Lax Hopf formula and Max-Plus properties of solutions to Hamilton Jacobi equations

Transkript

1 Nonlinear Differ. Equ. Appl. 20 (2013), c 2012 Springer Basel AG /13/ published online August 21, 2012 DOI /s Nonlinear Differential Equations and Applications NoDEA Lax Hopf formula and Max-Plus properties of solutions to Hamilton Jacobi equations Jean-Pierre Aubin Abstract. We state and prove a Lax Hopf formula characterizing viable capture basins of targets investigated in viability theory and derive a Max-Plus morphism of capture basins with respect to the target. Capture basins are used to define viability solutions to Hamilton Jacobi equations satisfying trajectory conditions (initial, boundary or Lagrangian conditions).the Max-Plus morphism property of Lax Hopf formula implies the fact that the solution associated with inf-convolution of trajectory conditions is the inf-convolution of the solutions for each trajectory condition. For instance, Lipschitz regularization or decreasing envelopes of trajectory condition imply the Lipschitz regulation or decreasing envelopes of the solutions. Keywords. Lax Hopf, Viability, Capture Basins, Inf-convolution, Fenchel transform, Lagrangian, Hamiltonian. Contents 1. Introduction Trajectory-valued Hamilton Jacobi problems Hamiltonians and Lagrangians A Lax Hopf formula for viable capture basins The viability solution Inf-convolution One-dimensional example 201 Acknowledgments 210 References Introduction Let X := R n be a finite dimensional vector space. Let us consider 1. a concave upper semicontinuous Hamiltonian h : p X h(p) R;

2 188 J.-P. Aubin NoDEA 2. an extended lower semicontinuous function c :(t, x) R + c(t, x) R {+ }, with which we associate the domain tube t C(t) defined by C(t) := {x such that c(t, x) < + } (1) which can be regarded as a thick trajectory. An extended function V :(t, x) R + K V (t, x) R is a solution of the trajectory-valued Hamilton Jacobi problem if: ( ) V (t, x) V (t, x) (i) t>0, x / C(t), = h t x (2) (ii) t 0, x C(t), V(t, x) c(t, x) The second condition encompasses Cauchy, Dirichlet, Lagrangian conditions as well as many other internal conditions. We call it a trajectory condition (see [2] for examples and applications to traffic management). The Lax Hopf formula proved for Cauchy problems extend to general trajectory-valued problems: we associate with the Hamiltonian h its Fenchel transform u, l(u) := d [0,T ] ξ C(d) sup [h(p) p, u ] p Dom(h) regarded as the Lagrangian. The Lax Hopf formula reads [ V (T,x) = inf inf c (d, ξ)+(t d)l ( x ξ T d Furthermore, let us introduce the inf-convolution v c of the trajectory condition c with any lower semicontinuous positively-homogeneous convex extended function v defined by (v c)(t, x) = inf(v(y)+c(t, x y)) y We shall prove that the solution to the Hamilton Jacobi equation associated with the inf-convoluted trajectory condition (t, x) (v c)(t, x) by v is the inf-convolution (t, x) (v V)(t, x)(t, x) of the solution (t, x) (t, x) V (t, x) to the Hamilton Jacobi equation associated with trajectory condition (t, x) c(t, x). Among some consequences, we may single out: 1. if we regularize the trajectory condition by taking its Lipschitz regularization which is the largest smaller λ-lipschitz function obtained by its Pasch Hausdorff envelope, obtained as an inf-convolution product by the function x λ x, we deduce that the solution is λ-lipschitz, and, actually, the Pasch Hausdorff envelope of the solution. 2. if we associate with the trajectory condition its decreasing envelope, which is the smallest larger decreasing function, obtained as an inf-convolution product by the indicator function of a cone, we deduce that the solution is decreasing, and, actually, the decreasing envelope of the solution. )]

3 Vol. 20 (2013) Lax Hopf formula and Max-Plus properties 189 The method used is derived from set-valued analysis and viability theory (see the books [1,6,7]), actually, from the concept of viable capture basin (see Definition 4.1, p. 191). We define Lax Hopf formulas for computing them under adequate assumptions (Theorem 4.4, p. 193). We next define the epigraph of the viability solution to the trajectory-valued Hamilton Jacobi problem as a viable capture basin of an auxiliary target under an auxiliary characteristic system. It is shown (see [4]) that viability solutions defined in this way are viscosity solutions in the weaker sense of Barron Jensen/Frankowska. The Lax Hopf formulas proved (in a very simple way) at the level of viable capture basins imply the announced results we were looking for. After defining trajectory-valued Hamilton Jacobi problems (Sect. 2) and introducing the Lagrangian (Sect. 3), we prove the Lax Hopf formulas characterizing viable capture basins (Sect. 3) for computing the Lax Hopf formula for the solution (Sect. 4). Section 5 is devoted to the inf-convolution morphism. The last section apply the above results to one-dimensional trajectory-valued Hamilton Jacobi problems. 2. Trajectory-valued Hamilton Jacobi problems Definition 2.1. (Hamiltonian and trajectory conditions)letx := R n be a finite dimensional vector space. We associate with a Hamiltonian h : R + X R + R the Hamilton Jacobi partial differential equation V (t, x) t = h ( V (t, x) x A trajectory condition is described by a lower semicontinuous function c : (t, x) R + c(t, x) R {+ }, with which we associate the domain tube t C(t) defined by C(t) := {x such that c(t, x) < + } (3) These general trajectory conditions cover the following classical and less classical examples: 1. Cauchy initial conditions x γ 0 (x). In this case, we set c 0 (0,x):=γ 0 (x) and c 0 (t, x) =+ whenever t>0. Then C(0) = Dom(γ 0 )andc(t) = ; 2. Dirichlet boundary conditions x K δ(t, x) defined on the boundary of the domain, with which we associate c δ (t, x) :=δ(t, x) ifx K and c δ (t, x) =+ whenever x K; 3. Lagrangian mobile conditions t γ i (t) during some time intervals ]τ i, τ i ], with which we associate c γi (t, γ i (t)) := δ i (t) andc γi (t, x) :=+ whenever x γ i (t); 4. prescribed upper bounds on solutions to Hamilton Jacobi equations; 5. Combined Lagrangian conditions γ i (t),i=1,...,p,c(t) = i such that τ i t τ i {γ i (t)} and the trajectory condition c(t, γ i (t)) is defined on the graphs (t, γ i (t)) t 0 of the evolution of each vehicle i. )

