiments are discussed. © 2001 Elsevier Science Inc. All rights reserved. AMS classification: 65F10 Keywords: Matrix equation; Positive definite solution; Iterative method

The general nonlinear matrix equation $X + A^*X^{-n} A = I$ is discussed ($n$ is a positive integer). Some necessary and sufficient conditions for existence a solution are given. Two methods for iterative computing a positive definite solution are investigated. Numerical experiments to illustrate the performance of the methods are reported.

## Projects

Project (1)