Time-Interval Stability (PTI)

• Time-Interval Stability (PTI) is a rigorous framework that defines stability and performance metrics over a finite or prescribed time interval.
• It employs Lyapunov-theoretic approaches, convex criteria, and dwell-time analysis to assess and ensure system performance in networked, sampled, and switched systems.
• PTI techniques enable practical evaluation and synthesis by translating infinite-horizon stability concepts into finite-time, algorithmically tractable conditions for robust control.

Time-Interval Stability (PTI) refers to the set of rigorous system-theoretic properties, stability, and performance measures framed explicitly with respect to a finite or prescribed time interval, rather than classical notions over infinite-horizon or purely asymptotic behavior. PTI is central to modern control and analysis of networked, sampled, switched, and finite-time systems, where stability, dissipativity, and performance must be characterized or certified over bounded intervals—for instance in dwell-time, maximally allowable transfer intervals (MATI), or “prescribed-time” convergence and input-to-state stability. This article presents formal definitions, modeling frameworks, Lyapunov-theoretic and convex computational tools, roles of scheduling protocols and delays, and algorithmic synthesis procedures that together comprise the methodology of time-interval (PTI) stability.

1. Notions and Definitions of Time-Interval Stability

PTI encompasses several formally distinct, but closely related, notions all centering on the behavior of dynamical or hybrid systems over a finite or prescribed interval $[t_0, T]$:

• Finite-Time Stability (FTS): The state, or input-output map, is bounded in norm on $[t_0, T]$ under admissible disturbances or initial conditions, possibly with prescribed thresholds (Tommasi et al., 2011).
• Input-Output Finite-Time Stability (IO-FTS): For a system with state $x(t)$, input $u(t)$, and output $y(t)$, IO-FTS with respect to class $\mathcal{W}$, output weighting $Q(t)\succ0$, and horizon $\Omega=[t_0,T]$ is defined by

$$u(\cdot)\in\mathcal{W} \implies y(t)^\top Q(t) y(t) < 1,\ \forall t\in [t_0, T].$$

This generalizes both $L_2$-gain and ISS to finite horizons and arbitrary weighting (Tommasi et al., 2011).

• Prescribed-Time Stability (PTS), PT-ISS, and MATI: PTS refers to convergence, or ISS property, within an a priori designer-specified time interval $[t_0, T]$0, with rates enforced via blow-up functions $[t_0, T]$1 as $[t_0, T]$2. PT-ISS generalizes ISS to require convergence (and gain attenuation) within $[t_0, T]$3, using Lyapunov functions whose decrease is accelerated by $[t_0, T]$4 (Krishnamurthy et al., 2024). The Maximally Allowable Transfer Interval (MATI) denotes the maximum admissible sampling interval $[t_0, T]$5 under which robust $[t_0, T]$6-gain or stability properties are still provable (Tolic et al., 2016).
• Dwell-Time/PTI for Switched/Impulsive Systems: Here, time-interval stability is formulated via the dwell time between switches/impulses; the minimal (or maximal) dwell time $[t_0, T]$7 is the smallest (or largest) interval allowed between discrete events without violating prescribed stability/performance levels (Briat, 2018, Zakwan et al., 2020).

Notion	Key Property	Interval of Interest
FTS	State/output boundedness	$[t_0, T]$8
IO-FTS	Input-output gain bound	$[t_0, T]$9
MATI	Max. data transfer interval for $x(t)$0-gain	$x(t)$1
PT-ISS/PT-C	Convergence within $x(t)$2 via blow-up function	$x(t)$3
Dwell-Time	Min/Max switching/impulse interval for stability	$x(t)$4

2. Lyapunov-Theoretic and Convex Criteria

The establishment of PTI stability fundamentally relies on Lyapunov and dissipation inequalities parameterized over finite intervals, transmission intervals, or via clock/timer variables:

• Finite-Interval Lyapunov Functions and LMIs: For LTV/state-space systems, IO-FTS is characterized by existence of a differentiable positive-definite $x(t)$5 on $x(t)$6 such that the dissipation LMI

$x(t)$7

holds for all $x(t)$8, with $x(t)$9 and associated constraints on jump scale at switching or impulse times (Tommasi et al., 2011).