4 190 J.-P. Aubin NoDEA Definition 2.2. (The trajectory-valued Hamilton Jacobi problem) The trajectory-valued Hamilton Jacobi problem takes into account the two above requirements on its solution V :(t, x) R + K V (t, x) R: ( ) V (t, x) V (t, x) (i) t>0, x / C(t), = h t x (4) (ii) t 0, x C(t), V(t, x) c(t, x) Since the Lagrangian conditions may and do take infinite values and are assumed only to be semicontinuous, partial differential equation techniques (see for instance [11]) may not work. The lack or regularity is not only motivated by mathematical search of generality, but here, by very concrete applications. 3. Hamiltonians and Lagrangians It is mathematically natural in the framework of duality theory in mechanics, in economics and in convex analysis to associate with a (concave) Hamiltonian h : X R a (convex) Lagrangian l : X := X R. Since Werner Fenchel, we know that there exists a bijective correspondence between lower semicontinuous convex functions defined on a vector space and their conjugate functions defined on the dual. Here, there is a slight adaptation to perform, since the Hamiltonian h is concave and the Lagrangian l is convex. The adaptation to this situation, tedious as it is, poses no problem, and the results we shall use are summarized in the following Lemma: Lemma 3.1. (Hamiltonians and Lagrangians) Assume that the Hamiltonian h is concave and upper semicontinuous. Then its Lagrangian l defined by u, l(u) := sup p Dom(h) [h(p) p, u ] (5) is lower semicontinuous and convex. The Hamiltonian is related to the Lagrangian by the relation p, h(p) = so that the Fenchel inequality inf [l(u)+ p, u ] (6) u Dom(l) u, p, p, u l(u) h( p) always holds true. Furthermore, u achieves the minimum in the minimization problem h(p) =l(u) + p, u if and only if u + h( p) and p achieves the maximum in l(u) := h(p) p, u if and only if p l(u). The three following statements are thus equivalent: (i) p, u l(u) h( p) (ii) u + h( p) (7) (iii) p l(u)

5 Vol. 20 (2013) Lax Hopf formula and Max-Plus properties 191 Moreover, whenever h i is upper semicontinuous and concave (or l i is lower semicontinuous and convex), i =1, 2, then h 1 h 2 if and only if l 1 l 2 (See [3] or[14] for convex analysis among many other books). Using extended functions (taking infinite values) is mandatory because solutions of Hamilton Jacobi partial differential equation may take infinite values and may be non differentiable. However, they are viscosity solutions in the weak sense of Barron Jensen/Frankowska solutions (see [8, 12, 13], which are lower semicontinuous viscosity solutions (see [9, 10]). More details can be found in [5] andthebook[6]. 4. A Lax Hopf formula for viable capture basins We shall define the epigraph of viability solution to a trajectory-valued Hamilton Jacobi problem as a viable capture basin, one of the main concepts studied in viability theory: Definition 4.1. (Capture basin of a target viable in an environment)letk X be an environment, C K a target in this environment and differential inclusion x (t) F (x(t)). The capture basin Capt F (K, C) is the subset of initial states x K from which starts at least one evolution x( ) to differential inclusion x (t) F (x(t)) such that, for some finite time t 0, this evolution 1. reaches the target C at time t at x(t ) C; 2. is viable in K on the interval [0,t ]( t [0,t ], x(t) K). For constant differential inclusions x (t) G, we can obtain simple formulas of the capture basins Capt G (K, C) when G is a closed convex subset, and even for non convex right hand sides G for which there exists a closed convex cone P such that G P is closed and convex. If H is a closed subset, we denote by R + H = λ 0 λh the cone spanned by the subset H We shall need the following Lemma 4.2. Let H be a compact subset and P aclosedconvexcone.ifco(h) P, then R + co(h) P = R + (co(h P )) is closed (8) Consequently, if G := H P is closed and convex, then R + co(h) P = R + G (9) Proof. First, we observe that R + co(h) P = R + co(h P ) because P is a closed convex cone. The subset H being assumed to be compact, its convex hull co(h) =co(h) is also compact. We have to prove that if co(h) P =, then R + (co(h) P ) is closed. For that purpose, let us take a sequence λ n > 0,x n co(h) such that the sequence y n := λ n x n p n co(h) P

6 192 J.-P. Aubin NoDEA converges to some y. Since x n ranges over a compact subset, a subsequence (again denoted by) x n converges to some x co(h). Next, let us prove that the sequence λ n is bounded. If not, there would exist a subsequence (again denoted by) λ n going to +. Dividing by λ n, we infer that x n = yn λ n + q n where q n := pn λ n P. Since yn λ n converges to 0, we infer that q n converges to q = x P, because P is closed. Hence x co(h) P, which is impossible. Therefore, the sequence λ n being bounded, there exists a subsequence (again denoted by) λ n converging to some λ 0. Consequently, p n = λ n x n y n converges to some p = λx y P,sothaty belongs to co(h) P,whichis then closed. Remark. By taking P := {0} and assuming that H is a compact convex subset which does not contain {0}, the cone R + H spanned by H is a closed convex cone. Such a subset H is called a sole of the closed convex cone R + H. We now state and prove the Lax Hopf formula for capture basins: Theorem 4.3. (Lax Hopf formula for capture basins) Assume that the target C is contained in the environment K. The capture basin satisfies the inclusion Capt F (K, C) K (C R + co(im(f ))) (10) If the set-valued map F (x) :=G is constant, then K (C R + G) Capt G (co(k),c) (11) Consequently, if K is a closed convex subset, C K is closed and G is a constant set-valued map with a closed convex image G, then the capture basin enjoys the Lax Hopf formula Capt G (K, C) = K (C R + G) (12) Proof. First, let us consider an element x Capt F (K, C). Then x belongs to K and there exist a solution x( ) to the differential inclusion x (t) F (x(t)) and T 0 such that x(t ) C and t [0,T], x(t) K. Hence x(t ) x 1 T F (x(t))dt 1 T Im(F )dt = co(im(f )) (13) T T 0 T 0 This implies that x = x(t ) T x(t ) x C T co(im(f )) C R T + co(im(f )) On the other hand, if the set-valued map F (x) G is constant, let us take x co(k) (C R + G). Hence, there exist g G, T 0andξ C such that x = ξ Tg. The evolution x( ) :t x(t) :=x + tg is a solution to differential equation x (t) =g G starting at x K and satisfying x(t ):=x + Tg = ξ C. It is viable in co(k). Indeed, since x K and since ξ = x + Tg C K, then x(t) :=x + tg = ( ) 1 t T x + t T (x + Tg) co(k). This means that x belongs to the capture basin Capt F (co(k),c). The last statement follows from inclusions (10), p. 192 and (11), p. 192 when K is assumed convex and the constant set-valued map G is closed and convex.

