Lorenzo Guglielmetti
email: lorenzo.guglielmetti@gmail.com

Bedaquiline for the treatment of multidrug-resistant tuberculosis: another missed opportunity?

5 years after its first approval by a stringent regulatory authority, the role of bedaquiline in the treatment of multidrug-resistant tuberculosis (MDR-TB) and extensively drug-resistant tuberculosis (XDR-TB) is well established, and acknowledged by existing guidance. However, in many regards, the global introduction of bedaquiline can hardly be considered as a success.

Bedaquiline, an innovative compound belonging to the diarylquinoline family, was developed in 2005 [START_REF] Andries | A diarylquinoline drug active on the ATP synthase of Mycobacterium tuberculosis[END_REF], and approved by the Food and Drug Administration and the European Medicines Agency in 2012 and 2013, respectively. The results of two phase IIb clinical trials have shown promising results when adding bedaquiline to an optimised background MDR-TB regimen [START_REF] Diacon | Multidrug-resistant tuberculosis and culture conversion with bedaquiline[END_REF][START_REF] Pym | Bedaquiline in the treatment of multidrug-and extensively drug-resistant tuberculosis[END_REF][START_REF] Diacon | The diarylquinoline TMC207 for multidrug-resistant tuberculosis[END_REF][START_REF] Diacon | Randomized pilot trial of eight weeks of bedaquiline (TMC207) treatment for multidrug-resistant tuberculosis: long-term outcome, tolerability, and effect on emergence of drug resistance[END_REF], and raised new hopes in a field, like drug-resistant TB treatment, which had not seen any relevant innovation for decades. However, except for a few countries leading the way in the early programmatic introduction of bedaquiline [START_REF] Guglielmetti | Examples of bedaquiline introduction for the management of multidrug-resistant tuberculosis in five countries[END_REF], the global roll-out of this drug has been inadequate. According to a report by the Drug-Resistant TB Scale-up Treatment Action Team (DR-TB STAT), no more than 8195 patients globally were reported to have received treatment with bedaquiline before March 2017, with the large majority of them (59.6%) being treated in only one country, South Africa [7]. The disproportion between these numbers and the need for bedaquiline is striking: according to estimates provided by a multicentre study that included MDR-TB patients from the World Health Organization's (WHO) Europe and Africa regions, approximately one-third of patients would be eligible for treatment with bedaquiline based on drug-resistance criteria [START_REF] Bonnet | Identification of patients who could benefit from bedaquiline or delamanid: a multisite MDR-TB cohort study[END_REF]. Moreover, the proportion of eligible patients exceeds two-thirds of all cases if we include those with baseline risk factors for poor outcome, as suggested by the recently updated WHO recommendations on the use of bedaquiline [START_REF]Report of the Guideline Development Group Meeting on the use of bedaquiline in the treatment of multidrug-resistant tuberculosis: a review of available evidence[END_REF].

Unfortunately, the slow pace of the introduction of bedaquiline has not prevented the appearance of strains with resistance to this drug [START_REF] Veziris | Rapid emergence of Mycobacterium tuberculosis bedaquiline resistance: lessons to avoid repeating past errors[END_REF][START_REF] Andries | Acquired resistance of Mycobacterium tuberculosis to bedaquiline[END_REF][START_REF] Bloemberg | Acquired resistance to bedaquiline and delamanid in therapy for tuberculosis[END_REF]. Although drug resistance has inevitably followed the introduction of new antibiotics in the history of antimicrobial chemotherapy, some of the practices that are currently established in the use of bedaquiline may arguably increase the risk of resistance selection [START_REF] Guglielmetti | Preventing acquired resistance to bedaquiline and delamanid in multidrug-resistant tuberculosis treatment requires optimal management[END_REF]. In most settings, bedaquiline is used for 24 weeks as a component of conventional regimens, which usually last 20-24 months. The standardised duration of bedaquiline treatment is recommended by the manufacturer according to the evidence provided by clinical trials where bedaquiline was given for 24 weeks. This recommendation does not take into account the specific needs of individual MDR-TB patients. The interruption of bedaquiline treatment may lead to the administration of unacceptably weak regimens in cases where no therapeutic alternatives are available. A recent study confirms these concerns: in the study, almost 20% of 53 MDR-TB patients who had achieved sputum culture conversion with bedaquiline-containing regimens experienced reversion to positive cultures after interrupting bedaquiline treatment at 24 weeks [START_REF] Kiria | Compassionate use of bedaquiline in Armenia and Georgia: end of treatment results[END_REF]. In addition, current recommendations warn against the concomitant use of the new drugs, bedaquiline and delamanid, on the grounds of a lack of evidence and the potential for a synergic increase in QT interval prolongation. Even if based on understandable safety concerns, an excess of prudence in this case may lead to prioritising the prevention of the virtual risk of cardiac toxicity over the well-documented risk of death associated with drug-resistant TB. Indeed, the association of bedaquiline and delamanid may represent the only chance of an effective treatment for XDR-TB cases with advanced resistance patterns or intolerance to multiple drugs. The sequential use of the new drugs, after a recommended wash-out period of 6 months when switching from bedaquiline to delamanid, contradicts the principles of anti-TB treatment and may increase the risk of acquisition of drug resistance, as exemplified by a case report in which resistance to bedaquiline and delamanid was acquired sequentially during treatment [START_REF] Bloemberg | Acquired resistance to bedaquiline and delamanid in therapy for tuberculosis[END_REF][START_REF] Somoskovi | A mutation associated with clofazimine and bedaquiline cross-resistance in MDR-TB following bedaquiline treatment[END_REF][START_REF] Hoffmann | Delamanid and bedaquiline resistance in Mycobacterium tuberculosis ancestral Beijing genotype causing extensively drug-resistant tuberculosis in a Tibetan refugee[END_REF].

The shortcomings in global access to bedaquiline are reflected by the paucity of available evidence [START_REF] Pontali | Bedaquiline and MDR-TB: a systematic and critical analysis of the evidence[END_REF][START_REF] Pontali | Multidrug-resistance tuberculosis and beyond: an updated analysis of the current evidence on bedaquiline[END_REF]. Only a few treatment cohorts including at least 10 patients have been published to date [START_REF] Skrahina | Bedaquiline in the multidrug-resistant tuberculosis treatment: Belarus experience[END_REF][START_REF] Guglielmetti | Compassionate use of bedaquiline for the treatment of multidrug-resistant and extensively drug-resistant tuberculosis: interim analysis of a French dohort[END_REF][START_REF] Guglielmetti | Is bedaquiline as effective as fluoroquinolones in the treatment of multidrug-resistant tuberculosis?[END_REF][START_REF] Ndjeka | Treatment of drug-resistant tuberculosis with bedaquiline in a high HIV prevalence setting: an interim cohort analysis[END_REF][START_REF] Udwadia | Compassionate use of bedaquiline in highly drug-resistant tuberculosis patients in Mumbai, India[END_REF], and only one reporting treatment outcomes [START_REF] Guglielmetti | Long-term outcome and safety of prolonged bedaquiline treatment for multidrug-resistant tuberculosis[END_REF]. Extended treatment duration with bedaquiline has so far been described in one small cohort and in a case report [START_REF] Guglielmetti | Long-term outcome and safety of prolonged bedaquiline treatment for multidrug-resistant tuberculosis[END_REF][START_REF] Lewis | First experience of effectiveness and safety of bedaquiline for 18 months within an optimised regimen for XDR-TB[END_REF], and only two case reports of concurrent treatment with bedaquiline and delamanid have been published [START_REF] Lachâtre | Bedaquiline plus delamanid for XDR tuberculosis[END_REF][START_REF] Tadolini | First case of extensively drug-resistant tuberculosis treated with both delamanid and bedaquiline[END_REF].

These research gaps emphasise the importance of the study by BORISOV et al. [START_REF] Borisov | Effectiveness and safety of bedaquiline-containing regimens in the treatment of MDR-and XDR-TB: a multicentre study[END_REF] presented in this issue of the European Respiratory Journal, which reports the results of the first large multicentre cohort of patients receiving bedaquiline-containing treatment. This retrospective cohort study involved 25 centres from 15 different countries, including settings with both high and low TB burden. Consecutive, culture-confirmed MDR-TB patients treated outside clinical trials with a regimen including bedaquiline in the period between 2008 and 2016 were included in the study, regardless of the timing of bedaquiline introduction. Treatment efficacy was assessed through sputum culture conversion rates and, for patients who ended treatment, the outcomes of the treatment. Treatment safety data was collected, including QT interval follow-up when available, although no standardised protocol to grade severity and seriousness of the adverse events, nor their causality link with bedaquiline, was implemented. Overall, 428 MDR-TB patients were included, mostly from South Africa (44%) and the Russian Federation (28%). The cohort includes high rates of XDR-TB (46%) and previously treated (78%) cases, and a total prevalence of HIV infection of 22% among those tested. Treatment regimens included, in more than half of cases, re-purposed drugs like linezolid and clofazimine. Sputum culture conversion rates were 57% and 81% at 2 and 3 months after anti-TB treatment commenced, respectively. Remarkably, 77% of the 130 patients in the 2008-2014 cohort, which included patients with ⩾2 years of follow-up, achieved favourable treatment outcomes. These figures appear to be consistent with the outcomes of patients starting treatment after 2014, although the results of these recent cohorts are preliminary and should be interpreted carefully. As expected with MDR-TB regimens, a high rate of adverse events occurred during treatment. Treatment with bedaquiline was interrupted in 12% of patients, and discontinued permanently in 6%: the reasons for discontinuation were not reported. QT interval corrected according to the Fridericia formula (QTcF) values of >500 ms were recorded in 24 out of 247 (9.7%) patients with available data. Among the patients who died and had available QT follow-up information, no one experienced QT prolongation, with the exception of one patient who already had >500 ms QTcF at baseline. One patient died after developing a severe ventricular arrhythmia, although his QTcF never exceeded 500 ms.

