Recent Trends in Differential Inclusions

Transcription

1 Recent Trends in Alberto Bressan Department of Mathematics, Penn State University (Aveiro, June 2016) (Aveiro, June 2016) 1 /

2 Two main topics ẋ F (x) differential inclusions with upper semicontinuous, non-convex right hand side extremal solutions ẋ extf (x), by Baire category or probability arguments (Aveiro, June 2016) 2 /

3 An approximate selection theorem Theorem (A.Cellina, Ann. Mat. Pura Appl., 1969) X is a compact metric space F : X R n is an upper semicontinuous multifunction with compact, convex values. Then for every ε > 0 there exists a continuous map f ε : X R n with the ( ) approximate selection property: Graph(f ε ) B Graph(F ), ε. F F 0 x x f ε (Aveiro, June 2016) 3 /

4 Application: Brouwer s theorem = Kakutani s theorem Ω R n is a compact, convex set Theorem 1 (Brouwer, 1910) Every continuous map f : Ω Ω has a fixed point x = f (x ) Theorem 2 (Kakutani, 1941) If F : Ω R n is an upper semicontinuous multifunction, with compact convex values F (x) Ω, then it has a fixed point x F (x ) (Aveiro, June 2016) 4 /

5 One-dimensional version of Brouwer s and Kakutani s theorems Brouwer 1910 Kakutani 1941 x * = f(x *) x * F(x *) no fixed point 1 F f F f ε 0 x* 1 0 x* 0 x Luitzen Egbertus Jan Brouwer (1910) Shizuo Kakutani (1941) (Aveiro, June 2016) 5 /

6 Maps with non-convex values Definition. A multifunction F : Ω R n is Cellina-approximable if for every ε > 0 there exists a continuous function f ε : Ω R n such that ( ) graph(f ε ) B graph(f ), ε Corollary (A.Cellina, 1969) G : Ω R n u.s.c. with compact convex values ψ : R n R n continuous Then the composed multifunction (possibly with non-convex values) x F (x) = ψ(g(x)) = {ψ(y) ; y G(x)} is Cellina-approximable. g ε is an approximate selection of G = f ε = ψ(g ε ) is an approximate selection of ψ(g) (Aveiro, June 2016) 6 /

7 The Cauchy problem for a differential inclusion Theorem (Zaremba, 1936) If G : R n K(R n ) is a bounded, u.s.c. multifunction with convex values, then the Cauchy problem ẋ G(x), x(0) = x has a solution. Counterexample (without convexity assumption) {1} x < 0 ẋ F (x) = { 1, 1} x = 0 { 1} x > 0 F x (Aveiro, June 2016) 7 /

8 The assumption that F is Cellina-approximable removes obvious topological obstructions Conjecture: If G : R n R n is a bounded, u.s.c. multifunction with convex values, and if ψ : R n R n is a smooth diffeomorphism, then the Cauchy problem ẋ ψ(g(x)), x(0) = x still has a solution. Hint for a proof? The multifunction F (x) = ψ(g(x)) is Cellina-approximable For every ε > 0, take an ε-approximate selection f ε and solve ẋ = f ε(x), x(0) = x Take the limit ε 0 (Aveiro, June 2016) 8 /

9 A counterexample (A.B., Rend. Sem. Mat. Padova, 1988) F (x 1, x 2 ). = φ(s). = s cos 1 s if s 0, φ(0) = 0 {(0, 1)} if x 2 > φ(x 1 ) {(0, 1)} if x 2 < φ(x 1 ) {(y 1, y 2 ) ; y 1 0, y y 2 2 = 1 } if x 2 = φ(x 1 ) x 2 φ x 1 (Aveiro, June 2016) 9 /

10 Sobolev regularity for multifunctions What goes wrong: F is discontinuous on a curve of infinite length Need: additional regularity. Definition (A.B., 2011) Let Ω R n be an open set. F : Ω R n is W 1,p -Cellina approximable if for every ε > 0 there exists a smooth function f ε : Ω R n such that ( ) graph(f ε ) B graph(f ), ε and f ε W 1,p = f ε L p + Df ε L p remains bounded uniformly w.r.t. ε (Aveiro, June 2016) 10 /