7 Vol. 20 (2013) Lax Hopf formula and Max-Plus properties 193 We infer from these definitions the following consequence of Lax Hopf formula (12), p. 192: Theorem 4.4. (Lax Hopf formula for the sum of two targets) Let P be a closed convex cone contained in K. Assume that H is a compact subset which does not contain 0 and satisfies co(h) P. Then, for any target C K, Capt H (K, C)+P Captco(H P )(K + P, C + P ) Capt H (K + P, C + P ) Consequently, if G := H P is closed and convex, if the closed convex environment K is equal to K + P and if the closed target C is equal to C + P, then Capt H (K, C) = Captco(H P )(K + P, C + P ) = Capt H (K, C)+P (14) Proof. Formula (10), p. 192 of Theorem 4.3, p. 192 implies that { CaptH (K, C)+P K (C R + co(h)) + P (K + P ) (C (R + co(h) P )) By formula (8), p. 191 of Lemma 4.2, p.191, we infer that R + co(h) P = R + co(h P ), and thus, by formula (12), p. 192, that Capt H (K, C)+P (K + P ) (C R + co(h P ))=Captco(H P )(K + P, C) On the other hand, by formula (11), p. 192, we obtain { Capt co(h P )(K + P, C) = (K + P ) (C R + (H P )) = (K + P ) (C + P R + (H)) Capt H (K + P, C + P ) We thus derived inclusions Capt H (K, C)+P Captco(H P )(K + P, C) Capt H (K + P, C + P ) Consequently, if K = K + P and C = C + P are convex, then Capt H (K, C) Capt H (K, C)+P Captco(H P )(K, C) Capt H (K, C) (15) which implies that equation (15), p. 193 holds true. We deduce that the map A Capt H (K, A) is a Max-Plus morphism: Theorem 4.5. (Max-Plus morphism) Let P be a closed convex cone contained in K and H a constant subset. Assume that H P is closed and convex and that K = K +P and C = K +P (where C K). Then the map A Capt H (K, A) satisfies {(i) CaptH (K, A B) = Capt H (K, A) Capt H (K, B) (16) (ii) Capt H (K, C + A) = Capt H (K, C)+A Proof. The first statement is obvious. The second one follows from the Lax Hopf formula. Observe first that if C = C +P, then, for any subset A, C +A = C + A + P because (C + A)+P =(C + P )+A = C + A. Therefore, formula (15), p. 193 of Theorem 4.4, p. 193 implies that Capt H (K, C + A) = Capt H P (K, C + A)

8 194 J.-P. Aubin NoDEA Since H P is closed and convex by assumption, formula (12), p. 192 implies that { CaptH P (K, C + A) = (C + A) R + (H P ) = (C R + (H P )) + A = Capt H P (K, C)+A and thus, by (15), p. 193 again, that Capt H (K, C + A) = Capt H (K, C)+A 5. The viability solution We associate with partial differential equation (4)(i), p. 190 and the Lagrangian l the characteristic system (i) τ (t) = 1 (ii) x (t) = u(t) (iii) y (17) (t) = l(u(t)) where u(t) Dom(l) controlled by celerities u( ). Set H := { 1} ( Graph(l)) R + X R. The characteristic system is the differential inclusion (τ (t),x (t),y (t)) H with constant right hand side. This will allow us to use the Lax Hopf formula for capture basins and its consequences exposed in Sect. 4, p. 191 because we shall define the solution of the trajectory-valued problem as a capture basin. The epigraph Ep(c) R + X R of an extended function c :(t, x) R + X c(t, x) R {+ } is the subset of triples (t, x, y) such that c(t, x) y. Epigraphs of functions are thus subsets such that Ep(c) = Ep(c)+{0} {0} R + = Ep(c)+P where P := {0} {0} R + Recall that an extended function is convex (resp. lower semicontinuous, positively homogeneous) if and only if its epigraph is convex (resp. closed, a cone). We take for environment K := R + X R and for target C := Ep(c). Definition 5.1. (Viability solution) Let us consider the epigraph Ep(c) of the function c. The viability solution (associated with the trajectory condition c) V (or V c when the reference to the trajectory condition is useful) to problem (4), p. 190 is defined by the following formula V (T,x) := inf y (18) (T,x,y) Capt (17) (R + X R,Ep(c)) We recall that the viability solution, when it is differentiable, is a solution to the Hamilton Jacobi equation satisfying the trajectory conditions. Otherwise, when it is not differentiable, but only lower semicontinuous, we can give a meaning to a solution as a solution in the Barron Jensen/Frankoska sense, using for that purpose subdifferential of lower semicontinuous functions defined in non-smooth analysis [5,6]. This is not that important for two reasons: all other properties of viability solutions that are proven in this paper

9 Vol. 20 (2013) Lax Hopf formula and Max-Plus properties 195 Figure 1. Viability solution. The figure illustrates Definition 5.1, p. 194 of the viability solution for a Lagrangian condition t c(t, γ(t)) defined on a nominal t γ(t) in the case when ν = 0. Its epigraph is the capture basin of the epigraph of the Lagrangian condition under system (17), p. 194, which is mentioned in the figure. The point (T,x,y)+( 1, u, l(u)) is represented as well as the element (T t,x t u, y t l(u)) Ep(c) where an evolution reaches the epigraph of the Lagrangian condition are derived directly from the properties of capture basins without using the concept of derivatives, usual or generalized. In particular, the fact that the viability solution is a solution to the Hamilton Jacobi equation derives from the tangential conditions characterization of viable-capture basins provided by the Viability Theorem (see [4] for details which are not duplicated in this study, for instance). For proving that the viability solution is lower semicontinuous, i.e., that the capture basin Capt (17) (R + R R, Ep(c)) characterizing it is closed. This will be easier to prove thanks to the Lax-Hop formula, which also provides a simpler formula of the viability solution. The independence of the Hamiltonian h(p) on(t, x) and the convexity of the Lagrangian imply the Lax Hopf formula for partial differential equations (Fig. 1). Theorem 5.2. (The Lax Hopf formula for the viability solution) Assume that the Hamiltonian h is concave and upper semicontinuous. Then the viability solution is lower semicontinuous and is given by the following formula: V (T,x) = inf inf d [0,T ] ξ C(d) [ c (d, ξ)+(t d)l ( x ξ T d )] (19)

