Differentiability properties for a class of non-Lipschitz functions and applications

Nella teoria del controllo e in quella delle soluzioni di viscosità, un ruolo importante viene rivestito dalle funzioni semiconcave o semiconvesse. Semiconcavità e semiconvessità sono considerate buone classi di regolarità intermedie tra Lipschitzianità e regolarità $C^1$. Le funzioni semiconvesse possono essere caratterizzate come perturbazioni quadratiche di funzioni convesse e, pur non essendo convesse, ereditano dalle funzioni convesse alcune proprietà di regolarità come ad esempio la locale Lipschitzianità e la differenziabilità due volte quasi ovunque. Tali funzioni non sono in generale lisce, tuttavia il loro epigrafico gode della seguente proprietà geometrica: esiste un intorno di esso dove la proiezione è ben definita, e la distanza dall'epigrafico risulta essere una funzione di classe $C^{1,1}$. Un problema naturale che sorge nella Teoria del Controllo è quello dello studio delle proprietà di regolarità della funzione tempo minimo $T(\cdot)$. In P. Cannarsa, C. Sinestrari, Convexity properties of the minimum time function, Calc. Var. 3 (1995), 273-298, viene provato che, per un problema lineare con bersaglio convesso, se $T(\cdot)$ è localmente Lipschitziana, allora è semiconvessa. Tuttavia, esempi anche semplici mostrano come il tempo minimo in generale non sia localmente Lipschitziano anche nel caso di problemi lineari con bersaglio convesso (ad esempio, il sistema $\ddot x\in[-1,1]$ con bersaglio $S=\{0\}$). Risulta naturale quindi la ricerca di una classe di funzioni dotate di proprietà di regolarità confrontabili con le funzioni semiconvesse, ma non necessariamente localmente Lipschitziane. Una scelta possibile è quella di richiedere che l'epigrafico sia un insieme con reach positivo nel senso di Federer. Insiemi di questo tipo sono stati studiati da diversi autori sia in dimensione finita che in dimensione infinita. Questa scelta costituisce una generalizzazione naturale della semiconvessità, in quanto funzioni di questo tipo risultano essere in generale non localmente Lipschitziane e, qualora si aggiunga l'ipotesi di locale Lipschitzianità, si ritrova la classe delle funzioni semiconvesse. Nella prima parte della tesi vengono studiate le proprietà degli insiemi con reach positivo in $\mathbb R^n$. Si prova, tra gli altri risultati, che la loro frontiera è $\mathcal H^{n-1}$-rettificabile e che il loro perimetro è localmente finito. Si approfondisce quindi il confronto tra alcuni oggetti derivanti dall'analisi nonsmooth con altri derivanti dalla teoria geometrica della misura, in particolare, dalla teoria di De Giorgi sugli insiemi di perimetro finito. Tali proprietà vengono successivamente utilizzate nello studio delle funzioni. Si prova che una funzione $f$ semicontinua inferiormente soddisfacente alla condizione di epigrafico con reach positivo, gode delle seguenti proprietà di regolarità: (i) $f$ è strettamente differenziabile $\mathcal L^{n}$-q.o.; (ii) per $\mathcal L^{n}$-q.o. $x$, esiste $\delta =\delta (x)>0$ tale che $f_{|B(x,\delta)}$ sia Lipschitziana e semiconvessa; (iii) $f$ è due volte differenziabile $\mathcal L^{n}$-q.o. Nelle sezioni successive, viene studiato in dettaglio la struttura dell'insieme singolare degli insiemi con reach positivo, e dell'insieme $\Sigma$ dei punti di non differenziabilità per funzioni con epigrafico che goda di questa proprietà. In particolare, viene studiato il problema della propagazione delle singolarità, ossia dello studio delle condizioni sufficienti affinchè esista una curva Lipschitziana costituita interamente da punti singolari, uscente da un punto di non differenziabilità. In seguito vengono stabilite alcune formule di rappresentazione per il cono normale all' epigrafico, che vengono utilizzate nel contesto delle soluzioni di viscosità. Si prova come una funzione continua con ipografico con reach positivo soddisfacente q.o. un'equazione di Hamilton-Jacobi $H(x,u,Du)=0$, sia in realtà una soluzione di viscosità, posto che $H$ sia convessa in $Du$ e continua. Nella parte finale della tesi, l'attenzione viene concentrata sulla funzione tempo