Some regularity results for a class of upper semicontinuous functions

Marigonda, Antonio; Nguyen, Khai T.; Vittone, Davide

doi:10.1512/iumj.2013.62.4896

Cited by 7 publications

(9 citation statements)

References 25 publications

(15 reference statements)

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“…However C may fail to enjoy the ρ-internal sphere condition (an example is given by taking a square in R 2 , where the internal sphere property fails at the vertices); 2. if K is has positive reach, then R d \ K enjoys the ρ-internal sphere condition, with ρ = reach K; 3. in general, if R d \ K enjoys just the ρ-internal sphere condition we have that K may not have positive reach, additional hypotheses are required (see references below); 4. if K is a compact set with C 1,1 boundary then both K and R d \ K have positive reach (possibly reach K = reach R d \ K). We refer the reader to [21] and [8] for further details and applications, and to [19] for a generalized version of these results.…”

Section: Small-time Attainability In Control Systemsmentioning

confidence: 99%

Controllability of Some Nonlinear Systems with Drift via Generalized Curvature Properties

Marigonda¹,

Rigo²

2015

SIAM J. Control Optim.

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We discuss the problem of local attainability for finite-dimensional nonlinear control systems with quite general assumptions on the target set. Special emphasis is given to control-affine systems with a possibly nontrivial drift term. To this end, we provide some sufficient conditions ensuring local attainability, which involve geometric properties both of the target itself (such as a notion of generalized curvature), and of the Lie algebra associated with the control system. The main technique used is a convenient representation formula for the power expansion of the distance function along the trajectories, made at points sufficiently near to the target set.We notice that Petrov's condition is a first order condition in the sense that it involves only admissible velocities.From another point of view, this condition requires d S to be a sort of Lyapunov function for the system. Indeed, from a geometrical point of view, Petrov's condition states that at every point x of a neighborhood of the target there exists an admissible control u x ∈ U such that the corresponding trajectory points sufficiently toward the target. Moreover, the scalar product between the admissible velocity f (x, u x ) and the gradient of the distance is uniformly bounded away from zero.Petrov's condition is very strong, even if it is weaker than full controllability, which requires that every initial state can be steered to any final state in finite time along admissible trajectories. Moreover, it can be also shown that Petrov's condition is equivalent to the Lipschitz continuity of the minimal time function T up to the boundary of S (see [23]). However, it is also very easy to give simple examples where it fails. For instance, in R 2 take S = {0} and (ẋ(t),ẏ(t)) = (y(t), u(t)), where u : R → [−1, 1] is measurable: Petrov's condition fails on the x-axis.1.3. Higher order condition for pointwise target. When Petrov's condition is not satisfied, i.e., the trajectories of the system do not approach the target at the first order, it is natural to search for higher order conditions, which involve higher order terms in a convenient expansion of the trajectory itself. These conditions will be related to some properties of the Lie algebra generated by the family of vector fields associated with the system (see [15] for a complete introduction).In the early 1960s, Kalman proved the following result. Assume that f is linear, i.e., f (x, u) = Ax + Bu, where A ∈ Mat n×n (R), B ∈ Mat n×m (R) are two constant matrices, and S = {0}. Then the following are equivalent: 1. the system is controllable to the equilibrium point 0, i.e., every point can be steered to the origin in finite time; 2. the matrix (B|AB|A 2 B| . . . |A n−1 B) has full rank (equals n). The second condition above is the celebrated Kalman rank condition, and implies the Hölder continuity of T , with exponent depending on the smallest 0 ≤ k ≤ n − 1 such that the matrixhas full rank.Later, in the 1970s, several generalizations, mainly concerning the case when target set S is an equilibrium po...

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Section: Small-time Attainability In Control Systemsmentioning

confidence: 99%

Controllability of Some Nonlinear Systems with Drift via Generalized Curvature Properties

Marigonda¹,

Rigo²

2015

SIAM J. Control Optim.

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“…We will refer to this property as N-regularity (see Definition 2). We state our first general result, whose main ideas were presented in our recent paper [14], for closed set K ⊂ R d+1 concerning the structure and dimension of the set K (j) of points on ∂K where the Fréchet normal cone to ∂K has dimension larger than or equal to j. This result generalizes a similar result proved by Federer for sets with positive reach.…”

Section: Introductionmentioning

confidence: 55%

BV Regularity and Differentiability Properties of a Class of Upper Semicontinuous Functions

Marigonda

Nguyen

Vittone

2014

Large-Scale Scientific Computing

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“…Therefore our rectifiability results for S imply that T is of class C 1,1 outside a closed H N −1 -rectifiable set. We observe that, for general functions whose epigraph satisfies a uniform external sphere condition, the Hausdorff dimension of the set of non-Lipschitz points was proved to be less or equal to n − 1/2, with an example showing the sharpness of the estimate (see [21,Theorem 1.3 and Proposition 7.3]). The present paper therefore improves that result, for the particular case of a minimum time function.…”

Section: Introductionmentioning

confidence: 95%

Non-Lipschitz points and the $${\textit{SBV}}$$ SBV regularity of the minimum time function

Colombo¹,

Nguyen²,

Nguyen³

2013

Calc. Var.

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This paper is devoted to the study of the Hausdorff dimension of the singular set of the minimum time function T under controllability conditions which do not imply the Lipschitz continuity of T . We consider first the case of normal linear control systems with constant coefficients in R N . We characterize points around which T is not Lipschitz as those which can be reached from the origin by an optimal trajectory (of the reversed dynamics) with vanishing minimized Hamiltonian. Linearity permits an explicit representation of such set, that we call S. Furthermore, we show that S is H N−1 -rectifiable with positive H N−1 -measure. Second, we consider a class of control-affine planar nonlinear systems satisfying a second order controllability condition: we characterize the set S in a neighborhood of the origin in a similar way and prove the H 1 -rectifiability of S and that H 1 (S) > 0. In both cases, T is known to have epigraph with positive reach, hence to be a locally BV function (see [13,15]). Since the Cantor part of DT must be concentrated in S, our analysis yields that T is SBV , i.e., the Cantor part of DT vanishes. Our results imply also that T is locally of class C 1,1 outside a H N−1 -rectifiable set. With small changes, our results are valid also in the case of multiple control input.

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Some regularity results for a class of upper semicontinuous functions

Cited by 7 publications

References 25 publications

Controllability of Some Nonlinear Systems with Drift via Generalized Curvature Properties

Controllability of Some Nonlinear Systems with Drift via Generalized Curvature Properties

BV Regularity and Differentiability Properties of a Class of Upper Semicontinuous Functions

Non-Lipschitz points and the $${\textit{SBV}}$$ SBV regularity of the minimum time function

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