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Remarks regarding the gap between continuous, Lipschitz, and differentiable storage functions for dissipation inequalities appearing in H1 control Lionel Rosier a and Eduardo D. Sontag b ;1 a Laboratoire d'Analyse Num'erique et EDP, Universit'e Paris 11, b^at. 425, F91405 Orsay Cedex, France b Dept. of Mathematics Rutgers University New Brunswick, NJ 08903 This paper deals with the regularity of solutions of the Hamilton-Jacobi Inequality which arises in H1 control. It shows by explicit counterexamples that there are gaps between existence of continuous and locally Lipschitz (positive definite and proper) solutions, and between Lipschitz and continuously differentiable ones. On the other hand, it is shown that it is always possible to smooth-out solutions, provided that an infinitesimal increase in gain is allowed. Key words: H1 control, storage functions, dissipation inequalities, Lyapunov functions, stability, viscosity solutions 1 Introduction The so-called "H1 control problem" is that of finding a state (or more generally, a measurement-based) feedback which stabilizes a given system, while satisfying an energy-gain (L2 operator norm) constraint. For linear systems, the problem has a long history, and an elegant solution was provided in the by now classical paper [9]. The nonlinear version of this problem has been the subject of intense research as well; see for instance [4,5,14,21] among many 1 E-mail: sontag@control.rutgers.edu. Research supported in part by US Air Force Grant F49620-98-1-0242 2 E-mail: Lionel.Rosier@math.u-psud.fr. others. A central role in these studies is played by a partial differential inequality, a Hamilton-Jacobi Inequality (HJI) which is satisfied by a "storage" or "energy" function V associated with the closed loop system. In this paper, we concern ourselves with the analysis of the HJI for a system already in closed-loop form, since we wish to make some remarks about the regularity of solutions of this equation. Moreover, in order to keep the discussion as simple as possible, we analyze the case of full state measurements, but similar conclusions could be drawn for the case when outputs are of interest. Thus, the main focus of this paper will be on systems, affine in inputs, of the following form: .x = g0(x) + mX i=1 u i gi(x) (1) for which states x evolve in Rn , and inputs u = (u1; : : : ; um) take values in Rm . We assume that the vector fields gi, i = 0; : : : ; m are locally Lipschitz in x. (These systems might be thought of having been obtained from a more general class of systems .x = f (x; u; v) after applying a stabilizing feedback v = k(x), but that interpretation is irrelevant to the results that we give.) In this introduction, we restrict attention to the above class of systems, but later we will also provide several results valid for more general systems, not necessarily affine in inputs, of the form .x = F (x; u) ; (2) with x(t) 2 Rn and u(t) 2 U . (As far as the definitions are concerned, we do not need impose any technical conditions on U and F .) We use j\Delta j to denote Euclidean norm in Rn or Rm . For systems (1), the HJI's of interest are inequalities which are often expressed as: rV (x) \Delta g0(x) + 14fl mX i=1 (rV (x) \Delta g i(x)) 2 + jxj2 ^ 0 ; (3) where fl ? 0 is the (square of the) "L2 gain" of the system. Inequality (3) for non-differentiable V must be interpreted in a generalized sense, as we discuss below. The V 's that arise, because of implicit stability considerations, must be proper and positive definite. Modulo some elementary calculus to do the implied maximization over u, the inequality (3) is equivalent to: rV (x) \Delta g0(x) + mX i=1 u i gi(x)! ^ \Gamma jxj 2 + fljuj2 ; (4) 2 understood as holding for all u. This latter form is preferred, because it makes sense for arbitrary (not necessarily input-affine) systems (2), and because the theory then makes a natural contact with the theory of dissipative systems developed by Willems and Moylan and Hill (see e.g. [12,13,21,22]). It is wellknown (and we review this fact in the Appendix) that (4) is equivalent to the following inequality: V (x(b)) \Gamma V (x(a)) ^ bZ a flju(t)j2 \Gamma jx(t)j2 dt (5) holding along all solutions (x(\Delta ); u(\Delta )) of (1) with x(\Delta ) absolutely continuous and u(\Delta ) measurable and essentially bounded, for each a ! b in the domain of the solution. Notice that (5) has the following consequence for trajectories which start at x(0) = 0, and if V is nonnegative: for all T ? 0 for which the solution is defined, it must hold that TZ 0 j x(t)j2 dt ^ fl bZ a j u(t)j2 dt : This means that the map u(\Delta ) 7! x(\Delta ) seen as a map between square-integrable functions, has operator norm ^ pfl. Conversely, the theory relates such operator norms back to the existence of V 's solving the differential inequality. We will not discuss this well-known material any further, but rather consider the differential inequality as our object of study. In general, it is not natural to impose the requirement that solutions to (4) should be smooth, so the HJI must be interpreted in the viscosity sense (a "viscosity supersolution"), see e.g. [8,10]), or, in what is an essentially equivalent manner, the proximal analysis formalism found in [7]. Under suitable controllability hypotheses, it does make sense to restrict attention to continuous V 's, see for instance [4]. Thus we will always assume in our study that V is at least continuous. The main results in this note address the gap between continuous and C1 solvability. We give examples which show (a) that there may exist even Lipschitzcontinuous (proper and positive definite) solutions, but no possible continuously differentiable ones, and (b) there may exist continuous (proper and positive definite) solutions but no possible locally Lipschitz continuous ones. On the other hand, we provide a smoothing result which shows that it is possible to pass from C0 to C1 solutions as long as an infinitesimal increase in the gain fl is allowed. A last section treats the special case of one-dimensional systems, where no gap exists. None of these facts is unexpected, of course, but it would appear that the corresponding counterexamples and proofs are not 3 available in the literature. 