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Splitting Algorithms, Modern Operator Theory, and Applications pp 243–267Cite as

A Glimpse at Pointwise Asymptotic Stability for Continuous-Time and Discrete-Time Dynamics

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Abstract

Given a dynamical system, pointwise asymptotic stability, also called semistability, of a set requires that every point in the set be a Lyapunov stable equilibrium, and that every solution converge to one of the equilibria in the set. This note provides examples of pointwise asymptotic stability related to optimization and states select results from the literature, focusing on necessary and sufficient Lyapunov and Lyapunov-like conditions for and robustness of this stability property. Background on the classical asymptotic stability is included.

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Notes

  1. 1.

    The set-valued terminology in this note follows [56]. In particular, a set-valued mapping \(F:{\mathbb R}^n\rightrightarrows {\mathbb R}^n\) associates to each \(x\in {\mathbb R}^n\), a subset \(F(x)\subset {\mathbb R}^n\).

  2. 2.

    \(F:{\mathbb R}^n\rightrightarrows {\mathbb R}^n\) is locally bounded if for every bounded set \(C\subset {\mathbb R}^n\), F(C) :=⋃x ∈ C F(x) is bounded.

  3. 3.

    \(F:{\mathbb R}^n\rightrightarrows {\mathbb R}^n\) is outer semicontinuous at \(x\in {\mathbb R}^n\) if for every x i → x and every convergent y i ∈ F(x i), limi→∞ y i ∈ F(x). F is outer semicontinuous if it is outer semicontinuous at every \(x\in {\mathbb R}^n\). If F is locally bounded and has closed (hence compact) values, outer semicontinuity at x is equivalent to a property of a set-valued F often called upper semicontinuity at x: for every ε > 0, there exists δ > 0 such that \(F(x+\delta {\mathbb B})\subset F(x)+\varepsilon {\mathbb B}\). Here, and in the remainder of this note, \({\mathbb B}\subset {\mathbb R}^n\) is a closed unit ball centered at 0; \(x+\delta {\mathbb B}\) is the closed ball of radius δ centered at x; and \(F(x)+\varepsilon {\mathbb B}\) is the Minkowski sum of F(x) and \(\varepsilon {\mathbb B}\), i.e., \(\{y+z\, |\, y\in F(x),\ z\in \varepsilon {\mathbb B}\}\).

  4. 4.

    \(\overline { \mathop {\mathrm {con}}} f(x+\delta {\mathbb B})\) is the closure of the convex hull of \(f(x+\delta {\mathbb B})\), i.e., of the smallest convex set containing \(f(x+\delta {\mathbb B})\).

  5. 5.

    A square matrix M is stable, or Hurwitz, if all of its eigenvalues have negative real parts. For such a matrix and a linear differential equation \(\dot {x}=Mx\), the origin is not just (Lyapunov) stable but also attractive, and hence asymptotically stable.

  6. 6.

    A set-valued mapping \(M:{\mathbb R}^n\rightrightarrows {\mathbb R}^n\) is monotone if for every \(x,x'\in {\mathbb R}^n\), every v ∈ M(x), v′∈ M(x′), one has (x − x′) ⋅ (v − v′) ≥ 0. It is maximal monotone if it is monotone and its graph, \(\{(x,v)\in {\mathbb R}^{2n}\, |\, v\in M(x)\}\), cannot be enlarged without violating monotonicity. In particular, a linear M given by M(x) = Lx is monotone if and only if L is positive semidefinite, and if such M is monotone then it is maximal monotone.

  7. 7.

    For a set-valued mapping \(M:{\mathbb R}^n\rightrightarrows {\mathbb R}^n\), its effective domain, denoted \({ \mathop {\mathrm {dom}} \nolimits } M\), is the set \(\{x\in {\mathbb R}^n\, |\, M(x)\neq \emptyset \}\).

  8. 8.

    A monotone \(M:{\mathbb R}^n\rightrightarrows {\mathbb R}^n\) is strictly monotone if for every \(x,x'\in {\mathbb R}^n\) with x ≠ x′, every v ∈ M(x), v′∈ M(x′), one has (x − x′) ⋅ (v − v′) > 0, and strongly monotone if there exists ρ > 0 such that, for every \(x,x'\in {\mathbb R}^n\), every v ∈ M(x), v′∈ M(x′), one has (x − x′) ⋅ (v − v′) ≥ ρ∥x − x′∥2.

  9. 9.

    In systems theory, a system where ∥ϕ(t) − ψ(t)∥ is eventually decreasing to 0, for all solutions, often with appropriately understood uniform decrease rate over ∥ϕ(0) − ψ(0)∥ is called incrementally stable, see [3] and the references therein, and contractive if ∥ϕ(t) − ψ(t)∥ is decreasing, often at an exponential rate, see [2]. For applications of the contractive property, not related to monotonicity of the dynamics, see the survey [2] and the references therein.

  10. 10.

    A function \(f:{\mathbb R}^n\to {\mathbb R}\cup \{\infty \}\) is proper if it is not identically equal to ∞ and lsc if, for every \(x\in {\mathbb R}^n\) and every x i → x, liminfi→∞ f(x i) ≥ f(x). A useful condition, equivalent to f being proper, lsc, and convex is that the epigraph of f, namely the set \(\{(x,r)\in {\mathbb R}^n\, |\, r=f(x)\}\) be nonempty, closed, and convex.

  11. 11.

    The normal cone to a closed and convex set \(C\subset {\mathbb R}^n\) at x ∈ C is \(N_C(x)=\{v\in {\mathbb R}^n\, |\, v\cdot (x'-x)\leq 0\ \forall x'\in C\}\).

  12. 12.

    For a function \(f:{\mathbb R}^n\to {\mathbb R}\cup \{\infty \}\), \({ \mathop {\mathrm {dom}} \nolimits } f\) is the effective domain of f, i.e., the set \(\{x\in {\mathbb R}^n\, |\, f(x)\in {\mathbb R}\}\).

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Acknowledgements

This work was partially supported by the Simons Foundation Grant 315326.

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  1. Department of Mathematics and Statistics, Loyola University Chicago, Chicago, IL, USA

    Rafal Goebel

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  1. Department of Mathematics, University of British Columbia, Kelowna, BC, Canada

    Heinz H. Bauschke

  2. School of IT & Mathematical Sciences, University of South Australia, Mawson Lakes, SA, Australia

    Regina S. Burachik

  3. Inst. Numerische & Angewandte Mathematik, Universität Göttingen, Göttingen, Niedersachsen, Germany

    D. Russell Luke

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Goebel, R. (2019). A Glimpse at Pointwise Asymptotic Stability for Continuous-Time and Discrete-Time Dynamics. In: Bauschke, H., Burachik, R., Luke, D. (eds) Splitting Algorithms, Modern Operator Theory, and Applications. Springer, Cham. https://doi.org/10.1007/978-3-030-25939-6_10

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