On Discontinuous Differential Equations

Transcription

1 On Discontinuous Differential Equations Alberto Bressan and Wen Shen S.I.S.S.A., Via Beirut 4, Trieste 3414 Italy. Department of Informatics, University of Oslo, P.O. Box 18 Blindern, N-316 Oslo, Norway. This article is published in the volume of proceedings Differential Inclusions and Optimal Control, J. Andres, L. Gorniewicz and P. Nistri Eds., Julius Schauder Center, Lecture Notes in Nonlinear Analysis , pp Introduction. Consider the Cauchy problem for an ordinary differential equation ẋ = gt, x, x = x, t [, T ]. 1.1 When g is continuous, the local existence of solutions is provided by Peano s theorem. Several existence and uniqueness results are nown also in the case of a discontinuous right hand side [7]. We recall here the classical theorem of Carathéodory [8]: Theorem A. Let g : [, T ] IR n IR n be a bounded function. i If the map t gt, x is measurable for each x and the map x gt, x is continuous for each t, then the Cauchy problem 1.1 has at least one solution. ii If the map t gt, x is measurable for each x and the map x gt, x is Lipschitz continuous for each t, with a uniform Lipschitz constant, then the Cauchy problem 1.1 has a unique solution, depending Lipschitz continuously on the initial data x. 1

2 By a solution of 1.1 we mean an absolutely continuous function x : [, T ] IR n such that xt = x + t g t, xt dt for all t [, T ]. 1.2 More recent results rely on the notions of directional continuity and of bounded directional variation of a vector field. More precisely, given a closed convex cone Γ IR m, we say that a possibly discontinuous map φ : IR m IR n is directionally continuous if at each point p IR m one has lim p p, p p Γ φp = φp. 1.3 We say that the map φ has bounded directional variation if { N sup φp i φp i 1 } ; N 1, p i p i 1 Γ for every i i=1 <. 1.4 The following existence and uniqueness results were proved in [1, 9] and in [2], respectively. Theorem B. Let gt, x L < M for all t, x. i Assume that g is directionally continuous in the direction of the cone Γ =. { t, x; x Mt } IR n Then the Cauchy problem 1.1 admits at least one solution. ii Assume that g has bounded directional variation in the direction of the cone Γ in 1.5. Then the Cauchy problem 1.1 has a unique solution, depending Lipschitz continuously on the initial data x. Further uniqueness results can be found in [3, 4]. Concerning existence, see also the interesting wor [1]. Aim of the present paper is to prove two theorems on the existence and the uniqueness of solutions to the autonomous Cauchy problem ẋ = fx, x = x IR m, t [, T ], 1.6 extending the classical results of Carathéodory. As a preliminary, we observe that the equation 1.1 can be rewritten as an autonomous problem on IR n+1, introducing the variable x = t and the vector field fx, x = 1, gx, x. Under the assumptions of Theorem A, the function f can jump across the hyperplanes of the form x = constant. These hyperplanes are certainly transversal to f. Namely, taing the inner product of f with their normal vector, one trivially has f n = 1, gx, x 1, 1. 2

3 The next theorem shows that this transversality condition is indeed the ey ingredient toward the existence of solutions. Theorem 1. Assume that f in 1.6 has the form where: fx =. F g 1 τ1 x, x,..., g N τn x, x, 1.7 i Each map τ i : IR m IR is continuously differentiable. Each g i : IR IR m IR is a Carathéodory function, i.e. measurable in t and continuous in x. Moreover, F : IR N IR m is continuous. ii For some compact set K IR m, at every point x there holds: fx K, τ i x z > for every z K. 1.8 Then the Cauchy problem 1.6 has at least one solution. Remar 1. The assumption ii can be easily recognized as a transversality condition. Indeed, by every trajectory of 1.6 satisfies the differential inclusion ẋ K. Hence by this trajectory must cross transversally any hypersurface of the form τ i x = constant. According to the definition 1.7, these are the surfaces across which f can jump. To guarantee the existence of solutions, some ind of transversality condition is necessary, as shown by obvious counterexample ẋ = { 1 if x <, 1 if x, x =. Our second result is concerned with the uniqueness and continuous dependence of solutions. Theorem 2. Assume that f in 1.6 has the form fx =. g τx, x, 1.9 where: i The function τ : IR m IR is continuously differentiable. The map g : IR IR m IR m is measurable w.r.t. t and uniformly Lipschitz continuous w.r.t. x. ii There exists a compact set K such that fx K, τx z > for every x IR m, z K. 1.1 Moreover, the gradient τ has bounded directional variation w.r.t. the cone Γ. = { λz;, z K }. λ 3

