On the Minimum Time Function Around the Origin

We deal with finite dimensional linear and nonlinear control systems. If the system is linear and autonomous and satisfies the classical normality assumption, we improve the well known result on the strict convexity of the reachable set from the origin by giving a polynomial estimate. The result is based on a careful analysis of the switching function. We extend this result to nonautonomous linear systems, provided the time dependent system is not too far from the autonomous system obtained by taking the time to be 0 in the dynamics. Using a linearization approach, we prove a bang-bang principle, valid in dimensions 2 and 3 for a class of nonlinear systems, affine and symmetric with respect to the control. Moreover we show that, for two dimensional systems, the reachable set from the origin satisfies the same polynomial strict convexity property as for the linearized dynamics, provided the nonlinearity is small enough. Finally, under the same assumptions we show that the epigraph of the minimum time function has positive reach, hence proving the first result of this type in a nonlinear setting. In all the above results, we require that the linearization at the origin be normal. We provide examples showing the sharpness of our assumptions.


Introduction
In the theory of autonomous linear control systems, the assumption of normality, i.e., a strong controllability assumption requiring that if each control component is used separately, then Kalman rank condition is satisfied, is well known. In particular (see [15,Sections 14,15,and 16]), normality implies that the control steering the origin to a point x in minimum time is unique and bang-bang; moreover the reachable set from the origin, R τ , is a strictly convex body for all times τ > 0. Simple examples, on the other hand, show that convexity of the reachable set is easily lost when passing to a nonlinear dynamics, even if the control covers all directions and appears linearly. In particular, the control system (1.1)   ẏ dynamics. We are not aware of any result of semiconvexity type valid for a nonlinear dynamics, where actually results of semiconcave type are more natural and easier to obtain, provided the target (or the dynamics) satisfy an inner ball condition (see [7,3,13,17,4,5]). Further results on the regularity of the minimum time function T , for two dimensional systems with a single input control, are described in [2,Chapter 3], where in particular, under generic assumptions, a characterization of smooth and nonsmooth points of the level sets of T is given. In this paper, we give a contribution to the understanding of the behavior of the minimum time function both for linear, autonomous and nonautonomous, and nonlinear control systems. We take the origin as the target (or as the source, for the reversed dynamics), hence going outside the realm of semiconcavity, and give results, under easily verifiable assumptions, on the following topics: • strict convexity of the reachable set from the origin • uniqueness of the optimal control • a nonlinear bang-bang principle • extending backwards optimal trajectories • positive reach of the epigraph of T .
Our method is based on new results on linear control systems satisfying the normality condition, and on linearization at the origin. The underlying idea, in fact, is requiring enough strength to such linearization, so that examples like (1.1) are ruled out. Our linear results hold in any space dimension, while the nonlinear part is confined to two or three dimensional spaces. Our arguments are based essentially on a careful analysis of the switching function, namely the function whose sign is expected to determine the optimal control, according to Pontryagin's Maximum Principle. The point is exactly showing that this sign is well defined, except for at most finitely many zeros. To this aim, the normality assumption is pivotal, as it permits to split any finite interval into finitely many sets, each one being a disjoint union of finitely many intervals, where the switching function or its derivatives are uniformly bounded away from zero (see Lemma 3.2). From this fact we are able to deduce a quantitative estimate on the strict convexity of the reachable set. More precisely (see Theorem 3.3), for a linear control system in R N we show that for all τ > 0 there exists a positive constant γ > 0 such that ζ, y − x ≤ −γ ζ y − x N for all x, y ∈ R τ , ζ ∈ N R τ (x) (here N R τ (x) denotes the normal cone to R τ at x). We show through an example that the exponent N is optimal. Section 3 is devoted to the above topic, together with an auxiliary study for a linear nonautonomous dynamics. The nonlinear part starts with a nonlinear bang-bang result (Section 4), valid up to dimension 3. We consider a nonlinear control system which is affine with respect to the control: if the linearization at the origin is normal, then every optimal control is bang-bang. The proof is based on Pontryagin's Maximum Principle: if the nonlinearity contains only parts which are of order larger or equal to the space dimension, then we are able to transfer to the switching function all the properties satisfied by the switching function of the dynamics linearized at the origin. This idea is at the basis also of the strict convexity of the reachable set for a nonlinear two dimensional dynamics (see Theorem 5.1) and of proving that all points close enough to the origin are optimal, i.e., any trajectory steering a point to the origin optimally can be extended backwards still remaining optimal (see Theorem 6.2). In this case, the difficulty is extending the optimal control: our analysis permits to predict backwards the sign of the switching function. Finally, we show under the same assumptions that the epigraph of the minimum time function T has positive reach, hence obtaining a rich bunch of regularity properties for T , as listed in Theorem 2.4. We also show through an example (see Example 5.3) that the assumptions on the nonlinear part cannot be avoided, while (1.1) shows that normality at the origin is essential. The restrictions on the space dimension for the nonlinear results will be explained after the relevant proofs (see Remarks 4.2 and 5.4). To our knowledge, the results we present here as well as most of the used methods do not trace back to previous literature. In particular, the nonlinear bang-bang results of Krener [16] and Sussmann [19] seem to be of a very different nature.

