Pursuit–Evasion Problems and Viscosity Solutions of Isaacs Equations

Soravia, Pierpaolo

doi:10.1137/0331027

Cited by 48 publications

(37 citation statements)

References 25 publications

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“…One difficulty stems from the fact that the problem belongs to the class of "free boundary problems". This kind of problems was studied for instance in [2,36,37] and the results from [37] play a crucial role in our analysis in order to cope with this problem, cf. Proposition 12.…”

Section: Introductionmentioning

confidence: 99%

Differential Games and Zubov's Method

Grüne¹,

Serea²

2011

SIAM J. Control Optim.

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Section: Introductionmentioning

confidence: 99%

Differential Games and Zubov's Method

Grüne¹,

Serea²

2011

SIAM J. Control Optim.

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“…The following characterization of w.c. stability in terms of w.c. Lyapunov functions extends some results by Soravia [28,29] to the case of unbounded disturbances. In particular it gives a formula for the construction of a Lyapunov function in the case of asymptotic w.c. stability.…”

Section: As Stabilizability Is Equivalent To Wc Stabilitymentioning

confidence: 66%

“…We remark that this property is stronger than stability in probability and pathwise stability, and in fact it is never verified by a nondegenerate process. Worst case stability for bounded disturbances was studied by Soravia [28,29] by means of Lyapunov functions satisfying first order partial differential inequalities in the viscosity sense, and he also treated the case of unbounded noise in the framework of H ∞ control [30,31]. In Section 3 we extend his results on worst case stability to unbounded disturbances and show that the existence of a w.c. Lyapunov function is equivalent to w.c. stability.…”

Section: Introductionmentioning

confidence: 98%

“…When l is positive definite, using (19) again, the properties of W and standard properties of the trajectories of the system, we get in addition that x(t; γ(β), β) → 0 as t → +∞, and this gives the w.c. asymptotic stability. For more details see [29] for the w.c. stability and [28] for the w.c. asymptotic stability. Conversely, given a w.c. stable differential system, a candidate Lyapunov function W is constructed as follows: it is the lower semicontinuous envelope W * of the lower value function…”

Section: Theorem 32 Assume (4) (6) Then the System (Ds) Is Wc Smentioning

confidence: 99%

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Almost sure properties of controlled diffusions and worst case properties of deterministic systems

Bardi

Cesaroni

2008

ESAIM: COCV

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Abstract.We compare a general controlled diffusion process with a deterministic system where a second controller drives the disturbance against the first controller. We show that the two models are equivalent with respect to two properties: the viability (or controlled invariance, or weak invariance) of closed smooth sets, and the existence of a smooth control Lyapunov function ensuring the stabilizability of the system at an equilibrium.Mathematics Subject Classification. 93D09, 93E15, 49L25, 49N70.

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“…We refer to [21,11,1,19] and references therein for an introduction and some results for deterministic two-player games with infinite horizon.…”

Section: Introductionmentioning

confidence: 99%

Minimal Time Problems With Moving Targets and Obstacles

Bokanowski

Zidani

2011

IFAC Proceedings Volumes

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Abstract. We consider minimal time problems governed by nonlinear systems under general time dependent state constraints and in the twoplayer games setting. In general, it is known that the characterization of the minimal time function, as well as the study of its regularity properties, is a difficult task in particular when no controllability assumption is made. In addition to these difficulties, we are interested here to the case when the target, the state constraints and the dynamics are allowed to be time-dependent.We introduce a particular "reachability" control problem, which has a supremum cost function but is free of state constraints. This auxiliary control problem allows to characterize easily the backward reachable sets, and then, the minimal time function, without assuming any controllability assumption. These techniques are linked to the well known level-set approaches. Partial results of the study have been published recently by the authors in SICON. Here, we generalize the method to more complex problems of moving target and obstacle problems.Our results can be used to deal with motion planning problems with obstacle avoidance.

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Pursuit–Evasion Problems and Viscosity Solutions of Isaacs Equations

Cited by 48 publications

References 25 publications

Differential Games and Zubov's Method

Differential Games and Zubov's Method

Almost sure properties of controlled diffusions and worst case properties of deterministic systems

Minimal Time Problems With Moving Targets and Obstacles

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