Semiconcave Functions Research Articles

Viscosity solutions of centralized control problems in measure spaces

This work focuses on a control problem in the Wasserstein space of probability measures over $\mathbb{R}^d$. Our aim is to link this control problem to a suitable Hamilton-Jacobi-Bellman (HJB) equation. We explore a notion of viscosity solution using test functions that are locally Lipschitz and locally semiconvex or semiconcave functions. This regularity allows to define a notion of viscosity and a Hamiltonian function relying on directional derivatives. Using a generalization of Ekeland's principle, we show that the corresponding HJB equation admits a comparison principle, and deduce that the value function is the unique solution in this viscosity sense. The PDE tools are developed in the general framework of Measure Differential Equations.

Topology of singular set of semiconcave function via Arnaud's theorem

We proved the (local) path-connectedness of certain subset of the singular set of semiconcave functions with linear modulus in general. The proof is based on a theorem by Marie-Claude Arnaud ([4]). We also gave a new proof of the theorem in time-dependent case.

On the extension problem for semiconcave functions with fractional modulus

Consider a locally Lipschitz function u on the closure of a possibly unbounded open subset Ω of Rn with nonempty boundary. Suppose u is (locally) semiconcave on Ω¯ with a fractional semiconcavity modulus. Is it possible to extend u in a neighborhood of any boundary point retaining the same semiconcavity modulus? We show that this is indeed the case and we give two applications of this extension property. First, we derive an approximation result for semiconcave functions on closed domains. Then, we use the above extension property to study the propagation of singularities of semiconcave functions at boundary points.

Volume estimates for Alexandrov spaces with convex boundaries

In this note, we estimate the upper bound of volume of closed positively or nonnegatively curved Alexandrov space $X$ with strictly convex boundary. We also discuss the equality case. In particular, the Boundary Conjecture holds when the volume upper bound is achieved. Our theorem also can be applied to Riemannian manifolds with non-smooth boundary, which generalizes Heintze and Karcher's classical volume comparison theorem. Our main tool is the gradient flow of semi-concave functions.

Short retractions of CAT(1) spaces

We construct short retractions of a CAT(1) space to its small convex subsets. This construction provides an alternative geometric description of an analytic tool introduced by Wilfrid Kendall. Our construction uses a tractrix flow which can be defined as a gradient flow for a family of functions of certain type. In an appendix we prove a general existence result for gradient flows of time-dependent locally Lipschitz semiconcave functions, which is of independent interest.

Necessary and sufficient condition for locally semiconcave practical control Lyapunov functions

Abstract: A semiconcave function plays an important role in nonlinear control theory. Nakamura et al. proposed a static discontinuous global feedback controller that is based on a locally semiconcave practical control Lyapunov function (LS-PCLF) for asymptotic stabilization of a nonlinear system defined on a manifold. Further, when comparing related researches, it is important to analyze the relationship between generalized differentials of the LS-PCLF. For the LS-PCLF, equivalent definitions using a directional subderivative and a disassembled differential have been proposed. However, an equivalent definition of the LS-PCLF using a reachable differential has not yet been established, and the original definition for a disassembled differential is not adequately formal. In this paper, we present a formal definition for a disassembled differential, review its basic properties, and present an equivalent definition for the LS-PCLF using a reachable differential. Through analysis, we confirm that a disassembled differential is a useful tool for analyzing the relationship between generalized differentials of the LS-PCLF.

Compensated convex transforms and geometric singularity\\ extraction from semiconvex functions

The upper and lower compensated convex transforms are `tight' one-sided approximations for a given function. We apply these transforms to the extraction of fine geometric singularities from general semiconvex/semiconcave functions and DC-functions in $\mathbb{R}^n$ (difference of convex functions). Well-known geometric examples of (locally) semiconcave functions include the Euclidean distance function and the Euclidean squared-distance function. For a locally semiconvex function $f$ with general modulus, we show that `locally' a point is singular (a non-differentiable point) if and only if it is a scale $1$-valley point, hence by using our method we can extract all fine singular points from a given semiconvex function. More precisely, if $f$ is a semiconvex function with general modulus and $x$ is a singular point, then locally the limit of the scaled valley transform exists at every point $x$ and can be calculated as $ \lim_{\lambda\to+\infty}\lambda V_\lambda (f)(x)=r_x^2/4$, where $r_x$ is the radius of the minimal bounding sphere of the (Frechet) subdifferential $\partial_- f(x)$ of the locally semiconvex $f$ and $V_\lambda (f)(x)$ is the valley transform at $x$. Thus the limit function $ \mathcal{V}_\infty(f)(x):=\lim_{\lambda\to+\infty}\lambda V_\lambda (f)(x)=r_x^2/4$ provides a `scale $1$-valley landscape function' of the singular set for a locally semiconvex function $f$. At the same time, the limit also provides an asymptotic expansion of the upper transform $C^u_\lambda(f)(x)$ when $\lambda$ approaches $+\infty$. For a locally semiconvex function $f$ with linear modulus we show further that the limit of the gradient of the upper compensated convex transform $ \lim_{\lambda\to+\infty}\nabla C^u_\lambda(f)(x)$ exists and equals the centre of the minimal bounding sphere of $\partial_- f(x)$. We also show that for a DC-function $f=g-h$, the scale $1$-edge transform, when $\lambda\to+\infty$, satisfies $ \liminf_{\lambda\to+\infty}\lambda E_\lambda (f)(x)\geq (r_{g,x}-r_{h,x})^2/4$, where $r_{g,x}$ and $r_{h,x}$ are the radii of the minimal bounding spheres of the subdifferentials $\partial_- g$ and $\partial_- h$ of the two convex functions $g$ and $h$ at $x$, respectively.