Nonsmooth analysis and sets with positive reach.
In all the paper, the space dimension N will be supposed larger or equal to 2. Let K ⊂ R N be closed. The distance function to K and the projection mapping onto K are defined respectively by The boundary of K is bdry K. Given x ∈ K and v ∈ R N , we say that v is a proximal normal to K at x (and we will denote this fact by equivalently v ∈ N K (x) if and only if there exists ρ > 0 such that π K (x + ρv) = {x}, and in this case we say that the proximal normal v is realized by a ball of radius ρ > 0. If K is convex then N K (x) coincides with the normal cone of Convex Analysis. The proximal subdifferential of a function f : Ω → R at a point x of its domain, ∂f (x), is the set of vectors v ∈ R N such that where epi(f ), the epigraph of f , is defined as For an introduction to nonsmooth analysis we make reference, e.g., to [10, Chapters 1 and 2]. We will make use of the following concepts due to Federer [14]: given an arbitrary set K ⊂ R N , we set the latter being defined for x ∈ K. We remark that reach(K, x) is continuous with respect to x ∈ K.
Definition 2.1. We say that a closed K ⊆ R N has positive reach if A locally closed set K has locally positive reach if reach(K, x) > 0 for every x ∈ K.
The positive reach property is actually an external sphere condition with (locally) uniform radius. More precisely, it holds (see [14, §4]): Proposition 2.2. Let K ⊂ R N be closed. Then K has positive reach if and only if there exists a continuous function ϕ : K → [0, +∞) such that the inequality holds for all x, y ∈ K and v ∈ N K (x).
We say that f : Ω → R is of class C 1,1 (Ω) (here Ω ⊆ R N is open) if its partial derivatives exist and are Lipschitz.
Remark 2.3. The case when ϕ(x) ≡ 0 in (2.1) is equivalent to the convexity of K, and (2.1) is in this sense a generalization of convexity. Furthermore, it is easy to see that if the boundary of K is the graph of a C 1,1 -function, then K has positive reach and ϕ in (2.1) can be taken as constant. Hence positive reach property generalizes C 1,1 -manifolds as well.
Lower semicontinuous functions whose epigraph has positive reach enjoy remarkable regularity properties, which are similar to properties of convex functions. In particular, the following result holds true (we state it for continuous functions for simplicity). The Lebesgue N -dimensional measure and the Hausdorff d-dimensional measure are denoted, respectively, by L N and H d .
be open, and let f : Ω → R be continuous, and such that epi(f ) has locally positive reach. Then there exists a sequence of sets Ω h ⊆ Ω such that Ω h is compact in Ω and (1) the union of Ω h covers L N -almost all Ω; (2) for all x ∈ h Ω h there exist δ = δ(x) > 0, L = L(x) > 0 such that f is Lipschitz on B(x, δ) with ratio L, and hence semiconvex on B(x, δ).

Consequently,
(3) f is a.e. Fréchet differentiable and admits a second order Taylor expansion around a.e. point of its domain.
Moreover, the set of points where the graph of f is nonsmooth has small Hausdorff dimension. More precisely, Finally, (5) f has locally bounded variation in Ω.
This result is essentially Theorem 5.1 and Proposition 7.1 in [11]. For properties of semiconvex/semiconcave functions we refer to [7].
