# Andrii MironchenkoAlpen-Adria-Universität Klagenfurt · Institute of Mathematics

Andrii Mironchenko

Dr. habil.

## About

109

Publications

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Introduction

Andrii Mironchenko currently works at the Department of Mathematics, University of Klagenfurt.
His research interests include stability and control of infinite-dimensional systems, nonlinear control theory, and hybrid systems.
More info at: http://www.mironchenko.com/index.php/en/

Additional affiliations

October 2014 - November 2020

December 2013 - June 2014

August 2012 - September 2014

## Publications

Publications (109)

We show that the existence of a non-coercive Lyapunov function is sufficient for uniform global asymptotic stability (UGAS) of infinite-dimensional systems with external disturbances provided the speed of decay is measured in terms of the norm of the state and an additional mild assumption is satisfied. For evolution equations in Banach spaces with...

We prove characterizations of ISS for a large class of infinite-dimensional control systems, including differential equations in Banach spaces, time-delay systems, ordinary differential equations, switched systems. These characterizations generalize well-known criteria of ISS, proved by Sontag and Wang in \cite{SoW96} for ODE systems. For the speci...

In a pedagogical but exhaustive manner, this survey reviews the main results on input-to-state stability (ISS) for infinite-dimensional systems. This property allows estimating the impact of inputs and initial conditions on both the intermediate values and the asymptotic bound on the solutions. ISS has unified the input-output and Lyapunov stabilit...

We prove nonlinear small-gain theorems for input-to-state stability of infinite heterogeneous networks, consisting of input-to-stable subsystems of possibly infinite dimension. Furthermore, we prove small-gain results for the uniform global stability of infinite networks. Our results extend available theorems for finite networks of finite- or infin...

We prove a superposition theorem for input-to-output stability (IOS) of a broad class of nonlinear infinite-dimensional systems with outputs including both continuous-time and discrete-time systems. It contains, as a special case, the superposition theorem for input-to-state stability (ISS) of infinite-dimensional systems from [1] and the IOS super...

Input-to-state stability (ISS) unifies global asymptotic stability with respect to variations of initial conditions with robustness with respect to external disturbances. First, we present Lyapunov characterizations for input-to-state stability as well as ISS superpositions theorems showing relations of ISS to other robust stability properties. Nex...

Input-to-state stability (ISS) unifies the stability and robustness in one notion, and serves as a basis for broad areas of nonlinear control theory. In this contribution, we covered the most fundamental facts in the infinite-dimensional ISS theory with a stress on Lyapunov methods. We consider various applications given by different classes of inf...

Adapting a counter-example recently proposed by J.L. Mancilla-Aguilar and H. Haimovich, we show here that, for time-delay systems, global asymptotic stability does not ensure that solutions converge uniformly to zero over bounded sets of initial states. Hence, the convergence might be arbitrarily slow even if initial states are confined to a bounde...

We consider time-delay systems with a finite number of delays in the state space
${\mathrm {L}}^{\infty }\times \mathbb {R} ^{\mathrm {n}}$
. In this framework, we show that forward completeness implies the bounded reachability sets property. This implication was recently shown by J.L. Mancilla-Aguilar and H. Haimovich to fail in the state space...

We derive conditions for well-posedness of semilinear evolution equations with unbounded input operators. Based on this, we provide sufficient conditions for such properties of the flow map as Lipschitz continuity, bounded-implies-continuation property, boundedness of reachability sets, etc. These properties represent a basic toolbox for stability...

Despite all advantages of the ISS framework, for some practical systems, input-to-state stability is too restrictive.

In this book, we have developed the ISS theory for systems of ordinary differential equations, covering both fundamental theoretical results and central applications of the ISS theory to nonlinear control and network analysis. Furthermore, we have investigated several properties closely related to ISS, most notably integral ISS, and discussed recen...

Stability analysis and control of ODE systems remain the backbone of much of systems theory.

In this chapter, we recall basic concepts and results of the theory of ordinary differential equations (ODEs) with measurable inputs (“Carathéodory theory”).

Stability analysis and control of nonlinear systems are a rather complex task, which becomes even more challenging for large-scale systems.

In the previous chapter, we have analyzed well-posedness and properties of reachability sets for nonlinear control systems of the form

An objective of a control theory is to influence the dynamics of a system to guarantee its desired behavior. Challenges arising in this way are manifold and include nonlinearity of a system, a need to ensure robustness (or reliability) of designed controllers in spite of actuator and observation errors, hidden (unmodeled) dynamics of a system, and...

