A Remark on Tonelli’s Calculus of Variations Текст научной статьи по специальности «Математика»

Russian Journal of Nonlinear Dynamics, 2023, vol. 19, no. 2, pp. 239-248. Full-texts are available at http://nd.ics.org.ru DOI: 10.20537/nd230501

MATHEMATICAL PROBLEMS OF NONLINEARITY

MSC 2010: 49J15, 49M25, 37J51

A Remark on Tonelli's Calculus of Variations

Kohei Soga

This paper provides a quite simple method of Tonelli's calculus of variations with positive definite and superlinear Lagrangians. The result complements the classical literature of calculus of variations before Tonelli's modern approach. Inspired by Euler's spirit, the proposed method employs finite-dimensional approximation of the exact action functional, whose minimizer is easily found as a solution of Euler's discretization of the exact Euler - Lagrange equation. The Euler-Cauchy polygonal line generated by the approximate minimizer converges to an exact smooth minimizing curve. This framework yields an elementary proof of the existence and regularity of minimizers within the family of smooth curves and hence, with a minor additional step, within the family of Lipschitz curves, without using modern functional analysis on absolutely continuous curves and lower semicontinuity of action functionals.

Keywords: Tonelli's calculus of variations, direct method, action minimizing, minimizing curve, regularity of minimizer, Euler method, Euler-Cauchy polygon

1. Introduction

Investigation of the existence and regularity of minimizing curves of action functionals is one of the most fundamental problems in classical mechanics, which leads to various applications in many fields. Leonida Tonelli established a general method of the problem, known as the direct method of calculus of variations, based on the properties of absolutely continuous curves and lower semicontinuity of action functionals. We revisit well-established Tonelli's calculus of variations with a Lagrangian L = L(x, t, {) satisfying the following conditions:

(L1) L: Rd x R x Rd ^ R, C2;

(L2) fpK®, t, 0 is positive definite for each (x, t, £) e Rd x R x Rd, where ^ = stands for the Hessian matrix of L with respect to the {-variable;

d2L

Received December 13, 2022 Accepted March 31, 2023

This work was supported by JSPS Grant-in-aid for Young Scientists No. 18K13443.

Kohei Soga

soga@math.keio.ac.jp

Department of Mathematics, Faculty of Science and Technology, Keio University 3-14-1 Hiyoshi, Kohoku-ku, Yokohama, 223-8522 Japan

(L3) L is uniformly superlinear, i.e., for each a ^ 0, there exists ba £ R such that L(x, t, £) ^ ^ a\(\ + ba for all (x, t, £) £ Rd x R x Rd;

(L4) the Euler - Lagrange flow 4>L generated by L is complete, i.e., is global in time.

11°

Note that is well-defined under (L1)-(L3), which is indirectly seen via the corresponding Hamiltonian flow stated in (H1)-(H4) below. Let w: Rd R be a given C0-function such that there exist constants a ^ 0 and /3 £ R for which w(x) ^ -a\x\ + /3 holds for all x £ Rd. Let AC ([0, t]; Rd) be the family of all absolutely continuous curves 7: [0, t] ^ Rd. For each t > 0, we consider the action functional

t

Lw : AC ([0, t]; Rd) ^ R U{+^}, Lw(7) := J L(1(s),s,1'(s)) ds + w(7(0)).

0

We write L0 if w = 0. Since L — b0 ^ 0 in Rd x R x Rd due to (L3), the integral in Lw (7) always makes sense, including its value The typical minimizing problems are

inf Lw (7), inf Lo(Y), (1-1)

Y^AC([0,t]; Rd), y(t)=x y^AG([0,t]; Rd), j(t)=x, j(0)=x

where x,x £ Rd are arbitrarily given end points. The first one in (1.1) is also called the Bolza problem. We refer to [5] for the history and earlier developments of minimization problems for Lw. Before Tonelli, the existence of minimizers was not directly stated, but necessary conditions of minimizers were widely discussed. Tonelli's theory states that

• under (L1)-(L4), each of the minimizing problems (1.1) admits at least one minimizing curve in AC([0, t];

• any minimizing curve of (1.1) has C2-regularity and solves on [0, t] the Euler - Lagrange equation generated by L.

We refer to [3] for a good review of Tonelli's theory; Chapter 3 of [4] for full details of the proof of Tonelli's theory; [6], [2] and [4] for important applications of Tonelli's theory to La-grangian/Hamiltonian dynamics, Hamilton-Jacobi equations and optimal control.