2.2. Control theory. We will consider control systems linear or nonlinear with respect to the space variable and affine and symmetric with respect to the control. More precisely, we will consider the linear control system a.e. y(0) = 0, where 1 ≤ M ≤ N and A ∈ M N ×N , B ∈ M N ×M , being possibly time dependent, and U = [−1, 1] M ∋ (u 1 , . . . , u M ) =: u, together with the nonlinear control system where F and G are suitable vector fields (the actual assumptions will be stated later). We will use also the notation B = (b 1 , . . . , b M ) or G = (g 1 , . . . , g M ), where each entry is an N -dimensional column. We denote by U ad , the set of admissible controls, i.e., all measurable functions u, such that u(s) ∈ U for a.e. s. For any u(·) ∈ U ad , the (unique, as it will follow from the assumptions on F and G) Carathéodory solution of (2.2) or of (2.3) is denoted by y u (·). In the linear case, so that the reachable set from 0 in time t can be described by It is well known that in the linear case the set R t is convex and compact (see, e.g., [15,Lemma 12.1]), while in the nonlinear case (2.3) R t := {y u (t) | u(·) ∈ U ad } is compact and not necessarily convex (see, e.g., [8,Chapter 10]). For a fixed x ∈ R N , we define where y x,u (·) denotes the solution ofẏ = F (y) + G(y)u such that y(0) = x. Of course, θ(x, u) ∈ [0, +∞], and θ(x, u) is the time taken for the trajectory y x,u (·) to reach 0, provided θ(x, u) < +∞. The minimum time T (x) to reach 0 from x is defined by Observe that the sublevel R t = {x : T (x) ≤ t} of T (·) equals the reachable set from the origin within the (same) time t for the reversed dynamicṡ Ifū is an admissible control steering x to the origin in the minimum time T (x) (i.e., an optimal control), then the Dynamic Programming Principle (see, e.g., Proposition 2.1, Chapter IV, in [1]) implies that T (·) is strictly increasing along the optimal trajectory y x,ū . Therefore, for all 0 < t < T (x) the point y x,ū (t) belongs to the boundary of R t . Pontryagin's Maximum Principle is a fundamental tool for the analysis of optimal control problems. We state it for points belonging to the boundary of reachable sets. In view of the previous remark, this will apply also to points belonging to optimal trajectories. We give first its linear version.
Theorem 2.5 (Pontryagin's Maximum Principle for linear systems). Consider the problem (2.2), fix T > 0, and supposex ∈ R T is realized by the controlū(·) ∈ U ad (i.e., yū(T ) =x). Then x ∈ bdry R T if and only if for some ζ ∈ N R T (x), ζ = 0, it holds where here N R T (y u (T )) denotes the Clarke normal cone.
Definition 2.7. We say that a control u is essentially determined by Pontryagin's Principle if for any u 1 satisfying iii) in Theorem 2.6 (for the adjoint curve λ associated with the trajectory y u ) one has u 1 (t) = u(t) a.e. in [0, T ].
In the following we will make extensive use of the classical concept of normality for linear systems, which we are now going to introduce.
The main classical result for normal linear systems is concerned with the reachable set. Theorem 2.9. Assume that the linear control system(2.2) is normal. Then the reachable set R T is strictly convex for any T > 0.
Proof. One can find a proof in [15], Sections 14 and 15. 3. Quantitative strict convexity of reachable sets: the linear case

Autonomous systems.
This subsection is devoted to improving the classical result on the strict convexity of reachable sets for normal linear control systems of the type (2.2). We will give an estimate for the boundary of reachable sets which implies a uniform (polynomial) strict convexity with an optimal exponent. In the first Lemma we define the switching function and begin studying its behavior under the normality assumption.

Using (3.4) we have that
On the other hand, and the proof is concluded.
The next Lemma is crucial for estimating the number of zeros of the switching function g (corresponding to the number of switching points of the optimal control associated with g) and for studying their multiplicity. We recall that the constant L was defined in (3.4). Lemma 3.2. Let A ∈ M N ×N and b ∈ R N be satisfying (3.1). Take ζ ∈ R N , ζ = 1, and fix T > 0. Let g(s), s ∈ [0, T ], be defined as in (3.2).
Then there exist disjoint sets I 0 , . . . , I N −1 and numbers N i , depending only on A, b, T and N such that and, for all i = 0, 1, . . . , N − 1, the set I i is the disjoint union of at most N i intervals. Moreover, for each i = 0, 1, . . . , N − 1, for all s ∈ I i , we have Proof. We proceed by induction for i from 0 to N − 1. Set Since J 0 is open, we can write it as the disjoint union of at most countably many open intervals, We assume, without loss of generality, that there are at least N such intervals. Fix now any number N ′ ≥ N , and take a subfamily of the intervals J k 0 consisting of at most N ′ elements. Without loss of generality, we can rearrange their indexes k so that Now, fix k and consider the N intervals (a 2k , a 2k+1 ), . . . , (a 2(k+N −1) , a 2(k+N )−1 ). Set, for j = 0, 1, . . . , N − 1, (a 2(k+j) , a 2(k+j)+1 ) := I − j , and, for j = 0, 1, . . . , N − 2 [a 2(k+j)+1 , a 2(k+j+1) ] := I + j . Observe that We are going to give a lower bound on |a 2(k+N )−1 − a 2k | which will turn out to be independent of both k and N ′ . From this fact it will follow automatically that the intervals (a 2k , a 2k+1 ) are nonempty only for finitely many k.
