The Aronsson equation, Lyapunov functions and local Lipschitz regularity of the minimum time function
aa r X i v : . [ m a t h . A P ] J u l The Aronsson equation, Lyapunov functions and local Lipschitzregularity of the minimum time function
Pierpaolo Soravia ∗ Dipartimento di MatematicaUniversit`a di Padova, via Trieste 63, 35121 Padova, Italy
Abstract
We define and study C − solutions of the Aronsson equation (AE), a second order quasi linearequation. We show that such super/subsolutions make the Hamiltonian monotone on the trajecto-ries of the closed loop Hamiltonian dynamics. We give a short, general proof that C − solutionsare absolutely minimizing functions. We discuss how C − supersolutions of (AE) become specialLyapunov functions of symmetric control systems, and allow to find continuous feedbacks drivingthe system to a target in finite time, except on a singular manifold. A consequence is a simpleproof that the corresponding minimum time function is locally Lipschitz continuous away fromthe singular manifold, despite classical results show that it should only be H¨older continuous un-less appropriate conditions hold. We provide two examples for H¨ormander and Grushin familiesof vector fields where we construct C − solutions (even classical) explicitly. Primary 49L20; Secondary 35F21, 35D40, 93B05.
In this note we want to describe a possible new, non standard way of using the Aronsson equation, asecond order partial differential equation, to obtain controllability properties of deterministic controlsystems. We investigate a symmetric control system (cid:26) ˙ x t = f ( x t , a t ) ,x = x o ∈ Ω , (1.1)where − f ( a, A ) ⊂ f ( x, A ) , A is a nonempty and compact subset of a metric space. We define theHamiltonian H ( x, p ) = max a ∈ A {− f ( x, a ) · p } , which is therefore nonnegative and positively one homogeneous in the adjoint variable, and we wantto drive the system to a target, temporarily we say the origin. We are interested in the relationship of(1.1) with the Aronsson equation (AE) −∇ ( H ( x, ∇ U ( x ))) · H p ( x, ∇ U ( x )) = 0 , ∗ email: [email protected]. U of (AE) and we consider a trajectory x t of the Hamiltonian dynamics ˙ x t = − H p ( x t , ∇ U ( x t )) , which is a closed loop dynamics for the original control system, we find out that (AE) can be rewrittenas ddt H ( x t , ∇ U ( x t )) = 0 . Therefore H ( x t , ∇ U ( x t )) is constant. This is a very desirable propery on the control system since itallows to use U as a control Lyapunov function, despite the presence of a possibly nonempty singularset H = { x : H ( x, ∇ U ( x )) = 0 } , (1.2)which possibly contains the origin. Indeed if x o is outside the singular set and U has a unique globalminimum at the origin, then the trajectory of the Hamiltonian dynamics will reach the origin in finitetime.In general however, several steps of this path break down. From one side, (AE) does not have C classical solutions in general. Even in the case where f = a , A = B (0) ⊂ R n is the closed unit ball, H ( p ) = | p | and (AE) becomes the well known infinity Laplace equation − ∆ U ( x ) ∇ U ( x ) · ∇ U ( x ) = 0 , solutions are not classical, although known regularity results show that they are C ,α . Thereforesolutions of (AE) have to be meant in some weak sense, as viscosity solutions. For generic viscositysolutions, we can find counterexamples to the fact that the Hamiltonian is constant along trajectoriesof the Hamiltonian dynamics, as we show later. For an introduction to the theory of viscosity solutionsin optimal control, we refer the reader to the book by Bardi, Capuzzo-Dolcetta [5].In this paper we will first characterize when, for a given super or subsolution of (AE) the Hamil-tonian is monotone on the trajectories of the Hamiltonian dynamics (e.g. satisfies the monotonicityproperty ). To this end we introduce the notion of C − super/subsolution and prove for them that theysatisfy the monitonicity property of the Hamiltonian. We emphasize the fact that not all viscositysolution that are C functions, are C − solutions according to our definition. Moreover, as a sideresult, we also show that our C − solutions are absolutely minimizing functions, i.e. local minimiz-ers of the functional that computes the L ∞ norm of the Hamiltonian. It is a well know equivalentproperty to being a viscosity