2 Definitions and Statements of Results Recall (cf. [7], Section 3.4) that a vector i 2 Rn is a viscosity subgradient of a function V : Rn ! R, at the point x 2 Rn , if lim infh! 0 1j hj [V (x + h) \Gamma V (x) \Gamma i \Delta h] * 0 (6) The (possibly empty) set of all viscosity subgradients of V at x is called the viscosity subdifferential, and is denoted @DV (x). Observe that, if the function V is differentiable at x, then @DV (x) = frV (x)g. Definition 1 Suppose given a system \Sigma as in (2), and a fl * 0. We say that a function V : Rn ! R*0 witnesses the gain fl if the following condition holds: i \Delta F (x; u) ^ \Gamma jxj2 + fljuj2 8i 2 @DV (x) (7) for all x 2 Rn n f0g and u 2 U . The set of all those continuous V which witness a given gain fl for the system \Sigma is denoted as W(\Sigma ; fl). \Lambda We recall that a continuous function V : Rn ! R*0 is said to be proper (or "radially unbounded") provided that every set of the form fx j V (x) ^ ag is compact, for each a ? 0 (equivalently, V (x) ! 1 as jxj ! 1), and is said to be positive definite if V (x) = 0 if and only if x = 0. We denote W1(\Sigma ; fl) := fV 2 W(\Sigma ; fl) j V is proper and positive definite g : When V is C1, condition (7) means simply that rV (x)\Delta F (x; u) ^ \Gamma jxj2+fljuj2 for all x 6= 0 and u. As is well-known, asking for a globally C1 such function which is also positive definite is overly restrictive. To see this, consider an input-affine system with n = 1, and take u = 0. The inequality would force the bound jxj2= jg0(x)j ^ jV 0(x)j for all x 6= 0. If V is positive definite, it must have a local minimum at zero, so V 0(0) = 0. Hence jxj2 = o(g0(x)) as x ! 0 ; which is too strong a constraint on g0 (one would not be able to study a system such as .x = \Gamma x3). Therefore, it is routine (see for instance [20]) to drop the requirement of differentiability at the origin. We will denote by C10 4 the set of continuous functions V : Rn ! R*0 whose restriction to Rn n f0g is continuously differentiable. The first result consists of an example which shows that there may exist a globally Lipschitz V 2 W1(\Sigma ; fl), but no C10 such function (in fact, not even merely a C10 proper, nor a C10 positive definite, function in W(\Sigma ; fl)). Specifically, take the following system \Sigma 1: .x1 = jx1j (\Gamma x1 + jx2j + u1) .x2 = x2 (\Gamma x1 \Gamma jx2j + u2) : Note that this system is of the form (1), with n = m = 2. Now consider the following function (basically, the L1 norm): V1(x) = 2 jx1j + 2 jx2j : This function is proper and positive definite, and globally Lipschitz. We prove in Section 3: Proposition 2 For the above system \Sigma 1, and unit gain fl = 1, we have V1 2W 1(\Sigma 1; 1). On the other hand, if V is any C10 function in W(\Sigma 1; 1), then V is not positive definite nor proper. Remark 3 We chose to give example \Sigma 1 because of its simplicity. However, at the cost of added complexity, one may easily provide similar examples which are such that the vector fields gi are not merely Lipschitz. For a C1 example, we may take .x1 = jx1x2j3 + x1 (\Gamma jx1j + u1) .x2 = \Gamma (x1x2)3 + x2 (\Gamma jx2j + u2) : The proof of the analog of Proposition 2, using the same function V1 and gain 1, is virtually the same. \Lambda The second result produces an example showing that there may exist a continuous V 2 W1(\Sigma ; fl), but every locally Lipschitz function in W(\Sigma ; fl) must be either nonproper or non-positive definite. Specifically, we will consider the following system \Sigma 3: .x1 = \Gamma x1 + x2 + u1 .x2 = 3x4=32 (\Gamma x1 \Gamma x2 + u2) 5 which is of the form (1), with n = m = 2. We will consider the following function: V3(x1; x2) := x21 + x2=32 which is proper, positive definite, and continuous. We prove in Section 3: Proposition 4 For the above system \Sigma 3, and unit gain fl = 1, we have V3 2 W1(\Sigma 3; 1). On the other hand, if V is any locally Lipschitz function in W(\Sigma 3; 1), then V is not positive definite nor proper. There is a general positive result as well. We show that, for any system \Sigma as in (1), and any fl ? 