We set now N 0 to be the number of nonempty intervals contributing to the union in (3.8), and recall that we have just proved that N 0 depends only on A, b, T and N , and actually Set I 0 = [0, T ]\J 0 and observe that we have completed the proof of the lemma for i = 0. After this step, we formulate our induction process. We are going to construct, for each i = 0, . . . , N − 1, two disjoint sets I i , J i with the following properties For i = 0 the above construction was already performed (take J −1 = (0, T )). Pick any i = 1, . . . , N − 2 (the case i = N − 1 will be treated separately) and assume that (ind1), . . . , (ind5) hold up to i − 1. We wish to show that the above statements hold for i as well. To this aim, consider the set as a disjoint union of at most countably many intervals J k i , k ∈ N (for simplicity of writing we drop the dependence on (a, b)). Assume without loss of generality that there are at most N − i intervals J k i , fix any number N ′′ ≥ N − i, and take any subfamily of {J k i } consisting of at most N ′′ intervals. We can write Pick any s 0 ∈ (a 2k , a 2(k+1) ). By arguing as for i = 0, we obtain on one hand and, for all the latter inequality being due to (3.5). Thus On the other hand, owing to (ind2) and (3.14), we obtain By combining the above inequality with (3.15) we now obtain We define We finally set I i to be the union of the I Finally we observe that J i is the union of at most N i open intervals.
The proof is concluded.
We are now going to prove the main result of this subsection. Theorem 3.3. Consider the linear control system Let R T be defined according to (2.4). Then for all T > 0 there exists a constant γ > 0, depending only on N, M, A, B, T such that for all x, y ∈ R T , for all ζ ∈ N R T (x), the inequality for all T > 0. Finally, the constants γ and γ ′ are bounded away from zero as T → 0 + .
By Lemma 3.2 there exist disjoint sets I 0 , I 1 , . . . , I N −1 and numbers N i such that [0, T ] = N −1 i=0 I i , each I i is the disjoint union of at most N i intervals and (3.6) holds. Observe that, in particular, it follows that g may vanish at most at finitely many times, and so, recalling (2.6), the controlū is piecewise constant and equal to either 1 or to −1. We rewrite We are now going to estimate separately the integrals I i |g(s)|K 1 (s)ds, for all i = 0, 1, . . . , N − 1.
For i = 0, we have Fix now i = 1, 2, . . . , N − 1, and write, recalling Lemma 3.2, We are now going to apply inductively Lemma 7 . Then the assumption (7.1) is satisfied with C = L N e − A T , thanks to (3.6), and Lemma 7.2 yields that for some point Let k = 1. By applying Lemma 7.2 on each of the two (possibly degenerate) intervals a ij , c 0 ij , By continuing the induction process until Recalling (3.23) and the above discussion, we have Recalling (3.24) and (3.25), and C = L N e − A T , we obtain from the above expression that wherec l ij is either c l ij or c l+1 ij , according to the two possibilities appearing in (3.25). Applying Lemma 7.1 to each summand of (3.26) we therefore obtain Thus, recalling that 0 ≤ K 1 (s) ≤ 1 a.e., Let now M > 1. Takex ∈ bdry R T together with an optimal controlū(·) = (ū 1 (·), . . . ,ū M (·)) steering the origin tox in the optimal time T , andȳ ∈ R T with a control u(·) = (u 1 (·), . . . , u M (·)) steering the origin toȳ in time T . Then, for each ζ ∈ N R T (x), ζ = 1, we can write Moreover, We now apply the same argument leading to (3.28) to each summand of the right hand side of (3.30). Therefore we obtain, using (3.31), that We conclude the proof of (3.19) by applying (3.32) and setting where C ′′ is a constant enjoying the same properties as C ′ . In order to prove the statement concerning the ball contained in R T , observe that the inequality (3.33) with u ≡ 0 andȳ = 0, taking into account that the controlū is necessarily bang-bang, becomes ζ from which (taking ζ = 1) we obtain The above inequality yields in particular that 0 belongs to the interior of R T . Since R T is convex and (3.34) holds for allx ∈ bdry R T , (3.20) follows. The last statement follows from (3.29) and the explicit expressions for γ and γ ′ . The proof is concluded.
Remark 3.4. The exponent N in (3.19) is optimal.
