A Gradient Flow Equation for Optimal Control Problems With End-point Cost

In this paper, we consider a control system of the form ẋ=F(x)u\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\dot x = F(x)u$\end{document}, linear in the control variable u. Given a fixed starting point, we study a finite-horizon optimal control problem, where we want to minimize a weighted sum of an end-point cost and the squared 2-norm of the control. This functional induces a gradient flow on the Hilbert space of admissible controls, and we prove a convergence result by means of the Lojasiewicz-Simon inequality. Finally, we show that, if we let the weight of the end-point cost tend to infinity, the resulting family of functionals is Γ-convergent, and it turns out that the limiting problem consists in joining the starting point and a minimizer of the end-point cost with a horizontal length-minimizer path.


Introduction
In this paper, we consider a control system of the forṁ x = F (x)u, (1.1) where F : R n → R n×k is a Lipschitz-continuous function, and u ∈ R k is the control variable. If k ≤ n, for every x ∈ R n , we may think of the columns {F i (x)} i=1,...,k of the matrix F (x) as an ortho-normal frame of vectors, defining a sub-Riemannian structure on R n . For a thorough introduction to the topic, we refer the reader to the monograph [4]. In our framework, U := L 2 ([0, 1], R k ) will be the space of admissible controls, equipped with A. Scagliotti ascaglio@sissa.it 1 the usual Hilbert space structure. Given a base-point x 0 ∈ R n , for every u ∈ U , we consider the absolutely continuous trajectory x u : [0, 1] → R n that solves ẋ u (s) = F (x u (s))u(s) for a.e. s ∈ [0, 1], x u (0) = x 0 . (1.2) For every β > 0 and x 0 ∈ R n , we define the the functional F β : U → R + as follows: where a : R n → R + is a non-negative C 1 -regular function, and x u : [0, 1] → R n is the solution of Eq. 1.2 corresponding to the control u ∈ U . In this paper we want to investigate the gradient flow induced by the functional F β on the Hilbert space U , i.e., the evolution equation (1.4) where G β : U → U is the vector field on the Hilbert space U that represents the differential dF β : U → U * through the Riesz's isometry. In other words, for every u ∈ U , we denote by d u F β : U → R the differential of F β at u, and G β [u] is defined as the only element of U such that the identity G β [u], v L 2 = d u F β (v) (1.5) holds for every v ∈ U . In order to avoid confusion, we use different letters to denote the time variable in the control system Eq. 1.2 and in the evolution equation Eq. 1.4. Namely, the variable s ∈ [0, 1] will be exclusively used for the control system Eq. 1.2, while t ∈ [0, +∞) will be employed only for the gradient flow Eq. 1.4 and the corresponding trajectories. Moreover, when dealing with operators taking values in a space of functions, we express the argument using the square brackets.
The first part of the paper is devoted to the formulation of the gradient flow equation Eq. 1.4. In particular, we first study the differentiability of the functional F β : U → R + , then we introduce the vector field G β : U → U as the representation of its differential, and finally we show that, under suitable assumptions, G β is locally Lipschitz-continuous. As a matter of fact, it turns out that Eq. 1.4 can be treated as an infinite-dimensional ODE, and we prove that, for every initial datum U 0 = u 0 , the gradient flow equation Eq. 1.4 admits a unique continuously differentiable solution U : [0, +∞) → U . In the central part of this contribution, we focus on the asymptotic behavior of the curves that solve Eq. 1.4. The main result states that, if the application F : R n → R n×k that defines the linear-control system Eq. 1.1 is real-analytic as well as the function a : R n → R + that provides the end-point term in Eq. 1.3, then, for every u 0 ∈ H 1 ([0, 1], R k ) ⊂ U , the curve t → U t that solves the gradient flow equation Eq. 1.4 with initial datum U 0 = u 0 satisfies lim t→+∞ ||U t − u ∞ || L 2 = 0, (1.6) where u ∞ ∈ U is a critical point for F β . To establish this fact we first show that the functional F β satisfies the Lojasiewicz-Simon inequality. Finally, in the last part of this work, we prove a Γ -convergence result concerning the family of functionals (F β ) β∈R + . In particular, we show that, when β → +∞, the limiting problem consists in minimizing the L 2 -norm of the controls that steer the initial point x 0 to the set {x ∈ R n : a(x) = 0}. This fact can be applied, for example, to approximate the problem of finding a sub-Riemannian length-minimizer curve that joins two assigned points. We report below in detail the organization of the sections. In Section 2, we introduce the linear-control system Eq. 1.1 and we establish some preliminary results that will be used throughout the paper. In particular, in Subsection 2.2, we focus on the first variation of a trajectory when a perturbation of the corresponding control occurs. In Subsection 2.3, we study the second variation of the trajectories at the final evolution instant.
In Section 3, we prove that, for every intial datum u 0 ∈ U , the evolution equation Eq. 1.4 gives a well-defined Cauchy problem whose solutions exist for every t ≥ 0. To see that, we use the results obtained in Subsection 2.2 to introduce the vector field G β : U → U satisfying Eq. 1.5 and to prove that it is Lipschitz-continuous when restricted to the bounded subsets of U . Combining this fact with the theory of ODEs in Banach spaces (see, e.g., [10]), it descends that, for every choice of the initial datum U 0 = u 0 , the evolution equation Eq. 1.4 admits a unique and locally defined solution U : [0, α) → U , with α > 0. Using the particular structure of the gradient flow Eq. 1.4, we finally manage to extend these solutions for every positive time.
In Section 4, we show that, if the Cauchy datum u 0 has Sobolev regularity (i.e., u 0 ∈ H m ([0, 1], R k ) ⊂ U for some positive integer m), then the curve t → U t that solves Eq. 1.4 and satisfies U 0 = u 0 is pre-compact in U . The key-observation lies in the fact that, under suitable regularity assumptions on F : R n → R n×k and a : R n → R + , the Sobolev space H m ([0, 1], R k ) is invariant for the gradient flow Eq. 1.4. Moreover, we obtain that, when the Cauchy datum belongs to H m ([0, 1], R k ), the curve t → U t that solves Eq. 1.4 is bounded in the H m -norm.
In Section 5, we establish the Lojasiewicz-Simon inequality for the functional F β : U → R + , under the assumption that F : R n → R n×k and a : R n → R + are real-analytic. We recall that the first result on the Lojasiewicz inequality dates back to 1963, when in [11] Lojasiewicz proved that, if f : R d → R is a real-analytic function, then for every x ∈ R d there exist γ ∈ (1, 2], C > 0 and r > 0 such that for every y ∈ R d satisfying |y − r| 2 < r. This kind of inequalities are ubiquitous in several branches of Mathematics. For example, as suggested by Lojasiewicz in [11], Eq. 1.7 can be employed to study the convergence of the solutions oḟ Another important application can be found in [12], where Polyak studied the convergence of the gradient descent algorithm for strongly convex functions using a particular instance of Eq. 1.7, which is sometimes called Polyak-Lojasiewicz inequality. In [13], Simon extended Eq. 1.7 to real-analytic functionals defined on Hilbert spaces, and he employed it to establish convergence results for evolution equations. For further details, see also the lecture notes [14]. The infinite-dimensional version of Eq. 1.7 is often called Lojasiewicz-Simon inequality. For a complete survey on the topic, we refer the reader to the paper [7]. Following this approach, the Lojasiewicz-Simon inequality for the functional F β is the cornerstone for the convergence result of the subsequent section. In Section 6, we prove that, if the Cauchy datum belongs to H m ([0, 1], R k ) for an integer m ≥ 1, the corresponding gradient flow trajectory converges to a critical point of F β . This result requires that both F : R n → R n×k and a : R n → R + are real-analytic. Indeed, we use the Lojasiewicz-Simon inequality for F β : U → R + to show that the solutions of Eq. 1.4 with Sobolev-regular initial datum have finite length. This fact immediately yields Eq. 1. 6. In Section 7, we study the behavior of the minimization problem Eq. 1.3 when the positive parameter β tends to infinity. We address this problem using the tools of the Γconvergence (see [8] for a complete introduction to the subject). In particular, we consider U ρ := {u ∈ U : ||u|| L 2 ≤ ρ} and we equip it with the topology of the weak convergence of U . For every β > 0, we introduce the restrictions F β ρ := F β | U ρ , and we show that there exists a functional F ρ : U ρ → R + ∪ {+∞} such that the family F β ρ β∈R + Γ -converges to F ρ as β → +∞. In the case a : R n → R + admits a unique point x 1 ∈ R n such that a(x 1 ) = 0, then the limiting problem of minimizing the functional F ρ consists in finding (if it exists) a control u ∈ U ρ with minimal L 2 -norm such that the corresponding curve x u : [0, 1] → R n defined by Eq. 1.2 satisfies x u (1) = x 1 . The final result of Section 7 guarantees that the minimizers of F β ρ provide L 2 -strong approximations of the minimizers of F ρ .