• Razumikhin-Type and Small-Gain Arguments: In sampled/distributed systems, Lyapunov–Razumikhin conditions built over inter-transmission intervals enforce decrease properties as long as

$u(t)$0

Jump conditions depend on scheduling protocol regularity, tightly linking MATI to $u(t)$1-gain via small-gain inequalities (Tolic et al., 2016).

• Blow-Up-Weighted Certificates: For PT-ISS and prescribed-time convergence, Lyapunov certificates $u(t)$2 or $u(t)$3 satisfy

$u(t)$4

with $u(t)$5 as $u(t)$6, imposing exponential decay in a warped time domain (Krishnamurthy et al., 2024).

• Timer/Clock-Variable and Dwell-Time LMIs: In switched/impulsive settings, Lyapunov functionals $u(t)$7, parameterized by a timer $u(t)$8, are coupled via flow (intra-interval) and jump (inter-interval) LMIs, which guarantee decrease on every interval of length at least $u(t)$9 (Briat, 2018, Zakwan et al., 2020).

3. Maximally Allowable Transfer Interval (MATI) and Impulsive Modeling

Time-interval stability in networked, sampled-data, or impulsive systems is quantitatively described using MATI, where the goal is to guarantee robust $y(t)$0-performance as a function of the transfer (sampling) interval. The closed-loop is modeled as an impulsive delay system,

$y(t)$1

with state $y(t)$2 and disturbance stack $y(t)$3, interspersed by jump/sampling instants $y(t)$4 such that $y(t)$5.

The MATI $y(t)$6 is algorithmically computed via:

1. Computation of error-flow gain $y(t)$7 and protocol-dependent constants $y(t)$8 for the chosen UGES protocol (e.g., Round Robin (RR), Try-Once-Discard (TOD)).
2. Searching over $y(t)$9, for $\mathcal{W}$0 such that the inequalities

$\mathcal{W}$1

with $\mathcal{W}$2 the $\mathcal{W}$3-gain of the nominal system, are satisfied.

$\mathcal{W}$4

This $\mathcal{W}$5 is the maximal sampling interval for which the desired $\mathcal{W}$6-performance holds (Tolic et al., 2016).

4. Dwell-Time and Finite-Interval Analysis of Switched/Impulsive Systems

For switched or impulsive systems subject to dwell-time constraints, time-interval stability is guaranteed by convex tests over the set of admissible intervals:

• Clock-Dependent Lyapunov-Krasovskii Functionals: For LPV systems with time-varying or constant delay and piecewise-constant switching, the existence of state- and parameter-dependent functionals $\mathcal{W}$7 satisfying families of LMIs (across $\mathcal{W}$8 and all parameter pairs) ensures stability under all switching laws adhering to a minimal dwell-time $\mathcal{W}$9 (Zakwan et al., 2020).
• Infinite-Dimensional Linear Programs and Sum-of-Squares (SOS) Relaxation: Dwell-time stability for positive impulsive systems is encoded by infinite-dimensional LPs in timer-dependent multipliers, with practical tractable relaxations via SOS polynomial optimization, facilitating algorithmic certification for range, minimal, maximal, or constant dwell-time scenarios (Briat, 2018).
• Worst-Case System Reductions: Under invertible or diagonal uncertainty structure, dwell-time stability can often be checked on a “worst-case” system by eliminating uncertain scalings, further streamlining verification (Briat, 2018).

Stability Method	Key Mathematical Tool	Reference
Lyapunov-Razumikhin for impulses	Impulsive-delay inequalities, small gain	(Tolic et al., 2016)
Clock-dependent LK functionals	Parameterized LMIs in $Q(t)\succ0$0	(Zakwan et al., 2020)
SOS/LP relaxation	Convex positivity of timer-dependent polynomials	(Briat, 2018)

5. Prescribed-Time Stability and Blow-Up Lyapunov Techniques

In prescribed-time (PT) stability, convergence and disturbance attenuation are imposed a priori within a designer-chosen time $Q(t)\succ0$1 via use of time-varying “blow-up” functions $Q(t)\succ0$2:

• $Q(t)\succ0$3 is a blow-up function if $Q(t)\succ0$4 as $Q(t)\succ0$5 and $Q(t)\succ0$6 as $Q(t)\succ0$7.
• Lyapunov certificates $Q(t)\succ0$8 must satisfy

$Q(t)\succ0$9

(cascade/feedback interconnections allow $\Omega=[t_0,T]$0 or $\Omega=[t_0,T]$1 cross-terms).