In actual applications mentioned above, it is often enough to consider minimization in the smooth class or in the continuous piecewise smooth class. The Lipschitz class would be also natural, since Lw becomes R-valued. Nevertheless, Tonelli's approach requires the absolutely continuous class. We briefly recall Tonelli's method together with the reason why the absolutely continuous class is technically necessary, apart from the point that minimization in a wider class of curves would be mathematically more interesting. The first important fact in Tonelli's approach is that the sequence {Yj}jeN C {7 £ AC ([0, t]; Rd) | 7(t) = such that

Lw (7j) ^ A , r inf. „ Lw (Y) as J ^^

~ieAC([0,f\; Rd), y(t)=x

is uniformly bounded and equicontinuous, providing a uniformly convergent subsequence and its limit 7* £ C0 ([0, t]; Rd). The next important fact is that the sequence {7'j}jeN is uniformly integrable, which implies that 7* belongs to AC ([0, t]; Rd) and 7'j converges to 7*' in

the weak-L1 topology. Then the lower semicontinuity of Cw with respect to the C0-topology resulting from (L2) shows

Lw (Y*) < liminf Cw ),

3 ^^

to conclude that y* is a minimizing curve of (1.1). In this argument, it is not a priori clear if 7* is Lipschitz or not, even if we start with the Lipschitz class instead of AC ([0, t]; Rd). In a subset of the Lipschitz class with an artificial bound of Lipschitz constants, Tonelli's approach may find a Lipschitz minimizing curve with the same artificial bound. However, variations around the minimizer cannot necessarily be taken in all directions and the Euler - Lagrange equation fails to be derived (one does not a priori know if a Lipschitz constant of the minimizer is strictly smaller than the artificial bound). Tonelli's approach based on compactness of a minimizing sequence and the lower semicontinuity is powerful enough to extend the existence result to the case with Lagrangians out of (L1)-(L4): see, e.g., [1] for more details.

Higher regularity of 7* is easily seen if there are additional growth conditions of the derivatives dxL, d^L that guarantee integrability of dxL(y(s), s, y' (s)) and d^L(y(s), s, y'(s)) over [0, t]

for y G AC ([0, t]; Rd). In this case, we easily obtain the vanishing Gâteaux derivative of Lw at y *, which gives the Euler - Lagrange equation in the integral form. This leads to the conclusion that y * has C 2-regularity and satisfies the Euler - Lagrange equation. Only with the assumptions (L1)-(L4), one can still obtain the same regularity result through a classical result by Weierstrass: if t > 0 and \x — x\ are sufficiently small, the second problem of (1.1) admits the unique C2-minimizer satisfying the Euler - Lagrange equation. See the Appendix of [6] and Chapter 3 of [4] for more details.

In this paper, we propose a completely different approach to find minimizing curves of Lw directly within the smooth class, which immediately leads to results on the minimization within the Lipschitz class. This is an attempt to complement the missing part of the classical calculus of variations (i. e., existence of minimizers) only by means of classical techniques. We emphasize that the proposed method does not reduce the significance of Tonelli's approach, also because it works only with Lagrangians satisfying (L1)-(L4). However, the proof of the existence and regularity of minimizers for (1.1) becomes much simpler. Furthermore, it would be interesting to observe that Euler's classical idea to find minimizers (see Chapter 2 of [5]) seems almost successful and general enough.

The key idea of the proposed method is to introduce a finite-dimensional problem corresponding to (1.1), i.e., we discretize [0, t] and deal with the Riemann sum of the integral in Lw. The finite-dimensional problem is merely minimization of a smooth function defined in Rd' with d'G N becoming arbitrarily large for the limit of approximation but finite in each step. Then we easily obtain a minimizing point with the vanishing derivative. Furthermore, the vanishing derivative yields a finite difference equation that is Euler's discretization of the exact Euler - Lagrange equation generated by L; the minimizing point solves the discrete Euler-Lagrange equation with a priori boundedness. Due to (L1)-(L3), we have the Legendre transform H of L. The discrete Euler - Lagrange equation is transformed to Euler's discretization of Hamilton's equations generated by H. It is straightforward to see that the Euler-Cauchy polygonal line generated by the solution of discrete Hamilton's equations converges to a C2-cur-ve (y*(s), p(s)) solving exact Hamilton's equations; y*(s) is a C2-minimizer of Lw within the smooth class. The essential point of finite-dimensional approximation is that one can immediately obtain the approximate Euler - Lagrange equation with nice estimates on the discrete level without being bothered by regularity issues.