11 Existence of solutions ẋ F (x), x(0) = 0 Theorem (A.B. - R.DeForest, Rocky Mountain J.Math, 2011) (special volume dedicated to Lloyd Kenneth Jackson) Let F : R 2 R 2 be a bounded multifunction with closed graph. Assume that F is Cellina W 1,1 -approximable. Then the Cauchy problem admits a Caratheodory solution, defined for all times t R. Proof. (i) Can assume F (x) {v R 2 ; v = 1}. (ii) Take a sequence of continuous approximate selections f ε, solve ẋ ε (t) = f ε (x ε (t)) x ε (0) = 0 (iii) As ε 0, take a convergent subsequence x εk (t) x(t). The arc-length reparameterization s x(s) of this limit trajectory is the desired solution (Aveiro, June 2016) 11 /

12 ẋ ε (t) = f ε (x ε (t)) x ε (0) = 0 The assumption F is W 1,1 -Cellina approximable implies the uniform BV bound Df ε dx C Q On a fixed interval [0, T ], taking a subsequence this guarantees that the trajectories x εk ( ) converge to a well defined curve x( ), not reduced to a point. x 1 x 2 x 1 x k 0 x k x 2 0 BAD GOOD (Aveiro, June 2016) 12 /

13 Lemma. Let f : Q R 2 be a smooth vector field on the unit square, with f (x) 1. Then every trajectory of the ODE ẋ = f (x) starting inside the unit square Q reaches the boundary of Q within time T. = Df L 1 (Q) x 2 p (s) k p (s) k 1 J 2 0 Λ (s) k s J 1 x 1 Indeed, the total variation of f on Q can be bounded below in terms of the length of a trajectory remaining inside Q. (Aveiro, June 2016) 13 /

14 Remarks and open problems ẋ F (x), x(0) = 0 R n (CP) The existence of a continuous approximate selection f ε removes some topological obstructions, but does not guarantee existence of solutions Question: Let F : R n R n be a bounded multifunction with closed graph, which is W 1,p -Cellina approximable. For which values of p does this guarantee the existence of a solution? p > n = TRUE (F has a Hölder continuous selection) p < n 1 = FALSE (W 1,p (R n ) contains functions which have jumps on a curve with infinite length) (Aveiro, June 2016) 14 /

15 Extremal trajectories via Baire category x F (x) Hausdorff continuous multifunction with closed, convex values F : solutions to ẋ F (x), x(0) = 0 F ext : solutions to ẋ ext F (x), x(0) = 0 ext F (x) = extreme points of F (x) Note: to solve ẋ G(x) not convex-valued, consider the convex closure F (x). = cog(x). Then ext F (x) G(x) (Aveiro, June 2016) 15 /

16 An example (A.Cellina, Atti Accad. Lincei, 1980) In the Banach space C([0, T ]), the set F ext of solutions to ẋ { 1, 1} x(0) = 0 is residual in the set F of all solutions to residual ẋ(t) [ 1, 1] x(0) = 0 contains a countable intersection of open dense subsets 0 T Research program on & problems in Calculus of Variations with lack of convexity (1980 ) (i) Prove that a convexified problem has a nonempty, closed set of solutions. (ii) Prove that almost all these solutions (in the sense of Baire category) are extremal, hence provide solutions to the original problem. (Aveiro, June 2016) 16 /

17 A basic technique For any compact, convex set Ω R n, define ϕ(ω, y) ϕ(ω, y) {. 1 = max 0 θ(s) y 2 ds : θ : [0, 1] Ω,. = if y / Ω 1 0 } θ(s) ds = y ϕ(ω, y ) Ω y The function ϕ is upper semicontinuous in Ω, y, and concave in y. ϕ(ω, y) gives a quantitative measure of how much the point y is not extremal ϕ(ω, y) = 0 y extω (Aveiro, June 2016) 17 /

18 Relation with Baire category ϕ(ω, y) = 0 y extω A solution of ẋ F (x) is also a solution to ẋ ext F (x) if T 0 ( ) ϕ F (x(t)), ẋ(t) dt = 0 To prove that F ext F is a subset of second Baire category, it suffices to show that for every ε > 0 F ε = { x F ; is an open, dense subset of F T 0 ( ) } ϕ F (x(t)), ẋ(t) dt < ε (Aveiro, June 2016) 18 /

19 The Baire category approach F. De Blasi and G. Pianigiani, The Baire category method in existence problems for a class of multivalued differential equations with nonconvex right-hand side. Funkcial. Ekvac F. De Blasi and G. Pianigiani, Differential inclusions in Banach spaces. J. Differential Equations F. De Blasi and G. Pianigiani, Non-convex-valued differential inclusions in Banach spaces. J. Math. Anal. Appl F. De Blasi and G. Pianigiani, Baire category and boundary value problems for ordinary and partial differential inclusions under Carathodory assumptions. J. Differential Equations A. Tolstonogov, Differential inclusions in a Banach space. Kluwer, (Aveiro, June 2016) 19 /

