State-space fading memory

State-space fading memory Gustav e Bainier a, ∗ , Antoine Chaillet b , Rodolphe Sepulchre c,d , Alessio Franci a,e, ∗ a Dept. of Electrical Engineering and Computer Science, University of Liège , 10 allée de la Découverte, Liège, 4000, Belgium b Université P aris-Saclay , CNRS, CentraleSupélec, Labor atoir e des signaux et systèmes, Gif-sur-Yvette, 91190, F rance c KU Leuven, Department of Electrical Engineering (ESA T), ST ADIUS Center for Dynamical Systems, Signal Pr ocessing and Data Analytics, KasteelP ark Ar enberg 10, Leuven, 3001, Belgium d Department of Engineering, University of Cambridge , T rumpington Str eet, Cambridge, CB2 1PZ, United Kingdom e WEL Resear ch Institute, W avr e, 1300, Belgium Abstract The fading-memory (FM) property captures the progressi ve loss of influence of past inputs on a system’ s current output and has originally been formalized by Bo yd and Chua in an operator-theoretic framework. Despite its importance for systems approxima- tion, reservoir computing, and recurrent neural networks, its connection with state-space notions of nonlinear stability , especially incremental ones, remains understudied. This paper introduces a state-space definition of FM. In state-space, FM can be interpreted as an extension of incremental input-to-output stability ( δ IOS) that explicitly incorporates a memory kernel upper-bounding the decay of past input di ff erences. It is also closely related to Boyd and Chua’ s FM definition, with the sole di ff erence of requiring uniform, instead of general, continuity of the memory functional with respect to an input-fading norm. W e demonstrate that incre- mental input-to-state stability ( δ ISS) implies FM semi-globally for time-in variant systems under an equibounded input assumption. Notably , Boyd and Chua’ s approximation theorems apply to delta-ISS state-space models. As a closing application, we show that, under mild assumptions, the state-space model of current-driv en memristors possess the FM property . K e ywor ds: Fading Memory, Incremental Input-to-State Stability, Incremental Input-to-Output Stability 1. Introduction 1.1. F ading memory and the repr esentation of stable systems Fading-Memory (FM), introduced by Stephen P . Boyd and Leon O. Chua, is an incremental stability property of input- output (I / O) systems, where the current outputs depend on past inputs, but the influence of older inputs gradually diminishes. Boyd and Chua showed that a large class of FM operators can be approximated arbitrarily well by a linear time-inv ariant state- space model followed by a nonlinear static readout: ˙ x ( t ) = A x ( t ) + Bu ( t ) , (1a) y ( t ) = g ( x ( t )) , (1b) with u and y the input and output signals respectively . The state matrix A is Hurwitz and the static readout g is smooth, ensuring both internal and e xternal stability [1]. This result e ff ecti vely provides a canonical representation of FM operators. Ho wev er , it applies in the operator -theoretic context and does not directly translate to nonlinear state-space systems theory . For linear systems, the situation is simpler: Boyd and Chua hav e established a con volution theorem, stating that Linear T ime-In v ariant (L TI) operators hav e FM if and only if they ad- mit a conv olution representation [1]. This makes FM an al- most automatic property of L TI systems in engineering prac- tice. L TI systems with FM enjo y well-kno wn I / O properties ∗ Corresponding author Email addr esses: gustave.bainier@uliege.be (Gustav e Bainier), afranci@uliege.be (Alessio Franci) such as Bounded-Input Bounded-Output (BIBO), Con ver ging- Input Con verging-Output (CICO), and Periodic-Input Periodic- Output (PIPO). In contrast, for nonlinear systems, the notions of represen- tation, internal stability , and external stability are largely de- coupled. For state-space nonlinear systems, classical asymp- totic stability concerns only the internal dynamics (con vergence to an equilibrium in the absence of input) and does not au- tomatically imply well-behaved I / O behaviors, nor vice-v ersa. Stronger stability notions are therefore required. One such a notion is incremental input-to-state stability ( δ ISS), which im- plies internal asymptotic stability , and I / O properties analogous to linear systems: namely , BIBO, CICO and PIPO [2, 3, 4]. Moreov er , δ ISS is preserv ed under I / O interconnections satisfy- ing a small-gain condition, making it particularly po werful for analyzing lar ge-scale nonlinear systems [2]. Howe ver , unlike FM systems, δ ISS does not readily provide a canonical repre- sentation theorem similar to Boyd and Chua’ s approximation result. A state-space stability notion that recovers Boyd and Chua’ s FM is needed in order to apply their approximation result to nonlinear state-space models. This paper introduces this miss- ing state-space FM notion. Rather unsurprisingly , state-space FM closely relates to the notion of δ ISS. W e show that, under equibounded inputs, δ ISS implies the state-space FM property semi-globally , o ff ering a simple pathway to generalize Boyd and Chua’ s representation result to the state-space setting. Our broader objective is to establish state-space FM as a nonlinear analog of the stable con volution property in L TI system theory . 1.2. Context of the study Giv en the widespread nature of the FM property , sev eral at- tempts ha ve been made to establish its abstract and rigorous formalization. T o the best of the authors’ knowledge, the first rigorous formulation of the FM concept was introduced by Bernard D. Coleman in a series of articles focused on appli- cations to continuum mechanics, specifically the viscoelastic and rheological beha vior of polymers [5, 6, 7]. While Cole- man’ s work had a significant impact on continuum mechan- ics and thermodynamics [8, 9], the concept of FM remained largely unnoticed in the control and signal-processing commu- nities, until it was re-discovered and re-formalized in 1985 by Boyd and Chua for I / O operators [1]. While Boyd and Chua’ s work focused on time-in v ariant operators and their approxima- tion, one of its long-standing corollary contribution is the con- volution theorem for L TI operators, as mentioned earlier in this introduction [10]. Although this early formalization of FM continues to be one of the most widely recognized, a series of alternativ e def- initions were also de veloped. The first of these appears in the work of Je ff S. Shamma, who characterized FM in terms of finite-memory oper ators [11], and subsequently introduced with Rongze Zhao a distinction between uniform and pointwise FM [12]. A drawback of the operator-theoretic definitions of FM is that the y typically require inputs to be defined over the entire negati ve time axis, infinitely into the past. This limitation motiv ated Irwin W . Sandberg to reformulate FM so that it ap- plies to operators defined on both discrete-time and continuous- time positiv e domains N and [0 , + ∞ ) [13, 14, 15]. Research e ff orts hav e also been dedicated to study operators with the FM property . Notable contributions include the ex- amination of linear FM operators [16], system identification of FM systems [17], and work clarifying the relationship be- tween FM and the conv entional concept of asymptotic stability for discrete-time systems [18]. T wo major research directions related to FM operators have emerged: the repr esentation ques- tion, i.e. which classes of systems can approximate any FM operator with arbitrary precision; and the universality question, i.e. what can the class of FM operators represent. Crucially , these