10 196 J.-P. Aubin NoDEA which can also be written in the following form: V (T,x) = inf inf [c (T s, x su)+sl(u)] (20) s [0,T ] u Dom(l) Proof. We apply Lax Hopf formula (see Theorem 4.3, p. 192) in the following context. The state space of variables (t, x, y) isx := R + X R, the subset H is equal to { 1}( Graph(l)) since the characteristic system (17), p. 194 can be written in the form (τ (t),x (t),y (t)) H The subset H is compact, thanks to Proposition 7.3, p.202, but not convex: we cannot use the Viability Theorem. However, setting P := {0} {0} R + (which is a closed convex cone), G := H P = { 1} ( Ep(l)) = { 1} ( Ep(l)) {0} {0} R + is a closed and convex, since the graph of l is compact and the function l is convex. We deduce that G := H P = { 1} ( Ep(l)) is closed and convex We exploit the fact that, setting P := {0} {0} R +, closed epigraphs Ep(V ) of extended functions satisfy Ep(V )=Ep(V)+R + X R +. Formula (15), p. 193 of Theorem 4.4, p. 193 stating that Capt H (K, C) = Capt H P (K, C) = Capt H (K, C)+P implies that the epigraph of the viability epigraphical solution satisfies Ep(V ) := Capt { 1} ( Graph(l)) (Ep(c)) = Capt { 1} ( Ep(l)) (Ep(c))=Capt { 1} ( Graph(l))(Ep(c)) + {0} {0} R + = Ep(c)+R + ({1} Ep(l)) Consequently, for any (T,x,y) Ep(V ), there exist (τ,ξ,η) Ep(c),u Dom(l) ands 0 such that (T,x,y) = (τ,ξ,η) s( 1, u, l(u) π) is such that y = η + sl(u)+π c(τ,ξ)+sl(u) = c(t s, x su)+sl(u) By taking the infimum over y V (T,x),s [0,T]andu Dom(l), we deduce that V (T,x) U(T,x) := inf inf [c (T s, x su)+sl(u)] s [0,T ] u Dom(l) Conversely, for any ε>0, there exist s ε [0,T]andu ε Dom(l), U(T,x) c (T s ε,x s ε u ε )+s ε l(u ε )+ε V (T,x)+ε so that, letting ε converge to 0, we derive that V (T,x)=U(T,x), i.e., formula (20), p By making the change of variables d = T s [0,T], where d can be regarded as a departure date whenever s [0,T] is regarded as a travel

11 Vol. 20 (2013) Lax Hopf formula and Max-Plus properties 197 time, and ξ = x (T d)u, regarded as a departure position, the Lax Hopf formula reads [ ( )] x ξ V (T,x) = inf inf c (d, ξ)+(t d)l (21) d [0,T ] ξ C(d) T d Since the capture basin is closed, we deduce from Definition 5.1, p. 194 that it is equal to the epigraph Ep(V ) := Capt H (Ep(c)) of the viability solution V to Hamilton Jacobi problem (4), p We shall use this Lax Hopf formula for deriving in Sect. 7, p. 201 explicit formulas and estimates of the viability solution for Cauchy and Lagrangian trajectory conditions. The question arises to know precisely the domains Dom(V (t, )) of the viability solution, i.e., the set of states x such that V (t, x) < + : we shall prove that it is couched in terms of the set-valued map C: Theorem 5.3. (Domain of the viability solution) Assume that the Hamiltonian h is a Hamiltonian defined in Definition 7.1, p.201. For any t 0, the domains of the viability solutions V (t, ) associated with the trajectory condition c are equal to Dom(V (t, )) = (C(t s)+su) (22) s [0,t], u Dom(l) If we assume furthermore that t 0, C(t) C(0) + tdom(l) (23) then Dom(V (t, )) = C(0) + tdom(l) (24) Proof. For any x Dom(V (t, )) and any ε>0, there exist s [0,t],u Dom(l) such that c(t s, x su)+sl(u) V (t, x)+ε < + so that x su C(t s), and thus, Dom(V (t, )) C(t s)+su (C(t s)+su) s [0,t], u Dom(l) Conversely, let us take x s [0,t], u Dom(l) (C(t s)+su), and thus, take s [0,t] and u Dom(l) such that x C(t s) +su. Therefore, c(t s, x su) < + is finite and V (t, x) c(t s, x su)+sl(u) < + so that x Dom(V )(t, ). As a particular case, C(0) + tdom(l) (C(t s)+su) = Dom(V (t, )) s [0,t], u Dom(l)

12 198 J.-P. Aubin NoDEA Assume now that the tube C satisfies (23), p Take any x Dom(V (t, )), with which we associate s [0,t],ξ C(t s) andu Dom(l) such that x = ξ + su thanks to (22), p Property 23) implies that ξ C(t s) C(0) + (t s)dom(l) so that there exists v Dom(l) such that ξ C(0) + (t s)v. Hence x C(0) + su +(t s)v. Since Dom(l) is convex and since u and v belong to it and t s 0, then su +(t s)u = tw where w Dom(l). Hence x belongs to C(0) + tdom(l) and thus, Dom(V (t, )) C(0) + tdom(l) This completes the characterization of the domain of the viability solution. Lemma 5.4. (Properties of the domain tube) Let us assume that property (23) holds true. Then, for any u Dom(l) such that (C(0) + tu) \ C(t) and for any x (C(0)+tu) C(t), there exists s [0,t] such that x C(t s)+su. Proof. Let u Dom(l) andt 0 such that there exists x (C(0) + tu) \ C(t). We introduce the subsets E(u) := {s [0,t] such that x su C(t s)} and E(u) + := {s [0,t] such that x su C(t s)}. Since the domain of the function c is assumed to be closed, i.e., that the graph of the tube C is closed, so are the subsets E(u) and E(u) +.By definition, [0,t]=E(u) E(u) +. We observe that 0 belongs to E(u) + since we have assumed that x C(t). On the hand, since x C(0)+tu, we deduce that t E(u). The interval [0,t] being connected and covered be the union of the two non empty closed subsets E(u) and E(u) +, we infer that there exists some s E(u) E(u) +, i.e., such that x su C(t s). This completes the proof of Lemma 5.4, p Inf-convolution We begin by recalling few words about inf-convolutions: Definition 6.1. (Inf-Convolution and decreasing envelopes of functions) The inf-convolution i u i of functions is defined by Ep( i u i ) := i Ep(u i ) One observe immediately that ( i u i )(x) inf x= i xi i u i (x i ) Equality in this formula holds true whenever the functions u i are lower semicontinuous and inf-compact. This implies that the function i (c i ) is lower semicontinuous and that i u i (x) := inf i xi=x i u i (x i )