20 Theorem (S. De Blasi, G. Pianigiani) Let F : R n R n be a Lipschitz continuous multifunction with compact, convex values. Then the set F ext of solutions to ẋ ext F (x), x(0) = 0 is residual in the set F C([0, T ]; R n ) of all solutions to ẋ F (x), x(0) = 0 residual contains a countable intersection of open dense subsets Existence results obtained via Baire category do not rely on any compactness assumption. In a Banach space, the set of solutions does not need to be precompact. (Aveiro, June 2016) 20 /

21 Probability vs. Baire Category Can a similar theory be developed, replacing Baire category with probability? Given a (Lipschitz) continuous, convex valued multifunction F, construct a probability distribution on the set of all solutions to with the following properties ẋ F (x) t [0, T ], x(0) = 0 (1) (i) with probability 1, one has ẋ extf (x) for a.e. t [0, T ] (ii) If S C 0 ([0, T ]) is an open set containing a solution of (1), then Prob.(S) > 0 (Aveiro, June 2016) 21 /

22 Random extremal solutions Theorem (A. Bressan and V. Staicu, NoDEA, 2016). Given a Lipschitz continuous, convex valued multifunction F, there exists a probability distribution on the set of all solutions to with the following properties ẋ F (x) t [0, T ], x(0) = 0 (1) (i) with probability 1, one has ẋ extf (x) for a.e. t [0, T ] (ii) If S C 0 ([0, T ]) is an open set containing a solution of (1), then Prob.(S) > 0 Conjecture If F is only continuous, the conclusion still holds, but with (ii) replaced by (iii) For every continuous selection f (x) F (x), if S C 0 ([0, T ]) is an open set containing all solutions to then Prob.(S) > 0 ẋ = f (x) t [0, T ], x(0) = 0 (Aveiro, June 2016) 22 /

23 ẋ F (x), x(0) = 0 R n An extremal solution can be constructed as limit of a sequence of approximate solutions. At each step of the construction, instead of making one particular choice, put some natural probability measure on the set of admissible choices. The resulting probability measure on the set of all solutions F is far from unique. Is there a canonical way to define such a probability measure? (Aveiro, June 2016) 23 /

24 A dual approach F (x) R n compact, convex, w R n extremal face: F w (x). = { } y F (x) ; y, w = max z, w z F (x) w w F (x) w F (x) F(x) w (Aveiro, June 2016) 24 /

25 A probability measure on the set of extremal solutions Given a continuous guiding function w : [0, T ] R n, call F w( ) C([0, T ]; R n ) the nonempty, compact set of solutions to ẋ(t) F w(t) (x(t)), x(0) = 0 (2) Conjecture (A. Bressan, 1999) Let F be Lipschitz continuous, with compact, convex values. Consider Brownian motion on the surface of the unit sphere in R n. Then For a.e. random path w( ), the Cauchy problem (2) has a unique solution. This solution satisfies ẋ extf (x). This construction yields a unique probability measure on F, with the desired properties. Theorem (G. Colombo and V. Goncharov, NoDEA, 2013). The conjecture is true in the case F (x) = {f 1 (x), f 2 (x)} R 2, with f 1, f 2 Lipschitz continuous. (Aveiro, June 2016) 25 /

26 Revisiting the dual approach, via Baire category Theorem (A.B., J. Differential Equations, 2013) Let F : R n R n be a continuous multifunction with compact, convex values. Then the set W = {w C([0, T ]; R n ) ; F w( ) F ext} is residual in C([0, T ]; R n ). For almost every w C([0, T ]; R n ) (in the sense of Baire Category), all solutions of ẋ(t) F w(t) (x(t)), x(0) = 0 (2) are also solutions of ẋ(t) ext F (x(t)), x(0) = 0 (3) = Filippov s theorem (Aveiro, June 2016) 26 /

27 Key step in the proof: for every ε > 0 the set W ε = { w C([0, T ]; R n ) ; T 0 ( ) } ϕ F (x(t)), ẋ(t) dt < ε for every x F w is open and dense in C([0, T ]; R n ). F w = set of solutions to ẋ(t) F w(t) (x(t)) w(t) w(t) F(x) F (x) (Aveiro, June 2016) 27 /

