The Bang-Bang theorem via Baire category. A Dual Approach

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1 The Bang-Bang theorem via Baire category A Dual Approach Alberto Bressan Marco Mazzola, and Khai T Nguyen (*) Department of Mathematics, Penn State University (**) Université Pierre et Marie Curie, Paris VI s: bressan@mathpsuedu, marcomazzola@imj-prgfr, ktn2@psuedu November 19, Introduction For t [,T], consider the Cauchy problem ẋ(t) F ( x(t) ), (11) x() =, (12) wherex F(x) IR n isabounded, Hausdorffcontinuous multifunction with compact convex values Call F C ( [,T]; IR n) the set of Carathéodory solutions of (11) Moreover, call F ext the set of trajectories of ẋ(t) extf(x(t)), (13) where the time derivative takes values within the set of extreme points of F(x) If F is Lipschitz continuous wrt the Hausdorff distance, starting with the seminal paper by Cellina [7], it is now well known that the set of extremal solutions F ext is a residual subset of F, ie it contains the intersection of countably many open dense subsets [9, 1] By an application of Baire s theorem, this implies that the set F ext is nonempty and every solution of (11) can be uniformly approximated by solutions of (13) In [3] an alternative approach was developed, still based on Baire category but from a dual point of view For every w IR n, consider the compact, convex subset of vectors in F(x) which maximize the inner product with w, namely F w (x) = y F(x); y,w = max y F(x) y,w (14) For each continuous path t w(t), the multifunction F w (t,x) = F w(t) (x) is upper semicontinuous with compact, convex values Hence the Cauchy problem ẋ(t) F w(t)( x(t) ), x() =, (15) 1

2 has a non-empty, compact set of solutions F w C([,T]; IR n ) The main result in [3] shows that for almost all functions w C([,T]; IR n ), in the Baire category sense, all solutions of (15) are also solutions of (13) Theorem 1 Let F be a bounded, Hausdorff continuous multifunction on IR n, with compact convex values Then the set W = w( ); F w F ext (16) is a residual subset of C([,T];,IR n ) Notice that here the Lipschitz continuity of F is not required Incidentally, this yields yet another proof of the classical theorem of Filippov [11], on the existence of solutions to differential inclusions with continuous, non-convex valued right hand side The purpose of the present paper is to explore whether a dual Baire category approach can be applied also to some boundary value problems (or variational problems) without convexity assumptions The basic setting is as follows Consider a non-convex problem P, and let S be the set of solutions to a suitable convexified problem P Under natural assumptions, S will be a nonempty, closed subset of a Banach space, hence a complete metric space Moreover, one can identify a set S ext S of extremal solutions which solve the original non-convex problem P Direct approach: Show that the set S ext of extremal solutions is residual in S Dual approach: Consider a family of constrained optimization problems min u( ) S Jw (u), (17) where the functional J w depends on an auxiliary function w, ranging in a Banach space W For each w W, call S w the set of minimizers Show that the set w W ; S w S ext is residual in W As a first step, we apply these ideas to derive an alternative proof of the classical bang-bang principle Namely, consider the linear control system in IR n ẋ(t) = A(t)x(t)+B(t)u(t), u(t) Ω (18) Here A(t) and B(t) are n n and n m matrices respectively, while Ω IR m is a compact convex set Together with (18) we consider a system where the control takes values in the extreme points: ẋ(t) = A(t)x(t)+B(t)u(t), u(t) extω (19) Given initial and terminal conditions x() =, x(t) = x, (11) assume that the boundary value problem (18)-(11) has a solution Then by the bang-bang theorem [6, 8, 12] the problem (19)-(11) has a solution as well 2

3 The standard proof of the bang-bang theorem relies on Lyapunov s theorem, providing the convexity of the range of a non-atomic vector measure An alternative approach, based on Baire category, was developed in [5] Call S = x : [,T] IR n ; x() =, x(t) = x, ẋ(t) = A(t)x(t)+B(t)u(t) for some measurable control u : [,T] Ω S ext = x : [,T] IR n ; x() =, x(t) = x, ẋ(t) = A(t)x(t)+B(t)u(t) As proved in [5], one has for some measurable control u : [,T] extω Theorem 2 Let A,B be bounded, measurable, matrix-valued functions, and let Ω IR m be a compact convex set Assume that S Then S is compact in C([,T]; IR n ) and S ext is a residual subset of S In particular, S ext is nonempty In this paper we develop a dual approach, also based on Baire category Let U be the set of all measurable functions u : [,T] Ω such that the solution to the Cauchy problem satisfies the terminal condition ẋ(t) = A(t)x(t) + B(t)u(t), x() = (111) x(t) = x (112) Throughout the following, we shall assume that U is non-empty Given any continuous function w C([,T]; IR m ), consider the constrained optimization problem min u U w(t), u(t) dt (113) We show that, for almost all continuous functions w C([,T];IR m ), in the sense of Baire category, the problem (113) has a unique minimizer This minimizer is a bang-bang control, ie it takes values within the set of extreme points of Ω Theorem 3 Let A,B be bounded, measurable, matrix-valued functions, and let Ω IR m be a compact convex set Let W C([,T]; IR m ) be the set of all continuous functions w such that the variational problem (113) has a unique minimizer, satisfying u(t) ext Ω for ae t [,T] Then W is residual in C([,T]; IR m ) A proof of the theorem will be given in the next section For the basic theory of multifunctions and differential inclusions we refer to the classical monograph [1] Definitions and basic properties of Young measures, mentioned in the proof, can be found in [14, 15, 16] 2 Proof of Theorem 3 1 Given the compact convex set Ω IR m, consider the function ϕ : IR m IR 3