questions helped to clarify key properties and limitations of sev eral machine learning architectures. In Boyd and Chua’ s original approximation theorem [1], it is showed, by virtue of the Stone-W eierstrass theorem, that (under mild assumptions) all time-in variant FM operators can be approximated by a finite V olterra series and by a state-space L TI system follo wed by a static nonlinear readout. Further results have established that discrete-time FM systems can be approximated by Recurrent Neural Networks (RNNs) [19], and bounds on the computa- tional capacity of discrete-time systems with the FM property hav e also been derived [20]. A clear articulation of FM’ s rela- tionship to related concepts in RNNs can be found in this re- cent study [21]. The FM property has also been in vestigated in the context of reservoir computing. Theoretically , reser- voir systems with FM have been established as univ ersal [22], though subsequent analysis refined this view , asserting that the FM property was not strictly necessary for reserv oir systems to achiev e uni versality [23]. Despite the recent dev elopment of a kernel-based definition of FM [24], the operator -theoretic notion of FM has attracted little attention from control theorists [25]. The control commu- nity has instead focused its interests on state-space definitions of FM [26]. The FM state-space notion that is the closest to its operator theoretic counterpart is input-to-state stability (ISS), a property which imposes that the state of the system is asymp- totically attracted to an equilibrium, up to a bounded error re- lated to the largest past input value [27]. In ISS systems, in- puts that con verge to zero will typically still steer the system tow ard its equilibrium, as past input v alues are progressi vely forgotten over time. This explicit FM property of ISS sys- tems was first introduced in [28] using the notion of e xp-ISS, and w as later generalized to input-to-state dynamical stability (ISDS) [29]. Ho wev er , the exact link between these notions and Boyd and Chua’ s original FM definition remains elusiv e. In particular , neither exp-ISS nor ISDS captures the inherently incr emental nature of Boyd and Chua’ s FM, which is character- ized by how mismatches in past inputs propagate to mismatches in current outputs. W e argue that such incremental state-space notion is missing from the literature, and that incremental input- to-output stability ( δ IOS) and δ ISS are much more natural can- didates to obtain a state-space equiv alent of Boyd and Chua’ s FM [30]. 1.3. Contributions After revie wing some preliminary definitions in Section 2, the new state-space definition of FM is proposed in Section 3. This ne w definition is sho wn to be closely related to both Bo yd and Chua’ s original operator-theoretic definition, and to the δ IOS property of state-space models. It is in fact the δ IOS prop- erty with the addition of an explicit memory kernel leading to the decay of past input values. The new definition is first mo- tiv ated through the example of a memristor in Section 3.1, and then properly introduced in Section 3.2. Some elementary re- sults are derived from the existence of the memory kernel in Section 3.3. The connection between our definition and Boyd and Chua’ s definition in terms of uniform and general continuity is pre- sented in Section 4. Is the introduced state-space FM equiv alent to the δ ISS prop- erty , and can Boyd and Chua’ s approximation theorem be ap- plied to δ ISS systems? Section 5 is dedicated to answering these two questions. After introducing the δ ISS property in Section 5.1, it is sho wed that δ ISS implies input-to-state FM on all compact sets of initial conditions for time-in variant systems with equibounded inputs in Section 5.2 (thus, the input-to-state FM holds semi-globally). Some settings where δ ISS does not imply input-to-state FM are discussed in Appendix A. The rep- resentation theorem obtained by applying Boyd and Chua’ s ap- proximation results to δ ISS state-space models is briefly stated in Section 5.3. Finally , we also exhibit simple conditions for the state-space model of a current-driven memristor to hav e FM in Section 6. 2 Namely , that the current remains bounded, that the internal dy- namic of the resistance is δ ISS, and that the memristance value is Lipschitz-continuous with respect to its internal state. Section 7 briefly concludes the paper and o ff ers se veral per- spectiv es for future works. 2. Notations A function α : [0 , + ∞ ) → [0 , + ∞ ) is of class K if it is con- tinuous, strictly increasing, and α (0) = 0. Moreover , it is of class K ∞ is it is of class K and lim r →∞ α ( r ) = + ∞ . A function β : [0 , + ∞ ) × [0 , + ∞ ) → [0 , + ∞ ) is of class K L if it is continuous, and such that for each s ∈ [0 , + ∞ ), β ( · , s ) ∈ K , and for each r ∈ [0 , + ∞ ), β ( r , · ) is nonincreasing on [0 , + ∞ ) with lim s → + ∞ β ( r , s ) = 0. 3. FM for state-space models 3.1. Motivating example: the curr ent-driven memristor A memristor is a two-terminal electronic de vice whose re- sistance, or memristance, M > 0 varies over time, typically depending on the history of current I that has flowed through it, pro viding its beha vior with a built-in memory of past ac- tivity [31]. For a memristor , Ohm’ s law is typically gi ven in state-space representation and reads: U ( t ) = M ( x ( t )) I ( t ) , (2a) ˙ x ( t ) = f ( x ( t ) , I ( t )) . (2b) Equation (2a) resembles the standard U = R I relationship of a Ohmic resistor, except that the resistance R is replaced by M , a memristance function, which depends on the internal state x of the memristor . The memory e ff ect of the memristance arises from the dynamics of the internal state variable (2b). In many memristor models the influence of past acti vity gradually di- minishes, and the memristor serves as a canonical e xample of a FM system [32]. While originally , Boyd an Chua defined FM through the con- tinuity of the system operator with respect to a fading norm [1], our formalization highlights the system’ s retained memory by relying on its input-to-output contraction properties. T o gain some intuition, consider two identical memristors dri ven by two input currents I a and I b that di ff er before some time t ∗ but become identical thereafter . Because the two de vices retain a memory of past inputs, their corresponding voltages U a and U b still di ff er for t ≥ t ∗ . Ho wev er , as time progresses, the influence of the earlier discrepancies between I a and I b fades away , and the two v oltages gradually con ver ge tow ard each other at a rate characterizing the temporal scale of the system’ s memory . The paper proposes this systematic output contraction in response to input contraction as an alternati v e way to formalize FM. This definition will ultimately be shown to be closely related to that of Bo yd and Chua, thus enabling the use of their original ap- proximation theorem [1]. In particular, if the memristor has FM, then its model can can be approximated by a state-space L TI system with a static nonlinear readout, e ff ectiv ely remov- ing the nonlinearity from the dynamical equation (2b) of the original state-space model. Practical conditions for this FM property to hold on the memristor are gi ven at the end of the paper (Section 6). 