minimo per un problema di controllo. Dapprima vengono studiate alcune condizioni di controllabilità per un problema non lineare con bersaglio non necessariamente convesso. Tali condizioni garantiscono l'Hölderianità del tempo minimo, indebolendo così l'ipotesi di Lipschitzianità necessaria per la semiconvessità nel risultato di Cannarsa e Sinestrari. Viene successivamente caratterizzata una classe di problemi lineari in cui la funzione tempo minimo ha epigrafico con reach positivo: si mostra che per un problema lineare con bersaglio convesso se il tempo minimo è continuo, il suo epigrafico ha reach positivo. Se ne deducono perciò nuove proprietà di regolarità in accordo con (i)-(iii). Infine vengono presentati alcuni problemi aperti, il principale dei quali è quello di identificare le classi di problemi di controllo non lineari in cui il tempo minimo abbia epigrafico (o ipografico) con reach positivo.In optimal control or in the theory of viscosity solutions of partial differential equations, semiconcave functions play an important role (see, e.g., the monographs, \cite{BCD} and \cite{CS}). As an example, we mention the fact that some classes of PDE's admit a unique semiconcave solution (see \cite[Chapter II]{BCD}). Semiconcavity, together with the dual concept of semiconvexity, is also considered a good regularity property, in between Lipschitz continuity and $\mathcal C^{1}$-regularity (see, again, \cite{BCD} and \cite{CS}). For example, the Euclidean distance from a closed set is semiconcave, and this is in a sense an optimal result, since in general this function is not smooth. Semiconvex functions are, essentially, quadratric perturbations of convex functions. Therefore, though being not necessarily convex, they inherit from convexity some regularity properties, such as local Lipschitzianity and a.e. double differentiability in the interior of their domain. Moreover, their epigraph may admit corners, as such functions are not necessarily smooth, but those corners may occur only downwards (we recall that this last property is usually called (Clarke) regularity in nonsmooth analysis). In the framework of Control Theory, the regularity of the minimal time function $T(\cdot)$ is a widely studied topic (see, e.g., \cite{bru,Vel,CS2, CS, car, Wyu, BCD} and references therein), under different viewpoints. It can be shown that, under suitable controllability assumptions, the time optimal function for a control problem is semiconcave (or semiconvex, see \cite[Chapter 8]{CS} and references therein). In particular, it is proved in \cite{CS} that with linear dynamics and convex target, $T(\cdot)$ is semiconvex provided the Petrov condition holds. The latter is equivalent to the Lipschitz continuity of $T(\cdot)$ near the target, and thus is a type of {\it strong} local controllability condition. Since $T(\cdot)$ is not necessarily convex (see p. 100 in \cite{Co}) even for a point-target, this is a natural regularity class for a linear minimum time problem. Classical examples, however, exhibit minimal time functions that are not locally Lipschitz even though the system is small time locally controllable (see, e.g., \cite[Example 2.7, p. 242]{BCD}). Therefore, it is natural to seek conditions that identify regularity properties of $T(\cdot)$ in situations where $T(\cdot)$ is not locally Lipschitz. In the first part of this Thesis (Chapter \ref{chapfunc} and Chapter \ref{chapdiff}), we study a class of functions which enjoy Clarke regularity, but are not necessarily locally Lipschitz continuous, yet being, among other things, a.e. twice differentiable. The simplest example illustrating our work is $f(x)=\sqrt{|x|}$, but less trivial functions indeed belong to this class, such as the minimum time to reach the origin for the double integrator (rocket railroad car model, see No. 14 in the Examples of $\varphi$-convex functions in \S \ref{gendef} below). This last example is in a sense the motivating point of our analysis. In fact, it suggests that the functions which are studied in the present thesis may be a reasonable candidate as a regularity paradigm for some optimal control problems, and therefore for