About Total Stability of a Class of Nonlinear Dynamic Systems Eventually Subject to Discrete Internal Delays

)is paper studies and investigates total stability results of a class of dynamic systems within a prescribed closed ball of the state space around the origin. )e class of systems under study includes unstructured nonlinearities subject to multiple higher-order Lipschitz-type conditions which influence the dynamics and which can be eventually interpreted as unstructured perturbations. )e results are also extended to the case of presence of multiple internal (i.e., in the state) point discrete delays. Some stability extensions are also discussed for the case when the systems are subject to forcing efforts by using links between the controllability and stabilizability concepts from control theory and the existence of stabilizing linear controls.)e results are based on the ad hoc use of Gronwall’s inequality.


Introduction
e study of the stability properties is of major interest in dynamic systems since it allows to investigate the existence and nature of the equilibrium points, their respective domains of attraction, and the eventual existence of oscillations dependent on the initial conditions or asymptotic oscillations independent on the initial conditions (limit cycles). Typical techniques of investigation of stability are those based on local and global Lyapunov stability methods and Bendixson-type theorems for investigation of the oscillatory and asymptotic oscillatory behaviors (see, for instance, [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16] and references therein). ere are also frequency-type methods of stability analysis, such as, Routh-Hurwitz criterion, Jury criterion, amplitude and phase Bode plots, Nyquist criterion, root locus, Tsypkin locus, and describing function analysis, which are very popular in the time-invariant case and also in systems including some specific types of separate nonlinearities in actuators or sensors such as, for instance, ban-bang (i.e., relay) or saturating devices (see, for instance, [4][5][6]). Also, the wellknown Gronwall inequality in its various versions can be adapted to stability studies in the time-domain (see, for instance, [1,2]).
On the other hand, it can be noticed that delays are present in a natural way in the real life and in many real processes such as, for instance, in some biological equations as, for instance, in epidemic models, in some generalizations of the Beverton-Holt equation, in prey-predator problems, or in the sunflower equation and in population growing and diffusion problems.
ey are also present in many engineering diffusion problems, information package transmission, queuing storages, teleoperation systems, chemical processes, and robotic systems. e study of stability of timedelay systems has received much attention in the last decades. For instance, a nonautonomous functional differential equation of the third order is considered in [7] with multiple deviating arguments. Asymptotic stability results and uniform boundedness of the solution results are obtained based on the ad hoc use of the Lyapunov-Krasovskii functional approach in the above paper. Other stability results for third-order differential equations are proved in [9,10]. Also, some sufficiency-type conditions of global stability and asymptotic stability results for time-varying systems with state-dependent parameterizations are obtained in [8]. On the other hand, other studies of stability and robustness of dynamic systems have been performed in [11][12][13][14] and related studies for the presence of oscillatory behaviors have been performed in [15,16] and references therein. e study of stabilization of time-delay dynamic systems and/or integrodifferential equations has been also addressed to more general classes of equations. For instance, the exponential decay of a first-order linear Volterra equation is investigated in [17] while the exponential stability of controlled nonlinear systems subject to time-varying delays has been discussed in [18]. It can be noticed that Volterra-type equations can be interpreted as being subject to a distributed delay on their previous evolution history. On the other hand, the solvability and optimal controls of a noninstantaneous impulsive neutral stochastic integer-differential equation forced by Brownian motion are addressed in [19]. It can be noticed that neutral differential equations are a class of retarded equations where the highestorder time-derivative is affected by delays contrarily to the case of nonneutral ones.
is paper gives some results on total stability of a class of nonlinear time-varying systems subject to combined multiple higher-order-type Lipschitz conditions. e stability results are of global type within some prescribed closed ball around the origin and are subject to conditions of exponential stability of the nonlinearity-free system counterpart and to maximum bounds of the various Lipschitz constants. Some applications are given for the control case. e above results are included in Section 2. On the other hand, Section 3 extends the results of Section 2 for the case of presence of multiple punctual constant delays which appear jointly in the linear dynamics of the system and in the nonlinear one which is also subject to combined multiple higher-order-type Lipschitz conditions.