0, W1(\Sigma ; fl) 6= ; implies that W1(\Sigma ; fl0) T C10 6= ; for each fl0 ? fl. In other words, it is always possible to smoothly approximate a proper positive definite continuous V by one that is continuously differentiable away from zero (actually, the proof provides an infinitely differentiable such approximation), provided that we allow a negligible increase in gain. This is summarized in the following statement, which we also prove in Section 3: Theorem 5 For any system \Sigma as in (1), inf ffl j W1(\Sigma ; fl) 6= ;g = inf nfl j W1(\Sigma ; fl) " C10 6= ;o : This result is significant in so far as the "inf" in question is the one of interest in H1 control problems. Remark 6 The result in Theorem 5 is stated for input-affine systems. It is false in general for arbitrary systems (2). One elegant statement can be made by considering systems of the following special form, for any p * 1: .x = g0(x) + mX i=1 u p i gi(x) (8) (again assuming that the vector fields gi, i = 0; : : : ; m are locally Lipschitz in x). We may interpret the powers up, for negative u, either as "jujp" or as "sign u jujp"; with either interpretation, we shall show, also in Section 3, that Theorem 5 holds (for any system of this form) if p ^ 2 and does not hold (for some systems of this form) if p ? 2. \Lambda Finally, we will analyze in Section 4 the special case m = n = 1. We will show that there is in that case no gap between the locally Lipschitz and the differentiable case, for systems affine in inputs, but that a gap reappears if we deal with systems that are not input-affine. 6 3 Proofs of Main Results We first prove Proposition 2 and Proposition 4. We then prove Theorem 5 (in somewhat more generality), and, finally, we justify the claims made in Remark 6. 3.1 Proof of Proposition 2 In order to verify that V1 2 W(\Sigma 1; 1), we compute its subgradients. At any point x = (x1; x2) 2 R2 with x1x2 6= 0, obviously @DV1(x) = frV1(x)g = (2 x1jx 1j ; x2j x2j!) ; so (7) becomes rV1(x) \Delta g0(x) + mX i=1 u igi(x)! = 2 x 1j x1j jx 1j (\Gamma x1 + jx2j + u1) +2 x2jx 2jx 2 (\Gamma x1 \Gamma jx2j + u2) = 2u1x1 + 2u2jx2j \Gamma 2x21 \Gamma 2x22^ j uj2 \Gamma jxj2 ; as desired. Suppose now that x1 = 0 and x2 6= 0. Then, it is an easy exercise with the definition of subgradients to see that @DV1(x) = [\Gamma 2; 2] \Theta (2 x2jx 2j ) ; so in this case, for any i = (i1; i2) 2 @DV1(x), we have: i \Delta g0(x) + mX i=1 u igi(x)! = i1 \Delta 0 + 2 x 2j x2jx2 (\Gamma jx2j + u2) = 2u2jx2j \Gamma 2x22^ j u2j2 \Gamma jx2j2 ^ juj2 \Gamma jxj2 ; again as desired. (The actual form of i1 turns out to be irrelevant.) The case (x1 6= 0; x2 = 0) is similar. So V1 2 W(\Sigma 1; 1). 7 We next show that, if W 2 W(\Sigma 1; 1) is of class C10 , then it cannot be proper nor positive definite. So, assume that rW (x) \Delta g0(x) + mX i=1 u igi(x)! ^ juj 2 \Gamma jxj2 (9) holds for all x 6= 0 and all u. Fix any number a ? 0. Consider any point of the form x = (x1; x2) = (a; x2), with x2 6= 0. With the special choice u1 = x1 and u2 = jx2j, we have jxj = juj, so (using subscripts to denote partial derivatives) inequality (9) reduces to: Wx1(a; x2) \Gamma (sign x2) Wx2 (a; x2) ^ 0 : When x2 ? 0 this gives Wx1(a; x2) \Gamma Wx2(a; x2) ^ 0, so taking the limit as x2 ! 0+, we conclude Wx1 (a; 0) \Gamma Wx2(a; 0) ^ 0 for all a ? 0. Arguing with negative x2 and taking x2 ! 0\Gamma , we get also Wx1(a; 0) + Wx2 (a; 0) ^ 0 for all a ? 0. We conclude that Wx1(a; 0) ^ 0 for all a ? 0. This means that W (\Delta ; 0) must be bounded above (and is nonnegative on R*0 ), so W cannot be proper. On the other hand, if W (0) = 0, this implies W (a; 0) j 0, so W cannot be positive definite either. This completes the proof of Proposition 2. 3.2 Proof of Proposition 4 The intuitive idea of the construction of \Sigma 2 and V2 is as follows. We start with eV2(~x1; ~x2) = ~x21 + ~x22, the square norm in R2 , (~x1; ~x2) denoting the canonical coordinates on R2 , and consider the harmonic oscillator motion, but with rescaled time so that the ~x1-axis consists of equilibria: eg(~x1; ~x2) = (~x2 ~x22; \Gamma ~x1 ~x22). Note that eV2 is an integral for the motions of g. Next, we make a change of coordinates which is a global homeomorphism but fails to be a diffeomorphism: ~x1 7! ~x1 =: x1; ~x2 7! ~x32 =: x2. In the new coordinates (x1; x2), V2, the transformation of eV2, is no more locally Lipschitz. However, the transformation g of eg is locally Lipschitz, and it is of course still true that V2 integrates g. The key fact is this: for any locally Lipschitz W which is nonincreasing along trajectories of g, W cannot be positive definite nor proper. (In other words, there cannot be any locally Lipschitz "weak Lyapunov function" for .x = g(x)). Finally, we add an input-dependent term to g which 8 provides the dissipation property. When juj = jxj, this dissipation property reduces to the nonincreasing property mentioned above, and hence leads to a contradiction for V locally Lipschitz. We define the following locally Lipschitz vector field g in R2 : g(x1; x2) = ` x2\Gamma 3x 1x 4=3 2 ' and note that: i \Delta g(x) = 0 8i 2 @DV2(x) ; 8x 2 R2 : (10) Indeed, pick any x = (x1; x2). If x2 6= 0 then necessarily i = (2x1; (2=3)x\Gamma 1=32 ), so the claim is clear. If instead x2 = 0, then g(x1; x2) = 0, so the claim holds as well. Note that system \Sigma 2 is of the form .x = f (x; u), where f (x; u) = g(x) + h(x; u), with h(x; u) := ` u1 \Gamma x13x4=3 2 (u2 \Gamma x2) ' : Now note that: i \Delta h(x; u) ^ juj2 \Gamma jxj2 8i 2 @DV2(x) ; 8x 2 R2 ; 8u 2 R2 : (11) To verify this, pick any x = (x1; x2). Take first the case x2 6= 0. Since i must be equal to (2x1; (2=3)x\Gamma 1=32 ), we have that i \Delta h(x; u) = 2x1u1 \Gamma 2x21 + 2x2u2 \Gamma 2x22 ^ juj2 \Gamma jxj2 : If instead x2 = 0, then i = (2x1; i2), for some i2, so i \Delta h(x; u) = 2x1u1 \Gamma 2x21 + i2 \Delta 0 ^ u21 \Gamma x21 ^ juj2 \Gamma jxj2 : Together with (10), we have thus that i \Delta f (x; u) ^ juj2 \Gamma jxj2, so we have proved that V2 2 W1(\Sigma 2; 1). Now we show