We derive converse Lyapunov theorems for input-to-state stability (ISS) of linear infinite-dimensional analytic systems. We show that input-to-state stability of a linear system does not imply existence of a coercive quadratic ISS Lyapunov function, even if the input operator is bounded. If, however, the semigroup is similar to a contraction semigr...

We consider two systems of two conservation laws that are defined on complementary spatial intervals and coupled by a moving interface as a single non-autonomous port-Hamiltonian system. We provide sufficient conditions that guarantee that this system is Kato-stable.

Input-to-state stability (ISS) allows estimating the impact of inputs and initial conditions on both the intermediate values and the asymptotic bound on the solutions. ISS has unified the input-output and Lyapunov stability theories and is a crucial property in the stability theory of control systems as well as for many applications whose dynamics...

We consider two systems of two conservation laws that are defined on complementary spatial intervals and coupled by an interface as a single port-Hamiltonian system. In case of a fixed interface position, we characterize the boundary and interface conditions for which the associated port-Hamiltonian operator generates a contraction semigroup. Furth...

We model two systems of two conservation laws defined on complementary spatial intervals and coupled by a moving interface as a single non-autonomous port-Hamiltonian system, and provide sufficient conditions for its Kato-stability. An example shows that these conditions are quite restrictive. The more general question under which conditions an evo...

We prove small-gain type criteria of exponential stability for positive linear discrete-time systems in ordered Banach spaces that have a normal and generating positive cone. Such criteria play an important role in the finite-dimensional systems theory but are rather unexplored in the infinite-dimensional setting, yet. Furthermore, we show that our...

We derive conditions for well-posedness of semilinear evolution equations with unbounded input operators and the corresponding boundary control systems. Based on this, we provide sufficient conditions for Lipschitz continuity of the flow map, bounded-implies-continuation property, boundedness of reachability sets, etc. These properties represent a...

We consider resolvent positive operators $A$ on ordered Banach spaces and seek for conditions that ensure that their spectral bound $s(A)$ is negative. Our main result characterizes $s(A) < 0$ in terms of so-called small-gain conditions that describe the behaviour of $Ax$ for positive vectors $x$. This is new even in case that the underlying space...

We show that boundedness of reachability sets for distributed parameter systems is equivalent to existence of a corresponding Lyapunov function, that increases at most exponentially along the trajectories. Next we show a similar characterization for the robust forward completeness property.

We study input-to-state stability (ISS) of discrete-time networks of infinitely many finite-dimensional subsystems. We develop a so-called relaxed small-gain theorem for ISS with respect to a closed set and show the necessity of the proposed small-gain condition in case of exponentially ISS infinite networks. Moreover, we study the well-posedness o...

We generalize a small-gain theorem for a network of infinitely many systems, recently developed in [Kawan et. al, IEEE TAC (2021)]. The generalized small-gain theorem addresses exponential input-to-state stability with respect to closed sets, which enables us to study diverse control-theoretic problems in a unified manner, and it also allows for ag...

In this paper, we show that an infinite network of input-to-state stable (ISS) subsystems, admitting ISS Lyapunov functions, itself admits an ISS Lyapunov function, provided that the couplings between the subsystems are sufficiently weak. The strength of the couplings is described in terms of the properties of an infinite-dimensional nonlinear posi...

We consider infinite heterogeneous networks, consisting of input-to-state stable subsystems of possibly infinite dimension. We show that the network is input-to-state stable, provided that the gain operator satisfies a certain small-gain condition. We show that for finite networks of nonlinear systems this condition is equivalent to the so-called s...

We prove new characterisations of exponential stability for positive linear discrete-time systems in ordered Banach spaces, in terms of small-gain conditions. Such conditions have played an important role in the finite-dimensional systems theory, but are relatively unexplored in the infinite-dimensional setting, yet. Our results are applicable to d...

This paper provides a Lyapunov-based small-gain theorem for input-to-state stability (ISS) of networks composed of infinitely many finite-dimensional systems. We model these networks on infinite-dimensional ℓ∞-type spaces. A crucial assumption in our results is that the internal Lyapunov gains, modeling the influence of the subsystems on each other...

We introduce the concept of non-uniform input-to-state stability for networks. It combines the uniform global stability with the uniform attractivity of any subnetwork while it allows for non-uniform convergence of all components. For an infinite network consisting of input-to-state stable subsystems, which do not necessarily have a uniform $\maths...