Overall, this collaborative study provides us with a wealth of evidence on the use of bedaquiline. First of all, the efficacy results are remarkable. The rate of favourable outcomes in the cohort who commenced treatment before 2014 is in line with the upper boundary of the confidence interval of the meta-analysis included in the latest WHO recommendations [START_REF]Report of the Guideline Development Group Meeting on the use of bedaquiline in the treatment of multidrug-resistant tuberculosis: a review of available evidence[END_REF], and is much higher than those reported in systematic reviews of MDR/XDR-TB patients treated without new and repurposed drugs [START_REF] Bonnet | Identification of patients who could benefit from bedaquiline or delamanid: a multisite MDR-TB cohort study[END_REF][START_REF] Falzon | Resistance to fluoroquinolones and second-line injectable drugs: impact on multidrug-resistant TB outcomes[END_REF][START_REF] Jacobson | Treatment outcomes among patients with extensively drug-resistant tuberculosis: systematic review and meta-analysis[END_REF]. Early culture conversion rates are also high, although the inclusion of patients starting bedaquiline at different time-points during treatment makes it difficult to single out the role of bedaquiline. Although prone to the shortcomings of retrospective, non-standardised data collection, safety results are reassuring. This is in line with other published reports [START_REF] Skrahina | Bedaquiline in the multidrug-resistant tuberculosis treatment: Belarus experience[END_REF][START_REF] Ndjeka | Treatment of drug-resistant tuberculosis with bedaquiline in a high HIV prevalence setting: an interim cohort analysis[END_REF][START_REF] Udwadia | Compassionate use of bedaquiline in highly drug-resistant tuberculosis patients in Mumbai, India[END_REF][START_REF] Guglielmetti | Long-term outcome and safety of prolonged bedaquiline treatment for multidrug-resistant tuberculosis[END_REF] and with the conclusions of the recently revised WHO recommendations, which downgraded the anticipated undesirable effects of bedaquiline from "large" to "moderate" [START_REF]Report of the Guideline Development Group Meeting on the use of bedaquiline in the treatment of multidrug-resistant tuberculosis: a review of available evidence[END_REF].

In this framework, the study by BORISOV et al. [START_REF] Borisov | Effectiveness and safety of bedaquiline-containing regimens in the treatment of MDR-and XDR-TB: a multicentre study[END_REF] represents a source of information that should not be overlooked. The authors' encouraging results bring additional evidence to support a systematic introduction of bedaquiline for patients who need it. At the same time, they highlight the urgent need for well-documented prospective observational cohorts of patients receiving bedaquiline in the years before results from phase III clinical trials will become available. The timely roll-out of bedaquiline will require, in parallel, increased attention to prevent drug resistance, including optimal, individualised patient management. The use of prolonged bedaquiline treatment and the combination of bedaquiline and delamanid are therapeutic options that should be considered whenever indicated. Criteria have already been proposed to guide the selection of such patients [START_REF] Guglielmetti | Long-term outcome and safety of prolonged bedaquiline treatment for multidrug-resistant tuberculosis[END_REF][START_REF] Matteelli | Compassionate and optimum use of new tuberculosis drugs[END_REF]. This is the time to scale-up and improve our efforts to avoid bedaquiline being remembered as another missed opportunity in the history of TB control.


Denis Efimov

Johannes Schiffer

On boundedness of solutions of periodic systems: a multivariable cell structure approach

Many dynamical systems are periodic with respect to several state variables. This periodicity typically leads to the coexistence of multiple invariant solutions (equilibria or limit cycles). As a consequence, while there are many classical techniques for analysis of boundedness and stability of such systems, most of these only permit to establish local properties. Motivated by this, a new sufficient criterion for global boundedness of solutions of such a class of nonlinear systems is presented. The proposed method is inspired by the cell structure approach developed by Leonov and Noldus and characterized by two main advances. First, the conventional cell structure framework is extended to the case of dynamics, which are periodic with respect to multiple states. Second, by introducing the notion of a Leonov function the usual definiteness requirements of standard Lyapunov functions are relaxed to sign-indefinite functions. Furthermore, it is shown that under (mild) additional conditions the existence of a Leonov function also ensures input-to-state stability (ISS), i.e., robustness with respect to exogenous perturbations. The performance of the proposed approach is demonstrated via the global analysis of boundedness of trajectories for a nonlinear system.

I. INTRODUCTION

In control systems and many related domains, such as mechanical, transportation, power and biological systems, stability is one of the most fundamental and sought-after properties of a dynamical system [START_REF] Bhatia | Stability Theory of Dynamical Systems[END_REF], [START_REF] Chetaev | The Stability of Motion[END_REF], [START_REF] Kh | Stability of nonlinear systems with non unique equilibrium[END_REF], [START_REF] Hahn | Stability of Motion[END_REF], [START_REF] Khalil | Nonlinear Systems[END_REF], [START_REF] Lakshmikantham | Stability analysis in terms of two measures[END_REF], [START_REF] Lyapunov | The general problem of the stability of motion[END_REF], [START_REF] Rumyantsev | Stability and stabilization of motion with respect to part of variables[END_REF]. Until today, most available approaches for stability analysis of nonlinear systems are based on Lyapunov theory [START_REF] Lyapunov | The general problem of the stability of motion[END_REF]. A key feature of a Lyapunov-based stability analysis is that boundedness and convergence properties of the system's solutions can be assessed without explicit computation of the latter. Instead, it suffices to verify a set of inequalities for the Lyapunov function and its time derivative, which is derived with respect to the system's equations. More precisely, the existence of a continuously differentiable (or at least Lipschitz continuous) Lyapunov function, which is positive definite with respect to an equilibrium (or an invariant set) and the time derivative of which is non-positive along the solutions of the system under investigation, is equivalent to stability of that equilibrium (or set). Similarly, instability of an equilibrium can be studied using the Chetaev function approach [START_REF] Chetaev | The Stability of Motion[END_REF], [START_REF] Efimov | On necessary conditions of instability and design of destabilizing controls[END_REF]. A Chetaev function may be sign-indefinite 1 with a negative or positive definite derivative.

The classical stability theory in the sense of Lyapunov has mainly focused on properties of a single isolated equilibrium point. Yet, in many (engineering) applications, such as biological and power systems or distributed optimization, the systems under study possess several equilibria or invariant sets (including hidden attractors [START_REF] Dudkowski | Hidden attractors in dynamical systems[END_REF]). The aim for a rigorous treatment of such systems has motivated the extension of Lyapunov-based stability theory to dynamical systems with several disjoint invariant sets [START_REF] Angeli | Detection of multistability, bifurcations and hysteresis in a large class of biological positivefeedback systems[END_REF], [START_REF] Nitecki | Filtrations, decompositions, and explosions[END_REF], [START_REF] Kh | Stability of nonlinear systems with non unique equilibrium[END_REF], [START_REF] Rantzer | A dual to Lyapunov's stability theorem[END_REF], [START_REF] Angeli | Multi-stability in monotone input/output systems[END_REF], [START_REF] Stan | Analysis of interconnected oscillators by dissipativity theory[END_REF], [START_REF] Efimov | Oscillatority of nonlinear systems with static feedback[END_REF], [START_REF] Efimov | Global Lyapunov analysis of multi-stable nonlinear systems[END_REF]. This extension is far from trivial and requires suitable modifications and relaxations of the employed stability notions, see, e.g., [START_REF] Efimov | Global Lyapunov analysis of multi-stable nonlinear systems[END_REF] and the subsequent developments in [START_REF] Angeli | On input-to-state stability with respect to decomposable invariant sets[END_REF], [START_REF] Angeli | Characterizations of input-to-state stability for systems with multiple invariant sets[END_REF] for an appropriate extension of the input-to-state stability (ISS) property in the presence of multiple equilibria. Further related results on robust stability analysis of systems with multiple equilibria are [START_REF] Angeli | An almost global notion of input-to-state stability[END_REF], [START_REF] Angeli | Stability robustness in the presence of exponentially unstable isolated equilibria[END_REF], [START_REF] Chaves | Bistable biological systems: A characterization through local compact input-to-state stability[END_REF].

