Viability Kernels and Capture Basins of Sets Under Differential Inclusions

Abstract. The problem of determining invariance kernels for planar single-input nonlinear systems is considered. If K is a closed set, its invariance kernel is the largest subset of K with the property of being positively invariant for arbitrary measurable input signals. It is shown that the boundary of the invariance kernel is a concatenation of solutions of two so-called extremal vector fields. Moreover, only the solutions through a finite number of special points are of interest. This result makes it possible to devise an algorithm which determines the invariance kernel of a simply connected set in a finite number of steps.

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“…We stress that the proof of this result is completely analogous to the proof of Theorem 4.18 in [4].…”

Section: Properties Of Extremal Solutionsmentioning

confidence: 61%

Invariance kernels of single-input planar nonlinear systems

Maggiore¹,

Rawn²,

Lehn³

2011

Proceedings of the 2011 American Control Conference

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“…(47). Although the practice of PS techniques requires that ϵ ≥ ϵ m , the theory allows ϵ to go to zero in the limit…”

Section: A Standard Pseudospectral Optimal Control Theorymentioning

confidence: 99%

“…(61) and (62)] that was implemented at Honeywell was a form of tychastic control (or "robust" control). Standard minimax optimal control problems are connected to the computation of viability kernels and capture basins [7,47,48], which form the central concepts in viability theory.…”

Section: Further Applications and Open Problems Inmentioning

confidence: 99%

Riemann–Stieltjes Optimal Control Problems for Uncertain Dynamic Systems

Ross

Proulx

Karpenko

et al. 2015

Journal of Guidance, Control, and Dynamics

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Motivated by uncertain parameters in nonlinear dynamic systems, we define a nonclassical optimal control problem where the cost functional is given by a Riemann-Stieltjes "functional of a functional." Using the properties of Riemann-Stieltjes sums, a minimum principle is generated from the limit of a semidiscretization. The optimal control minimizes a Riemann-Stieltjes integral of the Pontryagin Hamiltonian. The challenges associated with addressing the noncommutative operations of integration and minimization are addressed via cubature techniques leading to the concept of hyper-pseudospectral points. These ideas are then applied to address the practical uncertainties in control moment gyroscopes that drive an agile spacecraft. Ground test results conducted at Honeywell demonstrate the new principles. The Riemann-Stieltjes optimal control problem is a generalization of the unscented optimal control problem. It can be connected to many independently developed ideas across several disciplines: search theory, viability theory, quantum control and many other applications involving tychastic differential equations.

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“…Reachability analysis allows us to guarantee that the system with a given control law will always reach a target [9]. In this paper, we deal with the problem of computing the set X 0 of all initial states x 0 = x(0), of a discrete time system such that at timek the state x(k) is inside a given set Y.…”

Section: Introductionmentioning

confidence: 99%

Chain of Set Inversion Problems Application to reachability analysis

Desrochers

Jaulin

2017

IFAC-PapersOnLine

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To cite this version:Benoît Desrochers, Luc Jaulin. Chain of set inversion problems; Application to reachability analysis. Abstract-This paper deals with the set inversion problem X = f −1 (Y) in the case where f : R n → R m depends on a parameter vector p ∈ R q which is known to be inside a box [p]. We show that for a large class of problems, we can obtain an accurate approximation of the solution set, without bisecting in the pspace. To do this, symbolic methods are required to cast our initial problem into a chain of set-inversion problems, the links of which have some nice properties with respect to p. As an application, we consider the problem of computing the set of all initial states of an uncertain discrete-time state system that reach a target set Y in a given time.

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Viability Kernels and Capture Basins of Sets Under Differential Inclusions

Cited by 72 publications

References 34 publications

Invariance kernels of single-input planar nonlinear systems

Invariance kernels of single-input planar nonlinear systems

Riemann–Stieltjes Optimal Control Problems for Uncertain Dynamic Systems

Chain of Set Inversion Problems Application to reachability analysis

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