13 Vol. 20 (2013) Lax Hopf formula and Max-Plus properties 199 We see at once that the inf-convolution of convex functions is convex. The inf-convolution of lower semicontinuous is lower semicontinuous only under adequate conditions. Lemma 6.2. (Lower semicontinuity of inf-convolutions) Setting D := {(p) j=1,...,j } p X J j=1 X the inf-convolution J j=1 v j of lower semicontinuous v j functions is lower semicontinuous whenever assumption J 0 Int D + Dom(vj ) (25) j=1 holds true. Furthermore, if the functions v j are inf-compact, so is their weighted inf-convolution J j=1 v j and there exist J elements x j X j such that J x j = x and v(x) = J v j (x j ) j=1 In the case of two functions, we obtain (u v)(x) = inf(u(y)+v(x y)) y from which the name of the operation is derived (when inf y is replaced by for the usual convolution in analysis). We refer to monographs on convex y analysis (for instance, [3] or[14]) for more details. There are many examples of inf-convolution operators u v u by a function v. 1. Pasch Hausdorff envelopes (or Lipschitz regularization). This example is provided by the function x λ (x) :=λ x, the epigraph of which is a closed convex cone. We observe that a function u is λ-lipschitz if and only if u = λ u. The function λ u defined by j=1 (λ u)(x) := inf(u(y)+λ x y ) y is the called the λ-pasch Hausdorff envelope of u and can be regarded as the λ-lipschitz regularization of u. Indeed, (λ u)(x 1 ):=inf y (u(y)+λ x 1 x 2 + x 2 y ) inf y (u(y)+λ x 1 x 2 +x 2 y )+λ x 1 x 2 =(λ u)(x 2 )+λ x 1 x 2 is Lipschitz with constant λ; 2. Decreasing envelopes Let us consider a closed convex cone P X. It defines an order relation by setting y x if and only if y x + P.

14 200 J.-P. Aubin NoDEA On the other hand, let us associate with the cone P its indicator function ψ P defined by ψ P (x) =0ifx P and ψ P (x) =+ otherwise. Therefore, Ep(ψ P )=P R +. Consequently, Ep(ψ P u) = Ep(u)+P R + We observe that the two following statements are equivalent: u is decreasing in the sense that for any y x+p, then u(y) u(x) The epigraph of the function u satisfies Ep(u)+P R + = Ep(u) In this case, Dom(u) =Dom(u)+P.The function u defined by Ep(u ) := Ep(u)+P R + is called the P -decreasing envelope of the function u (or simply decreasing envelope if there is no ambiguity). The decreasing envelope u of u is larger than or equal to inf u(x p) u (x) anddom(u ) Dom(u)+P p P Equality inf u(x p) = u (x) anddom(u ) = Dom(u)+P (26) p P holds true whenever u is lower semicontinuous and inf-compact. For any family {u i } i I of extended functions, the decreasing envelope of the infimum is the infimum of their decreasing envelopes: (inf i I u i) = inf i I u i (27) Theorem 4.5,p.193 using the Lax Hopf formula, once translated in terms of epigraphs, implies very useful properties on viability solutions. The epigraph of the minimum u := min i I u i being obviously the union Ep(u) = i I Ep(u i) of their epigraphs and the epigraph of their inf-convolution u := i I u i being the sum Ep(u) = i I Ep(u i) of their epigraphs by Definition 6.1, p.198, we obtain Theorem 6.3. (The min inf-convolution morphism) Assume that the Hamiltonian h is a Hamiltonian defined in Definition 7.1, p Let us consider a finite family of trajectory conditions c i,i I. We denote their viability solutions by V ci. 1. The viability solution associated to the infimum min i (c i ) is the infimum of the viability solutions associated with trajectory condition c i : V mini(c i) = min(v ci ) i This allows us to study the contribution of each trajectory condition whenever the trajectory condition is their infimum. 2. The viability solution associated with the inf-convolution i c i of trajectory conditions defined by Ep( i c i ) := i Ep(c i )

15 Vol. 20 (2013) Lax Hopf formula and Max-Plus properties 201 is the inf-convolution of the viability solutions associated with trajectory condition c i : V i(c i) = i (V ci ) Proof. These formulas are consequences of Theorem 4.5, p.193 stating that the capture basin is a Max Plus morphism. 1. We recall that Ep(c mini ) = i I Ep(c i ) Since the union of the epigraphs of functions u i is the epigraph of their infimum, we deduce that if a finite number of trajectory conditions c i are given and denoting their viability solutions by V ci, we infer that V mini(c i) = min i (V ci ). 2. Since the sum of the epigraphs of functions u i is the epigraph of their inf-convolution i u i defined by i u i (x) := inf i xi=x i u i (x i ) we deduce that if a finite number of trajectory conditions c i are given and denoting their viability solutions by V ci, we infer that V i(c i) = i (V ci ). This completes the proof. 7. One-dimensional example We choose X := R and a concave Hamiltonians h : R R. We shall classify them according to characteristic parameters (ν,ν,ω,δ) where at p =0,h(0) = 0 and h (0) := ν 0; at p = ω, h(ω) =0andh (ω) := ν where ν 0 (when ω =+, we assume that it is concave increasing and that lim p + h(p) =δ<+ and set ν = 0); we set δ := max p [0,ω] h(p) and denote by [β,β ] the interval on which h achieves its maximum. Definition 7.1. (Hamiltonian) We associate with scalars ν 0, ν 0, ω> 0andδ > 0 the class of Hamiltonians h : R R associated with those parameters which is any concave function h satisfying h(p) := ν p if p ], 0] h(p) [0,δ] if p [0,ω] (28) h(p) ν (ω p) if p [ω, + [ and { (i) h(0) = 0 and h (0) = ν (ii) h(ω) = 0 and h (ω) = ν (29)