In order to extend the minimization to the Lipschitz class, we use the standard fact that each Lipschitz curve is approximated by a series of smooth curves with bounded derivatives. Then Lebesgue's dominated convergence theorem concludes that a minimizing curve within the smooth class is also minimizing among all Lipschitz curves. It is not difficult to see that any other minimizing curve in the Lipschitz class (if it exists) necessarily has C2-regularity and solves the Euler - Lagrange equation. Note that, in the case of absolutely continuous curves, derivatives are unbounded in general and Lebesgue's dominated convergence theorem would not work; hence, our approach without any additional growth condition of L would fail in AC ([0, t]; Rd).

2. Minimization in the smooth class

We discuss minimization of Lw in the smooth class as simply as possible. We proceed in a slightly wider class than C1 ([0, t]; Rd). Let Cpw ([0, t]; Rd) be the family of all continuous piecewise smooth curves; namely, Cpw ([0, t]; Rd) is the family of all continuous curves 7: [0, t] — — Rd such that each 7 has a finite division 0 = t0 < t1 < ■ ■ ■ < Tm = t with m = m(7) ^ 1 for which 7 is C1 within each interval [tz_ 1, t1 ]. Note that each 7 £ Cpw ([0, t]; Rd) is Lipschitz. Calculus of variations for Lw in Cpw ([0, t]; Rd) is the minimum requirement in analysis of viscosity solutions (or the value functions) to the Hamilton-Jacobi equation generated by H and weak KAM theory based on the Lax-Oleinik type operator generated by L. Reasoning in Cpw ([0, t]; Rd) is essentially the same as that in C1 ([0, t]; Rd) (in both cases, one does not need the technique of mollification and Lebesgue's dominated convergence theorem). Note that our reasoning works also in C2 ([0, t]; Rd). In this section, we discuss how to find a C2-minimizing curve directly within Cpw ([0, t]; Rd). The result of Section 3 shows that any minimizer of Cpw ([0, t]; Rd) necessarily has C2-regularity and solves the Euler - Lagrange equation.

We briefly refer to the Legendre transform H of L with respect to the {-variable:

H(x, t, p) = sup {p ■{ — L(x, t, {)}. (2.1)

d

Here, x ■ y = ^ x%y% for x, y £ Rd. It follows from (L1)-(L4) that i=1

(H1) H: Rd x R x Rd — R, C2;

(H2) t, p) is positive definite for each (x, t, p) e Rd x R x Rd;

(H3) H is locally uniformly superlinear with respect to p, i.e., for each a ^ 0 and K C Rd x R compact, there exists b'a K £ R such that H(x, t, p) ^ a\p\ + b'a K for all (x, t, p) £ K x Rd;

(H4) the Hamiltonian flow generated by H is complete.

Furthermore, since d^L(x, t, {) = p is invertible with respect to the {-variable for any p £ Rd due to (L1)-(L3), we have the C 1-map { = {(x, t, p) such that d^L(x, t, {(x, t, p)) = p, where the supremum of (2.1) is uniquely attained by { = {(x, t, p). This implies that

^-(.r, t, ■): Rd -)• Rd is bijection, =

dH . . dL dH dL

— [x, t,p)=£e>p= —{x, t, 0, t> P) = i' Z(x> p>>-

We refer to [2] and [4] for a detailed description of the Legendre transform and to Lemma 3.1 of [7] for a proof of (L1)-(L4) ^ (H1)-(H4).

2.1. Problem with one fixed end point and one free end point

For each t > 0 and x £ Rd, consider the minimizing problem with one fixed end point and one free end point:

, inf Cw (y). (2.2)

l^Clw ([0,t]; Rd ),y (t)=x

Claim. One can find a C2-minimizing curve that attains (2.2) and satisfies the Euler-Lagrange equation generated by L, without going through the argument in AC ([0, t]; Rd) and Weierstrass's theorem.