Semi-concave singularities and the Hamilton-Jacobi equation

We study the Cauchy problem for the Hamilton-Jacobi equation with a semiconcave initial condition. We prove an inequality between two types of weak solutions emanating from such an initial condition (the variational and the viscosity solution).We also give conditions for an explicit semi-concave function to be a viscosity solution. These conditions generalize the entropy inequality characterizing piecewise smooth solutions of scalar conservation laws in dimension one.

Asymptotic stabilization with locally semiconcave control Lyapunov functions on general manifolds

Asymptotic stabilization on noncontractible manifolds is a difficult control problem. If a configuration space is not a contractible manifold, we need to design a time-varying or discontinuous state feedback control for asymptotic stabilization at the desired equilibrium.For a system defined on Euclidean space, a discontinuous state feedback controller was proposed by Rifford with a semiconcave strict control Lyapunov function (CLF). However, it is difficult to apply Rifford’s controller to stabilization on general manifolds.In this paper, we restrict the assumption of semiconcavity of the CLF to the “local” one, and introduce the disassembled differential of locally semiconcave functions as a generalized derivative of nonsmooth functions. Further, we propose a Rifford–Sontag-type discontinuous static state feedback controller for asymptotic stabilization with the disassembled differential of the locally semiconcave practical CLF (LS-PCLF) by means of sample stability. The controller does not need to calculate limiting subderivative of the LS-PCLF.Moreover, we show that the LS-PCLF, obtained by the minimum projection method, has a special advantage with which one can easily design a controller in the case of the minimum projection method. Finally, we confirm the effectiveness of the proposed method through an example.

Some regularity results for a class of upper semicontinuous functions

We study regularity properties enjoyed by a class of real-valued upper semicontinuous functions f : R d ! R whose hypograph satis es a geometric property implying, for each point P on the boundary of hypo f, the existence of a sort of (uniform) subquadratic tangent hypersurface whose intersection with hypo f in a neighbourhood of P reduces to P. This geometric property generalizes both the concepts of semiconcave functions and functions whose hypograph has positive reach in the sense of Federer; the associated class of functions arises in the study of regularity properties for the minimum time function of certain classes of nonlinear control systems and di erential inclusions. We will prove that these functions share several regularity properties with semiconcave functions. In particular, they are locally BV and di erentiable a.e. Our approach consists in providing upper bounds for the dimension of the set of nondi erentiability points. Moreover, a ner classi cation of the singularities can be performed according to the dimension of the normal cone to the hypograph, thus generalizing a similar result proved by Federer for sets with positive reach. Techniques of nonsmooth analysis and geometric measure theory are used.

Stability of [formula omitted]-Lipschitz and co-Lipschitz maps in Gromov–Hausdorff topology

A submetry is a metric analogue of a Riemannian submersion, and an eϵ-Lipschitz and co-Lipschitz map is a metric analogue of an ϵ-Riemannian submersion. The stability of submetries from Alexandrov spaces to Riemannian manifolds in the Gromov–Hausdorff topology can be viewed as a parametrized version of Perelman’s stability theorem in Alexandrov geometry. In this paper, we will study the stability of eϵ-Lipschitz and co-Lipschitz maps. Our approach is based on controlled homotopy theory and semi-concave functions on Alexandrov spaces. As applications of our stability results, we generalize fiber bundle finiteness results on Riemannian submersions and partially generalize the stability of submetries.