solutions of some partial differential equations. The functions belonging to the class studied in this thesis can be characterized by the fact that their epigraph satisfies a kind of external sphere condition, with (locally) uniform radius. Sets with this property were deeply studied as generalizations of convex sets, mainly in connection with uniqueness of the metric projection and with smoothness of the distance function, both in finite \cite{Fe} and in infinite dimensions (see, e.g., \cite{CSW,PRT}). Numerous equivalent definitions of this property were given independently by various authors. Among them, we choose the denomination ``$\varphi$-convexity'', as it better emphasizes the connections with convexity that we want to analyze. $\varphi$-convex sets are known to enjoy, in a neighborhood, some properties that convex sets satisfy globally, the reason being the radius of the external sphere, which is locally bounded away from zero (and continuous) for $\varphi$-convex sets, while it is arbitrarily large for convex sets (see \S \ref{gendef} below). In particular, in Section \ref{fedreview}, we review Federer's work showing the equivalence between his notion of positive reach sets (cfr. \cite[\S 4]{Fe}) and the notion of $\varphi$-convexity for sets. In Section \ref{semicephi} we compare in detail the notions of $\varphi$-convexity of the epigraph and semiconvexity. Actually, semiconvex functions are identified -- within the class of locally Lipschitz functions -- by exactly this requirement on their epigraph. By dropping the local Lipschitz continuity we therefore make a generalization which seems to be natural. In Section \ref{defepiphi} is presented a definition of $\varphi$-convexity for functions (see \cite{MT}), which was introduced in connection with evolution equations driven by nonconvex functionals. We show that $\varphi$-convexity of the epigraph is a particular case of $\varphi$-convexity of functions, and provide examples of $\varphi$-convex functions without a $\varphi$-convex epigraph. In Section \ref{normals} we study the normal cone to $\varphi$-convex sets, comparing different notions of normals, both from nonsmooth analysis and geometric measure theory viewpoint. We show that a $\varphi$-convex set $K$ admits $\mathcal H^{n-1}$-a.e. on its boundary a unique unit (proximal) normal vector. Moreover, if $K$ is the closure of an open set satisfying a kind of nondegeneracy condition, we show that the reduced boundary (in the sense of De Giorgi) coincides $\mathcal H^{n-1}$-a.e. with the topological boundary, and the De Giorgi external normal coincides with the proximal unit normal. The sharpness of the nondegeneracy assumption is shown through an example. These regularity properties enlarge the range of analogies between $\varphi$-convex and convex sets. Our results are essentially based on the (local) uniqueness of the metric projection onto $\varphi$-convex sets, and use some methods taken from geometric measure theory. In some cases apparently new proofs of classical facts are given. For example, our argument, based on the area formula, for the $\mathcal H^{n-1}$-uniqueness of the unit normal vector can be applied to convex sets. In Chapter \ref{chapdiff} we study lower semicontinuous functions with $\varphi$-convex epigraph. The main properties of functions with $\varphi$-convex epigraph are studied in Sections \ref{secdiff}, \ref{sectwice} and \ref{secBV}. We show that a function $f$ satisfying this assumption has the following properties: \begin{enumerate} \item[(i)] $f$ is $\mathcal L^{n}$-a.e. (strictly) differentiable; \item[(ii)] for $\mathcal L^{n}$-a.e. $x$, there exists $\delta =\delta (x)>0$ such that $f_{|B(x,\delta)}$ is Lipschitz continuous and semiconvex; \item[(iii)] $f$ is $\mathcal L^{n}$-a.e. twice differentiable. \end{enumerate} Moreover, such functions are $BV_{\rm loc}$ in the interior of their domain, but their differential is not necessarily $BV_{\rm loc}$; moreover, they do not belong necessarily to Sobolev spaces like $W^{1,\infty}_{\rm loc}$ or $W^{2,1}_{\rm loc}$. We mention that regularity results, in particular double differentiability, for not necessarily Lipschitz functions were obtained in \cite{ccks} for viscosity solutions of uniformly elliptic second order PDE's. Our results appear to be of a different nature, as they are derived from regularity assumptions on the epigraph rather than from an equation. The results of this first part are contained in \cite{CM}.