Our claim is justified by reasoning with finite-dimensional approximation of Cw:

h := -p, tk := hk, K, k G N (we will send K —>• oo), K

y = (yo ,Vi,...,Vk-1) £ RdK, Vk £ Rd for k = 0, 1,...,K — 1, y' = (y'0,y'1,...,y'K-1), y'k-=Vk+l~Vk for k = 0, 1, ..., K — 1 with yK = x,

K -1

CK: RdK ^ R, CK(y):= ^ L(Vk, tk, vk)h + w(yo).

k=0

Step 1. We minimize CK in RdK. It follows from (L3) and the condition w(x) > — a\x\ + ft that, for a = a and any y £ RdK, we have the lower bound of CK as

K -1

E vk

K -1

CK(y) > ^ (a\yk\ + ba)h — a\yo\ + ft > a k=0

= a\x — y0\ + bat — a\y0\ + ft > —a\x\ + bat + ft > —m. For yx := (x, ..., x) £ RdK, we have

k=0

h + ba t — a\y0 \ + ft =

K -1

K

Cx := sup CK(yx) = sup V^ L(x, tk, 0)h + w(x) < (2.3)

keN KeN k=0

It is enough to minimize CK within the set Y^ := {y £ RdK \ C^w (y) ^ Cx}. It follows from (L3) that, for a = 1 + a and any y £ YjK, we have

K-1 K-1

-K/ xw

k=0 k=l>0

Cx > CK(y) \vk\h — a\x\ + b1+at + ft > \vk\h — a\x\ + b^t + ft >

> \yi\ — \x\—a\x\ + b1+at + ft, (2.4)

Cx > CK(y) > min , \vk\t — a\x\ + b1+at + ft. (2.5)

0<k<K-1

Hence, we obtain by (2.4),

< R1 := Cx + (1 + a)lxl-b1+at - /, Vl = 0, ..., K - 1, Vy £ Y^.

(2.6)

Hereafter, R1, R2, ... stand for some constants independent of K. Therefore, Y^ is a bounded

subset of M.dK and there exists y = y(K) e FrA such that.

C%m= inf c%(y).

yeRdK

It follows from (2.5) that we have k* = k*(K) such that

y'k*{K)

= min lv',1 and 0<k<K-1 k

v'k*(K)

^ R2 := (Cx + alxl - b1+at - /)t-1.

(2.7)

Step 2. We take variations around y and send K —> oo. Since £^,(y0, ..., y^-i) is a C1-funct.ion except for the y0 variable (w is only continuous), its minimum point y implies

9 „ K /_\ 9L , (dL _, dL , ,

(V) = —(y^ ífc, Vk)h ~ [ —(y^ ífc, Vk) ~ -Q^ÍVk-1» h-v Vk-i) ) = 0»

dy

dx

d£

Vk = 1, ..., K - 1.

(2.8)

The Legendre transform equivalently transforms (2.8) into (2.9):

dL

Zk ■= -Q^iVk, h; y'k) for k = 0,

z'k :=

zk Zk-l h

for k = 1,

y'k = h; zk), Vfc = 0, .

z'k = (Vk> h> Vfc = 1, .

., K - 1, ., K - 1, .., K - 1, .., K - 1.

(2.9)

Note that (2.9) is discretization of Hamilton's equations in terms of the Euler method. Due to (2.6) and (2.7), we have for any K £ N,

Vk*{K)

< R1, tk*(K) £ [0,t],

"k*(K)

^ R3 := sup

\x\^R1, O^s^t, \£\<R2

dL

(x, s, £)

Note that, if w £ C1, we can take variations also with respect to y0 and obtain an instant a priori bound of ~zQ. Since j (yk*^Ky tk*(K)> zk*(K)) } is a bounded sequence of Rd xRx Rd, we find a convergent subsequence (denoted with {Kj}j€N) with its limit (x*, t*, p*) satisfying \x* \ ^ R1, t* £ [0, t], \p*\ ^ R3. We assert that the Euler-Cauchy polygon method and (H4) yield the following fact: the polygonal line generated by the solution ~Zjk)}k-=o a'.-i °f (2-9) with K = = Kj converges uniformly to

(7*(s),p* (s)) := (x*, p*) on [0, t] as j — x>, where 7* satisfies the Euler - Lagrange equation with 7*(t) = x.