28 A revised conjecture (Brownian motion does not wiggle fast enough) w(t) ϕ(s) x F(x) x(t) 0 s Given a modulus of continuity ϕ : R + R + with ϕ(0) = 0, ϕ (s) > 0, ϕ (s) < 0 consider the Banach space C ϕ ([0, T ]; R n ) C([0, T ]; R n ), with norm u( ) C ϕ. = sup u(t) + t [0,T ] sup 0<s<t<T u(t) u(s) ϕ(t s) (Aveiro, June 2016) 28 /

29 Conjecture (A. Bressan, 2014) Let F : R n R n be a continuous multifunction with compact, convex values. Then there exists a modulus of continuity ϕ (depending on F ), such that the set {w C ϕ ([0, T ]; R n ) ; F w( ) F ext} is residual in C ϕ ([0, T ]; R n ) Let µ be a probability distribution on C([0, T ]; R n ) whose paths have almost surely modulus of regularity ϕ (but not better!) For every random path w( ), choose a solution x w ( ) F w Then the map w x w takes values inside F ext almost surely The push-forward of the probability measure on C ϕ through the map w x w is a probability distribution supported on F ext with the desired properties. (Aveiro, June 2016) 29 /

30 Boundary value problems - a Baire category approach Consider a non-convex problem P, and let F be the family of solutions to a suitable convexified problem P. F is a nonempty, closed subset of a Banach space, hence a complete metric space. One can identify a set F ext F of extremal solutions which solve the original non-convex problem P. Direct approach: Show that the set F ext of extremal solutions is residual in F. Dual approach: Consider a family of constrained optimization problems min u( ) F J w (u), where the functional J w depends on an an auxiliary function w, ranging in a Banach space W. - For each w W, call F w F the (nonempty) set of minimizers. - Show that the set {w W ; F w F ext } is residual in W. (Aveiro, June 2016) 30 /

31 The Bang-Bang Theorem ẋ = f (x) + G(x)u(t) u(t) Ω (1) ẋ = f (x) + G(x)u(t) u(t) extω (2) Problem: given a trajectory of (1), construct a trajectory of (2) with the same initial and terminal point. y x(t)=y(t) x x 0 For linear systems: ẋ(t) = A(t)x(t) + B(t)u(t) J.P.La Salle, The bang-bang principle. Automatic and Remote Control, (Aveiro, June 2016) 31 /

32 A Baire category approach to the bang-bang property ẋ(t) = A(t)x(t) + B(t)u(t), u(t) Ω (1) ẋ(t) = A(t)x(t) + B(t)u(t), u(t) ext Ω (2) x(0) = a, x(t ) = b (3) Theorem (A.B. & B.Piccoli, J. Differential Equations 1995) Let Ω R m be compact, convex, and let A, B be bounded measurable matrix-valued functions. The set S a,b of solutions of (1), (3) is a compact subset of C([0, T ]; R n ). Within S a,b, the trajectories which satisfy (2) are a residual subset. Some extensions to nonlinear concave multifunctions (Aveiro, June 2016) 32 /

33 The dual Baire category approach U a,b = { u : [0, T ] Ω ; } u steers the system from a to b ẋ(t) = A(t)x(t) + B(t)u(t), x(0) = a, x(t ) = b For any given guiding function w C([0, T ]; R m ) consider the optimization problem. max = u U a,b T Theorem (A.B., M. Mazzola, K. Nguyen, 2015) 0 w(t) u(t) dt (J w ) Let Ω R m be compact, convex, and let A, B be bounded measurable matrix-valued functions. Then W =. { } w C([0, T ]; R m ) ; all maximizers u( ) of (J w ) take values inside ext Ω is a residual subset of C([0, T ]; R m ) (Aveiro, June 2016) 33 /

34 Further questions Can one prove Lyapunov s theorem on the range of a vector measure, or the Krein-Milman theorem, using a dual Baire category approach? Can a dual Baire category approach be applied to problems in several space dimensions? V. I. Arkin and V. L. Levin, Convexity of values of vectors integrals, theorems on measurable choice and variational problems, Russian Math. Surveys 27 (1972), A.B. and F. Flores, On total differential inclusions. Rend. Sem. Mat. Univ. Padova 92 (1994), F. S. De Blasi and G. Pianigiani, On the Dirichlet problem for first order partial differential equations. A Baire category approach. Nonlinear Differential Equations Appl. 6 (1999), 13. (Aveiro, June 2016) /