that we cannot have a locally Lipschitz V 2 W1(\Sigma 2; 1). For any V 2 W(\Sigma 2; 1), setting u = x in the inequality i \Delta f (x; u) ^ juj2 \Gamma jxj2, and using that h(x; x) = 0, gives that i \Delta g(x) ^ 0 8i 2 @DV (x) ; 8x 2 R2 : (12) 9 We will show that no locally Lipschitz function V can satisfy such a property and be positive definite or proper. Suppose given such a V ; we will prove that, for any positive number a at which the locally Lipschitz function x1 7! V (x1; 0) is differentiable, necessarily Vx1(a; 0) ^ 0. This means that V decreases along the x1-axis, and the negative conclusion follows. Fix now any such a, and denote , := (a; 0)0. We will analyze the behavior of V in a neighborhood of ,, by considering a curve which approaches , along an orbit of g. Take the following parameterized curve: fl : [0; 1] ! R2 : t 7! ` at(a2 \Gamma (at)2)3=2 ' which has the property that fl(1) = fl0(1) = ,. Note also that, for each t 2 [0; 1], g(fl(t)) = ` (a 2 \Gamma (at)2)3=2\Gamma 3at(a2 \Gamma (at)2)2 ' = fi(t) fl0(t) where fi(t) = 1a (a2 \Gamma (at)2)3=2 ? 0 for all t ! 1. Lemma 7 The function t 2 [0; 1] 7! W := V (fl(t)) is nonincreasing. PROOF. Since W is locally Lipschitz, its derivative exists almost everywhere. We must prove that W 0(t) ^ 0 for almost all t. This will follow from the following statement, valid for all subgradients: j ^ 0 for all t0 2 [0; 1) and all j 2 @DW (t0) : (13) The idea of the proof is as follows. If rV (fl(t0)) exists, then W 0(t0) = rV (fl(t0)) fl0(t0) = 1fi(t 0) rV (fl(t 0)) g(fl(t0)) ^ 0 where the last inequality follows from (12). However, there is no reason for V to be differentiable at the points in the image of fl. So we need to apply the "approximate chain rule" for subgradients. Pick any t0 2 [0; 1) and j 2 @DW (t0). Take any " ? 0, and let 0 ! ffi ! " be such that jij jg(x) \Gamma g(fl(t0))j ^ "fi(t0) (14) for all i 2 @DV (x) and all x such that jx \Gamma fl(t0)j ! ffi (using continuity of g, and noting that all such i are bounded by a Lipschitz constant for V on a ball 10 of radius ffi about fl(t0)) and also so that jij jfl0(t1) \Gamma fl0(t0)j ^ " (15) whenever i 2 @DV (x) for any x as above and jt1 \Gamma t0j ! ffi (using now the fact that fl 2 C1). We apply to j and ffi the chain rule in [7], Theorem 2.2.5, but in its viscosity rather than subgradient form (the viscosity form follows from the version given in that reference by applying the approximation theorem in [7], Proposition 3.4.5). This tells us that there exist t1, x, i, and i0 such that jt1 \Gamma t0j ! ffi,j x \Gamma fl(t0)j ! ffi, i 2 @DV (x), and i0 2 @D(i \Delta fl)(t1), so that jfl(t1) \Gamma fl(t0)j ! " and jj \Gamma i0j ! ". Since fl is differentiable, the scalar map i \Delta fl is too, and hence i0 = (d=dt)(i \Delta fl)(t1) = i \Delta fl0(t1), so i0 = i \Delta fl0(t1) = i \Delta (fl0(t1) \Gamma fl0(t0)) + i \Delta fl0(t0) = i \Delta (fl0(t1) \Gamma fl0(t0)) + 1fi(t 0) i \Delta g(fl(t 0)) = ` + 1fi(t 0) i \Delta g(x) ; where ` = i \Delta (fl0(t1) \Gamma fl0(t0)) + 1fi(t 0) i \Delta (g(fl(t 0)) \Gamma g(x)) ; and therefore i0 ^ 2" by (12), (14), and (15). So j ^ 3". As " was arbitrary, we conclude j ^ 0. Since W (t) * W (1) = V (,) for all t ! 1, for any h * 0 the last term in the following expression is nonpositive: 1 h (V (a; 0) \Gamma V (a \Gamma ah; 0)) = (16)1 h (V (fl(1 \Gamma h)) \Gamma V (a \Gamma ah; 0)) + 1 h (W (1) \Gamma W (1 \Gamma h)) : Since V is locally Lipschitz, there is some constant L such that (evaluating fl(1 \Gamma h)): 1 h jV (fl(1 \Gamma h)) \Gamma V (a \Gamma ah; 0)j ^ La 3h 12 (2 \Gamma h)3=2 ! 0 11 as h ! 0+. We conclude, by taking limits as h ! 0+ in (16), that Vx1(a; 0) ^ 0, as claimed. 2 3.3 Proof of Theorem 5 In order to be able to justify Remark 6, we will prove Theorem 5 for a more general class of systems, namely those of the following general form: .x = g0(x) + mX i=1 '(u i) gi(x) (17) where '(r) = jrjp or '(r) = (sign r) jrjp (interpreting '(0) = 0), and p is a fixed real number in the closed interval [1; 2]. Note that Theorem 5 as stated would correspond to p = 1 and the second choice of '. We still suppose that states x evolve in Rn , and inputs u = (u1; : : : ; um) take values in Rm , and that the vector fields gi, i = 0; : : : ; m are locally Lipschitz in x. The proof will be based upon the following general technical fact: Lemma 8 Assume that we are given: - an open subset O of Rn ; - a continuous function ff : O ! R?0 ; - a continuous function fi : O ! R*0 ; - an " ? 0; - a continuous function V : O ! R?0 satisfying i \Delta g0(x) + mX i=1 '(u i) gi(x)! ^ \Gamma ff(x) + fi(x) juj 2 8i 2 @ DV (x)(18) for all x 2 O and u 2 U . Then, there exists a smooth W : O ! R such that jV (x) \Gamma W (x)j ^ 12 V (x) (19) for all x 2 O, and rW (x) \Delta g0(x) + mX i=1 '(u i) gi(x)! ^ \Gamma ff(x) + [(1 + ")fi(x) + "] juj 2 (20) for all x 2 O and u 2 U . 12 Before proving the lemma, we explain how to obtain Theorem 5 as a corollary. We pick a system \Sigma as in (1), and any fl0 ? fl ? 0. We suppose given a continuous proper and positive definite V 2 W(\Sigma ; fl), and need to show the existence of some W 2 W(\Sigma ; fl0) which is C1 on O = Rn n f0g, in addition to being proper and positive definite. We pick, in Lemma 8, fi(x) j fl, ff(x) =j xj2, and any " ? 