7 Stability of discrete-time systems
The notions of the previous chapter for an autonomous discrete-time system \(x^{+} = f(k, x)\) (definition 94), under the same names. Discrete time is simpler in one respect: trajectories always exist and are unique (lemma 109, lemma 110), so every notion quantifies over trajectories on \([k_0, \infty )\) and no forward completeness is needed anywhere. Exponential estimates are geometric, \(\rho ^{k - k_0}\) with \(0 \le \rho {\lt} 1\), in place of \(e^{-\gamma (t - t_0)}\).
7.1 Definitions
The equilibrium \(x_{\mathrm{eq}}\) of an autonomous discrete-time system \(x^{+} = f(k, x)\) is stable when for every \(\varepsilon {\gt} 0\) and \(k_0 \in \mathbb {N}\) there is \(\delta = \delta (\varepsilon , k_0) {\gt} 0\) such that every trajectory on \([k_0, \infty )\) with \(\| x(k_0) - x_{\mathrm{eq}}\| {\lt} \delta \) satisfies \(\| x(k) - x_{\mathrm{eq}}\| {\lt} \varepsilon \) for all \(k \ge k_0\).
The counterpart of definition 275. Trajectories on rays suffice: a discrete-time trajectory exists from every state for all later times (lemma 109).
Reference: Khalil, Nonlinear Systems (3rd ed.), Definition 4.4; Jungers and van de Wouw (2026), Definition 2.
The equilibrium is unstable when it is not stable (definition 328).
The equilibrium is uniformly stable when \(\delta \) in definition 328 can be chosen independently of \(k_0\).
The equilibrium is globally uniformly stable when it is uniformly stable (definition 330) with a choice of \(\delta (\varepsilon )\) satisfying \(\delta (\varepsilon ) \to \infty \) as \(\varepsilon \to \infty \).
The equilibrium is locally asymptotically stable when it is stable (definition 328) and for every \(k_0\) there is \(c = c(k_0) {\gt} 0\) such that every trajectory on \([k_0, \infty )\) with \(\| x(k_0) - x_{\mathrm{eq}}\| {\lt} c\) satisfies \(x(k) \to x_{\mathrm{eq}}\). Unlike definition 279 no forward completeness is asked for: discrete-time trajectories always exist.
The equilibrium is globally asymptotically stable when it is stable (definition 328) and every trajectory on \([k_0, \infty )\) satisfies \(x(k) \to x_{\mathrm{eq}}\).
Trajectories starting within \(c\) of \(x_{\mathrm{eq}}\) have a uniform convergence time: for every \(\eta {\gt} 0\) there is \(T = T(\eta ) \in \mathbb {N}\) such that every trajectory on \([k_0, \infty )\) with \(\| x(k_0) - x_{\mathrm{eq}}\| {\lt} c\) satisfies \(\| x(k) - x_{\mathrm{eq}}\| {\lt} \eta \) for \(k \ge k_0 + T\).
The equilibrium is uniformly asymptotically stable when it is uniformly stable (definition 330) and there is \(c {\gt} 0\) from which convergence has a uniform convergence time (definition 334).
The equilibrium is globally uniformly asymptotically stable when it is globally uniformly stable (definition 331) and every \(c {\gt} 0\) admits a uniform convergence time (definition 334).
The equilibrium is locally exponentially stable, or geometrically stable, when there are \(c, C {\gt} 0\) and \(0 \le \rho {\lt} 1\) such that every trajectory on \([k_0, \infty )\) with \(\| x(k_0) - x_{\mathrm{eq}}\| {\lt} c\) satisfies
The counterpart of definition 284, with \(\rho = e^{-\gamma }\).
Reference: Jungers and van de Wouw (2026), Definition 2, estimate (6).
The equilibrium is globally exponentially stable when the estimate of definition 337 holds for every initial state, with no radius \(c\).
Ultimate boundedness is a property of the system near a point that need not be an equilibrium: trajectories end up, and stay, within a prescribed distance of it.
Trajectories are uniformly ultimately bounded with ultimate bound \(b\) around \(x_c\) when there is \(c {\gt} 0\) such that for every \(0 {\lt} a {\lt} c\) there is \(T = T(a, b) \in \mathbb {N}\) with
for every trajectory on \([k_0, \infty )\). The point \(x_c\) need not be an equilibrium.
Reference: Khalil, Nonlinear Systems (3rd ed.), Definition 4.6; Jungers and van de Wouw (2026), Definition 3.
Trajectories are globally uniformly ultimately bounded with ultimate bound \(b\) around \(x_c\) when definition 339 holds for every \(a {\gt} 0\), with no radius \(c\).
7.2 Implications
If \(x_{\mathrm{eq}}\) is stable (definition 328), then \(f(k, x_{\mathrm{eq}}) = x_{\mathrm{eq}}\) for every \(k\).
The trajectory from \(x_{\mathrm{eq}}\) at time \(k\) stays within every \(\varepsilon {\gt} 0\) of \(x_{\mathrm{eq}}\), so its value at \(k + 1\), which is \(f(k, x_{\mathrm{eq}})\), is at distance \(0\).