A special case of systems often exhibiting multiple equilibria is represented by dynamics, which is periodic with respect to one or several of its states. Paramount examples of such systems are the forced nonlinear pendulum [START_REF] Efimov | Robustness of delayed multistable systems with application to droop-controlled inverter-based microgrids[END_REF], [START_REF] Forni | Differential analysis of nonlinear systems: Revisiting the pendulum example[END_REF], power systems [START_REF] Ortega | Transient stabilization of multimachine power systems with nontrivial transfer conductances[END_REF], [START_REF] Schiffer | On stability of a distributed averaging PI frequency and active power controlled differential-algebraic power system model[END_REF], [START_REF] Dörfler | Synchronization and transient stability in power networks and non-uniform Kuramoto oscillators[END_REF], [START_REF] Trip | An internal model approach to (optimal) frequency regulation in power grids with time-varying voltages[END_REF] and microgrids [START_REF] Schiffer | Conditions for stability of droop-controlled inverter-based microgrids[END_REF], [START_REF] De | Bregman storage functions for microgrid control[END_REF], phase-locked loops [START_REF] Leonov | Analytical method for computation of phase-detector characteristic[END_REF], [START_REF] Leonov | Hold-in, pull-in, and lock-in ranges of pll circuits: Rigorous mathematical definitions and limitations of classical theory[END_REF] and complex networks of oscillators [START_REF] Stan | Analysis of interconnected oscillators by dissipativity theory[END_REF], [START_REF] Dörfler | Synchronization in complex networks of phase oscillators: A survey[END_REF], [START_REF] Tang | Synchronization in complex networks and its application-a survey of recent advances and challenges[END_REF]. Yet unfortunately, the application of the aforementioned existing results to periodic systems is in most cases very challenging. The main reason for this is the technical difficulty of constructing suitable Lyapunov functions. For example, if some of the states of the system are periodic (e.g., they evolve on the circle), in order to be able to apply the approach proposed in [START_REF] Angeli | Characterizations of input-to-state stability for systems with multiple invariant sets[END_REF] the corresponding Lyapunov function also has to be periodic with respect to these states, which is a severe requirement. As a consequence, available results are often limited to local stability claims, see [START_REF] Dörfler | Synchronization in complex networks of phase oscillators: A survey[END_REF].

Motivated by this wide range of potential applications, we consider a special class of systems, which possess periodic right-hand sides with respect to several state components and the state of which evolves in R n . For such systems, the two main contributions of the present paper are: to derive sufficient conditions for global boundedness of trajectories via the concept of a Leonov function and to show that under (mild) additional conditions the existence of a Leonov function readily implies input-to-state stability (ISS), i.e., robustness with respect to exogenous perturbations. We remark that in the context of systems with multiple equilibria, boundedness of trajectories is the natural counterpart to stability of a single equilibrium point in the classical Lyapunov theory.

With regard to boundedness of solutions, our approach is inspired by the cell structure framework proposed in [START_REF] Leonov | On the boundedness of the trajectories of phase systems[END_REF] (and later in [START_REF] Noldus | New direct Lyapunov-type method for studying synchronization problems[END_REF]) and further developed in [START_REF] Kh | Stability of nonlinear systems with non unique equilibrium[END_REF], [START_REF] Yakubovich | Stability of stationary sets in control systems with discontinuous nonlinearities[END_REF] for systems, the dynamics of which are periodic with respect to a scalar state variable. The following observation lies at the core of the cell structure approach of Leonov and collaborators. At first, exploiting the periodicity of the dynamics allows to construct -via a function, which is sign-indefinite in the periodic state -unbounded forward invariant sets, which in addition are globally attractive. Second, the intersection of these unbounded forward invariant sets consists of globally attractive compact forward invariant and isolated sets, i.e., cells, which gives the name to this method.

The restriction to periodicity of the dynamics with respect to a scalar state variable is essential to obtain forward invariant sets in [START_REF] Leonov | On the boundedness of the trajectories of phase systems[END_REF], [START_REF] Noldus | New direct Lyapunov-type method for studying synchronization problems[END_REF], since in the scalar case the individual cells are only connected via equilibria. But, clearly, this restriction is a severe limitation in the analysis of complex networks with periodic dynamics. Therefore, in the present paper we extend the approach of [START_REF] Leonov | On the boundedness of the trajectories of phase systems[END_REF], [START_REF] Noldus | New direct Lyapunov-type method for studying synchronization problems[END_REF] to systems the dynamics of which are periodic with respect to multiple state variables by proposing a multivariable cell structure approach.

As noted in [START_REF] Noldus | New direct Lyapunov-type method for studying synchronization problems[END_REF], such extension is far from trivial. In fact, while as done in [START_REF] Leonov | On the boundedness of the trajectories of phase systems[END_REF], [START_REF] Noldus | New direct Lyapunov-type method for studying synchronization problems[END_REF], we also exploit the periodicity of the dynamics to construct (unbounded) forward invariant sets, in the multivariable setting significant additional care has to be taken to obtain compact forward invariant sets, i.e., cells. This technical difficulty arises from the fact that in the multivariable case, the cells can, in general, not only be connected via equilibria. This challenge is overcome in the present paper by introducing the concept of a Leonov function. Loosely speaking, a Leonov function is sign-indefinite with respect to the periodic states and radially unbounded with respect to the non-periodic states. Furthermore, a Leonov function is negative definite with respect to the distance to a set, which separates the equilibria of the system. This last requirement is a major difference compared to the functions employed in [START_REF] Leonov | On the boundedness of the trajectories of phase systems[END_REF], [START_REF] Noldus | New direct Lyapunov-type method for studying synchronization problems[END_REF] and is key to establish our main result.

Besides boundedness of solutions, robustness with respect to exogenous disturbances is often a highly desirable property in many control engineering applications. Therefore, a second contribution of the paper is to provide additional conditions under which the existence of a Leonov function also implies the ISS property. This claim is established by leveraging recent results in [START_REF] Efimov | A relaxed characterization of iss for periodic systems with multiple invariant sets[END_REF], where an extension of the ISS theory from [START_REF] Angeli | Characterizations of input-to-state stability for systems with multiple invariant sets[END_REF] to periodic systems has been proposed. Furthermore, the presented results are tested via application to a complex nonlinear system.

We anticipate that the proposed multivariable cell structure approach may prove useful in a variety of scenarios and applications. For instance, the original cell structure approach has recently been combined in [START_REF] Barabanov | Conditions for almost global attractivity of a synchronous generator connected to an infinite bus[END_REF] with LaSalle's invariance principle and in [START_REF] Schiffer | An input-to-state stability approach to verify almost global stability of a synchronous-machine-infinite-bus system[END_REF] with the ISS result of [START_REF] Efimov | A relaxed characterization of iss for periodic systems with multiple invariant sets[END_REF] to establish global properties of a synchronous machine infinite bus system. The extension of this analysis to the multi-machine case by using the methods derived in the present paper is under current investigation. Furthermore, the existence of a compact forward invariant set is a prerequisite for many methods that allow to assess the asymptotic behavior of a system's trajectories. Such methods comprise the classical LaSalle invariance principle [START_REF] Van Der Schaft | L2-Gain and Passivity Techniques in Nonlinear Control[END_REF], but also contraction [START_REF] Forni | A differential lyapunov framework for contraction analysis[END_REF], [START_REF] Lohmiller | On contraction analysis of nonlinear systems[END_REF], respectively convergence [START_REF] Pavlov | Convergent dynamics, a tribute to boris pavlovich demidovich[END_REF], or differential positivity [START_REF] Forni | Differential positivity on compact sets[END_REF].

The presented analysis builds upon our previous work [START_REF] Efimov | A new criterion for boundedness of solutions for a class of periodic systems[END_REF], where the preliminary formulations of the main results of this paper are given without proofs. Furthermore, the extension in [START_REF] Efimov | A relaxed characterization of iss for periodic systems with multiple invariant sets[END_REF] has been derived for systems on manifolds and, thus, does not allow to establish boundedness of trajectories in R n .

The remainder of this paper is structured as follows. Preliminaries and the theory from [START_REF] Kh | Stability of nonlinear systems with non unique equilibrium[END_REF], [START_REF] Yakubovich | Stability of stationary sets in control systems with discontinuous nonlinearities[END_REF] are given in Section II. The problem statement is presented in Section III with the main results in Section IV. Robustness with respect to external inputs is investigated in Section V. The efficacy of the presented approach is illustrated in Section VI.

II. PRELIMINARIES

Denote the sets of real and integer numbers by R and Z, respectively, and let

R + = {s ∈ R : s ≥ 0}, Z + = Z ∩ R + .
Let the map f (x) : R n → R n be of class C 1 , f (0) = 0, and consider a nonlinear system of the following form:

ẋ(t) = f (x(t)) ∀t ≥ 0 (1)
with state x(t) ∈ R n . We denote by X(t, x) the uniquely defined solution of (1) at time t fulfilling X(0, x) = x. A set S ⊂ M is invariant for the system (1) if X(t, x) ∈ S for all t ∈ R and for all x ∈ S; for x ∈ R n the point y ∈ R n belongs to its ω-limit (α-limit) set if there is a sequence t i , lim i→+∞ t i = +∞, such that lim i→+∞ X(t i , x) = y (lim i→+∞ X(-t i , x; 0) = y); for any x ∈ R n its αand ωlimit sets are invariant [START_REF] Guckenheimer | Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields[END_REF]. We define the distance from a point x ∈ R n to the set S ⊂ R n as |x| S = inf a∈S |x -a|, where |x| = |x| {0} for x ∈ R n is a usual Euclidean norm of a vector x ∈ R n . Furthermore, we introduce the vector norm

|x| ∞ = max 1≤i≤n |x i |, then |x| ∞ ≤ |x| ≤ √ n|x| ≤ n|x| ∞ .
A continuous function α : R + → R + belongs to class K if α(0) = 0 and the function is strictly increasing. The function α : R + → R + belongs to the class K ∞ if α ∈ K and its supremum is infinity. A continuous function β : R + × R + → R + belongs to the class KL if β(•, t) ∈ K ∞ for each fixed t ∈ R + and lim t→+∞ β(s, t) = 0 for each fixed s ∈ R + .