Framework and Preliminary Results
In this paper, we consider control systems on R n with linear dependence in the control variable u ∈ R k , i.e., of the formẋ = F (x)u, (2.1) where F : R n → R n×k is a Lipschitz-continuous function. We use the notation F i for i = 1, . . . , k to indicate the vector fields on R n obtained by taking the columns of F , and we denote by L > 0 the Lipschitz constant of these vector fields, i.e., we set We immediately observe that Eq. 2.2 implies that the vector fields F 1 , . . . , F k have sublinear growth, i.e., there exists C > 0 such that for every x ∈ R n . Moreover, for every i = 1, . . . , k, if F i is differentiable at y ∈ R n , then from Eq. 2.2 we deduce that ∂F i (y) ∂x 2 ≤ L. (2.4) We define U := L 2 ([0, 1], R k ) as the space of admissible controls, and we endow U with the usual Hilbert space structure, induced by the scalar product Given x 0 ∈ R n , for every u ∈ U , let x u : [0, 1] → R n be the absolutely continuous curve that solves the following Cauchy problem: We recall that, under the condition Eq. 2.2, the existence and uniqueness of the solution of Eq. 2.6 is guaranteed by Carathéodory Theorem (see, e.g., [9,Theorem 5.3]). We insist on the fact that in this paper the Cauchy datum x 0 ∈ R n is assumed to be assigned.
In the remainder of this section, we introduce auxiliary results that will be useful in the other sections. In Subsection 2.1, we recall some results concerning Sobolev spaces in onedimensional domains. In Sections 2.2 and 2.3, we investigate the properties of the solutions of Eq. 2.6.

Sobolev Spaces in One Dimension
In this subsection, we recall some results for one-dimensional Sobolev spaces. Since in this paper we work only in Hilbert spaces, we shall restrict our attention to the Sobolev exponent p = 2, i.e., we shall state the results for the Sobolev spaces H m := W m,2 with m ≥ 1. For a complete discussion on the topic, the reader is referred to [6,Chapter 8]. Throughout the paper we use the convention H 0 := L 2 . For every m ≥ 1, the function We recall that a linear and continuous application T : E 1 → E 2 between two Banach spaces E 1 , E 2 is compact if, for every bounded set B ⊂ E 1 , the image T (B) is pre-compact with respect to the strong topology of E 2 . In the following result, we list three classical compact inclusions.
Finally, we recall the notion of weak convergence. For every m ≥ 0 (we set

General Properties of the Linear-Control System Eq. 2.1
In this subsection, we investigate basic properties of the solutions of Eq. 2.6, with a particular focus on the relation between the admissible control u ∈ U and the corresponding trajectory x u . We postpone the most technical proofs of this subsection to Appendix A. We recall that, for every u ∈ U := L 2 ([0, 1], R k ), the following inequality holds: We first show that, for every admissible control u ∈ U , the corresponding solution of Eq. 2.6 is bounded in the C 0 -norm. In our framework, given a continuous function f : where C > 0 is the constant of sub-linear growth prescribed by Eq. 2.3.
Proof This estimate follows from Eq. 2.3 as a direct application of Grönwall inequality.
In the following proposition, we prove that the solution of the Cauchy problem Eq. 2.6 has a continuous dependence on the admissible control.

Proposition 2.3
Let us consider u, v ∈ U and let x u , x u+v : [0, 1] → R n be the solutions of the Cauchy problem Eq. 2.6 corresponding, respectively, to the controls u and u + v. Then, for every R > 0 there exists L R > 0 such that the inequality The previous result shows that the map u → x u is Lipschitz-continuous when restricted to any bounded set of the space of admissible controls U . We remark that Proposition 2.3 holds under the sole assumption that the controlled vector fields F 1 , . . . , F k : R n → R n are Lipschitz-continuous. In the next result, by requiring that the controlled vector fields are C 1 -regular, we compute the first order variation of the solution of Eq. 2.6 resulting from a perturbation in the control.

Proposition 2.4
Let us assume that the vector fields F 1 , . . . , F k defining the control system Eq. 2.6 are C 1 -regular. For every u, v ∈ U , for every ε ∈ (0, 1], let x u , x u+εv : [0, 1] → R n be the solutions of Eq. 2.6 corresponding, respectively, to the admissible controls u and u + εv. Then, we have that where y v u : [0, 1] → R n is the solution of the following affine system: Proof See Appendix B.
Using the curve M u : [0, 1] → R n×n defined by Eq. 2.16, we can rewrite the solution of the affine system Eq. 2.14 for the first-order variation of the trajectory. Indeed, for every u, v ∈ U , a direct computation shows that the function y v u : [0, 1] → R n that solves Eq. 2.14 can be expressed as where, for every j = 1, . . . , n, the function g j s,u : [0, 1] → R k is defined as Moreover, Eq. 2.18 shows that the differential D u P s : U → R n is linear, and Lemma 2.6 implies that it is continuous. The representation follows as well from Eq. 2.18.

Remark 2.8
In the previous proof we used Lemma 2.6 to deduce for every u ∈ U the continuity of the linear operator D u P s : U → R n . Actually, Lemma 2.6 is slightly more informative, since it implies that for every R > 0 there exists C R > 0 such that for every v ∈ U and for every u ∈ U such that ||u|| L 2 ≤ R. As a matter of fact, we deduce that ||g j s,u || L 2 ≤ C R (2.25) for every j = 1, . . . , n, for every s ∈ [0, 1] and for every u ∈ U such that ||u|| L 2 ≤ R.

Remark 2.9
It is interesting to observe that, for every s ∈ (0, 1] and for every u ∈ U , the function g j s,u : [0, 1] → R k that provides the representation the j th component of D u P s is absolutely continuous on the interval [0, s], being the product of absolutely continuous matrix-valued curves. Indeed, on one hand, τ → F (x u (τ )) is absolutely continuous, being the composition of a C 1 -regular function with the absolutely continuous curve τ → x u (τ ) (see, e.g., [6,Corollary 8.11]). On the other hand, τ → M −1 u (τ ) is absolutely continuous as well, since it can be expressed as the solution of a linear system (see Eq. A.8).
We now prove that for every s ∈ [0, 1] the differential of the end-point map u → D u P s is Lipschitz-continuous on the bounded subsets of U . This result requires further regularity assumptions on the controlled vector fields. We first establish an auxiliary result concerning the matrix-valued curve that solves Eq. 2.16. and

Lemma 2.10
Proof See Appendix A.
We are now in position to prove the regularity result on the differential of the end-point map.

Proposition 2.11
Let us assume that the vector fields F 1 , . . . , F k defining the control system Eq. 2.6 are C 2 -regular. Then, for every R > 0, there exists L R > 0 such that, for every u, w ∈ U satisfying ||u|| L 2 , ||w|| L 2 ≤ R, the following inequality holds for every s ∈ [0, 1] and for every v ∈ U .
Proof See Appendix A.

