Practical Stabilization of Switched Affine Systems under Dwell Time: From Polytopic to p-Norm Lyapunov Functions

I. INTRODUCTION

• Under dwell-time constraints, switched affine systems with no common equilibrium at the desired point can generally achieve only practical stabilization, with trajectories converging to and remaining in a desired neighborhood.Existing approaches largely use quadratic Lyapunov functions, while practical target regions may be specified by separate bounds on state variables.Source: §I · p. 1
• The weighted p-norm family recovers the quadratic case at p = 2 and approaches the polytopic case as p grows, providing a smooth route from polytopic certificates to functions usable in switching strategies.The target-region motivation is that separate state bounds define polytopes, whereas quadratic sublevel sets are ellipsoidal.Source: §I · pp. 1–2

A. Problem Statement

• The objective is a state-dependent switching strategy that guarantees UGAS under arbitrarily fast switching and practical UGAS under dwell-time constraints.The analysis focuses on the origin; results for another operating point follow by a change of variables.Source: A. Problem Statement · p. 2

B. Weighted p-norm Lyapunov Functions

• For every finite p > 1, the paper characterizes Hurwitz matrices by the existence of weighted p-norm Lyapunov functions, extending the classical quadratic Lyapunov equivalence constructively.The proof uses Jordan decomposition, and the resulting matrix construction can be chosen independently of p.Source: B. Weighted p-norm Lyapunov Functions · p. 3

C. Hybrid Control Strategy

• For weighted p-norm Lyapunov functions of the average system, at least one subsystem has a strictly negative directional derivative at every nonzero state, enabling steepest-decrease mode selection.The corresponding property is guaranteed for quadratic functions but may fail for polytopic functions.Source: C. Hybrid Control Strategy · pp. 3–4
• The switching law uses flow and jump sets with a design parameter η ∈ (0, 1), and jumps select a mode meeting a stricter Lyapunov-decrease condition than flow requires.The resulting hybrid system has a Lyapunov function that does not increase across jumps and decreases along flows outside the origin.Source: C. Hybrid Control Strategy · p. 4
• The proposed hybrid strategy renders the origin UGAS under the stated assumption, using a mode whose Lyapunov derivative is strictly stronger than the flow condition at jumps.The jump places nonzero post-jump states in the interior of the flow set.Source: C. Hybrid Control Strategy · p. 4
• The steepest-decrease result does not extend directly to polytopic Lyapunov functions without further assumptions, because the required derivative property is not guaranteed for that class.This qualification distinguishes the smooth weighted p-norm strategy from a direct polytopic substitution.Source: C. Hybrid Control Strategy · pp. 3–4

III. PRACTICAL STABILIZATION UNDER DWELL-TIME

• Dwell-time constraints motivate replacing UGAS of the origin with practical stability, in which solutions converge to and remain within a neighborhood whose size decreases as minimum dwell time is reduced.The proposed space regularization inhibits switching near the origin; time regularization enforces a minimum dwell time using an auxiliary timer state.Source: §III · pp. 4–5
• Both regularization approaches characterize practical convergence regions as sublevel sets of the weighted p-norm Lyapunov function.The approach adapts the two regularization techniques introduced in.Source: §III · p. 5

A. Space Regularization

• For any ε > 0, the space-regularized system makes Aε = { (x, σ) : VR,p(x) ≤ ε } UGAS and guarantees a minimum dwell time T(ε) between consecutive switches.An initial jump may still occur within [0, T(ε)).Source: A. Space Regularization · p. 5
• Increasing ε enlarges the flow set and minimum dwell time, but also enlarges the convergence set Aε.Source: A. Space Regularization · p. 5

B. Time Regularization

• Time regularization augments the state with a timer and forces flow for at least T time units after each jump, regardless of the Lyapunov derivative condition.After the timer reaches T, the original flow and jump conditions resume; T can be chosen small enough to keep the Lyapunov derivative negative during these post-jump intervals.Source: B. Time Regularization · p. 5
• For T in (0, TM), Theorem III.4 provides an initial-condition-dependent bound and ensures t ≥ (j − 1)T along solutions.The result also supplies a strictly increasing ε(T) that tends to zero as T tends to zero.Source: B. Time Regularization · p. 5
• When τ0 < T, the initial forced flow may cause temporary exit from AT, so AT is not UGAS; choosing τ0 ≥ T removes this overshoot term from the bound.The overshoot is attributed to time regularization, not to the proof.Source: B. Time Regularization · p. 5