0 such that (1 + ")fl + " ! fl0. The Lemma then applies to the restriction of V to O. We obtain a W as in the Lemma, and extend it from O to Rn by defining W (0) := 0. The approximation property (19) insures that W is proper and positive definite, and it is continuous at zero as well. The proof of Lemma 8 will be based upon a reduction to the following result, which we quote from [16]: Lemma 9 Let \Sigma : .x = f (x; d) be a system, with x 2 X = Rn , and d 2 U , a compact metric space, so that f (x; d) is locally Lipschitz in x uniformly on d and jointly continuous in x and d. Assume that we are given: - an open subset O of X; - a continuous, nonnegative function V : O ! R satisfying i \Delta f (x; d) ^ \Theta (x; d) 8x 2 O; i 2 @DV (x); d 2 U (21) with some continuous function \Theta : O \Theta U ! R; - two positive, continuous functions \Upsilon 1 and \Upsilon 2 on O. Then, there exists a smooth ^V : O ! R such thatfififi V (x) \Gamma ^V (x)fififi ^ \Upsilon 1(x) 8x 2 O (22) and r ^V (x) \Delta f (x; d) ^ \Theta (x; d) + \Upsilon 2(x) 8 x 2 O; d 2 U : (23) \Lambda The proof given in [16] employs tools from nonsmooth analysis, borrowing in particular several simple facts from [6,7,19] in order to pass from continuous V to locally Lipschitz V , followed by a standard smoothing argument as given in [17]. (To be precise, the result is stated and proved in [16] using the language of proximal subgradients rather than viscosity subgradients. The proof, however, is the same in both cases.) 13 Proof of Lemma 8 We define f (x; d) := i1 \Gamma jdj2j g0(x) + mX i=1 '(d i) i1 \Gamma jdj 2j1\Gamma p 2 gi(x) with U = unit ball in Rm . (If p = 2, we interpret the coefficient of gi as simply '(di).) Note that this system satisfies the hypotheses of Lemma 9. We use the same O and V , and let \Theta (x; d) := \Gamma i1 \Gamma jdj2j ff(x) + jdj2 fi(x) : We need to verify that (21) indeed holds. Pick any x and i, and any d withj dj ^ 1. We treat separately the cases jdj = 1 and jdj ! 1. Case 1: jdj = 1. If p ! 2, then (21) holds trivially (because f (x; d) = 0 and \Theta (x; d) = fi(x) * 0). So suppose p = 2. We must verify that mX i=1 '(d i) i \Delta gi(x) ^ fi(x) (24) where '(r) = r2, or '(r) = r jrj, and d is a vector of unit norm. We are assuming that (18) holds, that is, i \Delta g0(x) + mX i=1 '(u i) i \Delta gi(x) ^ \Gamma ff(x) + fi(x) juj 2 (25) for all u. Now pick any u of the form (1=")d, with " ? 0, so "2'(ui) = '(di) for each i. Multiplying both sides of (25) by "2, we have that "2i \Delta g0(x) + mX i=1 '(d i) i \Delta gi(x) ^ \Gamma " 2ff(x) + fi(x) ; so taking limits as " ! 0 one obtains the desired inequality (24). Case 2: jdj ! 1. Introduce, for such a d, u = (u1; : : : ; um), where: ui := diq 1 \Gamma jdj2 i = 1; : : : ; m : 14 Observe that juj2 = jdj 2 1 \Gamma jdj2 ; 1 \Gamma jdj 2 = 1 1 + juj2 ; '(di) i1 \Gamma jdj 2j1\Gamma p 2 = '(ui) 1 + juj2 (26) for all i, so that f (x; d) = 11 + juj2 g0(x) + mX i=1 '(u i) gi(x)! and thus (18) implies that i \Delta f (x; d) ^ \Theta (x; d) for all i 2 @DV (x), as wanted. So, we apply Lemma 9 with \Upsilon 1(x) := 1 \Gamma ffi4 V (x) and \Upsilon 2(x) := ffi minf1; ff(x)g where ffi 2 (0; 1) is picked such that 1 1 \Gamma ffi ^ 1 + " and ffi 1 \Gamma ffi ^ min ae"; 1 4 oe : This provides a function ^V . Now fix any u 2 Rm , and introduce di := uiq 1 + juj2 i = 1; : : : ; m : Observe that the relations (26) hold, and hence also jdj2 = juj 2 1 + juj2 ! 1 holds. Furthermore, g0(x) + Pmi=1 '(ui) gi(x) = (1 + juj2)f (x; d). Therefore, (23) gives 15 r ^V (x) \Delta g0(x) + mX i=1 '(u i)gi(x)! ^ (1 + juj2) i\Gamma i1 \Gamma jdj2j ff(x) + jdj2 fi(x) + ffi minf1; ff(x)gj = \Gamma ff(x) + juj2 fi(x) + ffi minf1; ff(x)g + juj2 ffi minf1; ff(x)g^ \Gamma ff(x) + juj2 fi(x) + ffiff(x) + juj2 ffi = \Gamma (1 \Gamma ffi)ff(x) + juj2 (fi(x) + ffi) and if we now let W := 11 \Gamma ffi ^V we conclude that jV \Gamma W j ^ ffi1 \Gamma ffi V + 11 \Gamma ffi fififiV \Gamma ^V fififi ^ 12V ; and rW (x) \Delta g0(x) + mX i=1 '(u i) gi(x)! ^ \Gamma ff(x) + juj 2 fi(x) + ffi 1 \Gamma ffi^ \Gamma ff(x) + [(1 + ")fi(x) + "] juj2 as desired for the conclusions of Lemma 8. 2 Notice, incidentally, that W is smooth (C1), not merely C1. 3.4 Results for systems of form (8) The positive claim in Remark 6, for p ^ 2, has been proved already. To show that Theorem 5 does not extend to systems (17) when p ? 2, it will be convenient to introduce the vector field corresponding to what one might call an "L1 harmonic oscillator". This is the locally Lipschitz vector field on R2 defined by g(x1; x2) := ` jx1j x2\Gamma jx 2j x1 ' : (27) The flow of g leaves invariant the L1-balls around the origin, see Figure 1. Associated to g is the L1 norm, seen as a proper, positive definite Lipschitz 16 \Gamma \Gamma @ @ \Gamma \Gamma @ @\Gamma \Gamma ` @@R \Gamma \Gamma \Psi @@I Fig. 1. The vector field g function V1(x1; x2) := jx1j + jx2j : Except for the fact that we do not now need the factor "2", this function is as before. So @DV1(x) = f(x1=jx1j; x2=jx2j)g when x1x2 6= 0, @DV1(x) = [\Gamma 1; 1] \Theta fx2=jx2jg when x1 = 0 and x2 6= 0, and similarly if x2 = 0. Therefore, i \Delta g = 0 for all i 2 @DV1(x); x 2 Rn : (28) We also introduce the following vector field: g0(x1; x2) := ` \Gamma jx1j x1\Gamma jx 2j x2 ' : (29) For the same V1, i \Delta g0 = \Gamma jxj2 for all i 2 @DV1(x); x 2 Rn : (30) Now fix any p ? 2, and consider the following system \Sigma p, which is defined in terms of the above vector fields: .x = g0(x) + ju1jp g(x) \Gamma ju2jp g(x) (31) This is a system of type (8) with m = 2, g1 = g, and g2 = \Gamma g. In view of (28) and (30), we have that i \Delta (g0(x) + ju1jp g(x) \Gamma ju2jp g(x)) ^ \Gamma jxj2 for all x 2 R2 and u = (u1; u2) 2 R2 , and every i 2 @DV1(x). This means that V1 2 W1(\Sigma p; fl), for any fl ? 