4 defined by ϕ(y) 1 = max f(s) y 2 ds : f : [,1] Ω, 1 f(s)ds = y, with the provision that ϕ(y) = if y / Ω Otherwise stated, ϕ(y) is the maximum variance among all probability measures supported on Ω, whose barycenter is at y As proved in [2], ϕ is upper semicontinuous and concave on Ω Moreover, we have the equivalence ϕ(y) = y extω (21) A measurable control u : [,T] Ω takes values in extω for ae t iff ϕ(u(t))dt = (22) 2 The solution to the Cauchy problem (111) can be written as x(t) = t M(t,s)B(s)u(s)ds, where M(t,s) is the n n matrix fundamental solution to the linear homogeneous system ẋ = A(t)x In other words, t M(t,s) = A(t)M(t,s), M(s,s) = I n, where I n is the n n identity matrix Setting G(s) = M(T,s)B(s), for any w C([,T]; IR m ) the optimization problem (113) can be reformulated as follows (OP) Find u : [,T] Ω which minimizes the integral subject to J w (u) = w(t), u(t) dt (23) G(t)u(t)dt = x (24) 3 For any ε > and N 1, consider the set W ε,n C([,T]; IR m ) of all functions w with the following property If u(t) argmin ω, w(t) λg(t) (25) for some row vector λ = (λ 1,,λ n ) [ N,N] n and all t [,T], then ϕ(u(t))dt < ε (26) 4

5 In the next two steps we will prove that each W ε,n is open and dense, hence the intersection W = W ε,n (27) is residual in C([,T]; IR m ) ε>, N 1 4 We first show that each set W ε,n is open in C([,T]; IR m ) Indeed, consider a sequence (w i ) i 1 converging uniformly to w and such that w i / W ε,n for every i Then there exist sequences λ i [ N,N] n and u i : [,T] Ω satisfying and u i (t) argmin ω, w i (t) λ i G(t) for all t [,T] (28) ϕ(u i (t))dt ε (29) By taking subsequences, we can assume that λ i λ [ N,N] n and u i converges weakly in L 2 ([,T],IR m ) to some function u By the linearity of the functional in (25), and the convexity of the set Ω, this limit function u( ) satisfies (25), for ae t [,T] By the upper semicontinuity and the concavity of ϕ in Ω it follows Hence w / W ε,n, showing that W ε,n is open in C([,T]; IR m ) ϕ(u(t))dt ε (21) 5 In this and the next two steps we will prove that each W ε,n is dense in C([,T]; IR m ) Consider the function Φ : IR m IR m IR defined by Φ(v, w) = max ϕ(u) ; u argmin ω, w v (211) Observing that the map Φ is upper semicontinuous, for every η > we can consider the Lipschitz continuous approximation Φ η (v,w) = max Φ(v,w ) η v v η w w ; (v,w ) IR m IR m (212) Notice that for any η η it holds Φ η (v,w) Φ η (v,w), for all (v,w) IR m IR m (213) For any fixed v IR m, it is known that Φ(v,w) = for almost every w IR m Indeed (see [13]) Φ(v,w) = if and only if the minimum ψ(w) = min ω, w v is attained at a single point This is true if and only if the Lipschitz map z ψ(z) is differentiable at the point z = w By Rademacher s theorem, this is true for ae w IR n 5