In fact, consider the dynamics Let x 1 (·) be the solution corresponding to the control u ≡ 1. Fix s > 0 and let x s (·) be the solution corresponding to the control Fix any T > 0 and observe that x 1 (T ) = T N /N !, while, for 0 < s < T , Observe also that |x for a suitable positive constant γ. If s → T , then X s (T ) → X 1 (T ), and this shows that N is the smallest exponent allowed in (3.19).

Nonautonomous systems.
The following Lemma is a first step for studying the reachable sets in the case of nonlinear control systems by using the linearization approach which we design in this paper. We will prove that under the rank condition (normality) at 0 of the linear nonautonomous control system (3.35), the strict convexity of the reachable sets is preserved up to a sufficiently small time, provided A(t) and B(t) are not too far from A(0), B(0). Sufficient conditions for the validity of the assumptions if N = 2 will be given below. In Section 4 sufficient conditions in the case where A and B come from a linearization around a trajectory will also be given.
Let Observe that g and g 0 can be seen as the switching functions related to (3.35) and (3.36), respectively.
We state now an abstract result which permits to transfer to g some properties of g 0 and to establish the quantitative strict convexity estimate for the reachable set from the origin of (3.35). Sufficient conditions in order to apply the following lemma to suitable linearizations in dimension 2 and 3 will be given in Section 4.   Observe that condition (C 0 ) implies that t = 0 is a continuity point for A(·), so that A(0) in (C 0 ) is meaningful.
As an immediate corollary of Lemma 3.5 we obtain the following Proof. The argument developed in the proof of Theorem 3.3 can be used also in this case. Indeed, fixx ∈ bdry R τ together with a controlū(·) steering 0 tox in time τ and let ζ ∈ N R τ (x), ζ = 1, be such that (3.42)ū(t) = sign ζ, b(t)M (τ, t) for a.e. t ∈ (0, τ ) (here, as at the beginning of the proof of Theorem 3.3, we assume that B = b is a vector, i.e., the control is scalar).
Then the proof proceeds exactly as for Theorem 3.3, provided that g is given by so that for allȳ ∈ R τ one has is the control which steers the origin toȳ, and the sets I i are those appearing in the statement of Lemma 3.5.

A nonlinear bang bang principle in dimensions 2 and 3
Starting from the present section we will deal with nonlinear control systems, which are affine and symmetric with respect to the control. This section is devoted to giving sufficient conditions so that controls steering the origin to the boundary of the reachable set are always bang-bang, provided the final time is sufficiently small. More precisely, the following result holds.  Let R τ denote the reachable set of (4.1) at time τ > 0. Then there exists T > 0, depending only on DF (0), G(0), L, N , such that for every 0 < τ < T the following properties hold: (a) every admissible control u(·) such that the corresponding trajectory y u (·) of (4.1) at time τ belongs to the boundary of R τ is essentially determined by the curve λ(·), the solution of the adjoint equation through the identity (4.3) u(t) = sign λ(t), G(y u (t)) a.e.; (b) u is bang-bang, i.e., u(t) ∈ {−1, 1} M a.e., and is piecewise constant; (c) the maximum number of discontinuities of u depends only on DF (0), G(0), L, and N .
Proof. We consider first the case where M = 1, i.e., G(x) is a vector and the control u is scalar. Fix τ > 0 and an admissible control u such thatx := y u (τ ) ∈ bdry R τ . By the Maximum Principle (see Theorem 2.6) there exist ζ ∈ N R τ (x), ζ = 1, and an adjoint curve λ(·), a solution of (4.2), such that u satisfies (4.3). Proving (a), (b), and (c) amounts to showing that the switching function λ(t), G(y u (t)) vanishes at most finitely many times in [0, τ ] and the number of its zeros depends only on DF (0), G(0), L, and N . For convenience we rewrite the switching function in the following way. Let M ⊤ (·, ·) denote the matrix solution of (3.37), with A(t) = DF (y u (t)) + DG(y u (t))u(t), and set b(t) = G ⊤ (y u (t)), t ∈ [0, τ ]. Then Let also M ⊤ 0 (·, ·) be defined by (3.38), where A(0) = DF (0), and let g 0 be defined according to (3.40). We wish to apply Lemma 3.5 to the above introduced mappings g and g 0 . First of all, we compute g ′ and observe that it is continuous. Indeed, Moreover, if N = 3 g ′ is a.e. differentiable and we have .