We check our assertion. It follows from (H4) that (Y*(s),p*(s)) exists on the interval [0, t]. Due to the continuous dependency with respect to initial data, the curve

(y*(s), p*j(s)) := фнк <Aj) (///, /ч . гкчк]))

uniformly converges to (y*(s), p*(s)) on [0, t] as j ^ m. Hence, ^ [y* (s), s, p*(s)) f is

l\J 3 / J se[0, t]

contained in a bounded neighborhood B <z Rd x R x Rd of {(y*(s), s, p*(s))}s€[0 t] for all large j. The Taylor approximation yields with h-

dH

lj(h+i) - 7,-(ifc) = -— bjih), h, l'?1!,:) I' i + О {!>]), k <).....Kj - 1,

dH

Pj(tk+1) - Pj(tk) = —q^ (7,* (ifc+i), tk+1, Pj(h+1)) hj + ° k = 0, ..., Kj - 1,

where

O {hj^j is bounded by R4hj for any k and j ^m with some constant R4 > 0 determined

by the suprema of the first and second derivatives of H in B. Let y-k> z-k be the solution of (2.9)

Ujk = (ujki ùjk) '■= (Djk j(tk)> zjk —Pj(tk))>

with K = Kj and set

where Ujk*(K ) = 0. Let в > 0 be a common Lipschitz constant of dpH and dxH in B. As long as (Vjk, tk, Ijk) belongs to В (this is a priori the case at least for k close to k*(K■) for all large j), we have

and

Kfc+i - ujk\ < e\Ujk\hj + R^h), \ujtk+i - ujk\ < e\Uj,k+1\hj + R4hj,

1 + eh,

< r^lf.' '11- + ^ (1 + + R*hi> k >

1 + eh,

l^-il < т~ЩГг^ + R5h* ^ (1 + :u"'>]' '■>>" + k <

( R5hЛ ( R5 hA

[pj,k+il + -j/) < (1 + 3%) [\ujk\ + -j^j , k > к*(Щ),

( R5h~\ ( R5 hA

[Pjfi-il + ^ß1) < (1 + 3%) [\u3k\ + -^J, k^k*(K3),

\U3k\ + < (1 + m/ " * j + ^¡f), к > k*(K3)

R5h,

WI + < (1 + 3Oh^-* (| + * < k%K3).

It follows from (1 + v)l/v ^ e as v ^ 0+ (the convergence is from below) that

(1 + 3%)\k-k(Kj)\ < (1 + 3dh3)Kj = (1 + 3%)t/hj = {(1 + 3%)1/(3&hi^ < e

Therefore, we find a constant R6 > 0 independent of j for which

\Ujk\ < R6hj, Vk = k1,k1 + \,...,k2,

where k2 are such that, 0 ^ kt ^ k*(Kj) Y k2 Y - 1 and {(yjk, tk, zjk)}k C B.

Now it is clear that, tk, Zjk)}0<k<K _1 C B for all large j and the Euler-Cauchy polyg-

onal line generated by {(yjk, Zjkïïo^K — i wit,h VjK. = = zjK.-i uniformly converges

to (y* ( • ), p*(• )) as j ^ to. In particular, we have

max \y'ik — Y*'(ii-)l = m&x 04MK'-i 1 jk 1 o4MK'-1

j J

h,- zjk)~ ^ (Y *{tk), 11.- P*{tk))

^ 0 as j

Thus, denoting y ■ := (yj0, yji, ..., we obtain

j

k ■ _

¿w'iVj) ->■ Cw (Y*) as j —>• oo.

Step 3. We check that 7* attains (2.2). For each 7 £ Cpw ([0, t]; Rd) with 7(t) = x, there exists a division 0 = t0 < t1 < ■ ■ ■ < Tm-1 < Tm = t with m = m(7) ^ 1 for which 7 is C1 on each interval [t1-1, tJ. Let Kj and hj be the one given in Step 2. For each Kj, define yj :=

:= ^7(^0)1 Y(^i)) •••) Y^a'.-i))- Note that, each component, y'-k = of yj satisfies,

for any k with a quantity n(r) ^ 0 tending to 0 as r — 0+,

\yjk\ < R7 := suP \7X(s)\, if 3Ti £ (tk, tk+1); otherwise \yjk — 7'(tk)\ < n(hj).

se[0,t]

Let Aj be the set of k £ {0, 1, ..., Kj — 1} such that the interval (tk, tk+1) does not contain any Tl. We have

ifc+i

LW(y) - (Vj) < / \L(Y(s) s, V(s)) - L{yjk, tk, yjk)\ds +

j "k

+ R8(m — l)hj ^ 0 as j ^œ,

i8(m — ±)hj

K K K K

A„(y) - cw (7*) = cw{Y) - cw>(yj) + c^(y3) - Cw'iVj) + - cw (7*) ^

K K _

> 4„(y) - + - Cw (7*) —>■ 0 as j —>• 00,

which implies £w (7*) ^ £w(7).