6 We claim that for every δ > and K,M > there exists η > sufficiently large so that Φ η (v,w)dw < δ for all v IR m, v M (214) B(,K) Otherwise, let (v k,η k ) k 1 beasequenceof vectors inir m IR satisfying that lim k η k =, lim k v k = v and Φ η k (v k,w)dw δ for all k 1 (215) B(,K) From (213), for any η > there is K η > such that Φ η (v k,w)dw δ, for all k K η B(,K) The Lipschitz continuity of Φ η implies Φ η ( v,w)dw = lim k B(,K) Therefore, taking η to + we obtain B(,K) B(,K) Φ η (v k,w) δ Φ( v,w)dw δ (216) This yields a contradiction because Φ( v,w) = for almost every w 6 Choose a constant L > such that ϕ(u) L for all u Ω (217) Relying on Lusin s theorem, we can replace the measurable matrix-valued function G with a continuous function Ĝ such that ( t; meas ) Ĝ(t) G(t) < ε 4L (218) Let any radius ρ > and any w C([,T]; IR m ) be given By the previous step, we can find η > large enough so that Φ η( λĝ(t), w) dw ε B( w(t),ρ) 4T, (219) for every t [,T] and λ [ N,N] n Here and in the sequel, f dx denotes the average value of the function f over the set S From (219) we deduce Φ η (λĝ(t), w)dwdt ε B( w(t),ρ) 4 (22) Consider a sequence of continuous functions w ν C([,T]; IR m ), weakly converging to w, such that (i) w ν (t) B( w(t),ρ), for all ν 1 and t [,T], S 6

7 (ii) as ν, the limit is described by the family of Young measures µ t ; t [,T], where µ t is the probability measure uniformly distributed on the ball B( w(t),ρ) Since Φ η is continuous, this yields lim ν Φ η (λĝ(t), w ν(t))dt = η Φ (λĝ(t), w)dwdt B( w(t),ρ) Choosing w = w ν for some ν sufficiently large, in view of (22) we obtain Φ η (λĝ(t), w(t))dt ε 3 (221) In turn, by (217) and (218) and the obvious inequality Φ Φ η, this implies Φ(λG(t), w(t))dt L ε 4L + Φ η (λĝ(t), w(t))dt ε 4 + ε 3 (222) Since w w C ρ, and ρ > can be chosen arbitrarily small, this proves the density of W ε,n 7 The existence of a sequence (w ν ) ν 1 satisfying the properties (i)-(ii) in the previous step follows from a standard construction Consider a sequence of points y j B(,ρ) uniformly distributed on the ball centered at the origin with radius ρ That means lim ν 1 ν ν f(y j ) = j=1 B(,ρ) f dx, (223) for every continuous function f : IR m IR For a given ν 1, divide the interval [,T] into ν 2 subintervals, inserting the times t l = hl, with h = T/ν 2 Consider the piecewise continuous function z ν (t) = w(t)+y j if t [t l 1, t l ] with l = mν +j for some integer m Then choose a continuous function w ν such that ( ) meas t; w ν (t) z ν (t) < 1 ν This sequence satisfies the required properties 8 Now assume that w W We claim that the optimization problem (OP) has a unique solution u : [,T] extω Indeed, assume that u 1,u 2 are two distinct solutions Then u(t) = u 1(t)+u 2 (t) 2 is also a solution Set v(t) = u 1(t) u 2 (t) 2 and consider the reachable subspace Y = t G(t)v(t)θ(t)dt; θ : [,T] IR measurable IR n (224) Consider the control system on the product space IR Y, with variables (y,y) ẏ (t) = w(t), v(t) θ(t), ẏ(t) = G(t)v(t)θ(t) y () =, y() = 7

8 Then the control θ (t) provides an optimal solution to the problem minimize: y (T) subject to θ(t) [ 1,1], y(t) = The necessary conditions for optimality yield the existence of a nonzero vector λ = (λ,λ) IR Y such that ( ) θ (t) = arg min λ w(t),v(t) λ, G(t)v(t) θ θ [ 1,1] for ae t [,T] If λ =, then the non-zero vector λ Y has the property λ, G(t)v(t) = for ae t [,T] But this contradicts the definition (224) of Y If λ, by a normalization we can assume λ = 1 Then there exists a vector λ Y such that w(t), v(t) λ, G(t)v(t) = for ae t [,T] This implies while u(t) argmin ω, w(t) λg(t) ϕ(u(t))dt v(t) 2 dt > This contradicts the assumption w W, proving the theorem References [1] J P Aubin and A Cellina, Differential Inclusions, Springer, Berlin, 1984 [2] A Bressan, The most likely path of a differential inclusion, J Differential Equations 88 (199), [3] A Bressan, Extremal solutions to differential inclusions via Baire category: a dual approach J Differential Equations 255 (213), [4] A Bressan and G Colombo, Generalized Baire category and differential inclusions in Banach spaces, J Differential Equations 76 (1988), [5] A Bressan and B Piccoli, A Baire category approach to the bang-bang property J Differential Equations 116 (1995), [6] A Bressan and B Piccoli, Introduction to the Mathematical Theory of Control, AIMS Series in Applied Mathematics, Springfield Mo 27 [7] A Cellina, On the differential inclusion x [ 1,1], Atti Accad Naz Lincei Rend Cl Sci Fis Mat Natur 69 (198), 1 6 8

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