A. Boundedness of solutions of periodic systems

As outlined in Section I, the present paper is dedicated to the stability analysis of periodic systems [START_REF] Kh | Stability of nonlinear systems with non unique equilibrium[END_REF], [START_REF] Yakubovich | Stability of stationary sets in control systems with discontinuous nonlinearities[END_REF]. More precisely, we assume in the following that for the system (1) there exists ξ ∈ R n , ξ = 0, such that for all x ∈ R n f (x) = f (x + ξ).

Next, we recall a sufficient criterion derived in [START_REF] Leonov | On the boundedness of the trajectories of phase systems[END_REF], [START_REF] Kh | Stability of nonlinear systems with non unique equilibrium[END_REF], [START_REF] Yakubovich | Stability of stationary sets in control systems with discontinuous nonlinearities[END_REF], which allows to establish boundedness of solutions of periodic systems. To this end consider a special case of the system (1) given by f (x) = P x + bϕ(c x), where P ∈ R n×n is a singular matrix, c ∈ R n , b ∈ R n , ϕ : R ⇒ R is a ∆-periodical set-valued function, which is upper semicontinuous, with a nonempty, convex and closed set of values for any value of its argument. We note that a time-varying version of ϕ has been considered in [START_REF] Kh | Stability of nonlinear systems with non unique equilibrium[END_REF], [START_REF] Yakubovich | Stability of stationary sets in control systems with discontinuous nonlinearities[END_REF], but -to simplify the exposition -we restrict ourselves to the autonomous version of ϕ. Then under these restrictions and for any initial condition x 0 ∈ R n the system (1) has a solution X(t, x 0 ). Assume also that for all σ ∈ R\{0} and all φ ∈ ϕ(σ)

µ 1 ≤ φ σ ≤ µ 2 ; µ -1 1 µ -1 2 ϕ(0) = 0
for some µ 1 ∈ R∪{-∞} and µ 2 ∈ R∪{+∞}. The periodicity of ϕ implies that either µ 1 < 0, µ 2 > 0 or µ 1 = µ 2 = 0, and the latter case is excluded from consideration due to its triviality.

Theorem 1. [START_REF] Leonov | On the boundedness of the trajectories of phase systems[END_REF], [START_REF] Kh | Stability of nonlinear systems with non unique equilibrium[END_REF], [START_REF] Yakubovich | Stability of stationary sets in control systems with discontinuous nonlinearities[END_REF] Assume that there exists λ > 0 such that: 1) the matrix P + λI n , where I n ∈ R n×n is the identity matrix, has n -1 eigenvalues with negative real parts;

2) for all ω ∈ R

µ -1 1 µ -1 2 + (µ -1 1 + µ -1 2 )Reχ(iω -λ) + |χ(iω -λ)| 2 ≤ 0, where χ(s) = c T (P -sI n ) -1 b.
Then, for any initial condition x 0 ∈ R n the solution X(t, x 0 ) of the periodic system (1) is bounded for t ∈ [0, +∞).

Remark 1. Under the conditions of Theorem 1 the system (1) admits only a scalar periodic coordinate, i.e. the vector ξ has only one non-zero element and the remaining n -1 state variables are not periodic.

The proof of Theorem 1 (see Theorem 4.3.1 in [START_REF] Kh | Stability of nonlinear systems with non unique equilibrium[END_REF], or Theorem 4.7 in [START_REF] Yakubovich | Stability of stationary sets in control systems with discontinuous nonlinearities[END_REF]) is based on the fact that under the introduced conditions there is H = H ∈ R n×n (which has one negative and n -1 positive eigenvalues) such that for V 0 (x) = x Hx we have that dV 0 (x(t))/dt ≤ -2λV 0 (x(t)) for all t ∈ [0, +∞), which implies that the set

Ω 0 = {x ∈ R n : V 0 (x) ≤ 0} is forward invariant for (1), i.e. X(t, x 0 ) ∈ Ω 0 for all t ∈ [0, +∞) provided that x 0 ∈ Ω 0 . Next, introducing the functions V j (x) = V 0 (x -jδ) and sets Ω j = {x ∈ R n : V j (x) ≤ 0}
, where j is any integer and the vector δ ∈ R n satisfies the conditions δ = 0, P δ = 0 and c δ = ∆, by periodicity of f in (1) we obtain that dV j (x(t))/dt ≤ -2λV j (x(t)) for all t ∈ [0, +∞), then the sets Ω j are forward invariant for [START_REF] Angeli | An almost global notion of input-to-state stability[END_REF]. Finally, it is shown in [START_REF] Kh | Stability of nonlinear systems with non unique equilibrium[END_REF], [START_REF] Yakubovich | Stability of stationary sets in control systems with discontinuous nonlinearities[END_REF] that for any x 0 ∈ R n there is an index j 0 such that x 0 ∈ Γ j0 , where Γ j = Ω j ∩Ω -j ∩{x ∈ R n : |h x| ≤ j|h δ|} with h ∈ R n being the eigenvector of the matrix H corresponding to the negative eigenvalue. As it has been shown above X(t, x 0 ) ∈ Γ j0 for all t ∈ [0, +∞) (since it is true for Ω j0 ∩ Ω -j0 ). In addition the set Γ j0 is bounded, which was necessary to prove. In other words, an important observation of [START_REF] Leonov | On the boundedness of the trajectories of phase systems[END_REF], [START_REF] Kh | Stability of nonlinear systems with non unique equilibrium[END_REF], [START_REF] Yakubovich | Stability of stationary sets in control systems with discontinuous nonlinearities[END_REF] is that any intersection of the sets Ω j for all integers j forms a kind of cell cover of R n , where each cell is bounded and forward invariant. Therefore, this framework is commonly known as cell structure approach.

B. An illustrative example

The following example illustrates the main idea of the proof of Theorem 1.

Example 1. Suppose ( 1) is given by x = (z, θ) ∈ R 2 and f : R 2 → R 2 . Furthermore, suppose f is periodic in θ with period T > 0, i.e., for any integer j ∈ Z, we have that f (z, θ+ T j) = f (z, θ). Following Theorem 1, suppose that there exists a function V : R 2 → R, such that for some λ > 0,

V (0, 0) = 0, V (z, 0) > 0 ∀z ∈ R \ {0}, ∂V (x) ∂x f (x) ≤ -λV (x) ∀x ∈ R 2 .
By periodicity of f in T for any integer j

V (z, θ -T j) ≤ -λV (z, θ -T j) ∀(z, θ) ∈ R 2 . Denote V j (x) = V (z, θ -T j), then the invariant sets Ω j = {x ∈ R 2 : V j (x) ≤ 0}
are employed in Theorem 1 to establish boundedness of solutions for the system. These sets are illustrated in Fig. 1. More precisely, the set Ω 0 and a possible behavior of trajectories in its neighborhood are shown in Fig. 1,a, while the intersection of Ω j for j = -1, 0, 1 is plotted in Fig. 1,b. By construction, the common points in ∩ j∈Z Ω j form isolated invariant cells (the darkest rectangles in Fig. 1,b), to which all trajectories converge. Note that a passage from one cell to another is impossible since the individual cells are only connected through equilibria. In general, the boundary at which V j changes sign is not a straight line, but can be of an arbitrary shape. The straight lines are chosen here to ease the illustration of the main idea of Theorem 1 and also to motivate the notion of a cell structure.

III. PROBLEM STATEMENT

The functions proposed in [START_REF] Leonov | On the boundedness of the trajectories of phase systems[END_REF] for the analysis of boundedness of trajectories of the system (1) with a scalar periodic state are sign-indefinite with a sign-indefinite derivative. Clearly, such a relaxation of the definiteness of a Lyapunov function might simplify significantly its construction. Usually signindefinite functions with a sign-definite derivative are used to establish instability of an equilibrium of (1), e.g. Chetaev functions [START_REF] Chetaev | The Stability of Motion[END_REF], [START_REF] Efimov | On necessary conditions of instability and design of destabilizing controls[END_REF]. An important observation of [START_REF] Leonov | On the boundedness of the trajectories of phase systems[END_REF] is that the combination of "instability" and periodicity leads to boundedness of trajectories: under periodicity the existence of invariant solutions separating the domain of periodic variables, with probably repulsing trajectories around those invariants, implies the existence of a certain cell structure created by periodically repeated invariant solutions, which bounds the admissible behavior of the trajectories (see Example 1). A major restriction of the cell structure approach in [START_REF] Leonov | On the boundedness of the trajectories of phase systems[END_REF], [START_REF] Kh | Stability of nonlinear systems with non unique equilibrium[END_REF], [START_REF] Yakubovich | Stability of stationary sets in control systems with discontinuous nonlinearities[END_REF] is that it can only be applied in the case of a scalar periodic component in x(t). As a consequence, the main objective of this work is to extend that approach to a generic multidimensional case.

Inspired by [START_REF] Leonov | On the boundedness of the trajectories of phase systems[END_REF], [START_REF] Angeli | Characterizations of input-to-state stability for systems with multiple invariant sets[END_REF], an equivalent characterization of the ISS property for a periodic system on manifolds has been proposed in [START_REF] Efimov | A relaxed characterization of iss for periodic systems with multiple invariant sets[END_REF] in terms of ISS-Leonov functions, which are sign-indefinite. In the present work the concept of Leonov functions is further developed for (1) with x ∈ R n . More precisely, we assume the following:

Assumption 1. Let x = (z, θ) ∈ R n ,
where z ∈ R k and θ ∈ R q are two subsets of the state vector, n = k + q, k > 0 and q > 0. The vector field f in ( 1) is 2π-periodic with respect to θ.