4 Then the Cauchy problem 1.6 has a unique solution, depending on the initial data x in a locally Lipschitz continuous way. Remar 2. In the case where m = n + 1, x = x,..., x n and τx x, the above theorem reduces to part ii of Theorem A. Roughly speaing, in Carathéodory s theorem one allows jumps across the hyperplanes x = constant. On the other hand, in Theorem 2 we allow jumps across the hypersurfaces τx = constant, provided that these surfaces are transversal to the vector field f and the direction of their tangent planes does not wiggle too much. The reader should also notice that in Theorem 2 the assumption of bounded directional variation is placed on the gradient τ. This situation is quite different from part ii of Theorem B, where one assumes that the vector field f itself has bounded directional variation. Remar 3. In Theorems 1 and 2, the scalar functions τ, τ i were assumed to be C 1. This assumption simplifies some technical aspects of the proofs, but may liely be relaxed. We conjecture that the same results hold if τ, τ i are only assumed Lipschitz continuous, and the conditions 1.8, 1.1 are duly reformulated in terms of Clare generalized gradients [5]. Remar 4. If in Theorem 2 we drop the ey assumption that the directional variation of τ be bounded, then the uniqueness of solutions may fail. This will be illustrated by an example in the last section of this paper. On the other hand, the uniqueness result stated in [4] allows f to have discontinuities along a set of lines whose slopes have unbounded directional variation. However, the validity of this theorem relies on the very special structure of f, lined to the solution of a scalar conservation law. 2 - Proof of Theorem 1. It is not restrictive to assume that x =. Define the Picard operator u Pu Put. = t F g 1 τ1 us, us,, g N τn us, us ds. 2.1 We will prove that this operator is continuous on the compact set U. = { u : [, T ] IR m ; u =, ut us t s } K for all t > s. 2.2 Let ε > be given. Applying the theorem of Scorza-Dragoni [11] to each map g i, i = 1,..., N, we obtain the existence of a closed set J i with measir \ J i ε, 2.3 4

5 such that g i is continuous restricted to the set J i IR m. Define the closed set A. = { x IR m ; τ i x J i for every i = 1,..., N }. By 1.7, the composed map f is continuous restricted to A. Using the extension theorem of Dugundji [6, p. 188] we now construct a continuous map f : IR m K such that f = f on A. Call K = max z K z. By 2.2, every function u U thus taes values inside the closed ball X. = { x IR m ; x T K }. By and the continuity of the gradients τ i, there exists a strictly positive δ such that τ i x z δ > for all x X, z K, 2.4 because the sets X, K are compact. As a consequence, for each u U the maps t τ i ut from [, T ] into IR are strictly increasing. Namely, dτ i ut dt = τ i u δ >. 2.5 For a fixed u U, call I u [, T ] the set of times t such that ut / A, i.e.. } I u = {t [, T ]; τ i ut / Ji for some i = 1,, N. Because of 2.3 and 2.5, the measure of I u satisfies measi u Nε/δ. 2.6 To prove the continuity of P, call P the Picard operator corresponding to the function f, i.e. Put. = t f us ds. Clearly, P is continuous, hence for any fixed u U there exists a δ >, such that Pv Pu ε whenever v U, v u δ. 2.7 We now observe that the difference between the Picard operators P and P is small. Indeed, for every v U, 2.6 implies Together, 2.7 and 2.8 yield Pv Pv sup fx fx measiv 2 K Nε/δ. 2.8 x X Pv Pu Pv Pv + Pv Pu + Pu Pu ε + 4 K Nε/δ, 2.9 5

