Metric Regularity—a Survey Part 1. Theory

Ioffe, A. D.

doi:10.1017/s1446788715000701

Cited by 54 publications

(76 citation statements)

References 67 publications

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“…Let T : R ℓ → R n be a bounded linear operator. Then the Banach constant of T is defined as the quantity The Banach constant of a linear operator can be computed straightforwardly; see [10]. Combining Theorem 8, Lemma 10 with Definitions 5, 6, and 9 allows us to arrive at the following expressions for the exact bounds of linear openness and metric regularity of smooth mappings via the Banach constant.…”

Section: Linear Openness and Related Properties Of Nonlinear Mappingsmentioning

confidence: 99%

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A Variational Approach to Local Asymptotic and Exponential Stabilization of Nonlinear Systems

Christopherson

Jafari

Mordukhovich

2020

SN Oper. Res. Forum

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Local asymptotic stabilizability is a topic of great theoretical interest and practical importance. Broadly, if a system • x = f (x, u) is locally asymptotically stabilizable, we are guaranteed a feedback controller u(x) that forces convergence to an equilibrium for trajectories initialized sufficiently close to it. A necessary condition was given by Brockett: such controllers exist only when f is open at the equilibrium. Recently, Gupta, Jafari, Kipka and Mordukhovich [8] considered a modification to this condition, replacing Brockett's topological openness by the linear openness property of modern variational analysis. In this paper, we show that under the linear openness assumption there is a local diffeomorphism of neighborhoods of the equilibirum on which the system is exponentially stabilizable by means of continuous stationary feedback laws. Introducing a transversality property and relating it to the above diffeomorphism, we prove that linear openness and transversality on a punctured neighborhood of an equilibrium is sufficient for local exponential stabilizability of systems with a rank deficient linearization.The main result goes beyond the usual Kalman and Hautus criteria for the existence of exponential stabilizing feedback laws, since it allows us to handle systems for which exponential stabilization is achieved through higher-order terms. However, it is implemented so far under a rather restrictive row-rank conditions. This suggests a twofold approach to the use of these properties: a point-wise version is enough to ensure stability via the linearization, while a local version is enough to overcome deficiencies in the linearization.Theorem 8. Let f be of class C 1 in a neighborhood U × V ⊆ R ℓ × R n of the origin. Then f enjoys the equivalent linear openness and metric regularity properties around the origin if and only if the Jacobian J f (0,0) is of full rank. Furthermore, the exact bounds of the linear openness and metric regularity of f at the origin are precisely given by, respectively, covf (0, 0) = min J f (0,0) * v v = 1 and regf (0, 0) = J f (0,0)

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Section: Linear Openness and Related Properties Of Nonlinear Mappingsmentioning

confidence: 99%

“…is locally exponentially stabilizable by means of continuous stationary feedback laws. Further, it suffices to choose h as in (8) and (9), and the trivial feedback law u(x) = 0 yields the desired exponential stability for (10).…”

Section: Exponential Stabilizability Of Control Systems Via Compositimentioning

confidence: 99%

A Variational Approach to Local Asymptotic and Exponential Stabilization of Nonlinear Systems

Christopherson

Jafari

Mordukhovich

2020

SN Oper. Res. Forum

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“…Then f is a bi-Lipschitz homeomorphism around x 0 . More precisely, for each 0 < α < Reg Jf (x 0 ) there is an neighborhood V of x 0 contained in U with the following properties: [19] or exact covering bound of f around x 0 [28]-denoted by covf (x 0 ). So, in this context, covf (x 0 ) ≥ Reg Jf (x 0 ).…”

Section: Pseudo-jacobians and Local Inverse Theoremsmentioning

confidence: 99%

Surjection and inversion for locally Lipschitz maps between Banach spaces

Gutú

Jaramillo

2019

Journal of Mathematical Analysis and Applications

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We study the global invertibility of non-smooth, locally Lipschitz maps between infinite-dimensional Banach spaces, using a kind of Palais-Smale condition. To this end, we consider the Chang version of the weighted Palais-Smale condition for locally Lipschitz functionals in terms of the Clarke subdifferential, as well as the notion of pseudo-Jacobians in the infinite-dimensional setting, which are the analog of the pseudo-Jacobian matrices defined by Jeyakumar and Luc. Using these notions, we derive our results about existence and uniqueness of solution for nonlinear equations. In particular, we give a version of the classical Hadamard integral condition for global invertibility in this context.2000 Mathematics Subject Classification. 47J07, 58E05, 49J52.

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“…It was defined for F −1 via inequality (9). This property is called pseudo-calmness in [14], while the term linear recession is used in [19].…”

Section: Introductionmentioning

confidence: 99%

“…In Section 5, we analyze an inexact Newton-type iteration for the case when the function f in (16) is not necessarily differentiable. Specifically, we introduce a mapping H : X ⇒ L(X, Y ) viewed as a generalized set-valued derivative of the function f , and consider the following iteration: Given an index k ∈ N 0 and a point x k ∈ X, choose any A k ∈ H(x k ) and then find x k+1 ∈ X satisfying (19) f…”

Section: Introductionmentioning

confidence: 99%

On semiregularity of mappings

Cibulka

Fabian

Kruger

2019

Journal of Mathematical Analysis and Applications

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There are two basic ways of weakening the definition of the well-known metric regularity property by fixing one of the points involved in the definition. The first resulting property is called metric subregularity and has attracted a lot of attention during the last decades. On the other hand, the latter property which we call semiregularity can be found under several names and the corresponding results are scattered in the literature. We provide a self-contained material gathering and extending the existing theory on the topic. We demonstrate a clear relationship with other regularity properties, for example, the equivalence with the so-called openness with a linear rate at the reference point is shown. In particular cases, we derive necessary and/or sufficient conditions of both primal and dual type. We illustrate the importance of semiregularity in the convergence analysis of an inexact Newton-type scheme for generalized equations with not necessarily differentiable single-valued part.

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Metric Regularity—a Survey Part 1. Theory

Cited by 54 publications

References 67 publications

A Variational Approach to Local Asymptotic and Exponential Stabilization of Nonlinear Systems

A Variational Approach to Local Asymptotic and Exponential Stabilization of Nonlinear Systems

Surjection and inversion for locally Lipschitz maps between Banach spaces

On semiregularity of mappings

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