Fix \(\varepsilon {\gt} 0\) and \(k_0\). If for every \(\delta {\gt} 0\) some trajectory on \([k_0, \infty )\) starts within \(\delta \) of \(x_{\mathrm{eq}}\) and reaches distance at least \(\varepsilon \) from it, then \(x_{\mathrm{eq}}\) is unstable (definition 329).
A uniformly stable equilibrium (definition 330) is stable (definition 328).
A globally asymptotically stable equilibrium (definition 333) is locally asymptotically stable (definition 332).
A uniformly asymptotically stable equilibrium (definition 335) is locally asymptotically stable (definition 332). Unlike lemma 289, no forward completeness is needed.
A globally uniformly asymptotically stable equilibrium (definition 336) is globally asymptotically stable (definition 333).
A locally exponentially stable equilibrium (definition 337) is uniformly stable (definition 330).
The estimate bounds the distance by \(C\) times the initial distance, so \(\delta = \min (c, \varepsilon / C)\) works for every \(k_0\).
A locally exponentially stable equilibrium (definition 337) is uniformly asymptotically stable (definition 335).
Uniform stability is lemma 347. From the ball of radius \(c\) the distance after \(T\) steps is below \(C c \rho ^{T}\), which is below any \(\eta {\gt} 0\) for \(T\) large, independently of \(k_0\).
A globally exponentially stable equilibrium (definition 338) is locally exponentially stable (definition 337).
A globally exponentially stable equilibrium (definition 338) is globally uniformly asymptotically stable (definition 336).
\(\delta (\varepsilon ) = \varepsilon / C\) witnesses global uniform stability, and from every ball the estimate gives a uniform convergence time as in lemma 348.
7.3 Lyapunov’s direct method
For a time-invariant system \(x^{+} = f(x)\) the decrease of a Lyapunov function along the system is the difference \(\Delta V(x) = V(f(x)) - V(x)\). Unlike continuous time, the theorems need continuity of \(f\) for a reason of their own: a trajectory can jump over a sphere in one step, so the first-exit argument of continuous time is replaced by keeping one step from a small ball inside the region where \(V\) is controlled.
Let \(E\) be proper, and let the system be \(x^{+} = f(x)\) with \(f(x_{\mathrm{eq}}) = x_{\mathrm{eq}}\) and \(f\) continuous at \(x_{\mathrm{eq}}\). Let \(\mathcal{D}\) be a neighbourhood of \(x_{\mathrm{eq}}\) and \(V\) positive definite on \(\mathcal{D}\) about \(x_{\mathrm{eq}}\) (definition 27), with
Then \(x_{\mathrm{eq}}\) is uniformly stable (definition 330).
Continuity of \(f\) at \(x_{\mathrm{eq}}\) is not among the hypotheses of the reference, but it cannot be dropped: on \(\mathbb {R}\), \(V(x) = x^2/(1 + x^4)\) and \(f(x) = 1/x^2\) for \(0 {\lt} |x| {\lt} 1\) (\(f(x) = x\) otherwise) satisfy everything else, yet \(x_0 = 10^{-3}\) jumps to \(x_1 = 10^{6}\).
Reference: Jungers and van de Wouw (2026), Theorem 2, row 1.
Given \(\varepsilon \), shrink it so that the closed ball of radius \(\varepsilon '\) lies in \(\mathcal{D}\), and choose \(s \le \varepsilon '\) so that \(f\) maps the ball of radius \(s\) into the ball of radius \(\varepsilon '\). On the compact annulus \(s \le \| x - x_{\mathrm{eq}}\| \le \varepsilon '\), \(V\) has a positive minimum \(m\); choose \(\delta {\lt} s\) with \(V {\lt} m\) on the ball of radius \(\delta \). By induction, a trajectory with \(\| x(k_0) - x_{\mathrm{eq}}\| {\lt} \delta \) keeps \(\| x(k) - x_{\mathrm{eq}}\| {\lt} s\) and \(V(x(k)) {\lt} m\): the next state is within \(\varepsilon '\) by the choice of \(s\), has \(V(x(k + 1)) \le V(x(k)) {\lt} m\), and so cannot lie in the annulus. Nothing depends on \(k_0\), since a trajectory from \(k_0\) is the iterate sequence of \(x(k_0)\) (lemma 111).
Let \(E\) be proper, and let the system be \(x^{+} = f(x)\) with \(f(x_{\mathrm{eq}}) = x_{\mathrm{eq}}\). Let \(\mathcal{D}\) be a neighbourhood of \(x_{\mathrm{eq}}\) on which \(f\) is continuous, and \(V\) positive definite on \(\mathcal{D}\) about \(x_{\mathrm{eq}}\) (definition 27), with
Then \(x_{\mathrm{eq}}\) is locally asymptotically stable (definition 332).
Continuity of \(f\) on \(\mathcal{D}\) is not among the hypotheses of the reference, but it cannot be dropped. With \(V(x) = |x|\) on \(\mathbb {R}\), let \(f\) be odd with \(f(0) = 0\) and \(f(x) = (2^{-n-1} + x)/2\) on each interval \((2^{-n-1}, 2^{-n}]\): then \(\Delta V {\lt} 0\) away from \(0\), but a trajectory starting in \((2^{-n-1}, 2^{-n}]\) converges to \(2^{-n-1}\), at every scale.