Second Differential of the End-point Map
In this subsection, we study the second-order variation of the end-point map P s : U → R n defined in Eq. 2.20. The main results reported here will be stated in the case s = 1, which corresponds to the final evolution instant of the control system Eq. 2.6. However, they can be extended (with minor adjustments) also in the case s ∈ (0, 1). Similarly as done in Subsection 2.2, we show that, under proper regularity assumptions on the controlled vector fields F 1 , . . . , F k , the end-point map P 1 : U → R n is C 2 -regular. Therefore, for every u ∈ U , we can consider the second differential D 2 u P 1 : U × U → R n , which turns out to be a bilinear and symmetric operator. For every ν ∈ R n , we provide a representation of the bilinear form ν ·D 2 u P 1 : U ×U → R, and we prove that it is a compact self-adjoint operator. Before proceeding, we introduce some notations. We set V := L 2 ([0, 1], R n ), and we equip it with the usual Hilbert space structure. In order to avoid confusion, in the present subsection, we denote by || · || U and || · || V the norms of the Hilbert spaces U and V, respectively. We use a similar convention for the respective scalar products, too. Moreover, given an application R : U → V, for every u ∈ U , we use the notation R[u] ∈ V to denote the image of u through R. Then, for s ∈ [0, 1], we write R[u](s) ∈ R n to refer to the value of (a representative of) the function R[u] at the point s. More generally, we adopt this convention for every function-valued operator.
It is convenient to introduce a linear operator that will be useful to derive the expression of the second differential of the end-point map. Assuming that the controlled fields F 1 , . . . , F k are C 1 -regular, for every u ∈ U we define L u : U → V as follows: holds for every s ∈ [0, 1] and for every v ∈ U , and this shows that L u is a linear operator. Moreover, using the standard Hilbert space structure of U and of V, for every u ∈ U we can introduce the adjoint of L u , namely the linear operator L * u : for every v ∈ U and w ∈ V.
Remark 2. 12 We recall a result in functional analysis concerning the norm of the adjoint of a bounded linear operator. For further details, see [6,Remark 2.16]. Given two Banach spaces E 1 , E 2 , let L (E 1 , E 2 ) be the Banach space of the bounded linear operators from E 1 to E 2 , equipped with the norm induced by E 1 and E 2 . Let E * 1 , E * 2 be the dual spaces of E 1 , E 2 , respectively, and let L (E * 2 , E * 1 ) be defined as above. Therefore, if A ∈ L (E 1 , E 2 ), then the adjoint operator satisfies A * ∈ L (E * 2 , E * 1 ), and the following identity holds: . If E 1 , E 2 are Hilbert spaces, using the Riesz's isometry it is possible to write A * as an element of L (E 2 , E 1 ), and the identity of the norms is still satisfied.
We now show that L u and L * u are bounded and compact operators. In the next result, we study the local Lipschitz-continuity of the correspondence u → L u . Lemma 2.14 Let us assume that the vector fields F 1 , . . . , F k defining the control system Eq. 2.6 are C 2 -regular. Then, for every R > 0, there exists L R > 0 such that for every v ∈ U and for every u, w ∈ U such that ||u|| U , ||w|| U ≤ R.
Proof See Appendix B Remark 2.15 From Lemma 2.14 and Remark 2.12, it follows that the correspondence u → L * u is as well Lipschitz-continuous on the bounded sets of U .
If the vector fields F 1 , . . . , F k are C 2 -regular, we write ∂ 2 F 1 ∂x 2 , . . . , ∂ 2 F k ∂x 2 to denote their second differential. In the next result, we investigate the second-order variation of the solutions produced by the control system Eq. 2.6.

Proposition 2.16
Let us assume that the vector fields F 1 , . . . , F k defining the control system Eq. 2.6 are C 2 -regular. For every u, v, w ∈ U , for every ε ∈ (0, 1], let y v u , y v u+εw : [0, 1] → R n be the solutions of Eq. 2.14 corresponding to the first-order variation v and to the admissible controls u and u + εw, respectively. Therefore, we have that where z v,w u : [0, 1] → R n is the solution of the following affine system: where we used the linear operator L u : U → V defined in Eq. 2.29. From the previous expression it follows that, for every u, v, w ∈ U , the roles of v and w are interchangeable, i.e., for every s ∈ [0, 1] we have that z v,w u (s) = z w,v u (s). Moreover, we observe that, for every s ∈ [0, 1] and for every u ∈ U , z v,w u (s) is bilinear with respect to v and w.
We are now in position to introduce the second differential of the end-point map P s : U → R n defined in Eq. 2.20. In view of the applications in the forthcoming sections, we shall focus on the case s = 1, i.e., we consider the map P 1 : U → R n . Before proceeding, for every u ∈ U we define the symmetric and bilinear map B u : U × U → R n as follows where B u : U × U → R n is the bilinear, symmetric and bounded operator defined in Eq. 2.40.
Proof In view of Eq. 2.23, for every u, v, w ∈ U and for every ε ∈ (0, 1], we have that (1). Therefore, Eq. 2.41 follows directly from Eq. 2.33 and from Eq. 2.40. The symmetry and the bilinearity of B u : U ×U → R n descend from the observations in Remark 2.17. Finally, we have to show that, for every u ∈ U , there exists C > 0 such that In view of the previous result, for every u ∈ U , we use D 2 u P 1 : U × U → R n to denote the second differential of the end-point map P 1 : U → R n . Moreover, for every u, v, w ∈ U we have the following identities:

Remark 2.19
It is possible to prove that the correspondence u → D 2 u P 1 is continuous. In particular, under the further assumption that the controlled vector fields F 1 , . . . , F k are C 3regular, the application u → D 2 u P 1 is Lipschitz-continuous on the bounded subsets of U . Indeed, taking into account Eq. 2.42 and Eqs. 2.37-2.39, this fact follows from Lemma 2.10, from Lemma 2.14 and from the regularity of F 1 , . . . , F k .
For every ν ∈ R n and for every u ∈ U , we can consider the bilinear form ν · D 2 u P 1 : We conclude this section by showing that, using the scalar product of U , the bilinear form defined in Eq. 2.43 can be represented as a self-adjoint compact operator. Before proceeding, it is convenient to introduce two auxiliary linear operators. In this part we assume that the vector fields F 1 , . . . , F k are C 2 -regular. For every u ∈ U let us consider the application M ν u : U → V defined as follows: : R n × R n → R n is a symmetric and bilinear function. Hence, for every i = 1, . . . , k, for every u ∈ U , and for every τ ∈ [0, 1], we have that the application is a symmetric and bilinear form on R n . Therefore, for every i = 1, . . . , k, for every u ∈ U , and for every τ ∈ [0, 1], we introduce the symmetric matrix S ν,i u (τ ) ∈ R n×n that satisfies the identity for every η 1 , η 2 ∈ R n . We define the linear operator S ν u : In the next result, we prove that the linear operators introduced above are both continuous.

Lemma 2.20
Let us assume that the vector fields F 1 , . . . , F k defining the control system Eq. 2.6 are C 2 -regular. Therefore, for every u ∈ U and for every ν ∈ R n , the linear oper- We are now in position to represent the bilinear form ν · D 2 u P 1 : U × U → R through the scalar product of U . Indeed, recalling Eqs. 2.43 and 2.42, from Eqs. 2.37-2.39 for every u ∈ U , we obtain that We conclude this section by proving that N ν u : U → U is a bounded, compact, and selfadjoint operator.

Proposition 2.21
Let us assume that the vector fields F 1 , . . . , F k defining the control system Eq. 2.6 are C 2 -regular. For every u ∈ U and for every ν ∈ R n , let N ν u : U → U be the linear operator that represents the bilinear form ν · D 2 u P 1 : U × U → R through the identity Eq. 2.46. Then, N ν u is continuous, compact, and self-adjoint.
Proof We observe that the term L * u M ν u + (M ν u ) * L u at the right-hand side of Eq. 2.47 is continuous, since it is obtained as the sum and the composition of continuous linear operators, as shown in Lemma 2.13 and Lemma 2.20. Moreover, it is also compact, since both L u and L * u are, in virtue of Lemma 2.13. Finally, the fact that L * u M ν u + (M ν u ) * L u is self-adjoint is immediate. Let us consider the last term at the right-hand side of Eq. 2.47, i.e., L * u S ν u L u . We first observe that S ν u L u : U → V is continuous, owing to Lemma 2.20 and the inequality Eq. B.1. Recalling that L * u : V → U is compact, the composition L * u S ν u L u : U → U is compact as well. Once again, the operator is clearly self-adjoint.

Gradient Flow: Well-posedness and Global Definition
For every β > 0, we consider the functional F β : U → R + defined as follows: where a : R n → R + is a non-negative C 1 -regular function, and, for every u ∈ U , x u : [0, 1] → R n is the solution of the Cauchy problem Eq. 2.6 corresponding to the admissible control u ∈ U . In this section, we want to study the gradient flow induced by the functional F β on the Hilbert space U . In particular, we establish a result that guarantees existence, uniqueness and global definition of the solutions of the gradient flow equation associated to F β . In this section, we adopt the approach of the monograph [10], where the theory of ODEs in Banach spaces is developed. We start from the notion of differentiable curve with values in U . We stress that in the present paper the time variable t is exclusively employed for curves taking values in U . In particular, we recall that we use s ∈ [0, 1] to denote the time variable only in the control system Eq. 2.6 and in the related objects (e.g., admissible controls, controlled trajectories, etc.). Given a curve U : In this case, we use the notation ∂ t U t 0 := u. In the present section, we study the wellposedness in U of the evolution equation for every u, v ∈ U . More precisely, for every initial datum u 0 ∈ U we prove that there exists a curve t → U t that solves Eq. 3.3, that it is unique and that it is defined for every t ≥ 0. We first show that d u F β can be actually represented as an element of U , for every u ∈ U . We immediately observe that this problem reduces to study the differential of the end-point cost, i.e., the functional E : U → R + , defined as for every v ∈ U , where, for every j = 1, . . . , n, the function g j 1,u ∈ U is defined as in Eq. 2.22.
Proof See Appendix C.
Remark 3.2 Similarly as done in Remark 2.8, we can provide a uniform estimate of the norm of d u E when u varies on a bounded set. Indeed, recalling Lemma 2.2 and the fact that for every j = 1, . . . , n and for every u ∈ U such that ||u|| L 2 ≤ R. Combining the last inequality with Eqs. C.1 and 2.24, we deduce that there exists C R > 0 such that for every holds for every v ∈ U .