We are now going to estimate separately each summand of the above expression. First, we observe that, thanks to the assumptions (i) and (ii), we have where K 1 and K 2 are suitable constants depending only on DF (0), G(0), L, and τ . Gronwall's lemma therefore yields where the constant K 3 depends only on K 1 , K 2 . Therefore, there exists a constant K I , depending only on DF (0), G(0), L, and τ such that Assumptions (i) and (ii) yield in turn for a suitable constant K II depending only on DF (0), L, and τ , and where again K III and K IV depend only on DF (0), G(0), L, and τ . Therefore, summing (4.4), (4.5), (4.6), and (4.7), we obtain that there exist K and T ′ > 0, depending only on DF (0), G(0), and L, such that Let now N = 3, and observe that owing to assumption (iv) each summand I, II, III, IV , divided by t 2 , is bounded and a.e. differentiable, so that where the constant K ′ depends only on DF (0), G(0), and L and so does T ′ . Observe that all the above constants do not depend on ζ. Therefore, invoking assumption (iii), we can apply  The restriction N ≤ 3 depends on our method for comparing the switching function g for the nonautonomous system coming from the linearization along an optimal trajectory, with the switching function for autonomous system obtained by linearizing at the origin. This comparison requires higher order derivatives of g, whose existence we are not able to insure if N > 3.

5.
Quantitative strict convexity of reachable sets and uniqueness of optimal controls: the nonlinear two dimensional case In this subsection, we will show that, provided the linearization at 0 satisfies the normality condition and the nonlinear part is smooth and small enough, the reachable set is strictly convex up to a sufficiently small time.
Theorem 5.1. Consider the control system under the following assumptions (in the following M is either 1 or 2): u(·) = (u 1 (·), u M (·)) ∈ [−1, 1] M a.e., F : R 2 → R 2 , G : R 2 → M 2×M are of class C 1,1 (with Lipschitz constant L) and Let R τ denote the reachable set at time τ > 0 for (5.1). Then there exists T > 0, depending only on L, DF (0), G(0), with the following properties: (a) for every τ ≤ T and every x ∈ bdry R τ there exists one and only one admissible control u steering the origin to x in time τ (and u is bang-bang with finitely many switchings). (b) For every 0 < τ < T the reachable set R τ is strictly convex. More precisely, for every x 1 ∈ bdry R τ and x 2 ∈ R τ , for every ζ ∈ N P R τ (x 1 ), one has where γ is a positive constant depending only on L, DF (0), G(0). (c) There exist another time T ′ > T , depending only on L, DF (0), G(0), such that for every 0 < τ < T ′ the reachable set R τ has positive reach. More precisely, for every x 1 ∈ bdry R τ and x 2 ∈ R τ , for every ζ ∈ N P R τ (x 1 ), one has where γ ′ is a nonnegative constant depending only on L, DF (0), G(0). Proof. We begin proving the result for M = 1, i.e., for a scalar control. Fix τ > 0 and x 1 ∈ bdry R τ , together with an optimal control u 1 (·) steering 0 to x 1 and the associate trajectory x 1 (·). Take any x 2 ∈ R τ together with u 2 (·) steering 0 to x 2 and the associate trajectory x 2 (·), and set x(t) = x 2 (t) − x 1 (t). Then, for a.e. t ∈ [0, τ ], where w(t) = u 2 (t) − u 1 (t) and Let z(·) be the solution of the linear system which is defined by linearizing along the optimal trajectory x 1 (·): where A(t) = DF (x 1 (t)) + DG(x 1 (t))u 1 (t).
We remark (see (3.29)) that γ 1 (τ ) is bounded away from 0 as τ → 0 + . Moreover, one can see that z(t) = y 2 (t) − y 1 (t). Therefore Recalling (5.8) and (5.11), we obtain (5.13) ζ, From this inequality the uniqueness of the control steering the origin to x 1 in time τ follows immediately by contradiction. Setting T = min{T 0 , T 1 } and recalling (5.7) we obtain (5.2). The proof of the strict convexity is completed by applying Proposition 7.3. The proof of (5.3) is entirely analogous, where it suffices to take T ′ = T 0 .
We consider now the statement concerning the ball contained in the reachable set. To this aim, take u 2 ≡ 0 and set y 1 to be the solution of (5.5) with u 1 in place of w. Then (5.8) yields, for all t > 0, Recalling (3.41), we obtain from the previous inequality that for a suitable constantγ, which yields in particular that 0 belongs to the interior of R t , 0 ≤ t ≤ T . Since the above argument can be repeated for every point in the boundary of R t and the constant L 4 is independent of the reference point, the statement follows by recalling that we already proved that R t is convex for all 0 ≤ t ≤ T . The continuity of T follows easily from the the fact that reachable sets contain a ball (see, e.g., Propositions IV.1.2 and IV.1.6 in [1]).
In the case M = 2, it suffices to apply the above arguments to each control.
The following Remark follows immediately from the proof of Theorem 5.1.