IV. CONSTRUCTING THE p-NORM LYAPUNOV FUNCTION

• If ΨR,∞(x) = ∥Rx∥∞ describes the desired operating region and is a Lyapunov function for the average system, the method seeks a corresponding weighted p-norm Lyapunov function for sufficiently large p.Source: §IV · pp. 5–6

A. A Similarity Condition and Matrix Measures

• A sufficient certificate links A and R through an auxiliary matrix Γ, with μp(Γ) certifying that VR,p satisfies the Lyapunov condition.When R has full column rank, Γ can be parameterized using a free matrix Z.Source: A. A Similarity Condition and Matrix… · p. 6
• The conditions in Lemma IV.1 are sufficient, but not necessary in general, for VR,p to satisfy the Lyapunov condition.The paper refers to for cases where necessity also holds.Source: A. A Similarity Condition and Matrix… · p. 6
• Because μp lacks a closed-form expression for general p, the method first certifies R and Γ using the tractable μ∞ condition, then infers stability for sufficiently large p.Source: A. A Similarity Condition and Matrix… · p. 6

B. Step 1: Certifying a Polytopic Lyapunov Function

• For ΨR,∞(x) = ∥Rx∥∞, being a Lyapunov function for the average system is equivalent to the existence of Γ with μ∞(Γ) < 0 and ΓR = RAλ∗.The condition is presented as necessary and sufficient for the polytopic Lyapunov function.Source: B. Step 1: Certifying a Polytopic… · p. 6
• Given R and Γ satisfying the Step 1 certificate, Step 2 determines which p values make VR,p a Lyapunov function for the average system.Source: B. Step 1: Certifying a Polytopic… · p. 6

C. Step 2: Smoothing the Polytopic Lyapunov Function

• Matrix-measure interpolation bounds μp using extreme cases, avoiding direct evaluation of the generally intractable μp measure.If both μ2(Γ) and μ∞(Γ) are negative, then μp(Γ) is negative for every p > 2.Source: C. Step 2: Smoothing the Polytopic… · pp. 6–7
• When μ∞(Γ) < 0 and μ1(Γ) ≥ 0, Theorem IV.5 states that μp(Γ) < 0 for all p ∈ (p, ∞).Source: C. Step 2: Smoothing the Polytopic… · p. 7
• The required threshold p increases as the off-diagonal entries of M(Γ) become more asymmetric.Source: C. Step 2: Smoothing the Polytopic… · p. 7
• The two-step method turns a polytopic certificate into a valid weighted p-norm Lyapunov function for sufficiently large p, usable in the hybrid-control strategies with convergence regions approximating the prescribed polytope.The paper presents this as a smooth p-norm extension of the steepest-decrease strategy and a constructive route from polytopic certificates.Source: C. Step 2: Smoothing the Polytopic… · p. 7

V. EXAMPLES

• In this example, the polytopic Lyapunov function fails to stabilize the system, whereas weighted p-norm Lyapunov functions certify the average system for every p in (1, ∞).The example also notes that p has limited practical impact under unconstrained switching; its main benefit is under dwell-time constraints, where the convergence-region geometry depends on p.Source: §V · p. 7

A. Space Regularization

• Compared with quadratic p = 2, under space regularization with η = 0.5 across 500 initial conditions, minimum exterior dwell-time reaches 4.67× at p = 4 then declines; minimum interior reaches 1.55× at p = 4, improving until p ≈ 17 before declining.The mean exterior dwell-time increases by a factor of 3.79 at p = 4.Source: Figure 1 · p. 8, A. Space Regularization · p. 8

B. Time Regularization

• With time regularization, increasing the minimum dwell-time parameter lowers switching frequency but yields convergence to a larger neighborhood of the origin.The trajectories compare T1 = 0.06 with T2 = 0.03.Source: B. Time Regularization · p. 8

VI. CONCLUSION

• The paper generalizes the weighted p-norm Lyapunov characterization and gives a two-step construction from a polytopic certificate to a smooth valid function for sufficiently large p.The same matrix R is valid once the polytopic certificate µ∞(Γ) < 0 holds.Source: §VI · p. 8