Remark 3.3
We observe that, for every u, v ∈ U , we can rewrite Eq. 3.6 as follows where λ u : [0, 1] → (R n ) * is an absolutely continuous curve defined for every s ∈ [0, 1] by the relation where M u : [0, 1] → R n×n is defined as in Eq. 2.16, and ∇a(x u (1)) is understood as a row vector. Recalling that s → M −1 u (s) solves Eq. A.8, it turns out that s → λ u (s) is the solution of the following linear Cauchy problem: (3.10) Finally, Eq. 3.8 implies that, for every u ∈ U , we can represent d u E with the function for a.e. s ∈ [0, 1]. We observe that Eq. 3.7 and the Riesz's isometry imply that for every R > 0 there exists C R > 0 such that for every u ∈ U such that ||u|| L 2 ≤ R. We further underline that the representation h u : [0, 1] → R k of the differential d u E is actually absolutely continuous, similarly as observed in Remark 2.9 for the representations of the components of the differential of the end-point map.
Under the assumption that the controlled vector fields F 1 , . . . , F k and the function a : R n → R + are C 2 -regular, we can show that the differential u → d u E is Lipschitz-continuous on bounded sets.

Lemma 3.4
Let us assume that the vector fields F 1 , . . . , F k defining the control system Eq. 2.6 are C 2 -regular, as well as the function a : R n → R + designing the end-point cost. Then, for every R > 0 there exists L R > 0 such that Proof See Appendix C.
Remark 3.5 In Lemma 3.1 we have computed the Gateaux differential d u E of the functional E : U → R. The continuity of the map u → d u E implies that the Gateaux differential coincides with the Fréchet differential (see, e.g., [5, Theorem 1.9]).
Using Lemma 3.1 and Remark 3.3, we can provide an expression for the representation where h u : [0, 1] → R k is defined in Eq. 3.11. Before proving that the solution of the gradient flow Eq. 3.3 exists and is globally defined, we report the statement of a local existence and uniqueness result for the solution of ODEs in infinite-dimensional spaces.
Theorem 3.6 Let (E, || · || E ) be a Banach space, and, for every u 0 ∈ E and R > 0, let For every t 0 ∈ R, let us consider the following Cauchy problem: Then, setting α := R M , the equation Eq. 3.15 admits a unique and continuously differentiable solution t → U t , which is defined for every t ∈ I : Proof This result descends directly from [10, Theorem 5.1.1].
In the following result, we show that, whenever it exists, any solution of Eq. 3.3 is bounded with respect to the L 2 -norm. Lemma 3.7 Let us assume that the vector fields F 1 , . . . , F k defining the control system Eq. 2.6 are C 2 -regular, as well as the function a : R n → R + designing the end-point cost. For every initial datum u 0 ∈ U , let U : [0, α) → U be a continuously differentiable solution of the Cauchy problem Eq. 3.3. Therefore, for every Proof Recalling Eq. 3.3 and using the fact that both F β : U → R + and t → U t are differentiable, we observe that for every t ∈ [0, α), and this immediately implies that for every t ∈ [0, α). Moreover, from the definition of the functional F β given in Eq. 3.1 and recalling that the end-point term is non-negative, it follows that 1 for every t ∈ [0, α). Finally, using Lemma 2.2 and the continuity of the terminal cost a : R n → R + , we deduce the thesis.
We are now in position to prove that the gradient flow equation Eq. 3.3 admits a unique and globally defined solution. Proof Let us fix the initial datum u 0 ∈ U , and let us set R := ||u 0 || L 2 . Let C R > 0 be the constant provided by Lemma 3.7. Let us introduce R := C R + 1 and let us consider We observe that, for everyū ∈ U such that ||ū|| L 2 ≤ C R , we have that Recalling that the vector field that generates the gradient flow Eq. 3.3 has the form G β [u] = u + βh u for every u ∈ U , from Eq. 3.12, we deduce that there exists M R > 0 such that for every u ∈ B R (0). On the other hand, Lemma 3.4 implies that there exists L R > 0 such that for every u 1 , u 2 ∈ B R (0). Recalling the inclusion Eqs. 3.17, 3.18, and 3.19 guarantee that the hypotheses of Theorem 3.6 are satisfied in the ball B 1 (ū), for everyū satisfying ||ū|| L 2 ≤ C R . This implies that, for every t 0 ∈ R, the evolution equation admits a unique and continuously differentiable solution defined in the interval In particular, if we choose t 0 = 0 andū = u 0 in Eq. 3.20, we deduce that the gradient flow equation Eq. 3.3 with initial datum U 0 = u 0 admits a unique and continuously differentiable solution t → U t defined in the interval [0, α]. We shall now prove that we can extend this local solution to every positive time. In virtue of Lemma 3.7, we obtain that the local solution t → U t satisfies Since Lemma 3.7 guarantees that Eq. 3.21 holds whenever the solution t → U t exists, we can repeat recursively the argument and we can extend the domain of the solution to the whole half-line [0, +∞).
We observe that Theorem 3.6 suggests that the solution of the gradient flow equation Eq. 3.3 could be defined also for negative times. In the following result we investigate this fact.