In fact, put λ(t) in place of ζ in (5.12). Then the first summand can be estimated in the same way, while the upper bound on the second summand, namely the analogue of (5.11), can be obtained through the same arguments leading to (5.11).
We conclude this section with a counterexample showing the sharpness of assumption (iii) in Theorem 5.1.

Remark 5.
3. An example of a two dimensional nonlinear control system satisfying assumptions (i) and (ii) of Theorem 5.1 such that the reachable set R τ is not convex for all τ > 0. Consider the control system (5.14) 1), and observe that the assumptions (i) and (ii) of Theorem 5.1 are satisfied, while (iii) is not. The Hamiltonian for this system is and Pontryagin's Maximum Principle states that ifx(·) is an optimal trajectory corresponding to the controlū(·), then there exists a function λ = (λ 1 , λ 2 ), never vanishing, and a constant λ 0 ≤ 0 such that, for a.e. t, Therefore the function λ 1x2 (t) + λ 2 (t) has at most one zero, so that the optimal controlū is unique, bang-bang, and has at most one switching. Fix now τ > 0 and 0 < s < 1 and consider the control The trajectory of (5.14) emanating from the origin and corresponding to the control u 2 is, at time τ ,x 1 s := x 1 s (τ ) = s 2 τ 2 ,x 2 s := x 2 s (τ ) = τ (2s − 1). Simple computations show that any other bang-bang control with at most one switching cannot reach (x 1 s ,x 2 s ) at a time τ ′ < τ , for all 0 < s < 1. Thus u s is optimal. In particular, the curve γ(s) := (x 1 s ,x 2 s ) belongs to the boundary of the reachable set R τ . The unique unit normal to the curve γ(s) at s = 1/2 which points outside R τ is We compute: which implies that R τ is not convex. Observe however that R τ has positive reach at γ( 1 2 ) for every τ > 0, since Remark 5.4. On the assumption N = 2.
Motivations for the restriction N = 2 are twofold. First of all, our analysis is based on the switching function of the nonautonomous system obtained by linearizing around an optimal trajectory, and this method requires N ≤ 3 (see Remark 4.2). Second, the distance between trajectories of the nonlinear system (5.1) and of the linearized system (5.5) is of order two with respect to the control (see (5.8)), and this quadratic perturbation can be balanced by the strict convexity of the reachable set of the linearized system only if N = 2 (see (5.13)).

Further results for the nonlinear two dimensional case
This section is devoted to proving that the epigraph of the minimum time function has positive reach, under the assumptions of Theorem 5.1. To this aim, a results of optimal points, i.e., on points which are crossed by an optimal trajectory, is needed.

Optimal points.
The classical definition of optimal point reads as follows.
Definition 6.1. Let x ∈ R N \{0}. We say that x is optimal if and only if there exists a point x 1 such that T (x 1 ) > T (x) and a control u with the property that y x 1 ,u (·) steers x 1 to 0 in the optimal time T (x 1 ) and x = y x 1 ,u (T (x 1 ) − T (x)).
The following is the result on optimal points which will be used in the next subsection in order to ensure the positive reach of the epigraph of the minimum time function. It is based on the same estimates which lead to the strict convexity of the reachable set, and so it is restricted to two dimensional control systems. Theorem 6.2. Let N = 2 and let the assumptions of Theorem 5.1 be satisfied. Let T > 0 be such that, according to Theorem 5.1, for all 0 ≤ τ < T , the reachable set R τ of (5.1) satisfies (5.2) for all 0 < τ < T . Letx be such that T (x) < T . Thenx is an optimal point.
The second case is entirely analogous, by substituting 1 with −1.
Let us now drop the assumptionx ∈ bdry R τ . Since T is strictly decreasing along the optimal trajectoryx(·), and sox(t) ∈ bdry R τ −t for all 0 < t < τ , there exists a nontrivial adjoint vector λ(·) which uniquely determinesū(t) as in (6.2) up to the time τ . Thus the above argument can be applied also tox. If G is a 2 × 2 matrix, it suffices to perform the above construction for each column of G. The proof is concluded.
From Corollary 6.3 we obtain thatx(·) and λ(·) in the proof of Theorem 6.2 can be extended up to the time T . Therefore, we obtain the following further corollary. Proof. Let G be a vector and so the control u be scalar. Then Observe that the switching function g(t) = λ(t), G(x(t)) vanishes at most on a countable subset of [0, T ). Therefore, for a.e. t ∈ [0, T ), we have If G is a 2 × 2 matrix, it suffices to perform the above computation for each column. The proof is concluded.