Soul theorem for 4-dimensional topologically regular open nonnegatively curved Alexandrov spaces

In this paper, we study the topology of topologically regular 4-dimensional open nonnegatively curved Alexandrov spaces. These spaces occur naturally as the blow-up limits of compact Riemannian manifolds with lower curvature bound. These manifolds have also been studied by Yamaguchi in his preprint. Our main tools are gradient flows of semi-concave functions and critical point theory for distance functions, which have been used to study the 3 3 -dimensional collapsing theory in the paper by Cao and Ge. The results of this paper will be used in our future studies of collapsing 4-manifolds, which will be discussed elsewhere.

Exterior Sphere Condition and Time Optimal Control for Differential Inclusions

The minimum time function $T(\cdot)$ of smooth control systems is known to be locally semiconcave provided Petrov's controllability condition is satisfied. Moreover, such a regularity holds up to the boundary of the target under an inner ball assumption. We generalize this analysis to differential inclusions, replacing the above hypotheses with the continuity of $T(\cdot)$ near the target, and an inner ball property for the multifunction associated with the dynamics. In such a weakened setup, we prove that the hypograph of $T(\cdot)$ satisfies, locally, an exterior sphere condition. As is well known, this geometric property ensures most of the regularity results that hold for semiconcave functions, without assuming $T(\cdot)$ to be Lipschitz.

Ilmanen's Lemma on Insertion of $\mathbb C^{1,1} Functions

We give a proof of Ilmanen's lemma, which asserts that between a locally semi-convex and a locally semi-concave function it is possible to find a \mathbb C^{1,1} function.

Pseudographs and the Lax–Oleinik semi-group: a geometric and dynamical interpretation

Let be a Tonelli Hamiltonian defined on the cotangent bundle of a compact and connected manifold and let be a semi-concave function. If is the set of all the super-differentials of u and (φt) the Hamiltonian flow of H, we prove that for t > 0 small enough, is an exact Lagrangian Lipschitz graph.This provides a geometric interpretation/explanation of a regularization tool that was introduced by Bernard (2007 Ann. Sci. École Norm. Sup. 40 445–52) to prove the existence of C1,1 subsolutions.

Singular Dynamics for Semiconcave Functions

Semiconcave functions are a well-known class of nonsmooth functions that possess deep connections with optimization theory and nonlinear pde’s. Their singular sets exhibit interesting structures that we investigate in this paper. First, by an energy method, we analyze the curves along which the singularities of semiconcave solutions to Hamilton–Jacobi equations propagate—the socalled generalized characteristics. This part of the paper improves the main result in [P. Albano, P. Cannarsa, Propagation of singularities for solutions of nonlinear first order partial differential equations , Arch. Ration. Mech. Anal. 162 (2002), 1–23] and simplifies the construction therein. As applications, we recover some known results for gradient flows and conservation laws. Then we derive a simple dynamics for the propagation of singularities of general semiconcave functions. This part of the work is also used to study the singularities of generalized solutions to Monge–Ampère equations. We conclude with a global propagation result for the singularities of solutions e in weak KAM theory.

Extrinsic curvature of semiconvex subspaces in Alexandrov geometry

In Alexandrov spaces of curvature bounded either above (CBA) or below (CBB), we obtain extrinsic curvature bounds on subspaces associated with semiconcave functions. These subspaces play the role in singular geometry of submanifolds in Riemannian geometry, and arise naturally in many different places. For CBA spaces, we obtain new intrinsic curvature bounds on subspaces. For CBB spaces whose boundary is extrinsically curved, we strengthen Perelman’s concavity theorem for distance from the boundary, deriving corollaries on sharp diameter bounds, contractibility, and rigidity.

Refinement of the Benoist theorem on the size of Dini subdifferentials

Given a lower semicontinuous function $f:\mathbb R^n \rightarrow \mathbb R \cup$ {$+\infty$}, we prove that the set of points of $\mathbb R^n$ where the lower Dini subdifferential has convex dimension $k$ is countably $(n-k)$-rectifiable. In this way, we extend a theorem of Benoist(see [1, Theorem 3.3]), and as a corollary we obtain a classical result concerning the singular set of locally semiconcave functions.

Semiconcave Functions in Alexandrov’s Geometry

The following is a compilation of some techniques in Alexandrov's geometry which are directly connected to convexity.

On the existence of local smooth repulsive stabilizing feedbacks in dimension three

Given an affine control system in R 3 subject to the Hörmander's condition at the origin, we prove the existence of a local smooth repulsive stabilizing feedback at the origin. Our construction is based on the classical homogenization procedure, on the existence of a semiconcave control-Lyapunov function, and on the classification of singularities of semiconcave functions in dimension two.