Corollary 3.9 Under the same assumptions of Theorem 3.8, for every
Proof The fact that the solutions are defined for every positive time descends from Theorem 3.8. Recalling the expression of G β : U → U provided by Eq. 3.14, from Eq. 3.12 it follows that, for every R 2 > 0, there exists M R 2 such that On the other hand, in virtue of Lemma 3.4, we deduce that there exists L R 2 such that Therefore, the previous inequalities guarantee that the hypotheses of Theorem 3.6 are satisfied in B R (u 0 ), whenever ||u 0 || L 2 ≤ R 1 . Finally, in virtue of Theorem 3.6 and the inclusion B R (u 0 ) ⊂ B R 2 (0), we obtain the thesis with In Section 3, we considered the F β : U → R + defined in Eq. 3.1 and we proved that the gradient flow equation Eq. 3.3 induced on U by F β admits a unique solution U : [0, +∞) → U , for every Cauchy datum U 0 = u 0 ∈ U . The aim of the present section is to investigate the pre-compactness in U of the gradient flow trajectories t → U t . In order to do that, we first show that, under suitable regularity assumptions on the vector fields F 1 , . . . , F k and on the function a : R n → R + , for every t ≥ 0, the value of the solution U t ∈ U has the same Sobolev regularity as the initial datum u 0 . The key-fact is that, when F 1 , . . . , F k are C r -regular with r ≥ 2 and a : for the corresponding curves λ u , λ u+w : [0, 1] → (R n ) * the following inequality holds: Proof Recalling the definition of λ u given in Eq. 3.9, we have that , where x u : [0, 1] → R n is solution of Eq. 2.6 corresponding to the control u ∈ U . Lemma 2.2 implies that there exists C R > 0 such that |∇a(x u (1))| 2 ≤ C R for every u ∈ U such that ||u|| L 2 ≤ R. Combining this with Eq. 2.17, we deduce Eq. 4.1.
To prove Eq. 4.2, we first observe that the C 2 -regularity of a : R n → R + and Proposition 2.3 imply that, for every R > 0, there exists L R > 0 such that We recall the notion of Lie bracket of vector fields. Let G 1 , G 2 : R n → R n be two vector fields such that G 1 ∈ C r 1 (R n , R n ) and G 2 ∈ C r 2 (R n , R n ), with r 1 , r 2 ≥ 1, and let us set r := min(r 1 , r 2 ). Then, the Lie bracket of G 1 and G 2 is the vector field [G 1 , G 2 ] : R n → R n defined as follows: We observe that [G 1 , G 2 ] ∈ C r−1 (R n , R n ). In the following result, we establish some estimates for vector fields obtained via iterated Lie brackets.
. .] : R n → R n satisfies the following inequalities: for every x ∈ K, and for every x, y ∈ K.
Proof The thesis follows immediately from the fact that the vector field G is C 1 -regular.
The next result is the cornerstone this section. It concerns the regularity of the function h u : [0, 1] → R k introduced in Eq. 3.11. We recall that, for every u ∈ U , h u is the representation of the differential d u E through the scalar product of U , where the functional E : U → R + is defined as in Eq. 3.5. We recall the convention Proof It is sufficient to prove the thesis in the case m = r − 1, for every integer r ≥ 2. When r = 2, m = 1, we have to prove that, for every u ∈ U , the function h u : [0, 1] → R k is in H 1 . Recalling Eq. 3.11, we have that, for every j = 1, . . . , k, the j th component of h u is given by the product h j u (s) = λ u (s) · F j (x u (s)) for every s ∈ [0, 1], where λ u : [0, 1] → (R n ) * was defined in Eq. 3.9. Since both s → λ u (s) and s → F j (x u (s)) are in H 1 , then their product is in H 1 as well (see, e.g., [6,Corollary 8.10]). Therefore, since λ u : [0, 1] → (R n ) * solves Eq. 3.10, we can computė for every j = 1, . . . , k and for every u ∈ U such that ||u|| L 2 ≤ R. Finally, using Eq. 3.12, we obtain that Eq. 4.5 holds for r = 2, m = 1. To prove Eq. 4.6, we observe that, for every j = 1, . . . , k and for every u, w ∈ U we have holds for a.e. s ∈ [0, 1] and for every u, w ∈ U satisfying ||u|| L 2 , ||w|| L 2 ≤ R. Using Eq. 2.10, the previous inequality implies that there exists L R > 0 such that In this case, we use the notation ∂ t U t 0 := u.
It is important to observe that, for every order of derivation = 1, . . . , m, Eq. 4.15 implies that = 0, and we use the notation ∂ t U ( ) t 0 := u ( ) . In particular, for every = 1, . . . , m, it follows that In the next result, we study the following evolution equation for every t ∈ [0, α). This proves the thesis in the case r = 2, m = 1. Let us prove the induction step. We shall prove the thesis in the case r, m = r − 1. Let Using the inductive hypothesis and that ||u 0 || H m−1 ≤ ||u 0 || H m , for every R > 0 there exists C R > 0 such that, if ||u 0 || H m ≤ R, we have that for every t ∈ [0, α). Hence, it is sufficient to provide an upper bound to the semi-norm ||U (m) t || L 2 . Recalling Eq. 4.16, the same computation as before yields for every t ∈ [0, α). Combining Eq. 4.20 with Eq. 4.5, we obtain that there exists C 1 for every t ∈ [0, α). This yields Eq. 4.18 for the inductive case r, m = r − 1.
We are now in position to prove that the Cauchy problem Eq. 4.17 admits a unique and globally defined solution. The proof of the following result follows the lines of the proof of Theorem 3.8.   We now prove a pre-compactness result for the gradient flow trajectories. We recall that we use the convention H 0 = L 2 .

Lojasiewicz-Simon Inequality
In this section, we show that when the controlled vector fields F 1 , . . . , F k and the function a : R n → R + are real-analytic, then the cost functional F β : U → R + satisfies the Lojasiewicz-Simon inequality. This fact will be of crucial importance for the convergence proof of the next section. For a complete survey on the Lojasiewicz-Simon inequality, we refer the reader to the paper [7].
In this section, we prove that the functional F β : U → R + defined in Eq. 3.1 satisfies the Lojasiewicz-Simon inequality for every β > 0. We first show that, when the function a : R n → R + involved in the definition of the end-point cost Eq. 3.5 and the controlled vector fields F 1 , . . . , F k are real-analytic, F β is real-analytic as well, for every β > 0. We recall the notion of real-analytic application defined on a Banach space. For an introduction to the subject, see, for example, [15]. Definition 5.1 Let E 1 , E 2 be Banach spaces, and let us consider an application T : E 1 → E 2 . The function T is said to be real-analytic at e 0 ∈ E 1 if for every N ≥ 1 there exists a continuous and symmetric multi-linear application l N ∈ L ((E 1 ) N , E 2 ) and if there exists r > 0 such that, for every e ∈ E 1 satisfying ||e − e 0 || E 1 < r, we have In the next result, we provide the conditions that guarantee that F β : U → R is realanalytic.

Proposition 5.2
Let us assume that the vector fields F 1 , . . . , F k defining the control system Eq. 2.6 are real-analytic, as well as the function a : R n → R + designing the end-point cost Eq. 3.5. Therefore, for every β > 0, the functional F β : U → R + defined in Eq. 3.1 is real-analytic.
Proof Since F β (u) = 1 2 ||u|| L 2 + βE(u) for every u ∈ U , the proof reduces to show that the end-point cost E : U → R + is real-analytic. Recalling the definition of E given in Eq. 3.5 and the end-point map P 1 : U → R n introduced in Eq. 2.20, we have that the former can be expressed as the composition E = a • P 1 . In the proof of [4, Proposition 8.5] it is shown that P 1 is smooth as soon as F 1 , . . . , F k are C ∞ -regular, and the expression of the Taylor expansion of P 1 at every u ∈ U is provided. In [2, Proposition 2.1], it is proved that, when a : R n → R + and the controlled vector fields are real-analytic, the Taylor series of a • P 1 is actually convergent.
The previous result implies that the differential dF β : U → U * is real-analytic.

Corollary 5.3
Under the same assumptions as in Proposition 5.2, for every β > 0, the differential dF β : U → U * is real-analytic.
Another key-step in view of the Lojasiewicz-Simon inequality is the study of the Hessian of the functional F β : U → R + . In our framework, the Hessian of F β at a point u ∈ U is the bounded linear operator Hess u F β : U → U that satisfies the identity: In the next proposition we prove that, for every u ∈ U , Hess u F β has finite-dimensional kernel. We stress on the fact that, unlike the other results of the present section, we do not have to assume that F 1 , . . . , F k and a : R n → R + are real-analytic to study the kernel of Hess u F β .

Proposition 5.4
Let us assume that the vector fields F 1 , . . . , F k defining the control system Eq. 2.6 are C 2 -regular, as well as the function a : R n → R + defining the end-point cost Eq. 3.5. For every u ∈ U , let Hess u F β : U → U be the linear operator that represents the second differential d 2 u F β : U × U → R through the identity Eq. 5.1. Then, the the kernel of Hess u F β is finite-dimensional.
Proof For every u ∈ U , we have that Therefore, we are reduced to study the second differential of the endpoint cost E : U → R + . Recalling its definition in Eq. 3.5 and applying the chain-rule, we obtain that 2) where P 1 : U → R n is the end-point map defined in Eq. 2.20, and where the curve x u : [0, 1] → R n is the solution of Eq. 2.6 corresponding to the control u ∈ U . We recall that, for every y ∈ R n , we understand ∇a(y) as a row vector. Let us set ν u := (∇a(x u (1))) T and H u := ∇ 2 a(x u (1)), where H u : R n → R n is the self-adjoint linear operator associated to the Hessian of a : R n → R + at the point x u (1). Therefore, we can write for every v, w ∈ U , where D u P * 1 : R n → U is the adjoint of the differential D u P 1 : U → R n . Moreover, recalling the definition of the linear operator N ν u : U → U given in Eq. 2.46, we have that , w L 2 for every v, w ∈ U . Therefore, we obtain for every v, w ∈ U , where Hess u E : U → U is the linear operator that satisfies the identity: We observe that Hess u E is a self-adjoint compact operator. Indeed, N ν u u is self-adjoint and compact in virtue of Proposition 2.21, while D u P * 1 • H u • D u P 1 has finite-rank and it self-adjoint as well. Combining Eqs. 5.2 and 5.4, we deduce that Hess u F β = Id + βHess u E, (5.5) where Id : U → U is the identity. Finally, using the Fredholm alternative (see, e.g., [6, Theorem 6.6]), we deduce that the kernel of Hess u F β is finite-dimensional.
We are now in position to prove that the functional F β : U → R + satisfies the Lojasiewicz-Simon inequality. F 1 , . . . , F k defining the control system Eq. 2.6 are real-analytic, as well as the function a : R n → R + defining end-point cost Eq. 3.5. For every β > 0 and for every u ∈ U , there exists r > 0, C > 0 and γ ∈ (1,2] such that