3.2. Definition Consider the nonlinear state-space model ˙ x ( t ) = f ( x ( t ) , u ( t )) , (3a) y ( t ) = g ( x ( t )) , (3b) where x ( t ) ∈ R n x is the state of the system, u ∈ U + ⊂ L ∞ ([0 , + ∞ ) , R n u ) is its input (the + superscript in U + denotes the positiv e-time domain [0 , + ∞ )), and y ( t ) ∈ R n y is its output. It is assumed that (3a) is forw ard complete, meaning for all ini- tial condition x (0) ∈ X 0 ⊆ R n x and input u ∈ U + , the state trajectory x is uniquely defined by (3a) on the maximal interval of existence t ∈ [0 , + ∞ ), and moreover that g is continuous so y ∈ L ∞ ([0 , + ∞ ) , R n y ) for all u ∈ U + . Intuitiv ely , we propose that (3) has the FM property if: 1. The system is incrementally asymptotically output stable for all input signal u ∈ U + . This means that for any mis- match of initial conditions ∥ x a (0) − x b (0) ∥ , the trajecto- ries of the two outputs, y a and y b , considered with a com- mon input signal u ∈ U + , con verge toward each other uni- formly across all initial conditions. 2. The system is incrementally input-to-output stable (i.e. δ IOS). This means that for two distinct input signals u a and u b ∈ U + , y a and y b still con verge to each other, b ut only up to a mismatch that scales with the maximum past mismatch between u a and u b . 3. The system’ s output gradually forgets its past inputs ac- cording to a uniform memory mechanism. This means that the mismatch between y a and y b is computed by weight- ing past mismatches between u a and u b according to their temporal distance, with respect to a system-specific fad- ing memory k ernel . In particular , if u a and u b con ver ge asymptotically to one another , then so do y a and y b . Although these three properties were presented sequentially , it should be noted that 3) implies 2) which in turn implies 1). Con- sequently , a formalization of 3) alone would o ff er a complete characterization of FM. The memory kernel described in point 3) is defined by a function w : [0 , + ∞ ) → [0 , 1], with w ( ∆ t ) the decay weight associated to the input mismatch between u a and u b after a time ∆ t ≥ 0. T o ensure that the system exhibits FM, the discrepancies in the distant past must have progressively less influence on the present output behavior , as formalized in the following definition. Definition 1 (Memory kernel) . A function w : [0 , + ∞ ) → [0 , 1] is a memory kernel if it is continuous, nonincr easing, and satis- fies lim ∆ t → + ∞ w ( ∆ t ) = 0 . (4) The formal definition of our state-space FM property is now introduced: 3 Definition 2 (State-space FM) . The system (3) has w-FM if ther e exists β ∈ K L , γ ∈ K ∞ , and a memory kernel w satisfying Definition 1 such that for any two initial condi- tions, x a (0) , x b (0) ∈ X 0 , consider ed with two input trajectories, u a , u b ∈ U + , the corr esponding outputs satisfy for all t ≥ 0 : ∥ y a ( t ) − y b ( t ) ∥ ≤ β  ∥ x a (0) − x b (0) ∥ , t  (5) + γ  ess sup s ∈ [0 , t ] w ( t − s ) ∥ u a ( s ) − u b ( s ) ∥  . Remark 1. A qualitatively equivalent definition of FM is given by: ∥ y a ( t ) − y b ( t ) ∥ ≤ max  ˜ β  ∥ x a (0) − x b (0) ∥ , t  , (6) ˜ γ  ess sup s ∈ [0 , t ] w ( t − s ) ∥ u a ( s ) − u b ( s ) ∥   . Succinctly , (5) implies (6) with ( ˜ β, ˜ γ ) = (2 β, 2 γ ) , and (6) implies (5) with ( β, γ ) = ( ˜ β, ˜ γ ) . This is essentially a δ IOS property with the addition of a input-fading mechanism. Definition 3 (Input-to-state FM) . Succinctly , we define input- to-state FM as our state-space notion of FM with r espect to the output y = x . This is essentially a δ ISS property with the addition of a input-fading mechanism. 3.3. Elementary results using the memory kernel Similarly to most comparison functions in control, the mem- ory kernel w ( ∆ t ) provides an upper-bound on the influence of past inputs on the current output. It is therefore not uniquely defined. A memory kernel satisfying inequality (5) o ff ers an a priori conservative characterization of the system’ s memory constraints: the system may , in fact, be more forgetful than the memory kernel suggests. This ensures that the continuity and nonincreasing requirements of Definition 1 are obtained with- out loss of generality . This idea is formalized in the following proposition and corollary: Proposition 1 ( w 1 ≤ w 2 ⇒ ( w 1 -FM ⇒ w 2 -FM)) . Let the system (3) have w 1 -FM. The system (3) also has w 2 -FM for any mem- ory kernel w 2 : [0 , + ∞ ) → [0 , 1] satisfying lim ∆ t → + ∞ w 2 ( ∆ t ) = 0 and such that for all ∆ t ≥ 0 , w 1 ( ∆ t ) ≤ w 2 ( ∆ t ) . Pr oof. The proof is immediate by upper-bounding (5) in Defi- nition 2. Corollary 1. If (3) has w 1 -FM, with w 1 : [0 , + ∞ ) → [0 , 1] only satisfying condition (4) , then ther e exists a memory kernel w 2 satisfying Definition 1 such that the system (3) has w 2 -FM. Pr oof. From Proposition 1, it is su ffi cient to show that for all w 1 : [0 , + ∞ ) → [0 , 1] such that w 1 ( x ) → + ∞ 0, there e xists a continuous and nonincreasing w 2 : [0 , + ∞ ) → [0 , 1] such that w 2 ( x ) → + ∞ 0 and w 2 ≥ w 1 pointwise. This is, in fact, a usual result for comparison functions, see Lemma 2 of [33]. One can define ˜ w ( x ) ≜ sup y ≥ x w 1 ( x ) and obtain such w 2 from ˜ w through standard regularization. A common choice for the memory kernel is the exponentially decaying function w ( ∆ t ) = e − µ ∆ t , which arises naturally in ex- ponentially stable linear systems, such as the standard low-pass filter τ ˙ y ( t ) = − y ( t ) + u ( t ), for which the following FM inequality holds: ∥ ∆ y ( t ) ∥ ≤ e − t /τ ∥ ∆ y (0) ∥ + 2 ess sup s ∈ [0 , t ] e − ( t − s ) / 2 τ ∥ ∆ u ( s ) ∥ . (7) Similar to the properties of δ ISS systems (see Proposition 4.4 and Proposition 4.5 in [2]), for FM systems, con ver ging inputs lead to conv erging outputs (CICO), and periodic inputs lead to periodic outputs (PIPO). The first property is, ho we ver , more general than that of δ ISS systems, since it solely relies on the memory kernel and do not require specific state-space proper- ties (such as time-in v ariant or delay-free dynamics). Proposition 2 ( ∆ u ( t ) → 0 ⇒ ∆ y ( t ) → 0) . If the FM inequal- ity (5) stands and if u a and u b con ver ge asymptotically to one another , then y a and y b do as well. Pr oof. Let u a , u b ∈ U + be such that ∥ u a ( t ) − u b ( t ) ∥ → + ∞ 0. Since the input signals are in L ∞ , there exists M > 0 such that ess sup t ≥ 0 ∥ u a ( t ) − u b ( t ) ∥ < M . Moreover , for concision, let us introduce h ( t , s ) ≜ w ( t − s ) ∥ u a ( s ) − u b ( s ) ∥ . It is su ffi cient to show that lim t → + ∞ ess sup s ∈ [0 , t ] h ( t , s ) = 0 . (8) Let ε > 0. Since w ( t ) → + ∞ 0, there exists t 1 > 0 such that for all t ≥ t 1 , w ( t ) ≤ ε/ 2 M . Similarly , since ∥ u a ( t ) − u b ( t ) ∥ → + ∞ 0, there exists t 2 > 0 such that for all t ≥ t 2 , ∥ u a ( t ) − u b ( t ) ∥ ≤ ε/ 2. The following inequality holds: ess sup s ∈ [0 , t ] h ( t , s ) ≤ ess sup s ∈ [0 , t / 2] h ( t , s ) (9) + ess sup s ∈ [ t / 2 , t ] h ( t , s ) . On one hand, for all t ≥ 2 t 1 : ess sup s ∈ [0 , t / 2] h ( t , s ) ≤ ess sup s ∈ [0 , t / 2] w ( t − s ) M ≤ ε 2 . (10) On the other hand, for all t ≥ 2 t 2 : ess sup s ∈ [ t / 2 , t ] h ( t , s ) ≤ ess sup s ∈ [ t / 2 , t ] ∥ u a ( s ) − u b ( s ) ∥ ≤ ε 2 . (11) Overall, if t ≥ 2 max( t 1 , t 2 ), then ess sup s ∈ [0 , t ] h ( t , s ) ≤ ε , which demonstrates (8). Proposition 3 (Entrainment by periodic input) . Let the sys- tem (3) have be bounded-input bounded-state (BIBS) and have FM properties. If the input u is T -periodic, then the output y con ver ges to a T -periodic trajectory , independently of the ini- tial conditions. Pr oof. Let u ∈ U + . Since the input signal is in L ∞ , there exists M u > 0 such that ess sup t ≥ 0 ∥ u ( t ) ∥ < M u . Now consider the trajectory x of initial condition x (0) ∈ X 0 . The BIBO property implies that there e xists M x > 0 and t ∗ > 0 such that for all t ≥ t ∗ , ∥ x ( t ) ∥ < M x . Let t 1 = t ∗ + k 1 T + δ and t 2 = t ∗ + k 2 T + δ with k 1 , k 2 ∈ N and δ ∈ [0 , T ). Let us assume without loss of 4 generality that k 1 ≤ k 2 . An immediate application of the FM inequality (5) to the trajectories obtained with:        u a , 1 ( · ) ≜ u ( · + t ∗ ) x a , 1 (0) ≜ x ( t ∗ ) (12a) and: (12b)        u b , 1 ( · ) ≜ u ( · + t ∗ + ( k 2 − k 1 ) T ) [ = u ( · + t ∗ )] x b , 1 (0) ≜ x ( t ∗ + ( k 2 − k 1 ) T ) (12c) yields: sup δ ∈ [0 , T ) ∥ y ( t ∗ + k 1 T + δ ) − y ( t ∗ + k 2 T + δ ) ∥ (13) ≤ sup δ ∈ [0 , T ) β  ∥ x ( t ∗ ) − x ( t ∗ + ( k 2 − k 1 ) T ) ∥ , k 1 T + δ  + γ  ess sup s ∈ [ t ∗ , t 1 ] w ( t 1 − s ) ∥ u ( s ) − u ( s + ( k 2 − k 1 ) T ) ∥ | {z } = 0  ≤ β  2 M x , k 1 T  → k 1 → + ∞ 0 . (14) Therefore, gi ven that the output y ∈ L ∞ ([0 , + ∞ ) , R n y ), the se- quence ( y ( t ∗ + k T + · ) ) k ∈ N is a Cauchy sequence in the com- plete metric space L ∞ ([0 , T ) , R n y ), so it con ver ges to y ∗ ∈ L ∞ ([0 , T ) , R n y ). Now , let us show the uniqueness of y ∗ with respect to the initial condition by considering the trajectories associated with x a , 2 (0) and x b , 2 (0) ∈ X 0 . Again, the FM in- equality (5) yields for all k ∈ N : sup δ ∈ [0 , T ) ∥ y a , 2 ( t ∗ + k T + δ ) − y b , 2 ( t ∗ + k T + δ ) ∥ (15) ≤ β  ∥ x a , 2 ( t ∗ ) − x b , 2 ( t ∗ ) ∥ , k T  → k → + ∞ 0 . This shows that all output sequences ( y ( t ∗ + k T + · ) ) k ∈ N con- ver ge to the same limit y ∗ , regardless of the initial condi- tions. Finally , considering the T -periodic function y p ∈ L ∞ ([0 , + ∞ ) , R n y ) defined by y p ( t ) ≜ y ∗ ( t − ⌊ t / T ⌋ T ), this implies lim t → + ∞ ∥ y ( t ∗ + t ) − y p ( t ) ∥ = 0. Remark 2. If the system has input-to-state FM (Definition 3), Pr oposition 2 and Pr oposition 3 are exactly Proposition 4.4 and Pr oposition 4.5 of [2]. An inter esting cor ollary of Pr oposition 3 in this case is that, for constant inputs, all solutions con verge to an equilibrium. W e conclude this section with an evident, yet compelling, application of the memory kernel: it enables the deri v ation of explicit time-varying constraints on ∥ ∆ u ( t ) ∥ to achie ve a pre- scribed ∥ ∆ y ( t ) ∥ at a given time horizon t ∗ . This slightly re- fines the ultimate-bound property of δ IOS by replacing the static bounds on past perturbations with less conserv ati ve time- varying ones. Proposition 4. Let t ∗ , r > 0 , and let the system (3) have w-FM with w > 0 pointwise. If two inputs u a , u b ∈ U + satisfy the following constraint: ∥ u a ( t ) − u b ( t ) ∥ ≤        γ − 1 ( r ) / w ( t ∗ − t ) if t ∈ [0 , t ∗ ] γ − 1 ( r ) if t > t ∗ (16) then the mismatch between y a and y b at time t ≥ t ∗ satisfies: ∥ y a ( t ) − y b ( t ) ∥ ≤ β  ∥ x a (0) − x b (0) ∥ , t  + r . (17) Pr oof. Again, the proof is immediate by upper-bounding (5) in Definition 2, with t ≥ t ∗ . Succinctly: ess sup s ∈ [0 , t ] w ( t − s ) ∥ ∆ u ( s ) ∥ (18) ≤ max  ess sup s ∈ [0 , t ∗ ] w ( t − s ) ∥ ∆ u ( s ) ∥ , ess sup s ∈ [ t ∗ , t ] ∥ ∆ u ( s ) ∥  ≤ max ess sup s ∈ [0 , t ∗ ] w ( t − s ) w ( t ∗ − s ) γ − 1 ( r ) , γ − 1 ( r ) ! . (19) Since w is tak en nonincreasing, w ( t − s ) / w ( t ∗ − s ) ≤ 1 for all t ≥ t ∗ and s ∈ [0 , t ∗ ], so: ess sup s ∈ [0 , t ] w ( t − s ) ∥ ∆ u ( s ) ∥ ≤ γ − 1 ( r ) , (20) which concludes the proof. 4. Connection to Boyd and Chua’s FM Boyd and Chua’ s definition of FM w as dev eloped for time- in v ariant and causal I / O operators G : U → Y with U ⊂ L ∞ ( R , R n u ) and Y ⊂ L ∞ ( R , R n y ). In this context, the inputs and outputs are required to be defined over the entire time axis (in- cluding negativ e time), as recalled in the following definitions of time-in v ariance and causality . Definition 4 (T ime in variance) . Let us denote T τ : L ∞ → L ∞ the time translation defined by T τ f ( t ) ≜ f ( t − τ ) . The operator G : U → Y is time-in variant if for all u ∈ U and τ ∈ R : T τ u ∈ U , [ input set in variance ] (21) G ( T τ u ) = T τ G ( u ) . [ oper ator equivariance ] (22) Remark 3. The operator G : U → Y is equivariant with re- spect to the group of translation of the r eal line, so the time- in variance property is in fact a time-equivariance pr operty . However , this terminology is not commonly used by the liter - atur e [34]. Definition 5 (Causality) . The operator G : U → Y is time causal if for all u a , u b ∈ U and t 0 ∈ R : u a ( t ) = u b ( t ) , ∀ t ≤ t 0 ⇒ G ( u a )( t ) = G ( u b )( t ) , ∀ t ≤ t 0 . (23) Remark 4. Intuitively , this definition states that at each instant t 0 , the output is solely determined by the past of the input, not its futur e. It is relati vely well-known that this causality condition can be further simplified in the context of time-in variance [1]. Proposition 5 (Causality under time-inv ariance) . A time- in variant operator G : U → Y is time causal if for all u a , u b ∈ U : u a ( t ) = u b ( t ) , ∀ t ≤ 0 ⇒ G ( u a )( t ) = G ( u b )( t ) , ∀ t ≤ 0 . (24) Pr oof. Let t 0 ∈ R , and u a , u b ∈ U such that u a ( t ) = u b ( t ) for all t ≤ t 0 . This implies in particular T t 0 u a ( t ) = T t 0 u b ( t ) for all t ≤ 0. Now , by input set in v ariance (21), we hav e T t 0 u a , T t 0 u b ∈ U . Hence, (24) provides G ( T t 0 u a )( t ) = G ( T t 0 u b )( t ) for all t ≤ 0, which in turns provides, by operator equiv ariance (22), G ( u a )( t ) = G ( u b )( t ) for all t ≤ t 0 . This sho ws that G satisfies the definition (23) of time causality . 5 The properties of time-in v ariance and causality mean that G can solely be defined and studied through a memory functional , which maps its input before time t = 0 to its output at time t = 0 [1]. W e introduce the set U − ≜ { u − , u ∈ U } , where u − stands for the restriction of the function u to neg ativ e time. U + and u + are defined like wise for positiv e time. A definition for the memory functional of an operator is defined thereafter . Definition 6 (Memory functional) . Given the operator G : U → Y , the map F G : U − → R n y is a memory functional of G if for all u − ∈ U − , ther e exists ˜ u ∈ U such that u − = ˜ u − and F G ( u − ) = G ( ˜ u )(0) . At this stage, the definition of a memory functional remains underspecified: the memory functional of an operator is not uniquely defined, nor is the operator uniquely determined by its memory functional. While such memory functionals are not useful in themselves, the y serve to demonstrate that time- in v ariance and causality constitute the minimal assumptions re- quired to guarantee a con venient correspondence between an operator and its associated memory functional. This observ a- tion is purely rhetorical in nature. Proposition 6. If G : U → Y is causal, then G defines a unique memory functional F G : U − → R n y . Con versely , if G : U → Y is causal and time-in variant, then the memory functional F G : U − → R n y uniquely defines G. Pr oof. First, let us show that a causal G uniquely defines F G . Let F a G and F b G be two memory functionals associated with G . Let u − ∈ U − and take u a , u b ∈ U such that u − = u − a = u − b , F a G ( u − ) = G ( u a )(0) and F b G ( u − ) = G ( u b )(0). By definition of causality (23), G ( u a )( t ) = G ( u b )( t ) for all t ≤ 0, so in particular , G ( u a )(0) = G ( u b )(0), which implies that F a G ( u − a ) = F b G ( u − b ), and thus F a G ( u − ) = F b G ( u − ) for all u − ∈ U − , i.e. F a G = F b G . Con versely , let us show that if G is causal and time-in v ariant, then F G uniquely defines G . Let us take F G a and F G b the unique memory functionals associated with two time inv ariant and causal operators G a and G b . Assuming F G a = F G b , we hav e G a ( u )(0) = G b ( u )(0) for all u ∈ U . By the input set