6 for every v U with v u δ. Since ε > in 2.9 was arbitrary, this shows that the Picard operator u P u is continuous, mapping the compact set U into itself. By applying the Schauder fixed point theorem we thus obtain the existence of a solution to the Cauchy problem Proof of Theorem 2. For any given x IR m, the existence of a solution follows from Theorem 1. The main part of the proof consists in showing that, given a radius R > and any two solutions ẋt = f xt, x = x, ẏt = f yt, y = y, 3.1 with x, y R, one has the estimate xt yt C R x y t [, T ], 3.2 for some constant C depending only on f and R. The uniqueness of solutions is an obvious R consequence of 3.2. The proof is given in four steps. STEP 1. We first study the case where f, in addition to the assumptions i and ii in Theorem 2, is piecewise smooth. More precisely, we assume that f has the form fx = g x if τ τx < τ +1, 3.3 for some increasing sequence of times {τ ; Z}. Here the functions g have uniformly bounded C 1 norm, say with sup g x C, x, sup D x g x C x, for some constants C, C 1. Under these additional regularity assumptions, the uniqueness of solutions of 1.6 is clear. Our aim is to derive the uniform estimate 3.2 by studying the evolution of infinitesimal tangent vectors. Consider a one-parameter family of solutions ẋ ε t = f x ε t, x ε = x ε, 3.5 regarded as small perturbations of a reference solution x = x. Define the first order tangent vector vt = lim ε + 6 x ε t xt. 3.6 ε

7 Call t, Z, the times where the reference solution x crosses the hypersurfaces τx = τ. By 1.1, all these crossings are transversal. According to the standard theory of piecewise smooth differential equations [7, 8], if the limit 3.6 exists at time t =, then the tangent vector v is well defined for all t [, T ], t t, Z. The time evolution of v is governed by the linear equation vt = D x g xt vt for t ]t, t +1 [, 3.7 together with impulses at the crossing times t. To describe the linear impulse at time t, call. τ xt n = τ xt the unit normal vector to the surface τ = τ at the point xt. Moreover, define fig. 1 f. = lim t t + f xt = g xt,. f = lim f xt = g xt+1, 3.8 t t +1 v. = lim t t + vt,. ṽ = lim vt. 3.9 t t +1 ε x t n v v f v +1 τ x = τ xt + 1 f τ x = τ figure 1 With the above notations, an elementary computation shows that, at the crossing time t, the values vt + = v and vt = ṽ 1 satisfy the linear relation v = ṽ 1 + f f 1 ṽ 1 n f 1 n

8 Our next goal is to derive a priori bounds on the size of v. In the following, with the Landau symbol O1 we denote a quantity whose norm is uniformly bounded. The bound may depend on T, on the constants C, C 1 in 3.4 and δ > in 2.4, and on the total directional variation of τ, but not on the particular solution x in 3.1. Recalling 3.8, from 3.4 we deduce f f = g xt+1 g xt = O1t +1 t Moreover, recalling 3.9, from 3.7 we deduce ṽ v = vt +1 vt+ = O1t +1 t v In the following, we use the superscripts N and T to denote the components of a vector which are parallel and tangent to n, respectively. More precisely, we set v N = v n, v T = v v N n The same notations are used for f. In addition, for t ]t, t +1 [ we define w. = v N f N z. = v T f T w The quantities w and z are defined similarly. By 1.1, the quantities f N are uniformly positive. We thus have the estimates w = O1 v, z = O1 v, 3.15 v = v N + v T f N w + z + f T w = f w + z = O1 w + z Bounds on the size of v can thus be obtained from estimates on w and z. From 3.13 and 3.1 it follows w = v n f n 1 = f n [ ] ṽ 1 n + f n f n 1 n ṽ 1 f 1 n = ṽ 1 n f 1 n = w 1 + O1 ṽ 1 n n 1. In addition, using 3.11 and 3.12 we deduce w 1 w 1 = ṽ 1 n 1 v 1 n 1 f 1 n 1 + = O1t t 1 v 1. [ v 1 n 1 v ] 1 n 1 f 1 n 1 f 1 n