In other words, we suppose that the system (1) can be embedded into a manifold M = R k × S q by a simple projection of the variables θ on the sphere S q . For our subsequent derivations, it is convenient to introduce auxiliary sets:

W = (R k × S)
where S is a q-th sphere (a set topologically equivalent to S q ) such that

π ≤ |θ| ∞ < 2π ∀x = (z, θ) ∈ W, U = r∈Z+ U r , (2)
U r = {x = (z, θ) ∈ R n : z = 0, |θ| ∞ = 2rπ, f (x) = 0}.
Therefore, the set U includes all equilibria obtained by shifting the one at the origin using the property that, by assumption, f is 2π-periodic in θ. However, in general, the system (1) may possess other equilibria that do not belong to U.

Definition 1. We say that a C 1 function V : R n → R is a Leonov function for (1) if there exist a constant g ≥ 0, functions α ∈ K ∞ , ψ ∈ K and a continuous function λ : R → R satisfying λ(0) = 0 and λ(s)s > 0 for all s = 0, such that

α(|z|) -ψ(|θ|) -g ≤ V (x) ∀x = (z, θ) ∈ R n , inf x∈W V (x) > 0, sup x∈U V (x) ≤ 0 (3)
and the following dissipation inequality holds:

∂V (x) ∂x f (x) ≤ -λ (V (x)) ∀x ∈ R n . ( 4
)
Roughly speaking, the function V is radially unbounded with respect to the non-periodic variable z and negative definite with respect to the distance to the set W (it takes positive values on the set W and negative values at the origin and on the set U). Consequently, the conditions (3) can be equivalently formulated as

α(|z|) -ψ (|θ| S ) + g ≤ V (x) ∀x ∈ R n , sup x∈U V (x) ≤ 0
for some ψ ∈ K and g ≥ 0 with ψ (|0| S ) ≥ g (since V (0) ≤ 0). Note that by construction the set W cannot contain any equilibrium point of ( 1) since in such a case the derivative of V would be strictly negative at an equilibrium.

Remark 2. Chetaev's theorem about instability of a set can be formulated as follows [START_REF] Efimov | On necessary conditions of instability and design of destabilizing controls[END_REF]: for a C 1 function U : R n → R, with U (x) = 0 for all x ∈ Q, where Q ⊂ R n is a compact invariant set of (1), if there exists 0 > 0 such that U + ∩ B Q ( ) = ∅ for any ∈ (0, 0 ], where

U + = {x ∈ B Q ( 0 ) : U (x) > 0} and B Q (ε) = {x ∈ R n : |x| Q < ε}, and if ∂U (x) ∂x f (x) > 0 ∀x ∈ U + , (5)
then ( 1) is unstable with respect to Q with the region of repulsion B Q ( 0 ). Now, take U (x) = -V (x), where V : R n → R is a Leonov function for [START_REF] Angeli | An almost global notion of input-to-state stability[END_REF]. Then due to (4) the inequality

∂V (x)
∂x f (x) < 0 may be satisfied for V (x) < 0. In such a case the property (5) can be verified. In addition, if the set {x ∈ R n : V (x) = 0} (since V is continuous and sign indefinite according to (3) or ( 6) below, then this set is not empty) contains a compact and invariant set of ( 1), e.g. an equilibrium, then the existence of a Leonov function V implies instability of (1) at the set Q with the region of repulsion belonging to {x ∈ R n : V (x) ≤ 0}.

Remark 3. The appearance of the norm |•| ∞ in the formulation of a Leonov function is originated by the topology induced by periodicity of f in θ: replicating the equilibrium at the origin using periodicity in θ ∈ R q creates a set of equilibria located at intersections of the levels |θ| ∞ = 2jπ for j ∈ Z.

In order to design a Leonov function for an actual system, the following characterization of existence of a Leonov function may also be employed. This characterization is established by using upper and lower estimates in terms of class K functions, as often done in the conventional Lyapunov theory. Yet, it is only sufficient for existence of a Leonov function: Proposition 1. Let for a C 1 function V : R n → R there exist functions α 1 , α 2 ∈ K ∞ , σ 1 , σ 2 ∈ K, and a continuous function λ : R → R, λ(0) = 0 and λ(s)s > 0 for all s = 0, and scalars

0 < g 1 ≤ g 2 ≤ σ 2 (π), such that for all x = (z, θ) ∈ R n α 1 (|z|) -σ 1 (||θ| ∞ -π|) + g 1 ≤ V (x) (6) ≤ α 2 (|z|) -σ 2 (||θ| ∞ -π|) + g 2 ,
and (4) holds. Then V is a Leonov function for [START_REF] Angeli | An almost global notion of input-to-state stability[END_REF].

Proof. By assumption, g 1 ≤ g 2 ≤ σ 2 (π). Furthermore, note that at the origin the relations (6) take the form:

0 ≥ g 2 -σ 2 (π) ≥ V (0) ≥ g 1 -σ 1 (π).
Thus, g 1 ≤ σ 1 (π). The restriction g 1 > 0 ensures that on the set W = {x = (z, θ) ∈ R n : |θ| ∞ = π} (and in some of its vicinity) the function V takes positive values:

V (x) ≥ α 1 (|z|) + g 1 > 0 ∀x ∈ W, while g 2 ≤ σ 2 (π) implies that V (0) ≤ 0, and that V (x) ≤ g 2 -σ 2 (π) ≤ 0 ∀x ∈ U 1 .
Next, since σ 2 ∈ K, it is monotonically increasing, and

V (x) ≤ g 2 -σ 2 ((2r -1)π) ≤ g 2 -σ 2 (π) ≤ 0 ∀x ∈ U r
for all r ≥ 1. For any x ∈ R n we have from (6):

V (x) ≥ α 1 (|z|) -σ 1 (||θ| ∞ -π|) + g 1 ≥ α 1 (|z|) -σ1 (|θ|) -g,
where existence of some σ1 ∈ K and g ≥ 0 follows from the relations of the norms | • | and | • | ∞ , and the demonstrated property g 1 ≤ σ 1 (π) (i.e. V (0) ≤ 0). Therefore, all properties of (3) are verified.

IV. A MULTIVARIABLE CELL STRUCTURE APPROACH

This section is dedicated to the derivation of a multivariable cell structure approach based on the concept of a Leonov function. Our main result is presented in subsection IV-A. Next, we illustrate how the required conditions can be verified for a special case. Furthermore, we provide two relaxations of our main result, which already have been found very helpful in applications of the proposed cell structure framework, see [START_REF] Schiffer | Almost global synchronization in multi-machine power systems[END_REF].

A. Main result

Our main result is as follows. An illustration of the result is given in Example 2 below.

Theorem 2. If for the system (1) under Assumption 1 there exists a Leonov function, then for all x 0 ∈ R n the corresponding trajectories X(t, x 0 ) are bounded for all t ≥ 0.

Proof. Denote the set of negative values of V as

Ω = {x ∈ R n : V (x) ≤ 0}.
From the definition of the set Ω and the lower bound of the function V in (3) we obtain:

α(|z|) ≤ g + ψ(|θ|) ∀x = (z, θ) ∈ Ω.
Recall the sets U and W defined in [START_REF] Angeli | On input-to-state stability with respect to decomposable invariant sets[END_REF], and that |θ| ≤ q|θ| ∞ . Then from the definition of a Leonov function, see (3), the set U belongs to Ω, while W / ∈ Ω. Furthermore, the set Ω can be decomposed into two parts:

Ω = Ω ∪ Ω .
The first of these sets,

Ω ⊂ {x ∈ R n : |z| ≤ α -1 (g), |θ| ∞ < c}
is a compact set containing the origin for some c ∈ (π, 2π), i.e. 0 ⊂ Ω , with g ≤ g ≤ g +ψ(qc) (the constant g evaluates the upper bound on |z| after which the growing terms in z will force V to take positive values for any |θ| ∞ < c). The second set is defined as

Ω ⊂ {x ∈ R n : |θ| ∞ > c}.
Both these sets are non-empty since 0 ∈ Ω and U r ∈ Ω for r ≥ 1 due to (3). Furthermore, the set Ω is unbounded. In addition, the subsets Ω and Ω are separated by the set W (the definition of which determines the value of c).

Denote V (t) = V (X(t, x 0 )) for any x 0 ∈ R n , then under the conditions of the theorem we have:

V (t) + λ (V (t)) ≤ 0 ∀t ≥ 0.
Thus, clearly, V (t) is strictly decreasing while X(t, x 0 ) ∈ R n \ Ω, i.e. while V (t) > 0. Therefore, if x 0 ∈ Ω then X(t, x 0 ) ∈ Ω for all t ≥ 0 and the set Ω is forward invariant for (1). Conversely, if x 0 ∈ R n \ Ω then there exists 0 ≤ T x0 < +∞ such that X(T x0 , x 0 ) ∈ Ω and, by invariance, the trajectory remains in this set for all t ≥ T x0 . Thus, the set Ω is globally attractive and forward invariant. Note that this property does not imply any kind of stability since Ω may be unbounded (it could also be interpreted as instability of a set containing W with the domain of repulsion R n \ Ω).