16 202 J.-P. Aubin NoDEA Figure 2. Left This diagram displays the graphs of several Hamiltonians h (α,α,ω,δ). Right This diagram displays the graphs of the associated Lagrangians l (ν,ν,ω,δ) Proposition 7.2. (Examples of Hamiltonians and Lagrangians) Let us consider the trapezoidal Hamiltonian h (ν,ν,ω,δ) defined by ν p if p β h (ν,ν,ω,δ)(p) := δ if p [β,β ] (30) ν (ω p) if p β The associated Lagrangian l (ν,ν,ω,δ) associated with the trapezoidal Hamiltonian is equal to δ ν (ν u) if u [0,ν ] l (ν,ν,ω,δ)(u) = δ ων δ ν u if u [ ν (31), 0] + if u/ [ ν,ν ] It is piecewise affine (affine on [ ν, 0] and [0, +ν ]) and satisfies l(+ν )=0, l(0) = δ and l( ν )=ων. The Hamiltonians and Lagrangians l associated with the parameters ν 0, ν 0,ω >0 δ>0 share common properties (Fig. 2): Proposition 7.3. (Lagrangians) Let h be a concave Hamiltonian associated with the parameters ν > 0, ν 0, ω>0, δ δ. Then the associated convex Lagrangian l satisfies ων 0 l(u) (ν + ν ) (ν u) if u [0,ν ] ων ωu l(u) (ν + ν ) (ν u) if u [ ν, 0] + otherwise Therefore the function u l(u) is decreasing, satisfies 0 = l(ν ) max(0, ωu) l(u) ων (ν + ν ) (ν u) l( ν ) = ων

17 Vol. 20 (2013) Lax Hopf formula and Max-Plus properties 203 and the function u l(u) u the interval [ ν, 0[, and, consequently, satisfies 0 = l(ν ) if ν ω = l( ν ) ν is decreasing on the interval [0,ν [,increasingon l(u) u l(u) u 0 <u ν if ν u<0 Furthermore, l(0) = sup p h(p) =δ is the maximal flux and its subdifferential l(0) its critical interval. Proof. Proposition 7.2, p.202 implies that h (ν,ν,ω,0)(p) h(p) h (ν,ν,ω,δ) (p) We infer from the definition that max(0, ωu) = l (ν,ν,ω,0)(u) l(u) l (ν,ν,ω,δ) (u) = ων (ν + ν ) (ν u) by (31), p. 202 with δ := 0 and δ := δ respectively. Since h(0) = 0 and ν = h (0) (or ν + h(0)), we deduce from the Lemma 3.1, p.190 that for all u, 0,ν = h(0) l(ν ) h(0) l(u), which boils down to 0 = l(ν )=inf u l(u). In the case when ω > 0andν > 0 are strictly positive and finite, conditions h(ω) = 0 and ν = h (ω) (or ν + h(ω)) imply that ν,ω = h(ω) l( ν )and u [ ν,ν ], ω, u h(ω) l(u) = l(u) which can be written ν ω = l( ν )and u [ ν,ν ], ωu l(u) Since 0 = l(ν ) l(u) and since l is convex, we deduce that l is decreasing: take any u [ ν,ν ]andw := αu+(1 α)ν [u, ν ]forα [0, 1]. Therefore l(w) αl(u)+(1 α)l(ν )=αl(u) l(u). Hence, if w u, l(w) l(u). Since l is positive, then, for any u [0,ν [, we observe that 0= l(ν ) l(u) ν u. On the other hand, for any u ] ν, 0], we deduce from inequalities ωu l(u) that, by dividing by u > 0,ω = l( ν ) l(u) ν u since u 0. It is actually easy to check that the function u l(u) u is decreasing on the interval [0,ν [ and increasing on the interval [ ν, 0[. We derive from properties of the Hamiltonians associated with parameters ν 0, ν 0,ω > 0, δ > 0 exposed in Proposition 7.3, p.202 for providing lower and upper estimates of the viability solution to the Hamilton Jacobi problem (4), p. 190, the domains of the viability solutions x V (t, x), which are proven to be equal to C(0) + t[ ν,ν ] whenever we assume that for every t 0, C(t) C(0) + t[ ν,ν ]

18 204 J.-P. Aubin NoDEA couched only in terms of the trajectory condition c, its associated set-valued map C and the four characteristic parameters ν 0, ν 0,ω>0andδ>0. They are consequently valid for any Hamiltonian h associated with those parameters. Proposition 7.4. (Estimates of the viability solution) Proposition 7.5. (Estimates of the Moskowitz function) Let us associate with the function c and parameters ν 0, ν 0,ω > 0 andδ >0 the following subset { } E(t, x) := (s, u) [0,t] [ ν, +ν ] such that x C(t s)+su (32) Hence the viability traffic function to Moskowitz traffic problem (4), p.190 is given by V (t, x) := Let us set c(t, x) := and inf [c(t s, x su)+sl(u)] (33) (s,u) E(t,x) inf (c(t s, x su) + max(0, sωu)) (s,u) E(t,x) ( ων ) c(t, x) := inf c(t s, x su)+s (s,u) E(t,x) (ν + ν ) (ν u) Then, the viability traffic function V satisfies the estimates c(t, x) V (t, x) c(t, x) (34) Proof. Since the celerity flux function l satisfies ων max(0, ωu) l(u) (ν + ν ) (u ν ) thanks to Proposition 7.3, p.202, then inequality c(t, x) V (t, x) c(t, x) is straightforward from Lax Hopf formula. Let us associate with the function c and parameters ν 0, ν 0,ω >0 and δ>0 the following subset { } E(t, x) := (s, u) [0,t] [ ν, +ν ] such that x C(t s)+su (35) Hence the viability solution to Hamilton Jacobi trajectory-valued problem (4), p. 190 is given by V (t, x) := Let us set c(t, x) := and inf [c(t s, x su)+sl(u)] (36) (s,u) E(t,x) inf (c(t s, x su) + max(0, sωu)) (s,u) E(t,x) ( ων ) c(t, x) := inf c(t s, x su)+s (s,u) E(t,x) (ν + ν ) (ν u)