0. We now see that the equality in Theorem 5 cannot hold. In fact, we prove the following far stronger statement: Proposition 10 For all V 2 C10 , and all fl ? 0, V 62 W(\Sigma p; fl). 17 PROOF. Suppose that V 2 C10 and rV (x) \Delta (g0(x) + ju1jp g(x) \Gamma ju2jp g(x)) ^ \Gamma jxj2 + fl ju1j2 + fl ju2j2 (32) for all x 6= 0 and all u = (u1; u2). Fix any x = , 6= 0. Taking u = 0 gives thatr V (,) 6= 0. Since p ? 2, letting u2 = 0 and u1 ! +1 gives rV (,) \Delta g(,) ^ 0. On the other hand, u1 = 0 and u2 ! +1 gives \Gamma rV (,) \Delta g(,) ^ 0. So we conclude that rV (,) \Delta g(,) = 0 for all , 6= 0. Now fix an a ? 0 and suppose that , has the form (a; x2) with x2 6= 0. Then,r V (,) \Delta g(,) = 0 means that (sign x2) Vx1(a; x2) \Gamma Vx2(a; x2) = 0 Letting separately x2 ! 0+ and x2 ! 0\Gamma gives Vx1(a; 0) = Vx2(a; 0) = 0, contradicting rV (a; 0) 6= 0. A variation of this example is as follows. Fix again any p ? 2. Instead of (31), we consider now the following system \Sigma 0p: .x = g0(x) + sign u jujp g(x) (33) with m = 1. It is also true that V1 2 W1(\Sigma p; fl), for any fl ? 0, and Proposition 10 again holds, simply taking the separate limits as u ! +1 or u ! \Gamma 1 in its proof. Remark 11 There is nothing very special about the form (8). Mainly, we picked these systems in order to illustrate with a specific class when the theorem holds. But Theorem 5 holds also for a general class of systems of subquadratic growth in inputs. Specifically, we may consider systems of the general form .x = g(x; u), where g is jointly continuous in x and u, and satisfies, for some constant p 2 (0; 2) and some continuous function h : Rn ! R: (i) jg(x; u)j ^ h(x) (1 + jujp) for every x and u; (ii) jg(x; u) \Gamma g(y; u)j ^ cjx \Gamma yj (1 + jujp) for every u, and every x; y in some (arbitrary chosen) ball. The proof is the same as for systems of the form (8). One only needs to set f (x; d) = i1 \Gamma jdj2j g ix; d (1 \Gamma jdj2)\Gamma 1=2j provided jdj ! 1, and f (x; d) = 0 otherwise, and to replace everywhere g0(x)+P mi =1 '(ui)gi(x) by g(x; u). We omit the simple details. \Lambda 18 4 One-Dimensional Systems For one-dimensional systems, there is no gap between the locally Lipschitz and the C10 case: Proposition 12 Suppose given a system \Sigma as in (1), with n = 1. Assume that for some fl ? 0 there exists a locally Lipschitz V 2 W1(\Sigma ; fl). ThenW 1(\Sigma ; fl) T C10 6= ;. PROOF. Pick a V as in the statement of the proposition. By Rademacher's theorem there exist a zero measure (Borelian) set N ae R and a continuous positive function h such that x 62 N ) V 0(x) exists and jV 0(x)j ^ 12 h(x): (34) We shall construct W only on R*0 , the construction being similar on R^0 . If x 62 N , @DV (x) = fV 0(x)g, hence V 0(x) g0(x) + mX i=1 u igi(x)! ^ fl juj 2 \Gamma x2 8u 2 R: (35) Since the system .x = g0(x) is (globally) asymptotically stable, g0(x) ! 0 for each x ? 0. It follows from (34) and (35) (with u = 0) that 1 2h(x) * V 0(x) * x 2j g0(x)j ? 0 for x ? 0; x 62 N: (36) Set for any x ? 0 F (x) := (p * 0 : p g0(x) + mX i=1 u igi(x)! ^ fl juj 2 \Gamma x2 8u 2 R) : We claim that the (closed convex) set F (x) is nonempty for any x ? 0. Indeed, if x 62 N V 0(x) 2 F (x). If x 2 N we may pick a sequence (xn) in R?0 n N such that xn ! x as n ! 1. Since jV 0(xn)j ^ 12 h(xn) ^ c for some constant c, we may extract a subsequence of (V 0(xn)) which converges towards some p * 0. Clearly p 2 F (x). Let us set a := mX i=1 g i(x) 2; b := 4flg 0(x); c := 4flx 2 19 and \Delta (p) := ap2 + bp + c: It is a straightforward exercise to see that for any p * 0, p 2 F (x) () \Delta (p) ^ 0: (37) By the claim, \Delta (p) ^ 0 for some p, whereas \Delta (p) ! 1 as p ! \Gamma 1 (since b ! 0). It follows that \Delta has (at least) one real root, hence b2 \Gamma 4ac * 0. Set p(x) := 8?!?: h(x) if a = 0; min nh(x); \Gamma b+pb 2\Gamma 4ac 2a o if a 6= 0: (38) Note that \Gamma b = 4fl jg0(x)j ? 0 on x ? 0, so when a ! 0 the second expression in the minimum above becomes unbounded; from here, it follows that p is continuous on R?0 . We claim that V 0(x) ^ p(x) 8x 2 R?0 n N: (39) Pick any x 2 R?0 n N . If a = 0, then (39) follows from (36) and (38). If a 6= 0, since V 0(x) 2 F (x), we see that \Delta (V 0(x)) ^ 0 (by (37)), hence V 0(x) ^\Gamma b+pb2\Gamma 4ac 2a which, combined with (36) and (38), yields V 0(x) ^ p(x). We are now ready to define W on R*0 : we set W (x) = R x0 p(s) ds for any x * 0. Since 0 ^ p ^ h, W is a (well-defined) locally Lipschitz function on R*0 which is C1 away from 0. Integrating in (39) we get W (x) * V (x) ? 0 for any x ? 0 and limx!1 W (x) = 1. Finally we claim that \Delta (p(x)) ^ 0 8x 2 R?0 n N: (40) Pick any x 2 R?0 n N . If a = 0, then \Delta (p(x)) ^ \Delta (V 0(x)) (using b ! 0 and (39)) ^ 0 (by (37)): If a 6= 0, we are led to prove that p(x) 2 " \Gamma b \Gamma pb 2 \Gamma 4ac 2a ; \Gamma b + pb2 \Gamma 4ac 2a # : 20 Owing to the definition of p(x), we only have to prove h(x) * \Gamma b \Gamma pb 2 \Gamma 4ac 2a : But \Gamma b \Gamma pb2 \Gamma 4ac 2a = 2cj bj + pb2 \Gamma 4ac ^ 2cj bj = 2x2j g0(x)j ^ h(x); by (36). This completes the proof of (40). Note that (40) also holds true for any x ? 0, by continuity of \Delta ffi p. Using (37) it means that W 0(x) = p(x) 2 F (x) for all x ? 