Key results include:

• Cascade and feedback interconnections with such properties combine via small-gain/diagonal stability conditions to yield PT-convergence for the full interconnected system.
• PT-ISS extends classical ISS by requiring both convergence in $\Omega=[t_0,T]$2 and uniform bounds in bounded disturbance (Krishnamurthy et al., 2024).

6. Computational and Algorithmic Procedures

The practical verification and controller synthesis for PTI stability typically follow an explicitly algorithmic workflow:

1. Model the system in an impulsive, switched, or clock-dependent framework, with timer $\Omega=[t_0,T]$3 or block-sampled structure, as appropriate to protocol or network scenario (e.g., NCSs, LPV, positive/distributed-delay, etc.) (Tolic et al., 2016, Zakwan et al., 2020, Briat, 2018).
2. Formulate Lyapunov (or Lyapunov–Razumikhin/Lyapunov–Krasovskii) conditions dependent on the chosen interval, timer, and possibly parameter, with protocol-dependent jump/flow/jump-scaling inequalities.
3. Cast the verification conditions as parametric LMIs (for LTV/LPV), infinite-dimensional LPs (for positive/special structure), or small-gain inequalities (for compositional analysis, e.g., MATI, PT-ISS).
4. For dwell- or sampling-interval-dependent properties, search over interval parameter (e.g., $\Omega=[t_0,T]$4) to maximize allowable interval or minimize conservative gain (supremal $\Omega=[t_0,T]$5, maximal $\Omega=[t_0,T]$6, minimal $\Omega=[t_0,T]$7).
5. Where needed, employ gridded parameterization, polynomial/SOS relaxation, or explicit worst-case system reduction for tractable convex optimization.
6. Apply controller synthesis extensions (e.g., clock-dependent gain-scheduled feedback, observer design) via augmented LMI/LP frameworks (Zakwan et al., 2020).

7. Applications and Connections

PTI stability criteria have found application across domains:

• Networked Control Systems (NCS): Sampling- and transmission-interval stability (MATI) under UGES protocols, with explicit computation for RR, TOD (Tolic et al., 2016).
• Switched and Impulsive Systems: Dwell-time constrained stability and $\Omega=[t_0,T]$8 performance for uncertain, positive, or delayed systems (Briat, 2018).
• Delayed, LPV, and Sampled Systems: Clock/Krasovskii-based dwell-time LMI analysis and gain-scheduling synthesis (Zakwan et al., 2020).
• Prescribed-Time Stabilization: Uniform convergence or ISS by any $\Omega=[t_0,T]$9 for nonlinear, interconnected systems with polynomial-type time warping (Krishnamurthy et al., 2024).
• Stroboscopic/Torus Invariant Analysis: Set-based explicit invariant “tubes” for periodic/parametric ODEs, certifying interval invariance and limit-cycle capture (Jerray et al., 2020).
• Biological/Physiological Time Intervals: The concept of “Physiological Temperature Interval” (PTI) formalizes critical maintenance intervals in biological systems directly in terms of molecular and thermodynamic bounds (Bulavin et al., 2013).

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Time-interval stability thus forms an essential conceptual and computational backbone for finite-horizon analysis of deterministic, switched, sampled, delayed, and networked systems—enabling robust certification and synthesis when only interval-wise, rather than asymptotic, properties are meaningful or achievable. Each instantiation—MATI, IO-FTS, PT-ISS, dwell-time—translates infinite-horizon stability analogues into explicit, quantifiable, algorithmically tractable finite-time regimes.

References: (Tolic et al., 2016, Tommasi et al., 2011, Krishnamurthy et al., 2024, Briat, 2018, Zakwan et al., 2020, Jerray et al., 2020, Bulavin et al., 2013)