19 Vol. 20 (2013) Lax Hopf formula and Max-Plus properties 205 Then, the viability solution V satisfies the estimates Proof. Since the Lagrangian l satisfies c(t, x) V (t, x) c(t, x) (37) ων max(0, ωu) l(u) (ν + ν ) (u ν ) thanks to Proposition 7.3, p.202, then inequality c(t, x) V (t, x) c(t, x) is straightforward from Lax Hopf formula. Remark. Observe that inf c(t s, x su) c(t, x) (s,u) E(t,x) 1. the lower estimate c is the solution to the Hamilton Jacobi problem (4), p. 190 when the Hamiltonian is defined by: p [0,ω], h (ν,ν,ω,0)(p) = 0 the conjugate function of which is defined by u [ ν, +ν ], l (ν,ν,ω,0)(u) = max(0, ωu) 2. the upper estimate c is the solution to the Hamilton Jacobi problem (4), p. 190 when the Hamiltonian h is the Hamiltonian { ν h (ν,ν,ω,δ) (p) = p if p β ν (ω p) if p β the conjugate function of which is the Lagrangian defined by u [ ν, +ν ], l (ν,ν,ω,δ) (u) = ων (ν + ν ) (ν u) 3. by taking the values 0, ν and ν, we obtain the further estimate c(t, x) min inf c(t s, x sν ), (s,ν ) E(t,x) inf (s,0) E(t,x) inf (s, ν ) E(t,x) ν c(t s, x)+s ων ( (ν + ν ) c(t s, x + sν )+sων ) We provide some examples of trajectory conditions: 1. Cauchy initial conditions The Cauchy initial condition requires that at initial time t = 0, the initial viability solution γ 0 ( ) : x γ 0 (x) R {+ } is given. We extend the Cauchy trajectory condition by setting c(0,x):=γ 0 (x) andc(t, x) =+ whenever t>0. In this case, C(0) = Dom(γ 0 )andc(t) =. Since Ep(c) Ep(V ), we infer that x R, V(0,x) γ 0 (x)

20 206 J.-P. Aubin NoDEA Proposition 7.6. (Cauchy initial conditions) The viability solution to the Hamilton Jacobi problem (4), p.190 with Cauchy condition γ 0 ( ) satisfies actually condition x R, V(0,x) = γ 0 (x) The domains of its viability solutions are equal to t 0, Dom(V (t, )) = Dom(γ 0 )+t[ ν,ν ] The viability solution is equal to V (t, x) = inf u [ ν,ν ] (γ 0 (x tu)+tl(u)) and satisfies the estimates { infu [ ν,ν ] (γ 0 (x ( tu)+tmax(0, ωu)) V (t, ) x) ων inf u [ ν,ν ] γ 0 (x tu)+t (ν +ν ) (ν u) (38) Proof. Since 0 belongs to [ ν,ν ], condition t>0, C(t) = C(0) + t[ ν,ν ] holds true. Then the domains of the associated traffic profiles are equal to t 0, Dom(V )(t, ) = C(0) + t[ ν,ν ] = Dom(γ 0 )+t[ ν,ν ] thanks to Theorem 5.3, p.197. The estimates follow from Proposition 7.5, p Lagrangian conditions Lagrangian trajectory conditions are associated with (a) the trajectory of an increasing nominal evolution t γ(t) ofa given mobile, (b) a Lagrangian condition t c(t, γ(t)) defined on this trajectory. Definition 7.7. (Lagrangian Conditions on Nominal Evolution) Wesay that an evolution t γ(t) isconsistent with the controls ν and ν if it satisfies the speed limit t 0, γ (t) [ ν, +ν [ (39) We observe that a nominal evolution is consistent with the controls ν and ν if and only if t 0, s [0,t], sν γ(t) γ(t s) sν In this case, the associated tube C(t) :={γ(t)} satisfies property (23), so that the domain of the viability solution is equal to t 0, Dom(V (t, )) = [γ(0) tν,γ(0) + tν ] The subset E(t,x) defined by (35), p. 204 is equal to { } E(t, x) := (d, u) [0,t] [ ν, +ν x γ(t d) ] such that u = d

21 Vol. 20 (2013) Lax Hopf formula and Max-Plus properties 207 Proposition 7.8. (Lagrangian conditions) Assume that the nominal evolution is consistent with the controls ν and ν. Then the viability solution to partial differential equation (4), p.190 associated with Lagrangian trajectory condition t c(t, γ(t)) is defined by ( )] x γ(t d) [ c(t d, γ(t d)) + dl V (t, x) = inf d [0,t] d and satisfies the estimates { infd [0,t] (c(t ( d, γ(t d)) + max(0,ω(γ(t d) x))) V (t, x) ) inf d [0,t] c(t d, γ(t d)) + ων (ν +ν ) (sν + γ(t d) x) (40) (41) Proof. To say that (d, u) E(t, x) amounts to saying that E(t,x) is the graph of the map s x γ(t d) d. Since the Lagrangian trajectory condition is defined only of the graph of the nominal evolution, we deduce that c(t d, x du) is finite if and only if (d, u) E(t, x), in which case u := x γ(t d) d, so that general formula (36), p. 204 boils down to [ ( )] x γ(t d) V (t, x) = inf c(t d, γ(t d)) + dl d [0,t] d 3. Combined trajectory conditions We have listed particular trajectory conditions, the Cauchy conditions, as well as the Dirichlet, Eulerian and Lagrangian conditions. We now turn our attention to the combination of Cauchy and several Lagrangian conditions: the trajectory conditions involve a Cauchy condition c 0 such that c 0 (0,x)=γ 0 (x) andc 0 (t, x) =+ whenever t>0 and/orlagrangian conditions c i (t, γ i (t)) satisfying c i (t, x) =+ whenever x γ i (t),i I. We assume that these conditions satisfy { γi (0) Dom(γ i I, 0 ( )) γ i (t) [ ν, +ν (42) ] We introduce the combined trajectory condition c(t, x) := min [γ 0 (x), c i (t, x)] The domain C(t) :=Dom(c(t, )) of its profile is equal to Dom(γ 0 ) if t =0 C(t) = {γ i (t)} if t>0 i I The Min Inf-Convolution Morphism Theorem 6.3, p.200 implies Proposition 7.9. (The Min-Morphism property) Let us denote by V 0 the viability solution associated with the Cauchy condition c 0, V i the viability solution associated with the Lagrangian conditions c i V c the viability solution associated with the combined condition c.