Total Stability of Nonlinear Differential
Systems with Forcing Terms Subject to Multiple Combined Higher-Order Lipschitz Conditions e following result is two-fold since it relies on the uniqueness of the solution exhibiting, furthermore, global stability in a bounded region around the origin for a class of nonlinear time-varying differential systems under a set of Lipschitz-extended power-type perturbations of the state. A combined analysis supported by the Banach Contraction Principle and Bellman-Gronwall Lemma [1][2][3] is used to get the next main result. Theorem 1. Consider the following differential system: where A: [t 0 , ∞) ⟶ R n×n has bounded piecewise continuous entries and f i , g j : in the closed ball B r � x ∈ R n : ‖x‖ ≤ r { }, where ‖x‖ denotes some vector norm, and suppose that the following two assumptions hold: is exponentially stable on [t 0 , ∞) of stability abscissa − α < 0; that is, there exist real constants α > 0 and K ≥ 1 such that the fundamental matrix Ψ: e constants β i and c j , i � 1, 2, . . . , p and j � 1, 2, . . . , q, satisfy for some given r > 0. en, the following properties hold: (1) and (2) are globally Lyapunov stable in the closed ball of radius r centred at the origin.
for any given finite ‖x 0 ‖. (iii) Assume that A4 is restricted to a strict inequality, that is, for some given r > 0. en, there is a unique globally Lyapunov stable solution of (1) and (2) in the closed ball of radius r centred at the origin for t ≥ t 0 under Assumptions A1, A2, A3, and A5 satisfying Proof.
e solution of (1) and (2) for t ≥ t 0 is given by 2 International Journal of Differential Equations and, by using Assumptions A1 to A3, one gets provided that A4 holds subject to which is got from (9) for t � t 0 . en, Property (i) is proved.
On the other hand, since α > 0, one gets from A4 and (9) that for t ≥ t 0 . en, if ‖x 0 ‖ ≤ (r/K), one has from (9) that and one has then from (8) for any solutions which leads to is implies from the contraction mapping theorem that solution (8) is unique. From (11) and Assumptions A-A3, one gets International Journal of Differential Equations Also, one has from (15) for t ≥ t 0 , which is which implies (7) and proves Proposition 3. Some direct corollaries are as follows.

Corollary 2.
Consider the following controlled differential system: subject to initial conditions for all x ∈ R n and let us define u � sup x∈R n ∞ t 0 ‖u(τ, x)‖dτ. en, eorem 1 still holds by replacing (4) by and (5) by its corresponding strict inequality.
Proof. It follows directly by replacing (8) and (9) by and following similar arguments to those used to prove eorem 1. e following uniform-type result within a bounded ball of the solution of (1) and (2) follows from eorem 1: □ Corollary 4. Assume that A4 is modified by replacing (4) with where (25) en, the following properties hold under Assumptions A1-A3 and A4: given r ∈ B R � z ∈ R + : ‖z‖ ≤ R so that differential system (1) and (2) is globally Lyapunov stable in B R . (ii) Assume that A5 is modified by replacing (5) with en, there is a unique solution of (1) and (2) for t ≥ t 0 under Assumptions A1, A2, A3, and A5 satisfying Outline of Proof. e proof follows directly from eorem 1 by noting that (23) implies that and (24) implies that which makes (27) to be well-posed.

Total Stability of Nonlinear Differential Systems with Linear Constant Punctual Delays and Forcing Terms Subject to Combined Higher-Order Lipschitz Conditions
Now, consider the following unforced linear time-invariant differential system subject to a finite set of ϑ, in general, incommensurate constant point delays 0 < h 1 < h 2 < · · · < h ϑ � h. ese delays are commensurate in the particular case that h i � ih, i � 1, 2, . . . , ϑ (see, for instance, [3,7] and some references therein). In the sequel, fix the initial time instant to t 0 � 0 for simplicity of the presentation of the subsequent results: with A i ∈ R n×n , i � 0, 1, . . . , ϑ which is subject to any bounded piecewise continuous function of initial conditions φ: [t − h, 0] ⟶ R n with φ(0) � x 0 whose unique solution becomes to be International Journal of Differential Equations 5 where ‖e A 0 t ‖ ≤ Ke − αt for all t ≥ 0 and some real constants α and K ≥ 1 so that where φ 0 � sup − h≤τ≤0 ‖φ(τ)‖.
Above differential system (30) can be rewritten equivalently as follows after introducing the zero delay h 0 � 0 and the matrix A � ϑ i�0 A i : Note from (30) that the auxiliary differential system _ x(t) � A 0 x(t) describes the time-delay system as 1, 2, . . . , ϑ), which can be interpreted alternatively as the delay-free one when the contributions of delayed dynamics are zeroed described by A i � 0; i � 1, 2, . . . , ϑ. Also, note from (33) that the auxiliary differential system _ x(t) � Ax(t) describes the time-delay system as h i � 0 (i � 1, 2, . . . , ϑ). It follows that the solution may be equivalently defined by where ‖e At ‖ ≤ Ke − αt for all t ≥ 0 and some real constants α and K ≥ 1 so that where φ 0 � sup − h≤τ≤0 ‖φ(τ)‖. e following auxiliary result holds.