9 Together, 3.12, 3.17 and 3.18 yield w = w 1 + O1 { n n 1 + t t 1 } v Similar estimates can be obtained for the component z, namely [ z z 1 = [ = = v T ] [ f T w ṽ T 1 + f T ṽ T 1 ṽt 1 1 ṽ T 1 1 = O1 n n 1 ṽ 1, f T 1 1 w 1 ] ] [ ṽ T 1 1 f T 1 1 f T 1 w 1 f T w + f T w T 1 w + w 1 f 1 1 f T 1 w 1 ] 3.2 and z 1 z 1 = ṽ T 1 1 vt f T 1 1 = O1 t t 1 v 1. f T 1 1 w 1 + f T 1 1 w 1 w Therefore, z = z 1 + O1 { n n 1 + t t 1 } v Introducing the scalar quantity y y. = w + z, from 3.19, 3.22 and 3.16 we deduce 1 + O1 { n n 1 + t t 1 } y 1. By induction on, for any integers p < q we obtain q y q exp C =p+1 n n 1 + t t 1 y p, 3.23 for some constant C. Recalling the assumption on the directional variation of τ, we now have t t 1 T, n n 1 const From 3.23 and 3.24, since by the quantities y are uniformly equivalent to the corresponding norms vt, we finally obtain the estimate vt CR vs for every s < t T, 3.25 for some constant C R. Observe that C R may depend on R through the quantity δ. = min { τx z ; z K, x R + T K }

10 STEP 2. Relying on the uniform bounds 3.25 on tangent vectors, it is easy to derive the estimate 3.2 in the piecewise smooth case. Indeed, let the initial data x, y be given. We then construct a one-parameter family of solutions x θ : [, T ] IR m, satisfying ẋ θ t = f x θ t, x θ = θy + 1 θx, θ [, 1]. Defining the tangent vectors v θ t =. x θ+ε t x θ t lim, ε ε for all t [, T ], from 3.25 it follows 1 yt xt d 1 dθ xθ t dθ = v θ t 1 dθ C v θ dθ = C y x, 3.27 R R proving 3.2. In this piecewise smooth case, the evolution equation in 1.6 thus generates a uniformly Lipschitz continuous flow. In the following, to denote the unique solution of the Cauchy problem 1.6, we shall use the semigroup notation xt = S t x. We recall that, if w : [, T ] IR m is any Lipschitz function, one has the error estimate wt S t w T wt + h S h wt L lim inf dt, t [, T ], 3.28 h + h where L is the Lipschitz constant of the semigroup. In particular, if w solves the perturbed equation ẇt = f wt + et, 3.29 and satisfies the bounds wt R + t K, from 3.27 and 3.28 we deduce wt S t w C R t es ds t [, T ]. 3.3 STEP 3. The general case will be treated using an approximation procedure. Fix any radius R arbitrarily large, and define U. = { u : [, T ] IR m ; u R, ut us t s } K for all t > s Lemma 1. Let f, g, τ be as in Theorem 2, and let R and ε > be given. Then there exists a piecewise smooth function ˆf : IR m K of the form ˆfx = ĝ x if τ τx < τ +1,

11 with the following properties. Each ĝ is smooth, and its Lipschitz constant satisfies Lipĝ sup Lip gt, t IR Moreover, the Picard operators determined by f and ˆf are close, namely T sup ˆf ut f ut dt ε u U To construct ˆf, we first apply the theorem of Scorza-Dragoni [11] to the Carathéodory function g and obtain a closed set J with measir \ J ε, such that the restriction of g to J IR m is continuous. The complement of J is an open set, which can be written as a disjoint union of countably many open intervals, say ]a ν, b ν [, ν 1. We then define g t, x = { gt, x if t J, θgb ν, x + 1 θga ν, x if t = θb ν + 1 θa ν for some ν 1, < θ < By 3.35, the function g is continuous in t and Lipschitz continuous in x. More precisely sup t IR Moreover, calling K = max z K z, for any u U we have Lip g t, = sup Lip gt, t J T g τut, ut g τut, ut dt 2 K meas { t; τut / J } 2 K ε/δ, 3.37 where δ > is the constant in We now choose a small δ > and, for each Z, we. define τ = δ and let ĝ be a mollification of the function g δ,. We then define ĝt, x =. ĝ x if τ t < τ +1, and let ˆf be as in From 3.36 it thus follows If δ is sufficiently small and the mollification ernel is sufficiently close to the identity, this construction yields T g τut, ut ĝ τut, ut dt ε for all u U Since ε > was arbitrary, 3.37 and 3.38 together yield Lemma 1. STEP 4. We can now conclude the proof of Theorem 2. Let x, y be any two solutions, as in 3.1. To prove the estimates 3.2, let ε > be given, choose R. = max { x, y } and construct 11