0, i.e. W 0(x) g0(x) + mX i=1 u igi(x)! ^ fl juj 2 \Gamma x2 8u 2 R: The proof of the proposition is complete. We now show that, for systems not affine in inputs, there is again a gap between the Lipschitz and differentiable cases. In Proposition 2 we gave an example of a system \Sigma 1 and a Lipschitz V1 2 W1(\Sigma 1; 1), such that no V in W(\Sigma 1; 1) T C10 can be positive definite or proper. The system was twodimensional (n = 2) and had two-dimensional input (m = 2). We now provide a scalar (n = m = 1) system \Sigma 3 which has the following properties: (a) it admits a Lipschitz V3 2 W1(\Sigma 3; 1), but (b) any W in W(\Sigma 3; 1) must be nondifferentiable. Thus, the conclusions are even stronger than for \Sigma 1; on the other hand, \Sigma 3 is not affine in inputs. Before giving the form of \Sigma 3, we start by considering the following two functions '+ and '\Gamma : R2 ! R: '+(s; t) := max fmin f(s \Gamma t)=2; sg ; 0g and '\Gamma (s; t) := \Gamma max fmin f(\Gamma s \Gamma t)=2; \Gamma sg ; 0g : Both of these are Lipschitz, and '+(0; t) = '\Gamma (0; t) = 0 for all t. Thus, the function '(s; t) := (sign s) max aemin ae 12 (jsj \Gamma t); jsjoe ; 0oe 21 (with '(0; t) j 0) obtained by using '+ for s * 0 and '\Gamma for s ! 0 is also Lipschitz. As min n 12 (jsj \Gamma t); jsjo ^ jsj, it follows that j'(s; t)j 2 [0; jsj] for all (s; t). Since sign '(s; t) = sign s, this means that '(s) 2 [0; s] when s * 0 and '(s) 2 [s; 0] when s ^ 0. When t * jsj, min n 12 (jsj \Gamma t); jsjo = 12 (jsj \Gamma t), so '(s; t) = 0 if t * jsj : If t ^ \Gamma jsj then jsj = jsj =2 + jsj =2 ^ jsj =2 \Gamma t=2, so min n 12 (jsj \Gamma t); jsjo = jsj. Therefore '(s; t) = s if t ^ \Gamma jsj : The diagram in Figure 2 summarizes this information about the range of '. \Gamma \Gamma \Gamma \Gamma \Gamma \Gamma \Gamma \Gamma @@ @@ @@ @@ 0 [0; s][s; 0] s s = t s = \Gamma t Fig. 2. The range of ' We define now, for a; b * 0: (a; b) := 12 h'(b \Gamma a; b + a \Gamma 2) + (b \Gamma a)i : As jb \Gamma aj ^ b + a \Gamma 2 if and only if both a * 1 and b * 1, (a; b) = 12 (b \Gamma a) for such a; b. Similarly, the properties of ' also imply that (a; b) = b \Gamma a when a ^ 1 and b ^ 1, that (a; b) 2 [b \Gamma a; 12 (b \Gamma a)] when a * b, and that (a; b) 2 [ 12 (b \Gamma a); b \Gamma a] when a ^ b. We are now ready to specify \Sigma 3. We let: f (x; u) := (juj + x) (x; juj) for x * 0, and f (x; u) := x2 + juj (0; juj) 22 for x ! 0. This function is locally Lipschitz. Notice that (0; juj) * juj =2 * 0 for all u, so that f (x; u) * x2 for x ! 0. Observe the following properties, for all x * 0: (i) x ^ 1 & juj ^ 1 ) f (x; u) = u2 \Gamma x2. (ii) x ^ 1 & juj * 1 ) f (x; u) ^ u2 \Gamma x2. (iii) x * 1 & juj ^ 1 ) f (x; u) ^ 12 (u2 \Gamma x2). (iv) x * 1 & juj * 1 ) f (x; u) = 12 (u2 \Gamma x2). Finally, the Lipschitz function V3 shown in Figure 3. \Gamma \Gamma fifi fifi JJ JJ JJ Fig. 3. V3(x) := maxfjxj ; 2x \Gamma 1g We show that V3 2 W1(\Sigma 3; 1), that is, V 0(x)f (x; u) ^ u2 \Gamma x2 for all x 6= 0 and all u. For x ! 0 this is obvious, since V 0(x)f (x; u) = \Gamma f (x; u) ^ \Gamma x2 ^ u2 \Gamma x2 for all u. So we only need to analyze the case x ? 0. For any 0 ! x ! 1, V 0(x)f (x; u) = f (x; u) ^ u2 \Gamma x2 (properties i and ii) while for x ? 1 we have V 0(x)f (x; u) = 2f (x; u) ^ u2 \Gamma x2 as well (properties iii and iv). Finally, we deal with the nondifferentiability point x = 1. An easy calculation shows that @DV3(1) is the closed interval [1; 2]. Pick any ae 2 [1; 2]. When juj ^ 1, f (x; u) = u2 \Gamma x2 ^ 0, so ae * 1 implies aef (x; u) = ae(u2 \Gamma x2) ^ u2 \Gamma x2. Finally, if juj * 1, f (x; u) = 12 (u2 \Gamma x2) * 0, then ae ^ 2 implies aef (x; u) ^ u2 \Gamma x2 as well. We now show that any W in W(\Sigma 3; 1) must be non-differentiable. Suppose that there is some such W which is differentiable. Fix u = 1. For any x, we must have, since W 2 W(\Sigma 3; 1): W 0(x)f (x; 1) ^ 1 \Gamma x2 : When x 2 (0; 1), f (x; 1) = 1 \Gamma x2 ? 0, so W 0(x) ^ 1. On the other hand, for x ? 1 we have f (x; 1) = (1 \Gamma x2)=2 ! 0, so W 0(x) * 2 for such x. If W 2 C10 , this gives a contradiction as x ! 1+ and x ! 1\Gamma . However, continuity of W 0 is not needed for the contradiction, since we can argue as follows. By the mean value theorem, lim sup x!1 \Gamma W (x) \Gamma W (1) x \Gamma 1 ^ 1 ! 2 ^ lim infx!1+ W (x) \Gamma W (1) x \Gamma 1 ; which contradicts the existence of W 0(1). 2 23 A Viscosity and Integral Formulations We prove the equivalence between the integral and viscosity formulations. This equivalence is well-known, but we need to address a small technical point due to the fact that we only defined the viscosity condition at nonzero states x. (We did so simply for technical reasons, in working out the examples; the remarks given below show that it makes no difference whether we consider the origin or not.) We need the following simple fact first. We assume given a system .x = F (x; u), where F : Rn \Theta Rm ! Rn , as well as a closed set A ` Rn (in our application, the origin). and a continuous map w : Rn \Theta Rm ! R such that w(x; u) * 0 for all x 2 A (in our application,\Gamma j xj2 + fljuj2) By a solution of .x = F (x; u), given a measurable locally essentially bounded u : [a; b] ! Rm , we mean an absolutely continuous x : [a; b] ! Rn such that .x(t) = F (x(t); u(t)) for almost all t 2 [a; b]. We also assume given a continuous function V : Rn ! R with the property that V (x) = 0 for all x 2 A. Proposition 13 Suppose that the following property: V (b) \Gamma V (a) ^ bZ a w(x(s); u(s)) ds (A.1) holds for all solutions x : [a; b] ! Rn n A. Then, it also holds for all solutions. PROOF. We first prove the conclusion under the assumption that x(a) 2 A and x(b) 2 A. Since then V (a) = V (b) = 0, we need to prove that, for any such solution, R ba w(x(s); u(s)) ds * 0. Let I := ft 2 [a; b] j x(t) 2 Ag and J := ft 2 [a; b] j x(t) =2 Ag. Note thatZ I w(x(s); u(s)) ds = Z I w(0; u(s)) ds * 0 : Since J is open, there is a countable (or finite) sequence of disjoint open subintervals Jk = (ak; bk) of [a; b] such that J = S1k=1 Jk and ak; bk 2 I for all k. Pick any Jk. For all ff; fi such that ak ! ff ! fi ! bk, since the restriction of x(\Delta ) to [ff; fi] lies entirely outside A, we have that V (fi) \Gamma V (ff) ^ R fiff w(x(s); u(s)) ds, so taking limits as ff ! ak and fi ! bk, and using that V is continuous, we 24 conclude that 0 = 0 \Gamma 0 ^ RJk w(x(s); u(s)) ds. It follows that alsoZ J w(x(s); u(s)) ds = 1X k=1 ZJk w(x(s); u(s)) ds * 0 and we conclude that bZ a w(x(s); u(s)) ds = Z J w(x(s); u(s)) ds + Z I w(x(s); u(s)) ds * 0 ; as wanted. We now take the general case. Pick any solution defined on [a; b]. We assume without loss of generality that x(t) 2 A for some t 2 [a; b], since otherwise we are done by hypothesis. Let ff := minft j x(t) 2 Ag, fi := maxft j x(t) 2 Ag. On [a; ff), we have a trajectory that does not enter A, so, arguing as before by taking limits as we approach ff, R ffa w(x(s); u(s)) ds * V (ff) \Gamma V (a). Similarly on [fi; b]. The restriction to [ff; fi] is a trajectory which starts and ends in A, so the first case considered in our proof applies. In summary, V (b) \Gamma V (a) = (V (ff) \Gamma V (a)) + (V (fi) \Gamma V (ff)) + (V (b) \Gamma V (fi)) ^ ffZ a w(x(s); u(s)) ds + fiZ ff w(x(s); u(s)) ds + bZ fi w(x(s); u(s)) ds = bZ a w(x(s); u(s)) ds ; as desired. Note that the tempting "proof" which starts by writing V (b) \Gamma V (a) =R b a [dV (x(t))=dt] dt and splits into I and J will not work, since there is noreason for V (x(t)) to be absolutely continuous (and, indeed, we are interested in general continuous V 's in our main results). We now suppose that F and w are continuous, and locally Lipschitz on x uniformly with respect to u on compacts. We consider the scalar-valued map '(x; y) := V (x) \Gamma y on Rn \Theta R and introduce the new system .x = F (x; u), .y = w(x; u), which we write as .z = G(z; u) in terms of the state z = (x0; y)0 2 Rn+1 . 25 It is clear that inequality (A.1) holds for every solution x : [a; b] ! Rn n A if and only if '(z(b)) ^ '(z(a)) for every solution of the extended system .z = G(z; u) as a system on the state space O := (Rn n A) \Theta R. Moreover, as inputs are essentially bounded on finite intervals, this is in turn equivalent to asking that, for every given r ? 0, ' must decrease along all solutions of .z = G(z; u), again as a system in O but now with inputs satisfying ju(t)j ^ r for almost all t 2 [a; b], which is equivalent, by Filippov's Lemma, to having the corresponding decrease property for all solutions of the differential inclusions .z 2 Fr, for each r * 0, where Fr(z) := fG(z; u) j juj ^ rg : Now consider the convexification eFr(z) := co Fr(z). By standard relaxation results, e.g. [1], Corollary 10.4.5, any trajectory of .z 2 eFr(z) can be uniformly approximated by trajectories of .z 2 Fr(z), so, since ' is continuous, ' decreases along trajectories of .z 2 eFr(z) if it decreases along trajectories of .z 2 Fr(z) (and viceversa). In summary: inequality (A.1) holds, for every solution x : [a; b] ! Rn n A, if and only if, for each r ? 0 for every trajectory z : [a; b] ! O of the compact convex-valued and Lipschitz differential inclusion .z 2 eFr(z), it holds that '(z(b)) ^ '(z(a)). Using the characterization of strong invariance of differential inclusions given in [7], Theorem 4.6.3, this is equivalent 3 to the statement that i \Delta v ^ 0 for all z 2 O, all i 2 @D'(z), and all v 2 eFr(z), or equivalently (since the set fv j i \Delta v ^ 0g is convex) for all v 2 Fr(z). As every i 2 @D'(z), z = (x0; y)0, has the form i = (i01; \Gamma 1)0 with i1 2 @DV (x), and since every v 2 Fr(z) has the form (F (x; u); w(x; u)), the decrease property amounts to the requirement that i1 \Delta F (x; u) ^ w(x; u) for all x 2 Rn n A, all u 2 U , and every i1 2 @DV (x). In summary: Proposition 14 The following are equivalent: (i) Inequality (A.1) holds for every solution x : [a; b] ! Rn . (ii) i \Delta F (x; u) ^ w(x; u) for every x 2 Rn , u 2 U , and i 2 @DV (x). (iii) Inequality (A.1) holds for every solution x : [a; b] ! Rn n A. (iv) i \Delta F (x; u) ^ w(x; u) for every x 2 Rn n A, u 2 U , and i 2 @DV (x). PROOF. We already proved the equivalence of the last two statements, and 3 There is an additional assumption of linear growth on the inclusion, but this can be achieved by changing the inclusion to be bounded outside a large enough compact set, which depends on the solution being studied; also, the reference [7] uses "proximal" instead of "viscosity" subgradients, but the proof is basically the same. For another proof, see [15] and the discussion in [4]. 26 the first two are also equivalent (just take A = ;). By Proposition 13, the first and third statements are equivalent. Applying with V any positive definite continuous function (so V (0) = 0, in particular), and, w(x; u) = \Gamma jxj2 + fljuj2, and A = f0g, we conclude that (4) is equivalent to (5), as claimed in the introduction. 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