Reference: Jungers and van de Wouw (2026), Theorem 2, row 2.
Stability is theorem 351. For attractivity, choose a closed ball \(K \subseteq \mathcal{D}\) around \(x_{\mathrm{eq}}\) and, by stability, a \(\delta \) whose trajectories stay in \(K\). Along such a trajectory \(V(x(k))\) is nonincreasing and bounded below, so it converges to some \(L\). A cluster point \(y\) of the trajectory, which exists in the compact set \(K\), has \(V(y) = L\); by continuity of \(f\) at \(y\), \(x(k + 1) = f(x(k))\) clusters at \(f(y)\), so \(V(f(y)) = L\) too. Then \(\Delta V(y) = 0\), so \(y = x_{\mathrm{eq}}\). A sequence in a compact set whose only cluster point is \(x_{\mathrm{eq}}\) converges to it.
Let \(E\) be proper, and let the system be \(x^{+} = f(x)\) with \(f\) continuous and \(f(x_{\mathrm{eq}}) = x_{\mathrm{eq}}\). Let \(V\) be positive definite on \(E\) about \(x_{\mathrm{eq}}\) (definition 27), with compact sublevel sets \(\{ V \le c\} \) (equivalently, for continuous \(V\), radially unbounded: lemma 316), and
Then \(x_{\mathrm{eq}}\) is globally asymptotically stable (definition 333).
Continuity of \(f\) is not among the hypotheses of the reference but cannot be dropped: with \(V(x) = |x|\) on \(\mathbb {R}\), the map \(f(x) = x/2\) for \(|x| \le 1\), \(f(x) = \operatorname {sign}(x)(1 + |x|)/2\) otherwise, has \(\Delta V {\lt} 0\) away from \(0\), yet the trajectory from \(2\) converges to \(1\).
Reference: Jungers and van de Wouw (2026), Theorem 2, row 3.
Stability is theorem 351 with \(\mathcal{D} = E\). From any initial state, \(V(x(k)) \le V(x(k_0))\), so the trajectory stays in the compact sublevel set \(\{ V \le V(x(k_0))\} \), and the \(\omega \)-limit argument of theorem 352 applies there.
Let the system be \(x^{+} = f(x)\) with \(f(x_{\mathrm{eq}}) = x_{\mathrm{eq}}\) and \(f\) continuous at \(x_{\mathrm{eq}}\), and let \(\mathcal{D}\) be a neighbourhood of \(x_{\mathrm{eq}}\). Let \(0 {\lt} c_1 \le c_2\), \(p {\gt} 0\) and \(0 {\lt} c_3 {\lt} 1\), and let \(V : E \to \mathbb {R}\) satisfy, for every \(x \in \mathcal{D}\),
Then \(x_{\mathrm{eq}}\) is locally exponentially stable (definition 337).
Continuity of \(f\) at \(x_{\mathrm{eq}}\) is not among the hypotheses of the reference, but without it a trajectory can leave \(\mathcal{D}\) in a single step.
Reference: Jungers and van de Wouw (2026), Theorem 2, row 4.
Choose \(r_0 {\gt} 0\) so that the ball \(B(x_{\mathrm{eq}}, r_0)\) lies in \(\mathcal{D}\) and \(f\) maps it into \(\mathcal{D}\), and let \(C = (c_2/c_1)^{1/p} \ge 1\) and \(\delta = r_0 / (2C)\). By induction, a trajectory starting within \(\delta \) stays in \(B(x_{\mathrm{eq}}, r_0)\) and satisfies \(V(x(k)) \le (1 - c_3)^{k - k_0} V(x(k_0))\): the decrease applies at \(x(k) \in \mathcal{D}\), and \(x(k + 1) = f(x(k)) \in \mathcal{D}\), so the sandwich gives \(\| x(k + 1) - x_{\mathrm{eq}}\| \le C\delta = r_0/2\). The geometric bound with \(\rho = (1 - c_3)^{1/p}\) then follows as in theorem 355.
Let the system be \(x^{+} = f(x)\), let \(0 {\lt} c_1 \le c_2\), \(p {\gt} 0\) and \(0 {\lt} c_3 {\lt} 1\), and let \(V : E \to \mathbb {R}\) satisfy, for every \(x\),
Then \(x_{\mathrm{eq}}\) is globally exponentially stable (definition 338), with \(C = (c_2/c_1)^{1/p}\) and \(\rho = (1 - c_3)^{1/p}\). No continuity of \(f\) is needed.
Reference: Jungers and van de Wouw (2026), Theorem 2, row 4, on the whole space. They ask for \(p \ge 1\); \(p {\gt} 0\) suffices.
The decrease condition is \(V(f(x)) \le (1 - c_3) V(x)\), so by induction \(V(x(k)) \le (1 - c_3)^{k - k_0} V(x(k_0))\). Sandwiching,
and taking \(p\)-th roots gives the geometric bound.