in- variance (21), if u ∈ U , then for all τ ∈ R , we have T − τ u ∈ U , hence G a ( T − τ u )(0) = G b ( T − τ u )(0), which is to say , by the oper- ator equiv ariance (22), G a ( u )( τ ) = G b ( u )( τ ) for all u ∈ U and τ ∈ R , and thus G a = G b . Boyd and Chua’ s definition of FM relies on this memory functional: an operator G has FM if its memory functional F G is continuous with respect to the w -fading input norm, as for - malized hereafter . Definition 7 (FM operator) . Let G : U → Y be a causal and time-in variant oper ator . G has FM if ther e exists a memory kernel w satisfying Definition 1 suc h that for all u a , u b ∈ U − , the memory functional F G : U − → R n y satisfies: ∀ ε > 0 , ∃ δ > 0 , ess sup s ≤ 0 w ( − s ) ∥ u a ( s ) − u b ( s ) ∥ < δ (25) ⇒ ∥ F G ( u a ) − F G ( u b ) ∥ < ε. The state-space definition of FM introduced in Section 3.2 is slightly more contrived than that of Boyd and Chua. Succinctly , the operator theoretic equiv alent to our state-space FM defini- tion consists in imposing uniform continuity to the FM operator rather than solely continuity . Although imposing uniform conti- nuity is stronger than continuity , the two notions of FM become identical for well-behaved inputs , by which we mean inputs that are both Lipschitz-continuous and equibounded. Thus, we ar - gue that the two notions are fundamentally equi valent in prac- tice. Definition 8 (Uniform FM operator) . Let G : U → Y be a causal and time-in variant operator . G has uniform FM if ther e exists γ ∈ K ∞ and a memory k ernel w satisfying Defi- nition 1 such that for all u a , u b ∈ U − , the memory functional F G : U − → R n y satisfies: ∥ F G ( u a ) − F G ( u b ) ∥ ≤ γ  ess sup s ≤ 0 w ( − s ) ∥ u a ( s ) − u b ( s ) ∥  . (26) Remark 5. In this context, γ is called the modulus of continuity of the memory functional F G . Remark 6. Similarly to the memory kernel discussed in Sec- tion 3.3, the memory kernel introduced in the two definitions her eabove is taken continuous and nonincr easing without loss of generality . Proposition 7 (W ell beha ved U : (FM operator ⇔ Uniform FM operator)) . Let G : U → Y be a causal and time-in variant operator . If there e xists M , K > 0 such that: U ⊆ { u ∈ L ∞ : ∥ u ( t ) ∥ ≤ M , ∥ u ( t ) − u ( s ) ∥ ≤ K | t − s | , ∀ t , s ∈ R } , (27) then G has FM if and only if it has uniform FM. Pr oof. Follo wing the proof of Lemma 1 in Section 4 of [1], we can show that U − is contained in a compact set with respect to the w -fading input norm. The equi valence between FM and uni- form FM then follo ws from the Heine-Cantor theorem, which demonstrates the equi v alence between continuity and uniform continuity on compact sets. W e no w show the equi v alence between the operator-theoretic notion of uniform FM and the state-space notion of FM intro- duced earlier . T o this aim, we need to associate a fix ed initial condition x 0 ∈ X 0 of the state-space model (3) with a fixed negati ve-time input u 0 ∈ U − of the I / O operator G . The formal definition of a ( x 0 , u 0 )-realization is introduced hereafter . Definition 9 (( x 0 , u 0 )-realization) . The causal and time- in variant operator G : U → Y is a ( x 0 , u 0 ) -r ealization of the state-space model (3) if the ne gative-time input u 0 ∈ U − satis- fies: • for all u ∈ U + , the temporal concatenation u 0 ⋄ u is an element of U ; • the output y of (3) considered at x 0 ∈ X 0 and associated with u ∈ U + satisfies y ( t ) = G ( u 0 ⋄ u )( t ) for all t ≥ 0 . Remark 7. Note that G ( u 0 ⋄ u )( t ) = F G (( T − t ( u 0 ⋄ u )) − ) . 6 The equi valence between the operator-theoretic notion of uniform FM and the state-space notion of FM can then be stated as follows: Theorem 1 (Uniform FM operator ⇔ State-space FM) . Let G : U → Y be a ( x 0 , u 0 ) -r ealization of the state-space model (3) . The operator G has uniform w-FM (Definition 8) for all inputs of the form T − t ( u 0 ⋄ u ) , with u ∈ U + and t ≥ 0 , if and only if the state-space model (3) has w-FM (Definition 2) for X 0 = { x 0 } . Pr oof. Since for all u a , u b ∈ U + and t ≥ 0, we have: γ  ess sup s ≤ 0 w ( − s ) ∥ T − t ( u 0 ⋄ u a )( s ) − T − t ( u 0 ⋄ u b )( s ) ∥  (28) = γ  ess sup s ≤ t w ( t − s ) ∥ ( u 0 ⋄ u a )( s ) − ( u 0 ⋄ u b )( s ) ∥  = γ  ess sup s ∈ [0 , t ] w ( t − s ) ∥ u a ( s ) − u b ( s ) ∥  , (29) and ∥ y a ( t ) − y b ( t ) ∥ = ∥ F G (( T − t ( u 0 ⋄ u a )) − ) − F G (( T − t ( u 0 ⋄ u b )) − ) ∥ , (30) the inequalities (26) and (5) are equiv alent. Since the equiv alence is obtained at a fixed x 0 , it does not yet justify the K L -dissipation term β in Definition 2. An operator- theoretic justification for this term is provided hereafter, al- though the strict equiv alence between the two FM notions is lost. Succinctly , if the I / O operator G is a ( x 0 , u 0 )-realization of the state-space model (3) for a whole a set of initial conditions x 0 ∈ X 0 , with an appropriate negati ve-time control u 0 ∈ U − varying uniformly continuously with respect to the choice of x 0 , then a K L -dissipation term β can be recovered solely from the operator-theoretic Definition 8 of FM. Assumption 1. Let U − be equipped with the usual L ∞ norm. W e assume that ther e exists a uniformly continuous map ψ : X 0 → U − , with α ∈ K ∞ its modulus of continuity , such that for all x 0 ∈ X 0 , the causal and time-in variant oper ator G : U → Y is a ( x 0 , ψ ( x 0 )) -r ealization of the state-space model (3) consider ed at x 0 ∈ X 0 . Remark 8. In the case where X 0 is compact, the Heine-Cantor theor em guarantees that solely continuity of ψ is r equired. As an e vident example, we can consider once again the stan- dard low-pass filter τ ˙ y ( t ) = − y ( t ) + u ( t ) and its I / O operator realization G defined for all u ∈ U and t ∈ R by: G ( u )( t ) ≜ 1 τ Z t −∞ e − ( t − s ) /τ u ( s ) d s . (31) T aking ψ ( x 0 ) ≜ 1 ≤ 0 x 0 ov er the domain X 0 = R , with 1 ≤ 0 the constant ne gati ve-time function of v alue 1, we can easily v erify that ψ is uniformly continuous, and that for all u ∈ U + and t ≥ 0: G ( ψ ( x 0 ) ⋄ u )( t ) = e − t /τ x 0 + 1 τ Z t 0 e − ( t − s ) /τ u ( s ) d s = y ( t ) , (32) so Assumption 1 holds globally . Giv en this assumption, the K L -dissipation of initial condi- tions is recov ered as follows: Proposition 8 (Uniform FM operator ⇒ State-space FM) . Un- der Assumption 1, if G has uniform w-FM (Definition 8) for all inputs in S t ≥ 0 T − t ( ψ ( X 0 ) ⋄ U + ) , then the state-space model (3) has w-FM (Definition 2) on X 0 . Pr oof. Let x 0 , a , x 0 , b ∈ X 0 and u a , u b ∈ U + . The outputs of the state-space model (3) associated with these initial conditions and inputs are denoted y a and y b respectiv ely . Under Assump- tion 1, the following equality holds for all t ≥ 0: ∥ y a ( t ) − y b ( t ) ∥ (33) = ∥ F G (( T − t ( ψ ( x 0 , a ) ⋄ u a )) − ) − F G (( T − t ( ψ ( x 0 , b ) ⋄ u b )) − ) ∥ , thus, if G has uniform FM, then the following inequality stands for all t ≥ 0: ∥ y a ( t ) − y b ( t ) ∥ (34) ≤ γ  max  ess sup s ≤ 0 w ( t − s ) ∥ ψ ( x 0 , a )( s ) − ψ ( x 0 , b )( s ) ∥ , ess sup s ∈ (0 , t ] w ( t − s ) ∥ u a ( s ) − u b ( s ) ∥  . (35) By uniform continuity of ψ , the inequality ess sup s ≤ 0 ∥ ψ ( x 0 , a )( s ) − ψ ( x 0 , b )( s ) ∥ ≤ α ( ∥ x 0 , a − x 0 , b ∥ ) holds, hence, for all t ≥ 0: ∥ y a ( t ) − y b ( t ) ∥ (36) ≤ max  γ  ess sup s ≤ 0 w ( t − s ) α ( ∥ x 0 , a − x 0 , b ∥ )  , γ  ess sup s ∈ (0 , t ] w ( t − s ) ∥ u a ( s ) − u b ( s ) ∥  ≤ max  γ  α ( ∥ x 0 , a − x 0 , b ∥ )  , (37) γ  ess sup s ∈ (0 , t ] w ( t − s ) ∥ u a ( s ) − u b ( s ) ∥  . W ithout loss of generality , the memory kernel w is taken con- tinuous and