solution of (AE) at least when H is coercive or possibly in some CarnotCaratheodory spaces, but this fact is not completely understood in general. Therefore C − solutionappears to be an appropriate notion.We then prove that if (AE) admits a C − supersolution U having a unique minimum at the origin,then our control system can be driven to the origin in finite time with a continuous feedback, startingat every initial point outside the singular set H . If moreover U satisfies appropriate decay in a neigh-borhood of the origin only at points where the Hamiltonian H stays away from zero, then we showthat the corresponding minimun time function is locally Lipschitz continuous outside the singular set,despite the fact that even if the origin is small time locally attainable, then the minimum time functioncan only be proved to be H ¨older continuous in its domain, in general, under appropriate conditions.Thus the loss of regularity of the minimum time function is only concentrated at points in the singularset. Finally for two explicit well known examples, where the system has an H ¨ormander type, or aGrushin family of vector fields, we exibit two explicit not yet known classical solutions of (AE), their2auge functions, providing examples of smooth absolute minimizers for such systems and the proofthat their minimum time function is locally Lipschitz continuous outside the singular set. We remarkthe fact that neither in the general statement nor in the examples, the family of vector fields is eversupposed to span the whole space at the origin, therefore the classical sufficient attainability condi-tion ensuring that the minimum time function is locally Lipschitz continuous will not be satisfied ingeneral. Indeed in the explicit examples that we illustrate in Section 4, the minimum time function isknown to be locally only / − H ¨older continuous in its domain.Small time local attainability and regularity of the minimum time function is an important subjectin optimal control. Classical results by Petrov [22] show sufficient conditions for attainability at asingle point by requiring that the convex hull of the vector fields at the point contains the origin inits interior. Such result was later improved by Liverovskii [17] augmenting the vector fields withthe family of their Lie brackets, see also the paper by author [23]. More recently such results hadseveral extensions in the work by Krastanov and Quincampoix [16] and Marigonda, Rigo and Le[18, 19, 20]. Our regularity results rather go in the direction of those contained in two recent papersby Albano, Cannarsa and Scarinci [3, 4], where they show, by completely different methods, thatif a family of smooth vector fields satisfies the H ¨ormander condition, then the set where the localLipschitz continuity of the minimum time function fails is the union of singular trajectories, and thatit is analytic except on a subset of null measure. Our approach is instead more direct and comes asa consequence of constructing Lyapunov functions as C − supersolutions of the Aronsson equation.We finally mention the paper by Motta and Rampazzo [21] where the authors study higher orderhamiltonians obtained by adding iterated Lie brackets as additional vector fields, in order to proveglobal asymptotic controllability to a target. While we do not study asymptotic controllability in thispaper, their idea of constructing a higher order Hamiltonian may be seen complementary to ours,using instead the equation (AE).Equation (AE) was introduced by Aronsson [1], as the Euler Lagrange equation for absolute min-imizers, i.e. local minima of L ∞ functionals, typically the L ∞ norm of the gradient. There has beena lot of work in more recent years to develop that theory using viscosity solutions by authors likeJensen [14], Barron-Jensen-Wang [7], Juutinen [15], Crandall [12]. For the main results on the infin-ity Laplace equation, we refer the reader to the paper [2] and the references therein. For results forequation (AE) especially in the x dependent case, we also refer to the paper by the author [28] and thereferences therein, see also [27, 26]. In particular we mention that equation (AE) has been studied inCarnot groups by Bieske-Capogna [9], by Bieske [8] in the Grushin space, and by Wang [30] in thecase of C and homogeneous Hamiltonians with a Carnot Caratheodory