Theorem 5.5 Let us assume that the vector fields
for every v ∈ U such that ||v − u|| L 2 < r.
Proof If u ∈ U is not a critical point for F β , i.e., d u F β = 0, then there exists r 1 > 0 and κ > 0 such that On the other hand, by the continuity of F β , we deduce that there exists r 2 > 0 such that Combining the previous inequalities and taking r := min{r 1 , r 2 }, we deduce that, when d u F β = 0, Eq. 5.6 holds with γ = 2.
The inequality Eq. 5.6 in the case d u F β = 0 follows from [7,Corollary 3.11]. We shall now verify the assumptions of this result. First of all, [7, Hypothesis 3.2] is satisfied, being U an Hilbert space. Moreover, [7,Hypothesis 3.4] follows by choosing W = U * . In addition, we recall that dF β : U → U * is real-analytic in virtue of Corollary 5.3, and that Hess u F β has finite-dimensional kernel owing to Proposition 5.4. These facts imply that the conditions (1)-(4) of [7,Corollary 3.11] are verified if we set X = U and Y = U * .

Convergence of the Gradient Flow
In this section, we show that the gradient flow trajectory U : [0 + ∞) → U that solves Eq. 3.3 is convergent to a critical point of the functional F β : U → R, provided that the Cauchy datum U 0 = u 0 satisfies u 0 ∈ H 1 ([0, 1], R k ) ⊂ U . The Lojasiewicz-Simon inequality established in Theorem 5.5 will play a crucial role in the proof of the convergence result. Indeed, we use this inequality to show that the trajectories with Sobolev-regular initial datum have finite length. In order to satisfy the assumptions of Theorem 5.5, we need to assume throughout the section that the controlled vector fields F 1 , . . . , F k and the function a : R n → R + are real-analytic.
We first recall the notion of the Riemann integral of a curve that takes values in U . For general statements and further details, we refer the reader to [ We immediately observe that the following inequality holds: where ∂ t V θ is the derivative of the curve t → V t defined as in Eq. 3.2 and computed at the instant θ ∈ [a, b]. Finally, combining Eqs. 6.2 and 6.1, we deduce that We refer to the quantity at the right-hand side of Eq. 6.3 as the length of the continuously Let U : [0, +∞) → U be the solution of the gradient flow equation Eq. 3.3 with initial datum u 0 ∈ U . We say that u ∞ ∈ U is a limiting point for the curve t → U t if there exists a sequence (t j ) j ≥1 such that t j → +∞ and ||U t j − u ∞ || L 2 → 0 as j → ∞. In the next result, we study the length of t → U t in a neighborhood of a limiting point. Proof Let u ∞ ∈ U be a limiting point of t → U t , and let (t j ) j ≥1 be a sequence such that t j → +∞ and ||Ut j − u ∞ || L 2 → 0 as j → ∞. The same computation as in Eq. 3.16 implies that the functional F β : U → R + is decreasing along the trajectory t → U t , i.e., for every t ≥ t ≥ 0. In addition, using the continuity of F β , it follows that F β (Ut j ) → F β (u ∞ ) as j → ∞. Combining these facts, we have that for every t ≥ 0. Moreover, owing to Theorem 5.5, we deduce that there exist C > 0, γ ∈ (1, 2] and r > 0 such that for every v ∈ B r (u ∞ ). Let t 1 ≥ 0 be the infimum of the instants such that U t ∈ B r (u ∞ ), i.e., We observe that the set where we take the infimum is nonempty, in virtue of the convergence ||Ut j − u ∞ || L 2 → 0 as j → ∞. Then, there exists t 1 ∈ (t 1 , +∞] such that U t ∈ B r (u ∞ ) for every t ∈ (t 1 , t 1 ), and we take the supremum t 1 > t 1 such that the previous condition is satisfied, i.e., and We repeat this procedure (which terminates in a finite number of steps if and only if there exitst > 0 such that U t ∈ B r (u ∞ ) for every t ≥t), and we obtain a family of intervals Without loss of generality, we may assume that I is a set of infinite Lebesgue measure. Indeed, if this is not the case, we would have the thesis: || L 2 is bounded on the bounded subsets of U , as shown in Eq. 3.18. Therefore, we focus on the case when the Lebesgue measure of I is infinite. Let us introduce the following sequence: t j ), . . . , (6.8) where t 1 , t 1 , . . . are the extremes of the intervals {(t j , t j )} j =1,...,N constructed above. Finally, we define the function σ : [τ 0 , +∞) → [τ 0 , +∞) as follows: · · · (6.9) We observe that σ : [τ 0 , +∞) → [τ 0 , +∞) is piecewise affine and it is monotone increasing. In particular, we have that Moreover, from Eq. 6.8 and from the definition of the intervals t j , t j j ≥1 , it follows that for every t ∈ [τ 0 , +∞). Let us define the function g : [τ 0 , +∞) → R + as follows: where we used Eq. 6.6 to deduce that g is always non-negative. From Eq. 6.9, we obtain that the restriction g| (τ j ,τ j +1 ) is C 1 -regular, for every j ≥ 0. Therefore, using the fact thaṫ σ | (τ j ,τ j +1 ) ≡ 1, we computė for every t ∈ (τ j , τ j +1 ) and for every j ≥ 0. Recalling that G β : U → U is the Riesz's representation of the differential dF β : U → U * , it follows thaṫ for every t ∈ (τ j , τ j +1 ) and for every j ≥ 0. Moreover, owing to the Lojasiewicz-Simon inequality Eq. 6.7, from Eq. 6.11 we deduce thaṫ for every t ∈ (τ j , τ j +1 ) and for every j ≥ 0. Let h : [τ 0 , ∞) → [0, +∞) be the solution of the Cauchy problemḣ 15) whose expression is for every t ∈ [τ 0 , ∞). Using the fact that g| (τ 0 ,τ 1 ) is C 1 -regular, in view of Eq. 6.14, we deduce that for every t ∈ [τ 0 , τ 1 ). We shall now prove that the previous inequality holds for every t ∈ [τ 0 , +∞) using an inductive argument. Let us assume that Eq. 6.16 holds in the interval [τ 0 , τ j ), with j ≥ 1. From the definition of g, combining Eqs. 6.5 and 6.10, we obtain that Using that the restriction g| (τ j ,τ j +1 ) is C 1 -regular, in virtue of Eqs. 6.14, 6.15, and 6.17, we extend the the inequality Eq. 6.16 to the interval [τ 0 , τ j +1 ). This shows that Eq. 6.16 is satisfied for every t ∈ [τ 0 , +∞). We now prove that the portion of the trajectory that lies in B r (u ∞ ) is finite. We observe that where we recall that I = N j =1 t j , t j . For every j ≥ 1, in the interval (t j , t j ) we use the change of variable θ = σ (ϑ), where σ is defined in Eq. 6.9. Using Eqs. 6.8 and 6.9, we observe that σ −1 t j , t j = (τ j −1 , τ j ) and thatσ | (τ j −1 ,τ j ) ≡ 1. These facts yield for every j ≥ 1, where we used Eq. 6.13 in the last identity. Therefore, combining Eqs. 6.18 and 6.19, we deduce that Then, the thesis reduces to prove that the quantity at the right-hand side of Eq. 6.20 is finite. Let δ > 0 be a positive quantity whose value will be specified later. From the Cauchy-Schwarz inequality, it follows that On the other hand, for every j ≥ 1, using the integration by parts on each interval (τ 0 , τ 1 ), . . . , (τ j −1 , τ j ), we have that where we introduced the notation g(τ − i ) := lim ϑ→τ − i g(ϑ), and we used the first inequality of Eq. 6.17 and the fact that g is always non-negative. Finally, if the exponent γ in Eq. 6.7 satisfies γ = 2, we can choose any positive δ > 0. On the other hand, if γ ∈ (1, 2), we choose δ such that 0 < δ < 2γ −2 2−γ . This choice guarantees that that and therefore, in virtue of Eqs. 6.21 and 6.20, we deduce the thesis.
In the following corollary, we state an immediate (but important) consequence of Proposition 6.1.