nonincreasing, so we recover the FM inequality (6), with β ( r , t ) ≜ γ ( w ( t ) α ( r ) ) . The purpose of Proposition 8 is primarily rhetorical rather than practical. Although it may be di ffi cult to identify a set X 0 on which Assumption 1 holds in practice, its (most likely) ex- istence for a well-selected I / O operator realization of the state- space model (3) is su ffi cient to motiv ate the K L -dissipation of initial conditions in Definition 2. The reciprocal, howe ver , will turn out to be much more prac- tical, since it allows to access the operator-theoretic approxima- tion theorem of Boyd and Chua [1] from the state-space defini- tion of FM ov er a whole set of initial conditions. Ho wev er , once again because of the K L -dissipation of initial conditions, this con verse result must be handled carefully . Namely , we need the composition of the in verse map ψ − 1 : ψ ( X 0 ) → X 0 with the K L -dissipation β to itself satisfy a FM inequality . Assumption 2. W e assume that ther e exists a map ψ : X 0 → U − such that for all x 0 ∈ X 0 , the causal and time-in variant operator G : U → Y is a ( x 0 , ψ ( x 0 )) -r ealization of the state- space model (3) consider ed at x 0 ∈ X 0 . Mor eover we assume that there exist ˜ γ ∈ K ∞ and a memory kernel ˜ w satisfying Def- inition 1 such that the following FM inequality holds for all 7 x 0 , a , x 0 , b ∈ X 0 and t ≥ 0 : β ( ∥ x 0 , a − x 0 , b ∥ , t ) (38) ≤ ˜ γ  ess sup s ≤ 0 ˜ w ( t − s ) ∥ ψ ( x 0 , a )( s ) − ψ ( x 0 , b )( s ) ∥  . In the context of the lo w-pass filter τ ˙ y ( t ) = − y ( t ) + u ( t ) realized by the operator (31), one can consider ψ ( x 0 )( t ) ≜ 3 e t / 2 τ x 0 / 2, and verify Assumption with β ( r , t ) = e − t /τ r , ˜ γ ( r ) ≜ 2 r / 3 and ˜ w ( t ) = e − t / 2 τ . Proposition 9 (State-space FM ⇒ Uniform FM operator) . Un- der Assumption 2, if the state-space model (3) has w-FM (Def- inition 2) on X 0 , then G has uniform FM (Definition 8) for all inputs in S t ≥ 0 T − t ( ψ ( X 0 ) ⋄ U + ) . Pr oof. Let x 0 , a , x 0 , b ∈ X 0 and u a , u b ∈ U + . The outputs of the state-space model (3) associated with these initial conditions and inputs are denoted y a and y b respectiv ely . Under Assump- tion 1, the following equality holds for all t ≥ 0: ∥ y a ( t ) − y b ( t ) ∥ (39) = ∥ F G (( T − t ( ψ ( x 0 , a ) ⋄ u a )) − ) − F G (( T − t ( ψ ( x 0 , b ) ⋄ u b )) − ) ∥ , thus, if the state-space model (3) has w -FM, then the following inequality stands for all t ≥ 0: ∥ y a ( t ) − y b ( t ) ∥ (40) ≤ max  β ( ∥ x 0 , a − x 0 , b ∥ , t ) , γ  ess sup s ∈ [0 , t ] w ( t − s ) ∥ u a ( s ) − u b ( s ) ∥  . If Assumption 2 is verified, the first term can be upper -bounded as follows: β ( ∥ x 0 , a − x 0 , b ∥ , t ) (41) ≤ ˜ γ  ess sup s ≤ 0 ˜ w ( t − s ) ∥ ψ ( x 0 , a )( s ) − ψ ( x 0 , b )( s ) ∥  . Let us consider γ tot ∈ K ∞ and w tot a memory kernel, with γ tot ≜ max( ˜ γ, γ ) and w tot ≜ max( w , ˜ w ). This provides: ∥ F G (( T − t ( ψ ( x 0 , a ) ⋄ u a )) − ) − F G (( T − t ( ψ ( x 0 , b ) ⋄ u b )) − ) ∥ (42) ≤ γ tot  max(ess sup s ≤ 0 w tot ( t − s ) ∥ ψ ( x 0 , a )( s ) − ψ ( x 0 , b )( s ) ∥ , ess sup s ∈ [0 , t ] w tot ( t − s ) ∥ u a ( s ) − u b ( s ) ∥  ≤ γ tot  ess sup s ≤ 0 w tot ( − s ) (43) ∥ F G (( T − t ( ψ ( x 0 , a ) ⋄ u a )) − ) − F G (( T − t ( ψ ( x 0 , b ) ⋄ u b )) − ) ∥  . Thus G satisfies the uniform FM inequality of Definition 8 for all inputs in S t ≥ 0 T − t ( ψ ( X 0 ) ⋄ U + ). 5. Approximating δ ISS systems using FM 5.1. Introducing δ ISS for FM If the state-space model (3) has FM (Definition 2), then its operator realization at x 0 has uniform FM (Definition 8), so Boyd and Chua’ s approximation of FM operators holds, and the input-output behavior of (3) can be approximated arbitrar- ily well by a linear time-in v ariant state-space model follo wed by a nonlinear static readout [1]. Howe ver , practical conditions to prov e state-space FM remain to be inv estigated: to this end, we w ould lik e to show that the δ ISS property is su ffi cient to ob- tain state-space FM. W e recall the definition of a δ ISS system. Definition 10 ( δ ISS) . The system ˙ x ( t ) = f ( x ( t ) , u ( t )) is δ ISS if ther e exists β ∈ K L and γ ∈ K ∞ such that for any two initial conditions, x a (0) , x b (0) ∈ X 0 , consider ed with two input trajec- tories, u a , u b ∈ U + , the corr esponding state trajectories satisfy for all t ≥ 0 : ∥ x a ( t ) − x b ( t ) ∥ ≤ β  ∥ x a (0) − x b (0) ∥ , t  (44) + γ  ess sup s ∈ [0 , t ] ∥ u a ( s ) − u b ( s ) ∥  . It is clear , by definition, that input-to-state FM (Definition 3) implies the δ ISS property . Howe ver , we would like the opposite implication to hold as well. W e will see that, in practice, the con verse result holds semi-globally under a physically realistic assumption of equibounded inputs. The strength of this result is that it allows us to use already existing δ ISS criteria to ensure FM semi-globally , such as the following L yapunov condition: Theorem 2 ( δ ISS L yapunov conditions [2]) . The system ˙ x ( t ) = f ( x ( t ) , u ( t )) is globally δ ISS (i.e. δ ISS for X 0 = R n x ) for in- puts in U + = L ∞ ([0 , + ∞ ) , U ) if ther e exists a smooth Lya- punov function V : R n x × R n x → [0 , + ∞ ) such that there exists α 1 , α 2 , κ ∈ K ∞ and ρ positive-definite so for all x a , x b ∈ R n x and u a , u b ∈ U ⊆ R n y : α 1 ( ∥ x a − x b ∥ ) ≤ V ( x a , x b ) ≤ α 2 ( ∥ x a − x b ∥ ) , (45a) κ ( ∥ x a − x b ∥ ) ≥ ∥ u a − u b ∥ ⇒ ˙ V ( x a , x b ) ≤ − ρ ( ∥ x a − x b ∥ ) , (45b) wher e ˙ V stands for the Lie derivative of V along f . Mor eover , if the input value set U is compact, then the con verse also holds. Further su ffi cient conditions can be found in the literature to demonstrate the δ ISS property , such as contraction-based con- ditions (Theorem 37 in [35]), or a Killing vector field criterion in [36]. 5.2. δ ISS implies input-to-state FM for equibounded inputs Under an equibounded input assumption, a note from Seung- Jean Kim shows that the δ ISS property implies Boyd and Chua’ s notion of FM [37]. Since for well-behaved inputs, Boyd and Chua’ s notion of FM is equiv alent to the uniform FM no- tion for operators introduced in this paper (Proposition 7 of Section 4), this result provides a reasonable con verse in the operator-theoretic setting. Theorem 3 (W ell-behav ed U : ( δ ISS ⇒ Uniform FM opera- tor) [37]) . If the inputs of ˙ x ( t ) = f ( x ( t ) , u ( t )) are equibounded, Lipschitz-continuous, and the system is globally δ ISS, then for all x 0 ∈ R n x , given an operator G : U → Y such that G is a ( x 0 , u 0 ) -r ealization of the system, G has uniform FM (Defini- tion 8) for all inputs of the form T − t ( u 0 ⋄ u ) , with u ∈ U + and t ≥ 0 . 