• The appendix identifies separate lemmas for complex-eigenvalue and real-eigenvalue Jordan blocks, then combines their results through the blockwise construction.Source: Appendix A · pp. 8–10
• A block-diagonal Jordan matrix admits a constructive global p-norm Lyapunov function by combining Lyapunov functions for its individual blocks.The combined decay rate is the minimum of the blockwise rates.Source: Appendix A · pp. 8–9
• For any ε in (0, ζ) and p in (1, ∞), the function VR(ε,n,m),p(z) satisfies its stated condition with β := p(ζ − ε) > 0.Source: Appendix A · pp. 9–10
• For real Jordan blocks with eigenvalue s < 0, the construction uses ε < |s| and gives decay rate β = p(|s| − ε) > 0.Source: Appendix A · p. 10

APPENDIX B OUTER SEMICONTINUITY OF G

• The switching map G is outer semicontinuous: limits of selected modes remain among the minimizing modes at the limiting state.The proof uses continuity of the minimum of finitely many subsystem derivative functions and a contradiction argument.Source: Appendix B · pp. 10–11

APPENDIX C PROOFS OF THEOREMS III.1 AND III.4

• The regularization proofs bound variation in the Lyapunov derivative because both strategies permit the state to evolve where the Lyapunov function may increase.Source: Appendix C · p. 11

C.1 Instrumental Results

• The gradient of the weighted p-norm Lyapunov function is locally Hölder continuous, with exponent α = min{1, p − 1}.The supplied results treat p ∈ (1, 2] and p > 2 separately when bounding this regularity.Source: §C.1 · p. 11
• For affine subsystem trajectories, Lemma C.2 bounds the distance from the initial condition by integrating the dynamics and applying Grönwall’s inequality.The proof introduces the displacement ς(t) = ψ(t) − x and bounds its p-norm.Source: §C.1 · p. 11
• Auxiliary inequalities for powers of sums support the subsequent bounds on Lyapunov variation.The supplied proof uses concavity for p ∈ (0, 1] and Jensen’s inequality for p > 1.Source: §C.1 · pp. 11, 13
• Lemma C.4 establishes a state-independent K∞ bound for the variation of the time derivative of VR,p along subsystem trajectories.The bound’s comparison function depends on A, b, R, and p, but not on the initial state x.Source: §C.1 · pp. 11–12

C.2 Proof of Theorem III.1 (Space Regularization)

• A uniform time T(ε) exists such that the Lyapunov decrease condition holds over [0, T(ε)] for states in Dε.This uses continuity and a uniform bound on derivative variation along subsystem trajectories.Source: §C.2 · p. 13
• The proof uses VR,p to define Aε and W, then bounds W using the comparison between VS,p and VR,p.The certificate functions are comparable because R and S have full column rank.Source: §C.2 · p. 13
• Aε is UGAS for Hε: trajectories reach the region and remain there, with distance to it bounded by a KL function.The proof constructs a KL bound from exponential decay of W outside Aε and forward invariance once inside.Source: §C.2 · pp. 13–14

C.3 Proof of Theorem III.4 (Time Regularization)

• The timer enforces at least T time units between jumps, yielding t ≥ (j − 1)T along every solution.After each jump the timer resets to zero, and its rate is one until it reaches T.Source: §C.3 · p. 14
• Forced flow can increase VR,p, so the proof bounds that increase and subtracts ε(T) to control W.The tolerance ε(T) tends to zero as T tends to zero.Source: §C.3 · p. 14
• During regular flow, the gradient condition supports exponential decay, and the proof combines this with forced-flow bounds to obtain a KL estimate.The argument separates each flow interval into timer-forced and regular phases.Source: §C.3 · pp. 14–15

APPENDIX D PROOF OR LEMMA IV.3

• The weighted infinity norm is ΨR,∞(x) = ∥Rx∥∞ = maxi ˆRi:x.This expresses the norm as the maximum over the rows of the augmented matrix ˆR.Source: Appendix D · p. 15
• There exists a Metzler matrix H ∈ R^{2r×2r} satisfying H R̂ = R̂A and H1 ⪯ −β1; writing H in blocks and setting Γ := H11 − H12 gives ΓR = RA.Source: Appendix D · p. 15
• Conversely, if ΓR = RA and μ∞(Γ) < 0, the proof constructs a Metzler matrix H that satisfies the required inequalities.It sets β := −μ∞(Γ) > 0 and invokes the stated matrix theorem to conclude.Source: Appendix D · p. 16