22 208 J.-P. Aubin NoDEA Figure 3. Schematic view of the influence zones of two Lagrangian conditions λ( ) and μ. For a given (T,x), the figure displays the departure set (of initial states from which consistent evolutions arrive at x) and the set of their graphs ranging (triangle zone, convex hull of the deset and of (T,x). It contains a conflict zone of evolutions starting from λ(0) or from μ(0) and arriving at x Then V c (t, x) = min[v 0 (t, x),v i (t, x)] and is defined on Dom(V c (t, )) = Dom(γ 0 )+t[ ν,ν ]. It satisfies trajectory condition V c (t, x) = min [V 0 (t, x),v i (t, x)] c(t, x) := min [γ 0 (x), c i (t, x)] We deduce that if the traffic condition is increasing on time and decreasing in position, so is its associated traffic evolution (Fig. 3). Let us denote by P := R R + the order relation under which a traffic solution is decreasing: (t, x) (s, y) if and only if t s and y x. Therefore, the traffic function V is decreasing along this preorder if and only if Ep(V )=Ep(V )+P R +, i.e., if x 1 x 2 and t 1 t 2, then V (t 2,x 2 ) V (t 1,x 1 ). Its decreasing envelope is defined by V (t, x) = inf V (s, y) s t, y x and its epigraph is equal to Ep(V ):=Ep(V)+P R +. Theorem (Monotonicity property of the traffic function and its optimal viability traffic functions) Assume that the Hamiltonian h is a flux function defined in Definition 7.1, p.201. LetP := R + R + R +, c : t R +

23 Vol. 20 (2013) Lax Hopf formula and Max-Plus properties 209 Figure 4. Left Trajectory of a Lagrangian condition and domain of the viability solution. Right Trajectory of a the increasing envelopes of a Lagrangian condition and domain of the increasing envelope of the viability solution, equal to the viability solution of the increasing envelope of the Lagrangian condition R {+ } be a lower semicontinuous traffic condition, c its decreasing envelope, and V c and V c the viability traffic functions associated with the traffic conditions c and c. Therefore, the viability traffic function V c associated with the P -decreasing envelope c of the traffic condition c is the P -decreasing envelope (V c ) of the viability traffic function associated with the traffic condition c: (V c ) = V c (43) Consequently, the viability traffic function V c associated with the P -decreasing envelope c is decreasing in position and increasing in time. Proof. This a consequence of the second statement of Theorem 6.3, p.200 by taking c 1 c and c 2 := ψ R+ R +, the indicator of R + R +. Then c 1 c 2 = c and V c = V c c 2 =(V c ). The question which arises deals with the construction of decreasing traffic condition in position. Equation (26), p. 200 implies that the domain Dom(c )=:Graph(C ) satisfies Dom(c ) = Dom(c)+R R + When the domain of c is made of a single traffic condition γ( ) :t [τ γ,τ γ[ γ(t), we infer that the domain of c can be regarded as the epigraph of the function γ ( ) Hence, Ep(γ ) := Ep(γ)+R R + γ (t) := inf s t γ(s)

24 210 J.-P. Aubin NoDEA Its domain is the interval [0,τ γ[, it is equal to γ(t) :=inf s t γ(s) on the interval [0,τ γ], to γ(t) :=inf s t γ(s) on the interval [τ γ,τ γ[ and, if τ γ < +, to+ whenever t>τ γ (Fig. 4). In the case when γ is increasing on its domain [τ γ,τ γ[, extended to + elsewhere, we obtain the simple formula t [0,τ γ[, γ (t) := max(γ(τ γ),γ(t)) Acknowledgments This work was partially supported by the Commission of the European Communities under the 7th Framework Programme Marie Curie Initial Training Network (FP7-PEOPLE-2010-ITN), project SADCO, contract number References [1] Aubin, J.-P.: Viability Theory. Birkhäuser, Boston (1991) [2] Aubin, J.-P.: Macroscopic models: shifting from densities to Celerities. Appl. Math. Comput. (2010) [3] Aubin, J.-P.: Optima and Equilibria. Springer, Berlin (1993, 1998) [4] Aubin, J.-P.: Viability solutions to structured Hamilton Jacobi equations under constraints. SIAM J. Control Optim. 49, (2011). doi: / X [5] Aubin, J.-P., Bayen, A., et Saint-Pierre, P.: Dirichlet problems for some Hamilton Jacobi equations with inequality constraints. SIAM J. Control Optim. 47, (2008) [6] Aubin, J.-P., Bayen, A., Saint-Pierre, P.: Viability Theory. New Directions. Springer, Berlin (2011). [7] Aubin, J.-P., Frankowska, H.: Set-Valued Analysis. Birkhäuser, Boston (1990) [8] Barron, E.N., Jensen, R.: Semicontinuous viscosity solutions for Hamilton Jacobi equations with convex Hamiltonians. Comm. Partial Differ. Equ. 15, (1990) [9] Crandall, M.G., Evans, L.C., Lions, P.-L.: Some properties of viscosity solutions of Hamilton Jacobi equations. Trans. Am. Math. Soc. 282(2), (1984) [10] Crandall, M.G., Lions, P.-L.: Viscosity solutions of Hamilton Jacobi equations. Trans. Am. Math. Soc. 277(1), 1 42 (1983) [11] Evans, L.C.: Partial Differential Equations. American Mathematical Society, Providence (1998)

25 Vol. 20 (2013) Lax Hopf formula and Max-Plus properties 211 [12] Frankowska, H.: Lower semicontinuous solutions to Hamilton Jacobi Bellman equations. In: Proceedings of the 30th IEEE Conference on Decision and Control, Brighton, UK (1991) [13] Frankowska, H.: Lower semicontinuous solutions of Hamilton Jacobi Bellman equations. SIAM J. Control Optim. 31, (1993) [14] Rockafellar, R.T., Wets, R.: (1997) Variational Analysis. Springer, New York Jean-Pierre Aubin VIMADES (Viabilité, Marchés, Automatique et Décision) 14, rue Domat Paris France URL: Received: 17 October Accepted: 14 July 2012.