To establish stability, we exploit the periodicity of (1). Denote by j = [j 1 , . . . , j q ] a multi-index vector, where j i ∈ Z for all i = 1, . . . , q. Introduce the new variable x j = x-0 k 2πj

and the function V j (x) = V x -0 k 2πj for any such multi-index vector j (i.e. V 0 (x) = V (x)), where 0 k is the zero vector of dimension k. Hence

V j (x) = V j x j + 0 k 2πj = V (x j ).
Furthermore, by 2π-periodicity of f in θ,

ẋj (t) = f (x(t)) = f (x j (t)) ∀t ≥ 0.
Therefore, taking into account the properties substantiated for (1) and V , we obtain that the set {x j ∈ R n : V (x j ) ≤ 0} is globally attractive and forward invariant, which in the original coordinates x implies these properties for the set

Ω j = {x ∈ R n : V j (x) ≤ 0} = x ∈ R n : V x - 0 k 2πj ≤ 0) .
Using similar arguments as for Ω it is possible to show that Ω j = Ω j ∪ Ω j , where

Ω j ⊂ {x ∈ R n : |z| ≤ α -1 (g), |θ -2πj| ∞ < c}, Ω j ⊂ {x ∈ R n : |θ -2πj| ∞ > c}.
Since the sets Ω j are forward invariant, any trajectory X(t, x 0 ) has to belong asymptotically to the intersection of these sets:

j Ω j = j Ω j ∪ Ω j = j Ω j ∪ j Ω j ∪ Ξ,
where Ξ belongs to the union of all possible intersections of the form Ω j1 ∩ Ω j2 with j 1 = j 2 :

Ξ ⊂ j   Ω j ∩ k =j Ω k   .
Next, we show that j Ω j is composed of compact isolated subsets. At first, we note that since c ∈ (π, 2π), then j Ω j = j Θ j , where Θ j ⊂ Ω j is formed by compact isolated sets due to the fact that all Ω j possess the same property. Furthermore, by definition j Ω j = ∅ due to the fact that

Ω j ⊂ R n \ Υ j with Υ j = {x ∈ R n : |θ-2πj| ∞ ≤ c}, then j Ω j ⊂ j R n \ Υ j = R n \ j Υ j = R n \ R n = ∅. Since (0 k , 2πj 1 ) ∈ U ∩ Ω j1
for any j 1 by definition of both sets, and U \Ω j2 ⊂ Ω j2 due to c ∈ (π, 2π). These two facts imply that (0 k , 2πj 1 ) ∈ Ω j2 for any j 1 = j 2 . Consequently, Ω j1 ∩ Ω j2 = ∅ for any multiindex vectors j 1 = j 2 , which implies that Ξ ⊂ j Ω j ∩ R j since c ∈ (π, 2π), where

R j = {x ∈ R n : |θ -2πj| ∞ < π}.
Therefore, we have shown that

j Ω j ⊂   j Ω j ∩ R j   ∪   j Θ j   ,
where for each j the sets Ω j ∩ R j and Θ j are compact and isolated; for any neighboring Ω j1 ∩ R j1 ⊂ Ω j2 and Θ j2 ⊂ Ω j2 there is no intersection by construction (Ω j2 ∩Ω j2 = ∅). Hence, the compact sets in j Ω j form a kind of cell cover of {x ∈ R n : |z| ≤ α -1 (g)}.

Recall that for all multi-index vectors j the corresponding sets Ω j are globally attracting and forward invariant. Take any x 0 ∈ R n , then X(t, x 0 ) asymptotically enters and stays in a cell Ω j for some j. Hence, for any x 0 ∈ R n the corresponding solution X(t, x 0 ) is bounded for all t ≥ 0.

Example 2. For an illustration of the conditions of Theorem 2, let us consider an example of (1) with k = 1 and q = 2, then x = (z, θ) ∈ R 3 and f : R 3 → R 3 satisfies Assumption 1.

Assume that there is a Leonov function V : R n → R for some g ≥ 0, functions α ∈ K ∞ , ψ ∈ K and a continuous function λ : R → R satisfying (3) and ( 4) for all x = (z, θ) ∈ R × R 2 . Furthermore, let W = {x = (z, θ) ∈ 3 : |θ| ∞ = π}. Then a possible form of the set Ω = Ω 0 = Ω 0 ∪Ω 0 is shown in the left part of Fig. 2, where the compact set Ω 0 is represented by a red pyramid, while the unbounded set Ω 0 is just illustrated by its blue projection on the surface z = 0. The gray-colored area corresponds to the projection of the set {x ∈ R n : V (x) > 0} to the surface z = 0. This set also contains W and separates Ω 0 from Ω 0 . By applying periodicity and constructing the sets Ω j = Ω j ∪ Ω j , after their intersection we obtain the set of red pyramids shown in the right part of Fig. 2, which now form a kind of multivariable cell cover of the globally attracting and forward invariant set of (1).

B. Equivalent conditions and relaxations

Theorem 2 provides a very general result on application of the theory of Leonov functions. To illustrate and clarify the idea of the approach, equivalent conditions for a special case are developed below.

Corollary 1. Suppose that there exist C 1 functions W : R n → R and ϕ :

R n → R, functions α 1 , α 2 ∈ K ∞ , constants λ > 0 and a i ≥ a i > 0, c i ≥ c i ≥ 0, b i ≥ b i > 0, i = 1, . . . , q, such that for all x ∈ R n α 1 (|z|) ≤ W (x) ≤ α 2 (|z|), q i=1 a i θ 2 i -b i |θ i | + c i ≤ ϕ(x) ≤ q i=1 a i θ 2 i -b i |θ i | + c i , ∂V (x) ∂x f (x) + λV (x) ≤ 0,
where V (x) = W (x)ϕ(x) and x = (z, θ). Then X(t, x 0 ) of ( 1) is bounded for any x 0 ∈ R n and t ≥ 0 provided that

a min π + q i=1 c i π -1 < qb max < 2a min π + 0.5 q i=1 c i π -1 , q 2 a max π + q i=1 c i π -1 < b min < 2q 2 a max π + 0.5 q i=1 c i π -1 ,
where a min = min 1≤i≤q a i , b max = max 1≤i≤q b i , a max = max 1≤i≤q a i and b min = min 1≤i≤q b i .

Proof. The claim is established by applying Theorem 2 and Proposition 1. Note that ( 4) is satisfied by assumption. Moreover, under the standing hypothesis,

α 1 (|z|) -ϕ(x) ≤ V (x) ≤ α 2 (|z|) -ϕ(x).
Thus, by invoking Proposition 1, it suffices to show that there exist some σ 1 , σ 2 ∈ K and some 0 < g 1 ≤ g 2 ≤ σ 2 (π), such that

σ 2 (||θ| ∞ -π|) -g 2 ≤ ϕ(x) ≤ σ 1 (||θ| ∞ -π|) -g 1 . Note that q i=1 a i θ 2 i -b i |θ i | + c i ≤ q i=1 a max θ 2 i -b min |θ i | + c i = a max |θ| 2 + q i=1 -b min |θ i | + c i ≤ q 2 a max |θ| 2 ∞ -b min |θ| ∞ + q i=1 c i := f (|θ| ∞ ) and q i=1 a i θ 2 i -b i |θ i | + c i ≥ q i=1 a min θ 2 i -b max |θ i | + c i = a min |θ| 2 + q i=1 (-b max |θ i | + c i ) ≥ a min |θ| 2 ∞ -qb max |θ| ∞ + q i=1 c i := f (|θ| ∞ ).
Consequently,

f (|θ| ∞ ) ≤ ϕ(x) ≤ f (|θ| ∞ ).
Moreover, under the introduced restrictions the following properties are satisfied for f :

q i=1 c i = f (0) ≥ 0, a min π + q i=1 c i π -1 < qb max ⇒ f (π) < 0, 2a min π + 0.5 q i=1 c i π -1 > qb max ⇒ f (2π) > 0.
Furthermore, 2π 2 a min > q i=1 c i is a sufficient condition for existence of b max , which is satisfied due to the imposed assumptions. Similarly for f :

q i=1 c i = f (0) ≥ 0, q 2 a max π + q i=1 c i π -1 < b min ⇒ f (π) < 0, 2q 2 a max π + 0.5 q i=1 c i π -1 > b min ⇒ f (2π) > 0
and the inequality 2π 2 q 2 a max > q i=1 c i is satisfied by assumptions. From these properties of f and f we can conclude that the required functions σ 1 , σ 2 ∈ K and constants 0 < g 1 ≤ g 2 ≤ σ 2 (π) exist, completing the proof (for example, these functions can be selected in the form σ 1 (s) = 1 max{ 2 s, s 2 } and σ 2 (s) = 3 min{ 4 s, s 2 } for some i > 0 for i = 1, 2, 3, 4).

Another possibility to relax the conditions of Theorem 2 is based on the observation that, in the proof of the main result, the property (4) is used only for V (x) ≥ 0: Corollary 2. If for the system (1) under Assumption 1 there exists a Leonov function V : R n → R such that the inequality (4) is verified only for

x ∈ (R n \ Ω 0 ) ∩ (Z ∪ Ω ε ), where Ω ε := {x ∈ R n : V (x) ≤ ε} and Z = {x = (z, θ) ∈ R n :
|z| ≥ ζ} for some ε > 0 and ζ ∈ R + , then for all x 0 ∈ R n the corresponding trajectories X(t, x 0 ) are bounded for all t ≥ 0.