\\ Sections \ref{sing1} and \ref{sing2} study more in detail the structure of the singular set for $\varphi$-convex sets and of the set $\Sigma$ of nondifferentiability points for functions $f$ with $\varphi$-convex epigraph. We show that $\Sigma$ may be written as the union of $\Sigma_{\infty}$, the set where $f$ is not subdifferentiable, and $\Sigma_k$, the sets where the dimension of the (proximal) subdifferential of $f$ is at least $k$ ($k=1,\ldots ,n$), and $\Sigma_k$ is countably $\mathcal H^{n-k}$-rectifiable. This generalizes to this class of functions a result in \cite[\S 4.1]{CS} valid for semiconcave (-convex) functions (see also \cite{AA}). The set $\Sigma_{\infty}$ is not necessarily lower dimensionally rectifiable, as an example shows. The starting points are previous results valid for semiconvexity/semiconcavity which are described, e.g., in Chapter 4 of the book \cite{CS} (see also \cite{AAC}). In particular, in Section \ref{propsingul} we are concerned with {\em propagation} of singularities, which means giving sufficient conditions in order that at least a Lipschitz curve all made of singularities emanates from a nondifferentiability point. Differently from the semiconvex case, due to the lack of Lipschitz continuity, singularities may be of two different natures, i.e., either corners or ``vertical spots'' (i.e., points where both the sub- and the superdifferential are empty) in the epigraph. Therefore, both the type of results and the techniques appearing in \cite{CS} need an adaptation, including some preliminary work in nonsmooth analysis concerning new representations of normal cones as limiting cones. Furthermore, in Section \ref{appvisco} using these representations we prove that a continuous function with $\varphi$-convex hypograph which satisfies a.e. a Hamilton-Jacobi equation of the type $H(x,u,Du)=0$. is actually a viscosity solution, provided $H$ is convex in $Du$ and continuous. As a consequence, a generalization of the classical uniqueness result for semiconcave solutions by Kruzhkov (see \cite{kru}) is obtained. These results will appear in \cite{CM2}.\\ In Chapter \ref{chpmtfnc} we focus our attention on minimal time function for control problems. Since Petrov controllability condition is equivalent to Lipschitz continuity of the minimum time function up to the boundary of the target $S$ (see \cite{Vel}), a natural problem is to study weaker controllability assumption. From a geometrical point of view, Petrov condition states that at every point of a neighborhood of the target there exists an admissible control such that the corresponding trajectory points toward the target. Moreover, the angle between the vector field and the (generalized) gradient of the distance is uniformly bounded away from zero. So called "higher order conditions" for controllability, where the order means the number of Lie brackets of the vector fields, are also well known. Among these, in the linear case $\dot x=Ax+Bu$ with $S=\{0\}$ we recall the Kalman rank condition. This condition, which gives sufficient condition for controllability and imply H\"{o}lder continuity of $T$, is related to some properties of the Lie algebra generated by the family of vector fields associated to the system (see \cite[Chapter 1]{JUR} for a complete introduction). There are also some nonlinear version of this result if the target set is an equilibrium point for the system (cfr. \cite{BS}). In \cite[Theorem 1.18, p. 235]{BCD} there is a condition for H\"{o}lder continuity of $T$ in the case of nonlinear symmetric systems $\dot x=\sum u_ig_i(x)$ for a smooth target. The condition requires that if in a point $\bar x\in\partial S$ Petrov condition does not hold, there exists a vector field $F(\bar x)$ generated by bracket operations from the vector fields of the family $\mathcal F:=\{f(\cdot,u):\,u\in\mathcal U\}$ associated to the system such that: $$\langle F(\bar x),\nu(\bar x)\rangle<0$$ where $\nu(\bar x)$ is the normal unit vector to the target $S$ at $\bar x$.