In this paper, we show that an infinite network of input-to-state stable (ISS) subsystems, admitting ISS Lyapunov functions, itself admits an ISS Lyapunov function, provided that the couplings of the subsystems are sufficiently weak. The strength of the couplings is described in terms of the properties of an infinite-dimensional nonlinear positive...

This paper provides a Lyapunov-based small-gain theorem for input-to-state stability (ISS) of networks composed of infinitely many finite-dimensional systems. We model these networks on infinite-dimensional $\ell_{\infty}$-type spaces. A crucial assumption in our results is that the internal Lyapunov gains, modeling the influence of the subsystems...

This paper presents a small-gain theorem for networks composed of a countably infinite number of finite-dimensional subsystems. Assuming that each subsystem is exponentially input-to-state stable, we show that if the gain operator, collecting all the information about the internal Lyapunov gains, has a spectral radius less than one, the overall inf...

Motivated by the scalability problem in large networks, we study stability of a network of infinitely many finite-dimensional subsystems. We develop a so-called relaxed small-gain theorem for input-to-state stability (ISS) with respect to a closed set and show that every exponentially input-to-state stable system necessarily satisfies the proposed...

We prove new characterisations of exponential stability for positive linear discrete-time systems in ordered Banach spaces, in terms of small-gain conditions. Such conditions have played an important role in the finite-dimensional systems theory, but are relatively unexplored in the infinite-dimensional setting, yet. Our results apply to the large...

This paper presents a tight small-gain theorem for networks composed of infinitely many finite-dimensional subsystems. Assuming that each subsystem is exponentially input-to-state stable, we show that if the gain operator, collecting all the information about the internal Lyapunov gains, has a spectral radius less than one, the overall infinite net...

We introduce the concept of non-uniform input-to-state stability for networks, which combines the uniform global stability together with the uniform attractivity of any finite number of modes of the system, but which does not guarantee the uniform convergence of all modes. We show that given an infinite network of ISS subsystems, which do not have...

We derive a saturated feedback control, which locally stabilizes a linear reaction-diffusion equation. In contrast to most other works on this topic, we do not assume the Lyapunov stability of the uncontrolled system and consider general unstable systems. Using Lyapunov methods, we provide estimates for the region of attraction for the closed-loop...

Motivated by a paradigm shift towards a hyper-connected world, we develop a computationally tractable small-gain theorem for a network of infinitely many systems, termed as infinite networks. The proposed small-gain theorem addresses exponential input-to-state stability with respect to closed sets, which enables us to analyze diverse stability prob...

In this paper, we extend the ISS Lyapunov methodology to make it suitable for the analysis of ISS w.r.t. inputs from Lp-spaces. We show that the existence of a so-called Lp-ISS Lyapunov function implies Lp-ISS of a system. Also, we show that existence of a noncoercive Lp-ISS Lyapunov function implies Lp-ISS of a control system provided the flow map...

This paper presents a tight small-gain theorem for networks composed of infinitely many finite-dimensional subsystems. Assuming that each subsystem is exponentially input-to-state stable, we show that if the gain operator, collecting all the information about the internal Lyapunov gains, has a spectral radius less than one, the overall infinite net...

We consider an abstract class of infinite-dimensional dynamical systems with inputs. For this class, the significance of noncoercive Lyapunov functions is analyzed. It is shown that the existence of such Lyapunov functions implies integral-to-integral input-to-state stability. Assuming further regularity it is possible to conclude input-to-state st...

This paper presents a small-gain theorem for networks composed of a countably infinite number of finite-dimensional subsystems. Assuming that each subsystem is exponentially input-to-state stable, we show that if the gain operator, collecting all the information about the internal Lyapunov gains, has a spectral radius less than one, the overall inf...

In this paper we consider countable couplings of finite-dimensional input-to-state stable systems. We consider the whole interconnection as an infinite-dimensional system on the ∞ state space. We develop stability conditions of the small-gain type to guarantee that the whole system remains ISS and highlight the differences between finite and infini...

We derive a saturated feedback control, which locally stabilizes a linear reaction-diffusion equation. In contrast to most other works on this topic, we do not assume Lyapunov stability of the uncontrolled system, and consider general unstable systems. Using Lyapunov methods, we provide estimates for the region of attraction for the closed-loop sys...