2.2. Problem with two fixed end points

For each t > 0 and x, x £ Rd, consider the minimizing problem with two fixed end points:

inf £0(7). (2.10)

7ecpw([0,t]; Rd), Y(t)=x, Y(0)=x

We may obtain a minimizing curve of (2.10) in the same way as the previous subsection, where we fix y0 = x and take y = (y1, ..., yK_ 1) £ Rd(K-1) as free variables and use

Vxx := (7(ii), • • •, Y(^a'—1)) with y(s) := x +

in (2.3), instead of yx = (x, ..., x) G

We remark that there is another way to deal with (2.10): as one of the reviewers kindly pointed out, the problem (2.10) can be seen as a special case of (2.2) by allowing w to be lower semicontinuous and RU{+^}-valued except for w = everywhere. In fact, the demonstration in Subsection 2.1 still works with such w, where Cx in (2.3) is replaced by the straight line joining x and a point at which w is finite. If w is such that w(7) = 0 and w(y) = for all y = x, a minimizing curve of (2.2) is that of (2.10).

3. Minimization in the Lipschitz class

Let Lip ([0, t]; Rd) be the family of all Lipschitz continuous curves. Let 7* be a minimizing curve of (2.2). We show that any 7 e Lip ([0, t]; Rd) with 7(t) = x satisfies Lw (7*) ^ Lw(7). Let M > 0 be a constant such that |y'(s)| ^ M a. e. s e [0, t]. We extend 7 to be x for s > t and to be 7(0) for s < 0, and mollify it with the standard mollifier with a small parameter e > 0, obtaining a smooth curve 7£: [0, t] — Rd with 7s': [0, t] — [-M, M]d for each e > 0. We set

7e(s) := 7£(s) + (x - 7£(t)) £ + (7(0) - 7£(0)) ^ : [0, t] Rd.

Note that 7£(t) = x, y£(0) = 7(0) for each e > 0 and \\y£ — y\\co([0 t]-Rd) — 0, ||y£' — — 7'ILX([0 t];Rd) — 0 as e — 0+. Since sup IL (y£(s), s, 7£'(s))| is bounded independently

of e — 0+, Lebesgue's dominated convergence theorem yields lim Lw (y£) = Lw(7), where

we take a subsequence of e — 0+ so that Y£'(s) — Y'(s) a. e. s e [0, t] (this holds even when w is lower semicontinuous). Since y£ belongs to Cpw ([0, t]; R^ with 7£(t) = x, we have Lw (7*) ^ Lw (7£), which yields our assertion.

Finally, we check that any other possible minimizer of Lw in Lip ([0, t]; Rd) |7(t)=x has

C2-regularity and satisfies the Euler - Lagrange equation generated by L. Let 7 e

Lip([0, t]; Rd)

be such that 7(t) = x and Lw(7) = Lw (7*). Then, for any smooth curve v: [0, t] — Rd with v(t) = v(0) = 0, we have

/ \ -/(«)) " / r, i{r))dr U/(S)dS,

l t ( s

d „ . .

0=-£wh + ev)^ K........ j dx

0 I 0

which implies

s

dL , , s ,, „ f dL

/dL

1(t), t, 7;(t)) d,r = const, a.e. s e [0, t].

0 s

d

x

s

d

x

Since dgL(x, s, •) is invertible and f dxL(y(t), t, y'(t)) dr is continuous as a function of s e [0, t],

we see that y'(s) is continuous. Then f dxL(7(r), r, y'(t)) dr turns into C1 to conclude that 7

0x

has C2-regularity and satisfies the Euler - Lagrange equation.

We may deal with the problem with two fixed end points in the same way, where we extend 7 e Lip ([0, t]; Rd) to be x for s > t and to be 7 for s < 0, and take

7e(s) := 7£(s) + (x ~ 7 £(t)) £ + (i-Yem ^ : [0, t] ^ Rd.

Acknowledgments

The author thanks one of the reviewers for valuable comments in regard to Subsection 2.2.

Conflict of interest

The author declares that he has no conflicts of interest.

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