6.2. The epigraph of the minimum time function has positive reach. The present section is devoted to studying the "convexity type" of the minimum time function T (·), in the case where the dynamics satisfies a weak controllability condition, i.e., the function T (·) is merely continuous. The statement is two dimensional, since it is based on Theorem 6.2.
Theorem 6.5. Let N = 2 and let the assumptions of Theorem 5.1 hold. Let T be given by Theorem 5.1. Then for every 0 < τ < T the epigraph of the minimum time function T (·) on R τ has positive reach.
Corollary 6.6. Under the same assumptions of Theorem 6.5 the minimum time function T satisfies all the properties listed in Theorem 2.4.
Before beginning the proof of Theorem 6.5 we introduce the minimized Hamiltonian and study its sign. Definition 6.7. Let x, ζ ∈ R N . We define the minimized Hamiltonian for the control system in (5.1) as h(x, ζ) = ζ, F (x) + min u∈U ζ, G(x)u .
1 here N F R τ (x) denotes the Fréchet normal cone to R τ at x, i.e., all vectors v such that lim sup R τ ∋y→x v, (y − x)/ y − x ≤ 0 Proof. Letū(·) be an admissible control steering x to 0 in time τ , together with the associate trajectoryx(·). Then, for all 0 ≤ t ≤ τ the pointx(t) belongs to R τ , so that, by definition of Fréchet normal we have lim sup Observing that x(t) − x ≤ Kt for a suitable constant K, we have In other words, Let t n → 0 be a sequence such that lim n→∞ 1 tn tn 0ū (s)ds :=ũ exists. By the convexity of U ,ũ ∈ U , and so h(x, ζ) ≤ ζ, F (x) + ζ, G(x)ũ ≤ 0.
We are now ready to prove Theorem 6.5.
We claim that i.e., there exists a constant σ > 0 such that, for all y ∈ R N with 0 < T (y) < T and for all β ≥ T (y), we have where θ = h(x, ζ), and, moreover, σ is independent of x and ζ. (6.5) Indeed, we consider two cases: In the first case, y ∈ R T (x) , so that by Theorem 5.1 ζ, y − x ≤ 0.
On the other hand, (for a suitable constant K 6 given by Gronwall's Lemma).
In order to conclude the proof we observe that N P epi(T ) is pointed at every point (x, T (x)), x ∈ R τ , since it is easy to see that the projection of every (ζ, θ) ∈ N P epi(T ) (x, T (x)) onto R N is normal to the strictly convex set R τ . Therefore, we can apply Corollary 3.1 in [17], with Ω P = intR τ , which shows that epi(T ) has positive reach.

Appendix
This section is devoted to some technical lemmas which are used in the proof of the main results.      Then, either f has no zeros in (a, b) and then, for all s ∈ (a, b) either or there exists c ∈ (a, b) such that f (c) = 0 and then, for all s ∈ [a, b], The same conclusions hold if (7.1) is substituted by Proof. We treat the case where f is nondecreasing, while the other one can be handled by taking −f . So (7.1) now reads as If f has no zeros, we have two cases, namely f (s) > 0 for all s ∈ (a, b) or f (s) < 0 for all s ∈ (a, b).
For the first case In the second case, Therefore, Assume now that there exists c ∈ (a, b) such that f (c) = 0. Then, for all s ∈ [a, c] we have   Proposition 7.3. Let K ⊂ R N be compact and assume that there exist γ > 0 and p > 1 with the following property: for every x ∈ bdry K, there exists ζ = 0 such that for every y ∈ K one has (7.2) ζ, y − x ≤ −γ ζ y − x p .
Then K is convex (with nonempty interior) and, for each x ∈ bdry K, (7.2) is satisfied by all ζ ∈ N K (x).
Proof. We show first that K is strictly convex. To this aim, assume by contradiction that there exist points x 1 = x 2 ∈ K such that the segment ]x 1 , x 2 [ is not contained in the interior of K. Let 0 < t < 1 be such that x t = (1 − t)x 1 + tx 2 ∈ bdry K and let ζ = 0 be such that (7.2) holds with x t in place of x. Obviously, ζ, x 2 − x 1 = 0, and this is a contradiction. Since K is convex with nonempty interior, for each x ∈ bdry K the normal cone N K (x) is pointed and so it is the convex hull of its exposed rays (see [18]).
We now use Theorem 4.6 in [11] and see that for every unit vector w belonging to an exposed ray of N K (x), there exists a sequence x n → x such that N K (x n ) = R + w n , w n = 1 and w n → w.
Of course (7.2) holds with x n (resp., w n ) in place of x (resp., ζ), so that by passing to the limit, w also satisfies (7.2). By taking convex combinations, we conclude the proof.