Proposition 1. e following properties hold:
(i) Let A 0 be a stability matrix such that ‖e A 0 t ‖ ≤ Ke − αt for all t ≥ 0 and some real constants α > 0 and K ≥ 1 (with K being norm-dependent) and fix φ 0 � λr. If then ‖x(t)‖ ≤ r; ∀t ≥ 0, and thus, the system is globally Lyapunov stable. (ii) Let A � ϑ i�0 A i be a stability matrix such that ‖e At ‖ ≤ Ke − αt for all t ≥ 0 and some real constants α > 0 and K ≥ 1 (with K being norm-dependent) and fix φ 0 � λr. If then ‖x(t)‖ ≤ r; ∀t ≥ 0, and this the system is globally Lyapunov stable.
Proof. Note that λ ≤ 1 so that sup − h≤τ≤0 ‖x(τ)‖ � sup − h≤τ≤0 ‖φ(τ)‖ ≤ r and there exist some t 0 > 0 such that sup − h≤τ≤t 0 ‖x(τ)‖ ≤ r by continuity of the solution. Now, proceed by complete induction by using (32). Assume that there is a first time instant t 1 ( > 0) ∈ [t − (i + 1)h, t − ih) for some i(∈ Z) ∈ [0, max int(z: z ≤ (t 1 /h) − 1)] such that ‖x(t 1 )‖ > r. However, this gives a contradiction since then sup − h≤τ<t 1 ‖x(τ − h)‖ > r and then there is some t 2 ∈ (0, t 1 − h) which is the first time instant such that ‖x(t 2 )‖ > r. As a result, φ 0 ≤ (α + K ϑ i�1 ‖A i ‖(e αh i − 1))/(αK + K ϑ i�1 ‖A i ‖(e αh i − 1))r implies that sup t≥− h ‖x(t)‖ ≤ r which proves Property (i). e proof of Property (ii) is similar to that of Property (i) by using (35) instead of (32). □ Remark 1. Note that λ M and λ M are strictly decreasing functions of any delay on the positive real axis. is is seen easily by considering only one delay h � h 1 . A similar conclusion follows for λ M . is concern is that we can expect from intuition; that is, the increase in the delay sizes translates into a more strict sufficiently type constraint on the "smallness" constraint on the initial conditions to guarantee that the solution is kept under a certain prescribed closed ball through time.
Two sufficiently type conditions of global asymptotic Lyapunov stability at exponential rate of (30) are obtained in the next result.

Proposition 2.
e following properties hold: (i) Assume that A 0 is a stability matrix such that ‖e A 0 t ‖ ≤ Ke − αt for all t ≥ 0 and some real constants α > 0 and K ≥ 1.

en, differential system (30) is exponentially stable (i.e., for any given bounded piecewise continuous function of initial conditions
i�0 A i is a stability matrix such that ‖e At ‖ ≤ Ke − αt for all t ≥ 0 and some real constants α > 0 and K ≥ 1 so that the delay-free version of (30) is exponentially stable. en, differential system (30) is Proof. From the first inequality in (32), it follows that so that, from Gronwall Lemma [1][2][3], and then On the other hand, one obtains from the first inequality in (35) and Gronwall Lemma that

. , ϑ) have piecewise continuous and bounded entries. (A7) the unforced system _ z(t) � A 0 (t)z(t) is exponentially stable, that is, its fundamental matrix of
i�1 e αh i ‖A i ‖, then differential system (30) is globally asymptotically stable, with the solution trajectory being constrained to the ball ‖x(t)‖ ≤ r; ∀t > 0, and x(t) ⟶ 0 at exponential rate − (α − K ϑ i�1 e αh i ‖A i ‖) as t ⟶ ∞. (ii) Assume that Assumption A6 holds and, furthermore, ‖(e αh i − 1))K)r, then ‖x(t)‖ ≤ r; ∀t > 0. Also, if α > K ϑ i�1 e αh i ‖A i ‖, then differential system (30) is exponentially stable, with the solution trajectory being constrained to the ball ‖x(t)‖ ≤ r; ∀t > 0, so that e following main total stability result gives sufficiencytype global stability conditions of the delayed system in the presence of nonlinearities which generalize those considered in eorem 1.