Proof. Note that the inequality (4) does not allow to make any conclusion about the behavior of the Leonov function V (x) while x ∈ Ω = Ω 0 . Hence, limiting the validity of (4) to the set R n \ Ω is not restrictive at all. Next, it is necessary to take into consideration the set C ⊂ R n \ (Ω ε ∪ Z). Clearly on this set (4) is not obligatorily satisfied. But since Ω ∩ C = ∅ for any ε > 0 it follows that the set Ω = Ω ∪ Ω preserves its property of forward invariance for (1) and local attractiveness established in Theorem 2. Note that in the present case only local attractiveness can be claimed for Ω since the inequality (4) is not necessarily satisfied on C for V (x) > 0. Therefore, the system behavior is not specified there and a trajectory may stay in C (or travel between isolated subsets of C). Since

C ⊂ R n \ Ω, then C ⊂ {x = (z, θ) ∈ R n : c ≤ |θ| ∞ ≤ c} for some c ∈ (0, π) and c ∈ (π, 2π
) by definition of the set Ω (see the proof of Theorem 2, with c < c ≤ c where c ∈ (π, 2π) is used in the definition of the sets Ω and Ω ). Consequently, taking into account the set Z we obtain that

C ⊂ {x = (z, θ) ∈ R n : |z| < ζ, c ≤ |θ| ∞ ≤ c},
which is a bounded set. Next we show that any trajectory, which enters in C or even travels between isolated subsets of C, cannot leave a bounded domain containing C. Indeed, if a trajectory reaches either the set with |θ| ∞ > c or |θ| ∞ < c, then it is in the set Ω, which is forward invariant and the behavior of X(t, x 0 ) in this set is analyzed later. Therefore, assume that |θ| ∞ stays bounded by c and c. Then from the definition of a Leonov function and its lower bound (α(|z|)ψ(|θ|)g ≤ V (x) for all x = (z, θ) ∈ R n and some g ≥ 0, α ∈ K ∞ , ψ ∈ K) and also due to the imposed assumptions on the domain of validity of (4), there is a finite constant Ξ ≥ max{ζ, α -1 (ψ(qc) + g)} such that

V (x) > 0, ∂V (x) ∂x f (x) ≤ -λ (V (x)) provided that x ∈ {x = (z, θ) ∈ R n : |z| ≥ Ξ, c ≤ |θ| ∞ ≤ c},
which implies upper boundedness of V (x) in this set and consequently boundedness of z. Therefore, the union Ω ∪ Ω ∪ C only contains forward invariant and globally attracting sets for (1). In addition, all these sets are isolated, and if a trajectory X(t, x 0 ) enters and remains in C, then this implies boundedness of the trajectory (since the set C is bounded). Hence, the set Ω = Ω ∪ Ω preserves its meaning established in Theorem 2. Consequently, repeating the arguments of the proof of Theorem 2 (with introduction of the sets Ω j = Ω j ∪ Ω j ) leads to the conclusion that the trajectories X(t, x 0 ) are bounded for all x 0 ∈ R n and all t ≥ 0.

By definition Ω ⊂ Ω ε and since Ω = Ω ∪ Ω , then Ω ε = Ω ε ∪ Ω ε , where Ω ⊂ Ω ε and Ω ⊂ Ω ε . Therefore, the sets Ω ε and Ω ε are separated for a sufficiently small ε > 0. We can further relax the conditions of Corollary 2 by requiring (4) to be verified only on one of the sets Ω ε or Ω ε instead of Ω ε , which may be important in an application. Corollary 3. Suppose that for the system (1) under Assumption 1 there exists a Leonov function V : R n → R such that the inequality ( 4) is verified for one of these options:

a) x ∈ (R n \ Ω 0 ) ∩ (Z ∪ Ω ε ) and sup θ∈R q ψ(|θ|) < +∞, b) x ∈ (R n \ Ω 0 ) ∩ (Z ∪ Ω ε ), where Ω ε = Ω ε ∪ Ω ε = {x ∈ R n : V (x) ≤ ε} and Z = {x = (z, θ) ∈ R n : |z| ≥ ζ} for some ε > 0 and ζ ∈ R + .
Then for all x 0 ∈ R n the corresponding trajectories X(t, x 0 ) are bounded for all t ≥ 0.

Proof. By repeating the arguments of Corollary 2 we can substantiate local attractiveness and forward invariance of the sets Ω or Ω for the cases a) and b), respectively. In these cases we have no conclusion about the behavior of the solutions X(t, x 0 ) in the sets C ⊂ R n \ (Ω ε ∪ Z) or C ⊂ R n \ (Ω ε ∪ Z), correspondingly, which by construction admit the following restrictions:

C ⊂ {x = (z, θ) ∈ R n : |z| < ζ, |θ| ∞ ≥ c}, C ⊂ {x = (z, θ) ∈ R n : |z| < ζ, |θ| ∞ ≤ c}
for some c ∈ (π, 2π). The set C is obviously bounded and sup |θ|∞≤c ψ(θ) ≤ ψ(qc). The set C is unbounded in θ, but due to the additional restriction imposed on ψ there is ρ > 0 such that sup |θ|∞≥c ψ(θ) ≤ ρ. Hence, from the definition of a Leonov function and the imposed assumptions there are finite constants Ξ ≥ max{ζ, α -1 (ρ + g)} and Ξ ≥ max{ζ, α -1 (ψ(qc) + g)} for the cases a) and b), respectively, such that

V (x) > 0, ∂V (x) ∂x f (x) ≤ -λ (V (x)) provided that x ∈ {x = (z, θ) ∈ R n : |z| ≥ Ξ , |θ| ∞ ≥ c}, respectively that x ∈ {x = (z, θ) ∈ R n : |z| ≥ Ξ , |θ| ∞ ≤ c}.
This implies upper boundedness of V (x) in both sets C and C , and consequently boundedness of z:

α(|z|) ≤ V (x) + ψ(|θ|) + g ≤ V (x) + max{ρ, ψ(qc)} + g.
Therefore, in both cases a) and b), the unions Ω ∪ C and Ω ∪ C contain forward invariant and globally attracting sets for [START_REF] Angeli | An almost global notion of input-to-state stability[END_REF]. Next, the desired conclusion follows by repeating the steps of Theorem 2 (replacing in the arguments Ω with C or Ω with C ).

V. A RELATION WITH INPUT-TO-STATE STABILITY

In addition to stability, robustness properties with respect to exogenous disturbances are also of great relevance in many control applications. For the class of periodic systems considered in the present paper, robustness in terms of the ISS property has been established in [START_REF] Efimov | A relaxed characterization of iss for periodic systems with multiple invariant sets[END_REF] via the notion of an ISS-Leonov function. However, in [START_REF] Efimov | A relaxed characterization of iss for periodic systems with multiple invariant sets[END_REF] it is assumed that the states evolve on a manifold M , while in the present paper it is assumed that they evolve in R n .

We next show that a Leonov function as proposed in the present paper is closely related to an ISS-Leonov function as introduced in [START_REF] Efimov | A relaxed characterization of iss for periodic systems with multiple invariant sets[END_REF] and, thus, with minor additional requirements, existence of a Leonov function also allows to establish the ISS property for (1). To this end, it is more convenient to characterize a Leonov function via the conditions ( 4) and ( 6) rather than the more generic conditions in Definition 1. Furthermore, we consider the following system with an external input:

ẋ(t) = F (x(t), d(t)), (7)
where x(t) ∈ R n and d(t) ∈ R m input d(•) is a locally essentially bounded and measurable signal) for t ≥ 0, f (x) = F (x, 0) and Assumption 1 is satisfied. In such a case any solution X(t, x 0 , d) of ( 7) for an initial condition x 0 ∈ R n and an input d(•) can be embedded in a manifold M = R k × S q by a 2π-projection of the part of the state θ.

Recall from [START_REF] Efimov | A relaxed characterization of iss for periodic systems with multiple invariant sets[END_REF] that for a set A ⊂ R n , a C 1 function V : R n → R is a practical ISS-Leonov function for [START_REF] Barabanov | Conditions for almost global attractivity of a synchronous generator connected to an infinite bus[END_REF] if there exist functions ϑ 1 , ϑ 2 , σ, γ ∈ K ∞ , a continuous function λ : R → R, λ ∈ K ∞ for nonnegative arguments, and scalars r ≥ 0, ν ≥ 0 such that for all x = (z, θ) ∈ R n and d

∈ R m ϑ 1 (|x| A ) -σ(|θ|) ≤ V(x) ≤ ϑ 2 (|x| A + ν), (8)
and the following dissipation inequality holds:

∂V(x) ∂x F (x, d) ≤ -λ (V(x)) + γ(|d|) + r. (9)
If the latter inequality holds for r = 0, then V is said to be an ISS-Leonov function.

Assumption 2. Let a set A ⊂ R n contain all αand ωlimit sets of (1) and its projection on M be compact and decomposable (in the sense of [START_REF] Angeli | Characterizations of input-to-state stability for systems with multiple invariant sets[END_REF]).

It has been shown in [START_REF] Efimov | A relaxed characterization of iss for periodic systems with multiple invariant sets[END_REF] that under the hypothesis given in Assumption 2, the existence of a practical ISS-Leonov function is necessary and sufficient for ISS of ( 7) with respect to the set A. Then the following result can be formulated for a Leonov function.