In this paper, a class of abstract dynamical systems is considered which encompasses a wide range of nonlinear finite- and infinite-dimensional systems. We show that the existence of a non-coercive Lyapunov function without any further requirements on the flow of the forward complete system ensures an integral version of uniform global asymptotic s...

We prove a small-gain theorem for interconnections of $n$ nonlinear heterogeneous input-to-state stable control systems of a general nature, covering partial, delay and ordinary differential equations. Furthermore, for the same class of control systems we derive small-gain theorems for asymptotic gain, uniform global stability and weak input-to-sta...

We derive a saturated feedback control, which locally stabilizes a linear reaction-diffusion equation. In contrast to most other works on this topic, we do not assume Lyapunov stability of the uncontrolled system, and consider general unstable systems. Using Lyapunov methods, we provide estimates for the region of attraction for the closed-loop sys...

In this paper we consider countable couplings of finite-dimensional input-to-state stable systems. We consider the whole interconnection as an infinite-dimensional system on the ℓ∞ state space. We develop stability conditions of the small-gain type to guarantee that the whole system remains ISS and highlight the differences between finite and infin...

We prove a small-gain theorem for interconnections of n nonlinear heterogeneous input-to-state stable control systems of a general nature, covering partial, delay and ordinary differential equations. We use in this paper the summation formulation of the ISS property, but the method can be adapted to other formulations of the ISS concept as well. Th...

We consider an abstract class of infinite-dimensional dynamical systems with inputs. For this class the significance of noncoercive Lyapunov functions is analyzed. It is shown that the existence of such Lyapunov functions implies integral input-to-integral state stability. Assuming further regularity it is possible to conclude input-to-state stabil...

We introduce a monotonicity-based method for studying input-to-state stability (ISS) of nonlinear parabolic equations with boundary inputs. We first show that a monotone control system is ISS if and only if it is ISS w.r.t. constant inputs. Then we show by means of classical maximum principles that nonlinear parabolic equations with boundary distur...

In this paper, a class of abstract dynamical systems is considered which encompasses a wide range of nonlinear finite- and infinite-dimensional systems. We show that the existence of a non-coercive Lyapunov function without any further requirements on the flow of the forward complete system ensures an integral version of uniform global asymptotic s...

We introduce a monotonicity-based method for studying input-to-state stability (ISS) of nonlinear parabolic equations with boundary inputs. We first show that a monotone control system is ISS if and only if it is ISS w.r.t. constant inputs. Then we show by means of classical maximum principles that nonlinear parabolic equations with boundary distur...

In this paper a class of abstract dynamical systems is considered which encompasses a wide range of nonlinear finite-and infinite-dimensional systems. We show that the existence of a non-coercive Lyapunov function without any further requirements on the flow of the forward complete system ensures an integral version of uniform global asymptotic sta...

For a broad class of infinite-dimensional systems, we characterize input-to-state practical stability (ISpS) using the uniform limit property and in terms of input-to-state stability. We specialize our results to the systems with Lipschitz continuous flows and evolution equations in Banach spaces. Even for the special case of ordinary differential...

We prove that input-to-state stability (ISS) of nonlinear systems over Banach spaces is equivalent to existence of a coercive Lipschitz continuous ISS Lyapunov function for this system. For linear infinite-dimensional systems, we show that ISS is equivalent to existence of a non-coercive ISS Lyapunov function and provide two simpler constructions o...

We introduce a monotonicity-based method for studying input-to-state stability (ISS) of nonlinear parabolic equations with boundary inputs. We first show that a monotone control system is ISS if and only if it is ISS w.r.t. constant inputs. Then we show by means of classical maximum principles that nonlinear parabolic equations with boundary distur...

We show that practical uniform global asymptotic stability (pUGAS) is equivalent to existence of a bounded uniformly globally weakly attractive set. This result is valid for a wide class of robustly forward complete distributed parameter systems, including time-delay systems, switched systems, many classes of PDEs and evolution differential equatio...

We show that existence of a non-coercive Lyapunov function is sufficient for uniform global asymptotic stability (UGAS) of infinite-dimensional systems with external disturbances provided an additional mild assumption is fulfilled. For UGAS infinite-dimensional systems with external disturbances we derive a novel ‘integral’ construction of non-coer...

This paper studies stability of interconnections of hybrid dynamical systems, in the general scenario that the continuous or discrete dynamics of subsystems may have destabilizing effects. We analyze two existing methods of constructing Lyapunov functions for the interconnection based on candidate ISS Lyapunov functions for subsystems, small-gain c...