Remarks 2.
(1) It is convenient now to summarize at a first glance which are the basic ideas behind the above results concerning the studied time-delay system.
(2) Note that Proposition 2 gives two results of global asymptotic Lyapunov stability in the large (i.e., for any bounced piecewise continuous function of initial conditions). In this way, the solution trajectory converges asymptotically to the zero equilibrium point at exponential rate. e result is independent of the delay size type if the delays are small enough provided that either the auxiliary system with delay-International Journal of Differential Equations free dynamics of that with zero delay is exponentially stable. (3) Note also that Proposition 1 gives two results of global Lyapunov stability independent of the delay sizes for a certain range of values of the delays with the trajectory solution being constrained within a certain closed ball around the origin for t > 0 sufficiently small initial conditions provided that one of the above auxiliary unforced systems is exponentially stable. Proposition 3 states results on global stability and global asymptotic stability with the trajectory being constrained to as ball around the origin if one of the two mentioned auxiliary systems is exponentially stable. (4) eorem 2 gives two alternative results of solution uniqueness and global stability within a ball around the origin, based on either the stability of the matrix A 0 , corresponding to the system without delayed dynamics, or on the stability of the delay-free system of matrix dynamics A, in the case that systems (42) and (43) become forced under forcing functions with eventual delays which satisfy general nonlinear constraints based on extending those invoked in eorem 1 for the delay-free system. Both results are supported by the constraints that the stability of the matrices A 0 and A guarantee, respectively, the exponential stability of their respective unforced systems independent of the delay sizes within a range of values with a maximum bound.
In particular, Assumption A11 has a direct interpretation as follows: firstly, α > α 1 guarantees that the exponential stability of the unforced differential system _ z(t) � A 0 z(t), which describes the case of absence of delayed dynamics, under any finite bounded conditions z(0) � z 0 , guarantees that of system (30) for any bounded piecewise continuous function of initial conditions involving point delays for a certain maximum allowable size of the maximum delay. Secondly, if furthermore, in (46), α 2 is small enough to satisfy α 2 < α − α 1 , for given forcing functions f (·) , g (·) subject to Assumptions A9 and A10, i.e., such that Assumption A11, is also satisfied; then, the forcing system remains globally Lyapunov stable within the closed ball of radius r centred at the origin and satisfies (48). e alternative constraints invoked to get (49) are interpreted in a similar way based on the assumption of exponential stability under any finite initial conditions of the unforced delay-free differential system _ z(t) � Az(t), which describes the zerodelay case, that is, the case when h i � 0 for i � 1, 2, . . . , ϑ.
For the following discussion, recall that (F, G) is a controllable pair with F ∈ R n×n and G ∈ R m×n if and only if rank(G, FG, . . . , F n− 1 G) � n, equivalently, if and only if rank[sI n − F, G] � n for any s ∈ C which is not an eigenvalue of F (the Popov-Belevitch-Hautus controllability test). If (F, G) is controllable, then the spectrum of F + GK can be prefixed arbitrarily through the choice of K ∈ R m×n [4][5][6].
On the other hand, (F, G) is a stabilizable pair if and only if rank[sI n − F, G] � n for any s ∈ C with Res ≥ 0 which is not an eigenvalue of F (the Popov-Belevitch-Hautus stabilizability test). If (F, G) is stabilizable, then the spectrum of F + GK can be stabilized (but it cannot be either arbitrarily assigned or prefixed subject to a prefixed stability abscissa) through the choice of K ∈ R m×n so that, for some K ∈ R m×n , all the eigenvalues of F + GK can be allocated in the open complex left-hand-side plane Res < 0, [4,5].
It turns out that if (F, G) is controllable, then it is stabilizable, but the converse is not true. e subsequent results rely on the stabilization of (42) and (43) though a linear feedback control of the form u(t) � K 0 x(t).
Proof. Since (A 0 , B 0 ) is controllable, the spectrum (and then the stability abscissa) of any given matrix Λ 0 ∈ R n×n can be fixed by some existing K 0 ∈ R m×n which satisfies the matrix equality A 0 + B 0 K 0 � Λ 0 . So, for any given triple (a, β, c), it is possible to fix a stability abscissa (− α Λ 0 ) < 0 of some Λ 0 which satisfies Assumption A11, and then eorem 2 (i) holds.
Under stabilizability conditions, rather than controllability conditions, of the pair (A 0 , B 0 ), a weaker result than Corollary 5 is now given since the stability abscissa of A 0 + B 0 K 0 cannot be arbitrarily prefixed.

Data Availability
No data were used to support this study.

Conflicts of Interest
e author declares that he does not have any conflicts of interest.