structure.The structure of the paper is as follows. In Section 2 we introduce the problem and give a moti-vating example. In Section 3 we introduce C − solutions of (AE) and show for them some importantproperties: monotonicity of the Hamiltonian on the hamiltonian dynamics, an equivalent definitionand the fact that they are absolutely minimizing functions. In Section 4, we use C − solutions of (AE)as Lyapunov functions for nonlinear control systems and obtain local Lipschitz regularity of the mini-mum time function away from the singular set. In Section 5 we provide two new examples of explicitclassical solutions of (AE) in two important cases of nonlinear control systems where the results ofSection 4 apply. As we mentioned in the introduction, throughout the paper we consider the controlled dynamicalsystem (1.1) where Ω ⊂ R n is open, A is a nonempty, compact subset of some metric space, a · ∈ ∞ ((0 , + ∞ ); A ) and f : Ω × A → R n is a continuous function, continuously differentiable anduniformly Lipschitz continuous in the first group of variables, i.e. | f ( x , a ) − f ( x , a ) | ≤ L | x − x | for all x , x ∈ Ω , a ∈ A. We suppose moreover that f ( x, A ) is convex for every x ∈ Ω and that the system is symmetric, i.e. − f ( x, A ) ⊂ f ( x, A ) for all x ∈ R n and define the Hamiltonian H ( x, p ) = max a ∈ A {− f ( x, a ) · p } ∈ C (Ω × R n ) , (2.1)so that H ≥ and H ( x, − p ) = H ( x, p ) by symmetry. Notice that H is at least locally Lipschitzcontinuous, and H ( x, · ) is positively homogeneous of degree one by compactness of A . We will alsoassume that H is continuously differentiable on { ( x, p ) : ∈ Ω × R n : H ( x, p ) > } .The case we are mostly interested in the following sections is when f ( x, a ) = σ ( x ) a, σ : R n → M n × m (2.2)where M n × m is set of n × m matrices and A = B (0) ⊂ R m is the closed unit ball. In this case H ( x, p ) = | pσ ( x ) | .Given a smooth function U ∈ C (Ω) and x o ∈ Ω \H , where H is the singular set as in (1.2), weconsider the hamiltonian dynamics (cid:26) ˙ x t = − H p ( x t , ∇ U ( x t )) ,x = x o ∈ Ω , (2.3)where H p indicates the gradient of the Hamiltonian H = H ( x, p ) with respect to the group of adjoint variables p . Remark 2.1.
When the Hamiltonian H ( x, ∇ U ( x )) is differentiable, notice that for a x ∈ A such that − f ( x, a x ) · ∇ U ( x ) = H ( x, ∇ U ( x )) we have that − H p ( x, ∇ U ( x )) = f ( x, a x ) . Therefore trajectories of (2.3) are indeed trajectories of the system (1.1) and moreover (2.3) is a closedloop system of (1.1) with feedback a x . If in particular f ( x, a ) is as in (2.2), then, for | pσ ( x ) | 6 = 0 , H ( x, p ) = | pσ ( x ) | , H p ( x, p ) = σ ( x ) t σ ( x ) t pH ( x, p ) , a x = − t σ ( x ) ∇ U ( x ) H ( x, ∇ U ( x )) ∈ B (0) . Therefore in this case the feedback control is at least continuous on Ω \H and the closed loop systemalways has a well defined local solution starting out on that set.We want to discuss when H ( x t , ∇ U ( x t )) is monotone on a trajectory x t of (2.3). If we cancompute derivatives, then we need to discuss the sign of ddt H ( x t , ∇ U ( x t )) = ∇ ( H ( x t , ∇ U ( x t ))) · ˙ x t = −∇ ( H ( x t , ∇ U ( x t ))) · H p ( x t , ∇ U ( x t )) . Therefore a sufficient condition is that U ∈ C (Ω \H ) is a super or subsolution of the following pde − ∇ ( H ( x, ∇ U ( x ))) · H p ( x, ∇ U ( x )) = 0 , x ∈ Ω \H , (2.4)4hich is named Aronsson equation in the literature. Notice that H ( x t , ∇ U ( x t )) is actually constant if U is a classical solution of (2.4). The above computation is correct only under the supposed regularityon U and unfortunately if such regularity is not satisfied and we interpret super/subsolutions of (2.4)as viscosity solutions this is no longer true in general, as the following example shows. Notice thatif H is not differentiable at a point ( x o , ∇ U ( x o )) where H ( x o , ∇ U ( x o )) = 0 , then H p ( x o , ∇ U ( x o )) is multivalued, precisely the closed convex subgradient of the Lipschitz function H ( x o , · ) computedat ∇ U ( x o ) and contains the origin by the symmetry of the system. Therefore the dynamics (2.3) hasat least the constant solution also in this case. In some statements below it will be sometimes moreconvenient to look at (AE) for H in order to gain regularity at points where H vanishes. Example 2.2.