Corollary 6.2
Under the same assumptions as in Proposition 6.1, let the curve U : [0, +∞) → U be the solution of the Cauchy problem Eq. 3.3 with initial datum U 0 = u 0 . If u ∞ ∈ U is a limiting point for the curve t → U t , then the whole solution converges to u ∞ as t → ∞, i.e.,

Moreover, the length of the whole solution is finite.
Proof We prove the statement by contradiction. Let us assume that t → U t is not converging to u ∞ as t → ∞. Let B r (u ∞ ) be the neighborhood of u ∞ given by Proposition 6.1. Diminishing r > 0 if necessary, we can find two sequences {t j } j ≥0 and {t j } j ≥0 such that for every j ≥ 0 the following conditions hold: We observe that ∞ j =1 (t j , t j ) ⊂ I, where I := {t ≥ 0 : U t ∈ B r (u ∞ )}. Moreover, the inequality Eq. 6.3 and the previous conditions imply that for every j ≥ 0. However, this contradicts Eq. 6.4. Therefore, we deduce that ||U t − u ∞ || U → 0 as t → ∞. In particular, this means that there existst ≥ 0 such that U t ∈ B r (u ∞ ) for every t ≥t. This in turn implies that the whole trajectory has finite length, since We observe that in Corollary 6.2 we need to assume a priori that the solution of the Cauchy problem Eq. 3.3 admits a limiting point. However, for a general initial datum u 0 ∈ U we cannot prove that this is actually the case. On the other hand, if we assume more regularity on the Cauchy datum u 0 , we can use the compactness results proved in Section 4. We recall the notation H 0 ([0, 1], R k ) =: U . In addition, from Corollary 4.8, we deduce that {U t : t ≥ 0} is pre-compact with respect to the strong topology of H m−1 ([0, 1], R k ). Therefore, there exist u ∞ ∈ H m−1 ([0, 1], R k ) and a sequence (t j ) j ≥1 such that we have t j → +∞ and ||U t j − u ∞ || H m−1 → 0 as j → ∞. In particular, this implies that ||U t j − u ∞ || L 2 → 0 as j → ∞. In virtue of Corollary 6.2, we deduce that ||U t − u ∞ || L 2 → 0 as t → +∞. Using again the pre-compactness of the trajectory {U t : t ≥ 0} with respect to the strong topology of To conclude, we have to show that u ∞ ∈ H m ([0, 1], R k ). Owing to the compact inclusion Eq. 2.9 in Theorem 2.1, and recalling that the trajectory {U t : t ≥ 0} is pre-compact with respect to the weak topology of H m ([0, 1], R k ), the convergence Eq. 6.22 guarantees that u ∞ ∈ H m ([0, 1], R k ) and that U t H m u ∞ as t → +∞.
In the next result, we study the regularity of the limiting points of the gradient flow trajectories. Theorem 6.4 Let us assume that the vector fields F 1 , . . . , F k defining the control system Eq. 2.6 are real-analytic, as well as the function a : R n → R + designing the end-point cost. Let U : [0, +∞) → U be the solution of the Cauchy problem Eq. 3.3 with initial datum U 0 = u 0 , and let u ∞ ∈ U be any of its limiting points. Then, u ∞ is a critical point for the functional F β , i.e., d u ∞ F β = 0. Moreover, u ∞ ∈ H m ([0, 1], R k ) for every integer m ≥ 1.
Proof By Corollary 6.2, we have that the solution t → U t converges to u ∞ as t → +∞ with respect to the strong topology of U . Let us consider the radius r > 0 prescribed by Proposition 6.1. If d u ∞ F β = 0, taking a smaller r > 0 if necessary, we have that there exists ε > 0 such that ||d u F β || U * ≥ ε for every u ∈ B r (u ∞ ). Recalling that ||U t − u ∞ || U → 0 as t → +∞, then there existst ≥ 0 such that U t ∈ B r (u ∞ ) and for every t ≥t. On the other hand, this fact implies that ||∂ t U t || U = ||d U t F β || U * ≥ ε for every t ≥t, but this contradicts Eq. 6.4, i.e., the fact that the length of the trajectory is finite. Therefore, we deduce that d u ∞ F β = 0. As regards the regularity of u ∞ , we observe that d u ∞ F β = 0 implies that G β [u ∞ ] = 0, which in turn gives where the function h u ∞ : [0, 1] → R k is defined as in Eq. 3.11. Owing to Lemma 4.3, we deduce that the right-hand side of the previous equality has regularity H m+1 whenever u ∞ ∈ H m , for every integer m ≥ 0. Using a bootstrapping argument, this implies that u ∞ ∈ H m ([0, 1], R k ), for every integer m ≥ 1.  (1)).

Γ -convergence
In this section, we study the behavior of the functionals (F β ) β∈R + as β → +∞ using the tools of the Γ -convergence. More precisely, we show that the problem of minimizing the functional F β : U → R + converges as β → +∞ (in the sense of Γ -convergence) to a limiting minimization problem. A classical consequence of this fact is that the minimizers of the functionals (F β ) β∈R + can provide an approximation of the solutions of the limiting problem. Moreover, in the present case, the limiting functional has an important geometrical meaning, since it is related to the search of sub-Riemannian length-minimizing paths that connect an initial point to a target set. The results obtained in this section hold under mild regularity assumptions on the vector fields F 1 , . . . , F k and on the end-point cost a : R n → R + . Finally, for a complete introduction to the theory of Γ -convergence, we refer the reader to the monograph [8].
In this section, we shall work with the weak topology of the Hilbert space U := L 2 ([0, 1], R k ). We first establish a preliminary result. We consider a L 2 -weakly convergent sequence (u m ) m≥1 ⊂ U , and we study the convergence of the sequence (x m ) m≥1 , where, for every m ≥ 1, the curve x m : [0, 1] → R n is the solution of the Cauchy problem Eq. 2.6 corresponding to the admissible control u m .
Proof Being the sequence (u m ) m≥1 weakly convergent, we deduce that there exists R > 0 such that ||u m || L 2 ≤ R for every m ≥ 1. The estimate established in Lemma 2.2 implies that there exists C R > 0 such that Indeed, the convergence of the right-hand side is guaranteed by Eq. 7.3. On the other hand, for every j = 1, . . . , k, from Eq. 7.4 we deduce the strong convergence as → ∞ by the hypothesis. Finally, observing that Eq. 7.3 givesx(0) = x 0 , we deduce that This argument shows that x m H 1 x ∞ as m → ∞. Finally, the thesis follows using again the compact inclusion Eq. 2.8.
The standard theory of Γ -convergence requires the domain of the functionals to be a metric space, or, more generally, to be equipped with a first-countable topology (see [1,Chapter 12]). Since the weak topology of U is first-countable (and metrizable) only on the bounded subsets of U , we shall restrict the functionals (F β ) β∈R + to the set where ρ > 0. We set F β ρ := F β | U ρ , where F β : U → R + is defined in Eq. 3.1. Using Lemma 7.1 we deduce that for every β > 0 and ρ > 0 the functional F β ρ : U ρ → R + admits a minimizer.
Proof Let us set β > 0 and ρ > 0. If we show that F β ρ : U ρ → R + is sequentially coercive and sequentially lower semi-continuous, then the thesis will follow from the Direct Method of calculus of variations (see, e.g., [8,Theorem 1.15]). The sequential coercivity is immediate, since the domain U ρ is sequentially compact, for every ρ > 0. Let (u m ) m≥1 ⊂ U ρ be a sequence such that u m L 2 u ∞ as m → ∞. On one hand, in virtue of Lemma 7.1, we have that lim m→∞ a(x m (1)) = a(x ∞ (1)), Therefore, combining Eqs. 7.6 and 7.7, we deduce that the functional F β ρ is lower semicontinuous.
Before proceeding to the main result of the section, we recall the definition of Γ -convergence.