12 a function ˆf according to Lemma 1. According to STEP 1, the semigroup Ŝ generated by the evolution equation ẋ = ˆfx is Lipschitz continuous, more precisely Ŝt x Ŝty CR x y whenever x, y R, t [, T ], 3.39 for some constant C R not depending on ε. Define the quantities e x and e y as e x t. = f xt ˆf xt, e y t. = f yt ˆf yt, The functions x and y are thus solutions to ẋt = ˆf xt + e x t, x = x, ẏt = ˆf yt + e y t, y = y. By 3.39 we can now use 3.3 and deduce yt xt yt Ŝty + Ŝ t y Ŝtx + Ŝ t x xt C R t e y s ds + C R y x + C R 2C R ε + C R y x for every t [, T ]. Indeed, by 3.4 and 3.34 it follows T ey s ds < ε, Since ε was arbitrary, from 3.41 we deduce 3.2. T t [, T ]. 3.4 ex s ds < ε. t e x s ds A counterexample. In Theorem 2, the assumption that τ has bounded directional variation is essential for the uniqueness of the solutions, as shown by the following counterexample. Consider a function g : IR IR 2 IR 2 such that ] gt, x = gt =. 1, 1, if t 2 + 1, 1 ] 1, 1, if t ], 1 + 1, ], for any In the plane with coordinates x 1, x 2, define the sequences of points P, P, Q and Q by setting fig P =,, P. 1 =, 1, Q = 2 + 1, Q. = 12 1, ,

13 x 2 P P +1 Q P +1 P n x 1 Q figure 2 We can now construct a C 1 function τ : IR 2 IR with τx 1, λ > τx 1, 1 > for all x IR and such that, for every integer 1, 1 τx =., along the segment joining P, P, , along the segment joining Q, Q. 4.4 Letting fx =. g τx, and defining K =. co { 1, 1; 1, 1 } = { 1, λ; λ 1 }, all of the assumptions in Theorem 2 are satisfied, except the one on the directional variation of τ. Indeed, at all points P the gradient τ is parallel to the vector 1,. On the other hand, at each point Q this gradient is parallel to the vector 1, 1/ Since τ is continuous and never vanishes, its total variation in the direction of the cone Γ =. { } x 1, x 2 ; x 2 x 1 cannot be bounded. From the definitions it follows that, for each 1, fx. = { 1, 1 on the quadrilateral with vertices P, P, Q, Q, 1, 1 on the quadrilateral with vertices Q, Q, P +1, P

14 One can easily chec that the Cauchy problem on IR 2 ẋ = fx, x =, has two distinct solutions fig. 2. Namely, one solution passing through all the points Q, P, 1, and a second solution passing through all the points Q, P. Acnowledgment The second author is grateful to Department of Informatics, University of Oslo for supporting her visit at SISSA. References [1] A. Bressan, Directionally continuous selections and differential inclusions, Func. Evac , [2] A. Bressan, Unique solutions for a class of discontinuous differential equations, Proc. Amer. Math. Soc , [3] A. Bressan and G. Colombo, Existence and continuous dependence for discontinuous O.D.E. s, Boll. Un. Mat. Ital. 4-B 199, [4] A. Bressan and W. Shen, Uniqueness for discontinuous O.D.E. and conservation laws, Nonlinear Analysis, to appear. [5] F. H. Clare, Optimization and Nonsmooth Analysis, Wiley, [6] Dugundji, Topology, Allyn and Bacon, Boston [7] A. F. Filippov, Differential Equations with Discontinuous Right Hand Sides, Kluwer Acad. Publ., [8] P. Hartman, Ordinary Differential Equations, Birhäuser, 1982, Second Edition. [9] A. Pucci, Sistemi di equazioni differenziali con secondo membro discontinuo rispetto all incognita, Rend. Ist. Mat. Univ. Trieste, Vol. III 1971, ] W. Rzymowsi, Existence of solutions for a class of differential equations in IR n, J. Math. Anal. Appl., to appear. [11] G. Scorza-Dragoni, Un teorema sulle funzioni continue rispetto ad una e misurabili rispetto ad un altra variablie, Rend. Sem. Mat. Univ. Padova ,