8 W e now demonstrate a very similar result for our state- space notion of FM using a L yapunov con verse argument. The result is established using techniques similar in spirit to those employed in the proofs that ISS implies exp-ISS or ISDS [28, 29, 26]. Howe ver , the incremental nature of the in- vestigated properties necessitates restricting the analysis to a compact subset of the state space. Theorem 4 (Equibounded U + , compact X 0 : ( δ ISS ⇒ State-s- pace FM)) . Let X 0 be a compact set of initial conditions. If the inputs of ˙ x ( t ) = f ( x ( t ) , u ( t )) are equibounded and the system is globally δ ISS, then it has state-space FM on X 0 . Remark 9. Since a FM inequality holds for all compact sets of initial condition, one can say that the system has semi-global input-to-state FM (Definition 3). Pr oof. Let X 0 be a compact set and let M x > 0 be a constant such that for all x 0 ∈ X 0 , ∥ x 0 ∥ ≤ M x . Let U + be equibounded, so there exists M u > 0 such that the input signals take value in the compact and con vex set U ≜ { u ∈ R n u : ∥ u ∥ ≤ M u } ⊂ R n u containing the origin. Since the system is δ ISS, then for u ≜ 0 it is δ GAS, which implies the existence of a unique globally asymptotically stable equilibrium x ∗ [4]. Consequently , for all x 0 ∈ X 0 , the δ ISS inequality provides: ∥ x ( t ) − x ∗ ∥ ≤ β ( M x + ∥ x ∗ ∥ , t ) + γ ( M u ) . (46) The compact set X ≜ { x ∈ R n x : ∥ x − x ∗ ∥ ≤ β ( M x + ∥ x ∗ ∥ , 0) + γ ( M u ) } is such that all state trajectories x of initial condition x (0) ∈ X 0 remain bounded in X . The con verse L yapuno v result of Theorem 2 pro vides the existence of a smooth L ya- punov function V : R n x × R n x → [0 , + ∞ ) such that there e x- ists α 1 , α 2 , κ ∈ K ∞ and ρ positive-definite satisfying the condi- tions (45) [2]. According to Theorem 3.6.10 of [38] (see also Lemma A.34 of [39]), there exists α 3 ∈ K ∞ and λ > 0 such that W = α 3 ◦ V satisfies: α 3 ◦ α 1 ( ∥ x a − x b ∥ ) ≤ W ( x a , x b ) ≤ α 3 ◦ α 2 ( ∥ x a − x b ∥ ) , (47a) κ ( ∥ x a − x b ∥ ) ≥ ∥ u a − u b ∥ ⇒ ˙ W ( x a , x b ) ≤ − λ W ( x a , x b ) , (47b) with ˙ W ( x a , x b ) = ∂ W ∂ x a ( x a , x b ) f ( x a , u a ) + ∂ W ∂ x b ( x a , x b ) f ( x b , u b ) . (48) Let us consider the set F = { ( x a , x b , u a , u b ) : ∥ x a − x b ∥ ≤ κ − 1 ( ∥ u a − u b ∥ ) } ∩ ( X 2 × U 2 ) and the function µ 0 defined by: µ 0 ( s ) ≜ max ( x a , x b , u a , u b ) ∈F , ∥ u a − u b ∥≤ s max(0 , ˙ W ( x a , x b ) + λ W ( x a , x b )) . (49) Using Lemma A.40 of [39] with F defined pre viously , g ( x a , x b , u a , u b ) = ∥ u a − u b ∥ , P = [0 , + ∞ ), h ( r ) = r and f ( x a , x b , u a , u b ) = max(0 , ˙ W ( x a , x b ) + λ W ( x a , x b )) we see that µ 0 is continuous, nondecreasing, and that for all x a , x b ∈ X : ˙ W ( x a , x b ) ≤ − λ W ( x a , x b ) + µ ( ∥ u a − u b ∥ ) , (50) with µ ( r ) ≜ µ 0 ( r ) + r , so µ ∈ K ∞ . Let x a , x b be two state trajectories of initial conditions x a (0) , x b (0) ∈ X 0 and of in- puts u a , u b ∈ U + respectiv ely . Using Grönwall’ s inequality , this yields: W ( x a ( t ) , x b ( t )) (51) ≤ e − λ t W ( x a (0) , x b (0)) + Z t 0 e − λ ( t − s ) µ ( ∥ u a ( s ) − u b ( s ) ∥ ) d s ≤ e − λ t W ( x a (0) , x b (0)) + 2 λ sup s ∈ [0 , t ] e − λ ( t − s ) / 2 µ ( ∥ u a ( s ) − u b ( s ) ∥ ) (52) Now consider w ( t ) ≜ sup r ∈ (0 , 2 M u ] µ − 1 ( e − λ t / 2 µ ( r )) / r . Since e − λ t / 2 ∈ [0 , 1], then w ( t ) ∈ [0 , 1] as well, and lim t → + ∞ w ( t ) = 0. Moreov er , notice that the following inequality holds: e − λ ( t − s ) / 2 µ ( ∥ u a ( s ) − u b ( s ) ∥ ) ≤ µ ( w ( t − s ) ∥ u a ( s ) − u b ( s ) ∥ ) . (53) Consequently: W ( x a ( t ) , x b ( t )) ≤ e − λ t W ( x a (0) , x b (0)) + 2 λ µ sup s ∈ [0 , t ] w ( t − s ) ∥ u a ( s ) − u b ( s ) ∥ ! . (54) Finally , the following inequalities hold: ∥ x a ( t ) − x b ( t ) ∥ ≤ ( α 3 ◦ α 1 ) − 1 ( W ( x a ( t ) , x b ( t ))) (55) ≤ ( α 3 ◦ α 1 ) − 1  max  2 e − λ t W ( x a (0) , x b (0)) , (56) 4 λ µ sup s ∈ [0 , t ] w ( t − s ) ∥ u a ( s ) − u b ( s ) ∥ !!! ≤ max  ( α 3 ◦ α 1 ) − 1 (2 e − λ t ( α 3 ◦ α 2 )( ∥ x a (0) − x b (0) ∥ )) , (57) ( α 3 ◦ α 1 ) − 1 4 λ µ sup s ∈ [0 , t ] w ( t − s ) ∥ u a ( s ) − u b ( s ) ∥ !!! , so the FM inequality (6) is recov ered. Remark 10. Some settings wher e δ ISS does not imply input-to- state FM ar e discussed in the appendix. 5.3. δ ISS repr esentation theorem Theorem 5 ( δ ISS representation) . If the nonlinear state-space model (3) is globally δ ISS, and has equibounded and Lipschitz- continuous inputs, then its input-output behavior at a fixed x 0 can be appr oximated to an arbitrary de gree of accuracy by a cascaded system consisting of a linear time-in variant state- space model and a nonlinear static r eadout (1) . If Assumption 2 holds on a compact set of initial conditions, the approximation applies to this entir e set. Pr oof. Theorem 4 ensures that the system has input-to-state FM (Definition 3) semi-globally . Moving from the state-space realization to the operator perspecti ve, by Theorem 1, the oper- ator realization at a fixed initial condition x 0 necessarily inher- its uniform FM (Definition 8). Moreo ver , under Assumption 2, this result can be extended to an entire set of initial conditions using Proposition 9. The operator realization satisfies the re- quirements for Boyd and Chua’ s approximation theorem. This ensures that the input-to-state behavior of (3) can be approxi- mated to an arbitrary degree of accuracy by a cascaded system 9 consisting of a linear time-in v ariant state-space model and a nonlinear static readout [1]. By composition of nonlinear read- outs, the representation can be extended to the entire nonlinear state-space model (3). 6. FM of the current-driven memristor As a final application of the state-space notion of FM stud- ied in this paper , we come back to our motiv ating example, the current-driv en memristor , and we exhibit mild assumptions to ensure that it has FM. Proposition 10 (FM of the current-driv en memristor) . F or an equibounded input current I , the internal dynamics (2b) of the memristor has FM fr om I to x on compact sets of initial con- ditions if it is globally δ ISS. Mor eover , if the input current I is continuous and the memristance M is Lipschitz continuous, the memristor (2) also has FM fr om I to U on compact sets of initial conditions. Pr oof. Let I > 0 and λ > 0 denote the maximum of the signal I and the Lipschitz constant of M respectiv ely . Given that (2b) is delay-free, time-in variant, with equibounded input currents, and globally δ ISS, applying Theorem 4 provides the FM of (2b) for equibounded input currents I on the compact set of initial conditions X 0 , thus giv en a di ff erence of two state trajectories, the following FM inequality holds: ∥ ∆ x ( t ) ∥ ≤ β  ∥ ∆ x (0) ∥ , t  + γ  ess sup s ∈ [0 , t ] w ( t − s ) | ∆ I ( s ) |  . (58) Giv en that (2b) has equibounded input currents, δ ISS also im- plies for all X 0 compact, there e xists a compact X such that all state trajectories x ( t ) remain bounded in X , meaning we can also upper bound ∥ M ( x ( t )) ∥ by a constant M for all t ≥ 0. Then, the following inequalities hold: | ∆ U ( t ) | ≤ | M ( x a ( t )) I a ( t ) − M ( x b ( t )) I b ( t ) | (59) ≤ | M ( x a ( t )) | · | ∆ I ( t ) | + | I b ( t ) | · | M ( x a ( t )) − M ( x b ( t )) | (60) ≤ M | ∆ I ( t ) | + I λ ∥ ∆ x ∥ . (61) W ithout loss of generality , we take the memory kernel w with w (0) = 1, so by continuity of I and w , | ∆ I ( t ) | ≤ ess sup s ∈ [0 , t ] w ( t − s ) | ∆ I ( s ) | . Thus, the memristor (2) has FM from I ( t ) to U ( t ) with ˜ β ≜ I β and ˜ γ ( r ) ≜ Mr + γ ( r ). 