Definition 7.3
The family of functionals (F β ρ ) β∈R + is said to Γ -converge to a functional F ρ : U ρ → R + ∪ {+∞} with respect to the weak topology of U as β → +∞ if the following conditions hold: -for every u ∈ U there exists a sequence (u β ) β∈R + ⊂ U ρ called recovery sequence such that u β L 2 u as β → +∞ and such that lim sup If Eqs. 7.8 and 7.9 are satisfied, then we write F β ρ → Γ F ρ as β → +∞.
Remark 7.4 Let us assume that F β ρ → Γ F ρ as β → ∞, and let us consider a nondecreasing sequence (β m ) m≥1 such that β m → +∞ as m → ∞. For every u ∈ U ρ and for every sequence (u β m ) m≥1 ⊂ U ρ such that u β m L 2 u as m → ∞, we have that Indeed, it is sufficient to "embed" the sequence (u β m ) m≥1 into a sequence (u β ) β∈R + such that u β L 2 u as β → +∞, and to observe that Combining the last inequality with the lim inf condition Eq. 7.8, we obtain Eq. 7.10.
Let a : R n → R + be the non-negative function that defines the end-point cost, and let us assume that the set D := {x ∈ R n : a(x) = 0} is non-empty. Let us define the functional F ρ : U ρ → R ∪ {+∞} as follows: otherwise, (7.11) where x u : [0, 1] → R n is the solution of Eq. 2.6 corresponding to the control u.
Remark 7.5 A situation relevant for applications occurs when the set D is reduced to a single point, i.e., D = {x 1 } with x 1 ∈ R n . Indeed, in this case the minimization of the limiting functional F ρ is equivalent to find a horizontal energy-minimizing path that connect x 0 (i.e., the Cauchy datum of the control system Eq. 2.6) to x 1 . This in turn coincides with the problem of finding a sub-Riemannian length-minimizing curve that connect x 0 to x 1 (see [4,Lemma 3.64]).
We now prove the Γ -convergence result, i.e., we show that F β ρ → Γ F ρ as β → ∞ with respect to the weak topology of U . Theorem 7.6 Let us assume that the vector fields F 1 , . . . , F k defining the control system Eq. 2.6 satisfy the Lipschitz-continuity condition Eq. 2.2, and that the function a : R n → R + designing the end-point cost is continuous. Given ρ > 0, let us consider F β ρ : U ρ → R + with β > 0. Let F ρ : U ρ → R + ∪ {+∞} be defined as in Eq. 7.11. Then, the functionals (F β ρ ) β∈R + Γ -converge to F ρ as β → +∞ with respect to the weak topology of U . Remark 7.7 If ρ > 0 is not large enough, it may happen that no control in U ρ steers x 0 to D, i.e., x u (1) ∈ D for every u ∈ U ρ . In this case, the Γ -convergence result is still valid, and the Γ -limit satisfies F ρ ≡ +∞. We can easily avoid this uninteresting situation when system Eq. 2.1 is controllable. Indeed, using the controllability assumption, we deduce that there exists a controlũ ∈ U such that the corresponding trajectory xũ satisfies xũ(1) ∈ D.
Proof of Theorem 7. 6 We begin with the lim sup condition Eq. 7.9. If F ρ (u) = +∞, the inequality is trivially satisfied. Let us assume that F ρ (u) < +∞. Then, setting u β = u for every β > 0, we deduce that for every m ≥m. Recalling that β m → ∞ as m → ∞, the previous inequality yields i.e., that x u (1) ∈ D. This argument proves that, if u β L 2 u as β → ∞ and if the quantity at the right-hand side of Eq. 7.12 is finite, then the limiting control u steers x 0 to D. In particular, this shows that F ρ (u) < +∞, namely F ρ (u) = 1 2 ||u|| 2 L 2 . Finally, we observe that and this establishes the lim inf condition Eq. 7.8.
The next theorem motivates the interest in the Γ -convergence result just established. Indeed, we can investigate the asymptotic the behavior of the sequence (inf U ρ F β ρ ) β∈R + as β → +∞. Moreover, it turns out that the minimizers of F β ρ provide approximations of the minimizers of the limiting functional F ρ , with respect to the strong topology of L 2 . The first part of Theorem 7.8 holds for every Γ -convergent sequence of equi-coercive functionals (see, e.g., [8,Corollary 7.20]). On the other hand, the conclusion of the second part relies on the particular structure of (F β ) β∈R + . Theorem 7.8 Under the same assumptions of Theorem 7.6, given ρ > 0 we have that Moreover, under the further assumption that F ρ ≡ +∞, for every β > 0 letû β be a minimizer of F β ρ . Then, for every non-decreasing sequence (β m ) m≥1 such that β m → +∞ as m → ∞, (û β m ) m≥1 is pre-compact with respect to the strong topology of U ρ , and every limiting point of (û β m ) m≥1 is a minimizer of F ρ .
Proof For every β > 0 letû β be a minimizer of F β ρ , that exists in virtue of Proposition 7.2. Let us consider a non-decreasing sequence (β m ) m≥1 such that β m → +∞ as m → ∞ and such that Recalling that (û β m ) m≥1 ⊂ U ρ , we have that there existsû ∞ ∈ U ρ and a sub-sequence (β m j ) j ≥1 such thatû β m j L 2û ∞ as j → ∞. Since F β ρ → Γ F ρ as β → +∞, the inequality Eq. 7.10 derived in Remark 7.4 implies that where we used Eq. 7.15 in the last identity. On the other hand, for every u ∈ U ρ let (u β ) β∈R + be a recovery sequence for u, i.e., a sequence that satisfies the lim sup condition Eq. 7.9. Therefore, we have that From Eqs. 7.16 and 7.17, we deduce that Finally, setting u =û ∞ in Eq. 7.17, we obtain From Eqs. 7.18 and 7.19, it follows that Eq. 7.14 holds. We now focus on the second part of the thesis. For every β > 0 letû β be a minimizer of F β ρ , as before. Let (β m ) m≥1 be a non-decreasing sequence such that β m → +∞ as m → ∞, and let us consider (û β m ) m≥1 . Since (û β m ) m≥1 is L 2 -weakly pre-compact, there existsû ∈ U ρ and a sub-sequence (û β m j ) j ≥1 such thatû β m j L 2û as j → ∞. From the first part of the thesis, it descends thatû is a minimizer of F ρ . Indeed, in virtue of Eq. 7.10, we have that where we used F β m j ρ (û β m j ) = inf U ρ F β m j ρ and the identity Eq. 7.14. The previous relation guarantees that (7.20) To conclude, we have to show that Using the assumption F ρ ≡ +∞, from the minimality ofû we deduce that F ρ (û) = 1 2 ||û|| 2 L 2 . Hence, Eq. 7.20 implies that 1 2 where we used that F β ρ (u) ≥ 1 2 ||u|| 2 L 2 for every β > 0 and for every u ∈ U ρ . From Eq. 7.22 and from the weak convergenceû β m j L 2û as j → ∞, we deduce that Eq. 7.21 holds.

Conclusions
In this paper, we have considered an optimal control problem in a typical framework of sub-Riemannian geometry. In particular, we have studied the functional given by the weighted sum of the energy of the admissible trajectory (i.e., the squared 2-norm of the control) and of an end-point cost.
We have written the gradient flow induced by the functional on the Hilbert space of admissible controls. We have proved that, when the data of the problem are real-analytic, the gradient flow trajectories converge to stationary points of the functional as soon as the starting point has Sobolev regularity.
The Γ -convergence result bridges the functional considered in the first part of the paper with the problem of joining two assigned points with an admissible length-minimizer path. This fact may be of interest for designing methods to approximate sub-Riemannian length-minimizers. Indeed, a natural approach could be to project the gradient flow onto a proper finite-dimensional subspace of the space of admissible controls, and to minimize the weighted functional restricted to this subspace. We leave further development of these ideas for future work.

Appendix B: Proofs of Subsection 2.3
Proof (Lemma 2.13) It is sufficient to prove the statement for the operator L u : U → V. Indeed, if L u is bounded and compact, then L * u : V → U is as well Indeed, the boundedness of the adjoint descends from Remark 2.12, while the compactness from [6, Theorem 6.4]).
Using Lemma 2.6, we obtain that, for every u ∈ U , there exists C > 0 such that the following inequality holds ||L u [v]|| C 0 ≤ C||v|| U , (B.1) for every v ∈ U . Recalling the continuous inclusion C 0 ([0, 1], R n ) → V, we deduce that L u is a continuous linear operator. In view of Theorem 2.1, in order to prove that L u is compact, it is sufficient to prove that, for every u ∈ U , there exists C > 0 such that

Appendix C: Proofs of Section 3
Proof (Lemma 3.1) We observe that the functional E : U → R + is defined as the composition E = a • P 1 , where P 1 : U → R n is the end-point map defined in Eq. 2.20. Proposition 2.4 guarantees that the end-point map P 1 is Gateaux differentiable at every u ∈ U . Recalling that a : R n → R + is assumed to be C 1 , we deduce that, for every u ∈ U , E is Gateaux differentiable at u and that, for every v ∈ U , the following identity holds: where x u : [0, 1] → R n is the solution of Eq. 2.6 corresponding to the control u ∈ U .