We prove a novel Lyapunov-based small-gain theorem for networks of $ n \geq 2 $ hybrid systems which are not necessarily input-to-state stable. This result unifies and extends several small-gain theorems for hybrid and impulsive systems proposed in the last few years. We also show how average dwell-time (ADT) clocks and reverse ADT clocks can be us...

We show that existence of a non-coercive Lyapunov function is sufficient for uniform global asymptotic stability (UGAS) of infinite-dimensional systems with external disturbances provided an additional mild assumption is fulfilled. For UGAS infinite-dimensional systems with external disturbances we derive a novel 'integral' construction of non-coer...

We show by means of counterexamples that many characterizations of input-to-state stability (ISS) known for ODE systems are not valid for general differential equations in Banach spaces. Moreover, these notions or combinations of notions are not equivalent to each other, and can be classified into several groups according to the type and grade of n...

We prove that uniform global asymptotic stability of bilinear infinite-dimensional control systems is equivalent to their integral input-to-state stability. Next we present a method for construction of iISS Lyapunov functions for such systems if the state space is a Hilbert space. Unique issues arising due to infinite-dimensionality are highlighted...

We consider input-to-state stability (ISS) of nonlinear infinite dimensional impulsive systems with an emphasis on interconnections of such systems. Stability conditions as a combination of Lyapunov methods and dwell-time inequalities are provided. For stability of interconnections a further condition of a small-gain type comes into play. We illust...

We prove that uniform global asymptotic stability of bilinear infinite-dimensional control systems is equivalent to their integral input-to-state stability. Next we present a method for construction of iISS Lyapunov functions for such systems if the state space is a Hilbert space. Unique issues arising due to infinite-dimensionality are highlighted...

This paper is devoted to two issues. One is to provide Lyapunov-based tools
to establish integral input-to-state stability (iISS) and input-to-state
stability (ISS) for some classes of nonlinear parabolic equations. The other is
to provide a stability criterion for interconnections of iISS parabolic
systems. The results addressing the former proble...

We prove a novel Lyapunov-based small-gain theorem for interconnections of n hybrid systems, which are not necessarily input-to-state stable. This result unifies and extends several small-gain theorems for hybrid and impulsive systems, proposed in the last few years. Also we show how the average dwell-time (ADT) clocks and reverse ADT clocks can be...

For bilinear infinite-dimensional dynamical systems, we show the equivalence
between uniform global asymptotic stability and integral input-to-state
stability. We provide two proofs of this fact. One applies to general systems
over Banach spaces. The other is restricted to Hilbert spaces, but is more
constructive and results in an explicit form of...

We present a novel optimal allocation model for perennial plants, in which
assimilates are not allocated directly to vegetative or reproductive parts but
instead go first to a storage compartment from where they are then optimally
redistributed. We do not restrict considerations purely to periods favourable
for photosynthesis, as it was done in pub...

In this paper we propose a novel optimal allocation model for perennial plants. We consider not only favorable for photosynthesis periods, but analyze the whole life of a perennial plant. This provides more information about strategies of a plant during transitions between favorable and unfavorable seasons. One of predictions of our model is that a...

We investigate stabilizability of switched systems of differential-algebraic equations (DAEs). For such systems we introduce a parameterized family of switched ordinary differential equations that approximate the dynamic behavior of the switched DAE. A criterion for stabilizability of a switched DAE system using time-dependent switching is obtained...

We prove that impulsive systems, which possess an ISS Lyapunov function, are
ISS for time sequences satisfying the fixed dwell-time condition. If an ISS
Lyapunov function is the exponential one, we provide a stronger result, which
guarantees uniform ISS of the whole system over sequences satisfying the
generalized average dwell-time condition. Then...

In this paper we provide two small-gain theorems for impulsive systems. The first of them provides a construction of an ISS-Lyapunov function for interconnections of impulsive systems if ISS-Lyapunov functions for subsystems are given and a small-gain condition holds. If, in addition, these given ISS-Lyapunov functions are exponential then the seco...

We prove that impulsive systems, which possess an ISS Lyapunov function, are ISS for impulse time sequences, which satisfy the fixed dwell-time condition. If the ISS Lyapunov function is the exponential one, we provide stronger result, which guarantees uniform ISS of the whole system over sequences of impulse times, which satisfy the generalized av...