7. Conclusion & perspectives This work proposes a formulation of the FM property as an extension of the δ IOS property with an explicit memory kernel, with the aim of promoting FM as a core property of real-life nonlinear systems, much like the con volution property of L TI systems. This formulation bridges the operator-theoretic notion of FM with the incremental stability concepts from state-space theory . Although full equiv alence with the δ ISS property does not hold in general, we identified reasonable assumptions where FM can be recovered from δ ISS. The current-driven memristor example illustrates the practical rele v ance of the framew ork. For future work, global extensions of Theorem 3 and Theorem 4, eventually for inputs which are not necessarily Lipschitz-continuous and / or equibounded, remain to be found. It would also be v aluable to deri ve practical L yapunov condi- tions to guarantee global input-to-state FM. Further applica- tions of FM remains to be explored. In the context of intercon- nected systems [2]: what can be said about the memory kernel of the interconnection of two FM systems under a small-gain condition? In the context of dissipati vity theory [40]: could FM help to reduce the conceptual gap between the non-incremental and the (lack of) incremental properties of memristi ve systems? It would also be useful to de velop a state-space characterization of myopia [41], the spatial counterpart of FM, and to in vestigate how it can be integrated with the proposed state-space notion of FM for the study of partial di ff erential equations. As a final note, the L ∞ fading-norm used to define FM can be substituted by a generic L p fading-norm, which, for p = 1, creates a bridge between FM and the notion of incremental integral input-to- output stability ( δ iIOS) worth in vestigating [3]. References [1] S. Boyd, L. O. Chua, Fading memory and the problem of approximating nonlinear operators with V olterra se- ries, IEEE Transactions on Circuits and Systems 32 (11) (1985) 1150–1161. doi:10.1109/tcs.1985.1085649 . [2] D. Angeli, A L yapunov approach to incremental stability properties, IEEE T rans. Autom. Control. 47 (2002) 410– 421. doi:10.1109/9.989067 . [3] D. Angeli, Further results on incremental input-to-state stability , IEEE Transactions on Automatic Control 54 (6) (2009) 1386–1391. doi:10.1109/tac.2009.2015561 . [4] R. Kato, D. Astolfi, V . Andrieu, L. 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Xu, Uniform approximation of multidi- mensional myopic maps, IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications 44 (6) (1997) 477–500. doi:10.1109/81.585959 . [42] V . Hassani, T . Tjahjo widodo, T . N. Do, A survey on hys- teresis modeling, identification and control, Mechanical Systems and Signal Processing 49 (1–2) (2014) 209–233. doi:10.1016/j.ymssp.2014.04.012 . Appendix A. Some settings where δ ISS does not imply input-to-state FM Sev eral counterexamples demonstrate that the implication fails in the presence of delays or time-varying parameters. The discussion becomes particularly nuanced for time-in variant, delay-free systems, as we were unable to construct a globally valid counterexample, and can therefore only refute the impli- cation by restricting the set of initial conditions. In the absence of delay , the contradictions are obtained via an unbounded se- quence of input functions, causing the re verse implication to fail from a purely mathematical standpoint. Let us begin by showing that δ ISS does not imply FM when arbitrarily large input delays are allo wed. Counterexample 1 (Input delay: ( δ ISS ⇏ FM)) . The following system is δ ISS (at t 0 = 0 ), but it does not have input-to-state FM (at t 0 = 0 ). ˙ x ( t ) = − x ( t ) + Z min(1 , t ) 0 u ( s ) d s . (A.1) Pr oof. By linearity , the di ff erence between two trajectories sat- isfies the following inequalities: | ∆ x ( t ) | ≤ e − t | ∆ x (0) | + Z t 0 e s − t Z min(1 , s ) 0 | ∆ u ( r ) | d r d s (A.2) ≤ e − t | ∆ x (0) | + Z t 0 e s − t ess sup r ∈ [0 , 1] | ∆ u ( r ) | d s (A.3) ≤ e − t | ∆ x (0) | + ess sup s ∈ [0 , t ] | ∆ u ( s ) | , (A.4) hence the system is δ ISS. Howe ver , consider ∆ u ( t ) = 1 for t ≤ 1 and ∆ u ( t ) = 0 for t > 1. Notice that ∆ x ( t ) → + ∞ 1, so the CICO property of FM is not v erified (Proposition 2), hence the system does not hav e FM. Remark 11. Similar infinite memory e ff ects may arise in prac- tice, such as in hyster esis phenomena [42]. Remark 12. The delay of this counter example can be r emoved using a time-varying inte grator: ˙ x 1 ( t ) = 1 ≤ 1 ( t ) u ( t ) (A.5a) ˙ x 2 ( t ) = − x 2 ( t ) + x 1 ( t ) (A.5b) with 1 ≤ 1 ( t ) ≜ 1 if t ≤ 1 , and 1 ≤ 1 ( t ) ≜ 0 else. However , this state-augmented system is not δ ISS anymor e. More surprisingly , even in the absence of delays, time- varying parameters are su ffi cient to break the implication. The intuition behind the following counterexample is based on the following a veraging I / O map: y ( t ) = 1 t + 1 Z t 0 u ( s ) d s . (A.6) Although the relativ e influence of past inputs u on the output y diminishes o ver time, this decay occurs regardless of ho w far in the past the inputs were applied, and crucially , it is also van- ishing linearly . This av eraging process exhibits fragile stability , making the system lack a decaying memory mechanism, i.e. FM. By di ff erentiation, (A.6) admits a time-varying and delay- free state-space representation. Counterexample 2 (T ime-varying: ( δ ISS ⇏ FM)) . The fol- lowing system is δ ISS (at t 0 = 0 ), but it does not have input-to- state FM (at t 0 = 0 ). ˙ x ( t ) = 1 t + 1 ( − x ( t ) + u ( t ) ) . (A.7) Pr oof. By linearity , the di ff erence between tw o trajectories of (A.7) is easily determined: ∆ x ( t ) = e − R t 0 d s s + 1 ∆ x (0) + Z t 0 e − R t s dr r + 1 ∆ u ( s ) s + 1 d s (A.8) = e − ln( t + 1) ∆ x (0) + Z t 0 e    ln( s + 1) − ln( t + 1) ∆ u ( s )    s + 1 d s (A.9) = 1 t + 1 ∆ x (0) + Z t 0 ∆ u ( s ) d s ! , (A.10) so the following inequality holds, demonstrating that the system is δ ISS at t 0 = 0: | ∆ x ( t ) | ≤ 1 t + 1 | ∆ x (0) | + t t + 1 | {z } ≤ 1 ess sup s ∈ [0 , t ] | ∆ u ( s ) | . (A.11) The claim that (A.7) does not hav e input-to-state FM at t 0 = 0 is showed by contradiction. Assuming that (A.7) has input-to- state FM at t 0 = 0, there exists γ ∈ K ∞ and a memory kernel w : [0 , + ∞ ) → (0 , 1] satisfying Definition 1 (0 can be excluded thanks to Proposition 1) such that for all ∆ u ∈ L ∞ and ∆ x (0) = 0: | ∆ x ( t ) | =     R t 0 ∆ u ( s ) d s     t + 1 ≤ γ  ess sup s ∈ [0 , t ] w ( t − s ) | ∆ u ( s ) |  . (A.12) 12 Let us show that ( t + 1) − 1 R t 0 w ( s ) − 1 d s div erges as t → + ∞ . Let M > 0. It is showed that there exists t ∗ > 0 such that for all t ≥ t ∗ , ( t + 1) − 1 R t 0 w ( s ) − 1 d s ≥ M . Since w ( t ) is nonincreasing and w ( t ) → + ∞ 0 + , w ( t ) − 1 is nondecreasing and w ( t ) − 1 → + ∞ + ∞ , hence there exists t 1 > 0 such that for all t ≥ t 1 , w ( t ) − 1 ≥ 2 M . T ake t ∗ = 2 t 1 + 1. The following inequalities hold for all t ≥ t ∗ : 1 t + 1 Z t 0 d s w ( s ) ≥ 1 t + 1 Z t t 1 d s w ( s ) ≥ t − t 1 t + 1 2 M ≥ M , (A.13) thus demonstrating the div ergence of ( t + 1) − 1 R t 0 w ( s ) − 1 d s . In particular , this demonstrates that | ∆ x ( T ) | diver ges as T → + ∞ for inputs defined by ∆ u ( t ) ≜ w ( T − t ) − 1 for t ∈ [0 , T ]. Howe ver , the FM inequality (A.12) yields: | ∆ x ( T ) | ≤ γ  ess sup s ∈ [0 , T ] w ( T − s ) w ( T − s ) − 1  = γ (1) , (A.14) meaning γ (1) → + ∞ , a contradiction. Remark 13. In particular , this system pr ovides a time-varying counter example to the claim that a CICO pr operty is su ffi cient to guarantee input-to-state FM for a δ ISS system, which would otherwise appear to be a natural con verse of Pr oposition 2. Now what if the system is both delay-free and time-in variant, i.e. of the form ˙ x ( t ) = f ( x ( t ) , u ( t ))? A variation of the previous counterexample sho ws that restricting X 0 is su ffi cient to obtain a δ ISS system without FM in this context as well. Counterexample 3 (Restricting X 0 : ( δ ISS ⇏ FM)) . The fol- lowing system is δ ISS for X 0 = { 1 } × R , but it does not have input-to-state FM for X 0 = { 1 } × R . ˙ x 1 ( t ) = − x 1 ( t ) 2 , (A.15a) ˙ x 2 ( t ) = x 1 ( t )( − x 2 ( t ) + u ( t )) . (A.15b) Pr oof. If x 1 (0) = 1, then x 1 ( t ) = 1 t + 1 , and the rest of the proof follows from the pre vious counterexample. W e were unable to find a counterexample holding globally ( X 0 = R n x ). Thus, the question of whether global δ ISS implies global input-to-state FM for time-in variant, delay-free systems remains open. 13