Proposition 2. Consider a Leonov function V : R n → R satisfying the restrictions of Proposition 1. Let for V(x) = V (x) + τ with some τ ≥ 0, and assume that (4) can be strengthened to (9) with a radially unbounded function λ for positive arguments, and that the inequalities

ϑ 1 (|x| A ) -σ(|θ|) ≤ α 1 (|z|) -σ 1 (||θ| ∞ -π|) + g 1 + τ, ( 10
) α 2 (|z|) -σ 2 (||θ| ∞ -π|) + g 2 + τ ≤ ϑ 2 (|x| A + ν) (11)
are verified for all x = (z, θ) ∈ R n . Then V is a practical ISS-Leonov function for [START_REF] Barabanov | Conditions for almost global attractivity of a synchronous generator connected to an infinite bus[END_REF] and a set A ⊂ R n .

Proof. The result can be deduced by a straightforward application of Definition 1 and the restrictions given in propositions 1 and the current one.

Note that if {x = (z, θ) ∈ R n : z = 0} ⊂ A then there is τ ≥ 0 such that (10) is verified, and by taking ν > 0 large enough the fulfillment of the inequality [START_REF] De | Bregman storage functions for microgrid control[END_REF] can also be guaranteed.

Corollary 4. Let assumptions 1 and 2 be satisfied together with the hypothesis of Proposition 2. Then the system ( 7) is ISS with respect to the set A.

Proof. The result is a direct consequence of Proposition 2 and the main result of [START_REF] Efimov | A relaxed characterization of iss for periodic systems with multiple invariant sets[END_REF].

It is important to stress here that the ISS property is obtained on the manifold M only (when boundedness of the variable θ comes directly from the structure of M ). The fact that V is also a Leonov function as introduced in Definition 1 implies, via the claim of Theorem 2, that for d(t) ≡ 0 the solutions of (1) are bounded in R n .

VI. AN ACADEMIC EXAMPLE

We demonstrate the usefulness of the proposed approach by performing a global analysis of boundedness of solutions of the following system:

ż = F z + R sin(θ), (12)
θ = Dz + H sin(θ), where x = (z, θ) ∈ R n is the state, and F ∈ R k×k , R ∈ R k×q , D ∈ R q×k and H ∈ R q×q are constant matrices. Note that the system has an equilibrium at the origin.

Let the symbol diag(b) denote the diagonal matrix with a vector b ∈ R q + on the main diagonal. Then impose the following restrictions on the dynamics of (12): Assumption 3. There exist matrices P = P ∈ R k×k , Q = Q ∈ R k×k and a vector m ∈ R q + such that the following linear matrix inequalities are satisfied:

F P + P F = -Q, P > 0, Q > 0, 2P R = -D diag(m), diag(m)H + H diag(m) < 0.
Since Assumption 3 implies that the matrix F is stable, it is straightforward to establish boundedness of z ∈ R k . The result below demonstrates how the approach derived in Corollary 2 can be used to also infer global boundedness of the variables θ ∈ R q . Proposition 3. Let Assumption 3 be satisfied for the system [START_REF] Dörfler | Synchronization and transient stability in power networks and non-uniform Kuramoto oscillators[END_REF]. Then for all x 0 ∈ R n the solutions X(t, x 0 ) are bounded for all t ≥ 0.

Proof. Under the made hypothesis let us consider the function Note that under Assumption 3 there is a value of λ sufficiently small such that Q > λP , then ζ > 0 exists as needed (the functions υ i (θ i ) are globally bounded). The property

V (x) = z P z + q i=1 m i (1 -cos(θ i )) -c,
υ i (θ i ) > 0 ( 13
)
Figure 3. The levels of the function V (0, θ) for the system [START_REF] Dörfler | Synchronization and transient stability in power networks and non-uniform Kuramoto oscillators[END_REF] is valid for a small vicinity of the point θ i = π (the negative term -λ min (H) sin 2 (θ i ) of ( 13) equals zero for θ i = 0 or θ i = π, and their shifts on 2π, but the former is in Ω and only the latter point belongs to R n \ Ω by construction above). Furthermore, there also exists a sufficiently small λ > 0 such that this vicinity has no intersection with Ω. Indeed, λ[m i (1cos(θ i ))q -1 c] ≤ λ[2m iq -1 c] for all θ i ∈ [-π, π).

In addition, since λ min (H) > 0 by Assumption 3, then the inequality (13) can only be satisfied for θ i ∈ (πρ i , π + ρ i ) with ρ i = arcsin λ 2mi-q -1 c λmin(H) , whose value can be selected as small as required by decreasing λ, i.e. ε > 0 exists. The union of these restrictions means that the inequality (4) is not valid for πmax 1≤i≤q ρ i ≤ |θ| ∞ ≤ π + max 1≤i≤q ρ i and |z| < ζ with some ζ > 0 (for a sufficiently small λ). However, all conditions of Corollary 2 are satisfied. Then all conditions of Assumption 3 are satisfied for λ = 0.1.

The levels of the function V (x) for z = 0 and θ ∈ [-2π, 2π] 2 are presented in Fig. 3, the levels of the function s(θ) = λ[ q i=1 m i (1cos(θ i ))c]sin (θ)H sin(θ) corresponding to verification of (13) are shown in Fig. 4, where comparing the zones of positivity for V (0, θ) and s(θ) it is possible to check that the inequality (4) is verified for x ∈ (R n \ Ω) ∩ (Z ∪ Ω ε ) with some ε > 0. Illustrative simulation results for this system are given in Fig. 5. Clearly, the simulations confirm the efficiency of the proposed theory.

VII. CONCLUSIONS

By extending ideas of [START_REF] Leonov | On the boundedness of the trajectories of phase systems[END_REF], we have proposed a multivariable cell structure approach, which provides sufficient conditions to establish boundedness of trajectories of nonlinear Figure 5. Simulation results for the system (12) for several arbitrarily chosen initial conditions systems, the dynamics of which are periodic with respect to a part of the state vector. Key to the approach is the concept of a Leonov function. A Leonov function is in general signindefinite and not continuously differentiable on the manifold, on which the system dynamics can be projected. These characteristics represent significant relaxations compared to the usual requirements on a standard Lyapunov function [START_REF] Angeli | Characterizations of input-to-state stability for systems with multiple invariant sets[END_REF]. We have shown that for a class of periodical dynamical systems the existence of a Leonov function implies boundedness of all trajectories. Furthermore, we have shown that under (mild) additional conditions a Leonov function is also an ISS-Leonov function (as defined in [START_REF] Efimov | A relaxed characterization of iss for periodic systems with multiple invariant sets[END_REF]) and thus its existence can also guarantee robustness with respect to exogenous disturbances. The proposed approach is illustrated by providing a global analysis of a nonlinear system. Application of the proposed theory to power systems and microgrids (in particular, to the swing equation) is currently under investigation.

Figure 1 .

 1 Figure 1. Illustration of the sets Ω j for j ∈ {-1, 0, 1}

4 Figure 2 .

 42 Figure 2. Structure of the set Ω

m i ( 1 -

 1 where m i is the i th entry of the vector m and let us check the conditions of Corollary 2 for a Leonov function candidate V (x). Since 1cos(θ) ≥ 0, select α(s) = λ min (P )s 2 (here λ min (P ) denotes the smallest eigenvalue of the matrix P ), ψ as any function from class K and g = c. Then the lower bound on V (x) is verified for all x ∈ R n . Since cos(0) = 1 and it is a 2π-periodic function, the condition sup x∈U V (x) ≤ 0 is satisfied for any c ≥ 0. Select W = {x = (z, θ) ∈ R n : |θ| ∞ = p} for some p ∈ (π, 2π) and 0 ≤ c < (1cos(p)) inf 1≤i≤q m i . Then inf x∈W V (x) > 0, and all required in (3) properties hold.It remains to check the derivative of the Leonov function V (x) and the following upper estimate:∂V (x) ∂x f (x) ≤ -λV (x) ∀x ∈ (R n \ Ω 0 ) ∩ (Z ∪ Ω ε ),whereΩ ε = {x ∈ R n : V (x) ≤ ε} and Z = {x = (z, θ) ∈ R n : |z| ≥ ζ} for some λ > 0, ε > 0 and ζ > 0. Lengthy but straightforward calculations show that ∂V (x) ∂x f (x) + λV (x) ≤ -z (Q -λP )zsin (θ)H sin(θ) cos(θ i ))c] ≤ -z (Q -λP )z + q i=1 υ i (θ i ),where H = -0.5(diag(m)H + H diag(m)) > 0, υ i (θ i ) = λm i (1cos(θ i ))λ min (H) sin 2 (θ i )λq -1 c.

Example 3 .

 3 Select k = 1, q = 2 and F = -0.1, P = 1, Q = 0.2, R = -[0.2 0.05], D = [1 0.5] , m = [0.4 0.2] , c = 0.2, and

Figure 4 .

 4 Figure 4. The levels of the function s(θ) for the system (12)

This work was supported in part by the Government of Russian Federation (Grant 08-08) and the Ministry of Education and Science of Russian Federation (Project 14.Z50.31.0031).

1 sign-definite if V (0) = 0 and V (x) > \ {0}; and it is called sign-indefinite if V (x) takes both, positive and negative, values.