\\ Equivalently, there exists a constant $\mu>0$ such that for every point $\bar y\notin S$ in a neighborhood of $\bar x$ there exists a vector field $F(\bar y)$ generated by bracket operations from the vector field of $\mathcal F_{\bar y}$ such that: $$\langle F(\bar y),Dd_S(\bar y)\rangle<-\mu.$$ This condition can be viewed as a Petrov condition of higher order, and in fact it leads to H\"{o}lder continuity of $T$ and no longer to Lipschitz continuity, where the exponent of the modulus of continuity depends from the number of Lie brackets which are involved.\\ For a PDE's approach to the problem of regularity of $T$, see also \cite{So} where viscosity solution methods are used.\\ In Sections \ref{2thorder} and \ref{controllresult}, we study the question whether such condition can be extended to control systems with drift of the form: $$\dot x=f(x)+\sum_{i=1}^du_ig_i(x),\hspace{2cm}|u_i|\le 1,i=1...d.$$ Roughly speaking, we will require in a neighborhood of $S$ the existence of a vector field generated by bracket operation from vector fields of $\mathcal F$ pointing towards the target, but allowing the scalar product between $F(\bar y)$ and $Dd_S$ to vanish sufficiently slowly when $\bar y$ approaches to the target.\\ We state a second order condition, for general systems with drift, for a target $S$ satisfying some regularity conditions (automatically granted if we take $S$ of class $C^{1,1}$). This condition ensures the existence of an admissible trajectory steering every point $x$ of a neighborhood of $S$ (or a neighborhood relative to the reachable set) to $S$ in a finite amount of time $\tilde T(x)$, continuously depending on the starting point $x$. In order to make the required estimates, we need the expansion of the distance along such trajectory. This procedure involves the scalar product of the Lie brackets of the controlled vector fields $g_i$'s with $Dd_S$, and also the effect of a nontrivial drift. Such effect depends also on geometrical properties of the target itself, e.g. curvature, and in some situations can help in terms of controllability, for example when the target is the complementary of a convex set. The condition given provides a generalization of Kalman rank condition, \cite[Theorem 1.18, p. 235]{BCD} and \cite{BS}. This controllability result is contained in \cite{AM}.\\ In Section \ref{Tlincase}, a class of minimum time functions is shown to belong exactly to the class analyzed in Chapter \ref{chapfunc} and Chapter \ref{chapdiff}. Indeed, it is shown that the epigraph of $T(\cdot)$ is $\varphi$-convex, under suitable controllability assumptions. More precisely, we prove that, for a linear control problem with a convex target $S$, the epigraph of $T(\cdot)$ is $\varphi$-convex (Theorem \ref{Tphi}), provided $T$ is continuous. Our assumptions are satisfied in several situations, including, e.g., the case where the system fulfils the Kalman rank condition and the target is the origin. An example where Small Time Controllability does not hold, yet covered by Theorem \ref{Tphi}, is presented in Section \ref{subseccont}. This result is contained in \cite{CMW}. Our analysis depends on a representation formula for the normal cone to sublevel sets of $T$, which is proved using simple tools of convex analysis together with Pontryagin's Maximum Principle. The techniques used here are essentially linear, due to the repeated use of explicit formulas. The main difficulty to handle is the possibility of having points where both the subdifferential and the superdifferential of $T$ are empty, due to the lack of Lipschitz continuity. Finally, the regularity results obtained in the first part are applied to $T(\cdot)$, and the corresponding properties of $T$ are listed in Corollary \ref{regprop}.\\ Chapter \ref{open} is devoted to some problems which are still open. The main one is identifying classes of nonlinear minimal time problems where the minimal time functions has epi- or hypograph $\varphi$-convex