In the plane, suppose that H ( x, y, p x , p y ) = ( | p x | + | p y | ) / hence it is smooth andindependent of the state variables. In this case (AE) becomes the well known infinity Laplace equation − ∆ ∞ U ( x ) = − D U ( x ) ∇ U ( x ) · ∇ U ( x ) = 0 . It is easy to check that a viscosity solution of the equation is u ( x, y ) = | x | / − | y | / . The function u ∈ C , / ( R ) \ C . Among solutions of the Hamiltonian dynamics ( ˙ x t , ˙ y t ) = −∇ U ( x t , y t ) , we canfind the following two trajectories ( x (1) t , y (1) t ) = (cid:18) − t (cid:19) / , ! , ( x (2) t , y (2) t ) = , (cid:18) t (cid:19) / ! , defined in a neighborhood of t = 0 . Clearly the Hamiltonian along the two trajectories is H ( ∇ U ( x (1) t , y (1) t )) = 2 √ r − t, H ( ∇ U ( x (2) t , y (2) t )) = 2 √ r t, it is strictly decreasing in the first case, strictly increasing in the second but it is never constant.Therefore the remark that we made at the beginning fails in this example. In the next section we aregoing to understand the reason. Throughout this section, we consider a Hamiltonian not necessarily with the structure as in (2.1) butsatisfying the following: H : Ω × R n → R is continuous and H ( x, − p ) = H ( x, p ) ,H p ( x, p ) exists and is continuous for all ( x, p ) ∈ Ω × R n if H ( x, p ) > . (H1)We will also refer to the following property: H ( x, · ) is positively r > homogeneous, for all x ∈ Ω . (H2)Given U ∈ C (Ω) , the monotonicity of the Hamiltonian along trajectories of (2.3) is the object of thissection. It is a consequence of the following known general result. Proposition 3.1.
Let Ω ⊂ R n be an open set and F : Ω → R n be a continuous vector field. Thefollowing are equivalent:(i) V : Ω → R is a continuous viscosity solution of − F ( x ) · ∇ V ( x ) ≤ in Ω .5ii) The system ( V, F ) is forward weakly increasing, i.e. for every x o ∈ Ω , there is a solution ofthe differential equation ˙ x t = F ( x t ) , for t ∈ [0 , ε ) , x = x o such that V ( x s ) ≤ V ( x t ) for ≤ s ≤ t .Moreover the following are also equivalent(iii) V : Ω → R is a continuous viscosity solution of F ( x ) · ∇ V ( x ) ≥ in Ω .(iv) The system ( V, F ) is backward weakly increasing, i.e. for every x o ∈ Ω , there is a solution ofthe differential equation ˙ x t = F ( x t ) , for t ∈ ( − ε, , x = x o such that V ( x s ) ≤ V ( x t ) for s ≤ t ≤ . Corollary 3.2.
Let Ω ⊂ R n be an open set and F : Ω → R n be a continuous vector field. Thefollowing are equivalent:(i) V : Ω → R is a continuous viscosity solution of − F ( x ) · ∇ V ( x ) ≤ and of F ( x ) · ∇ V ( x ) ≥ in Ω .(ii) The system ( V, F ) is weakly increasing, i.e. for every x o ∈ Ω , there is a solution of thedifferential equation ˙ x t = F ( x t ) , for t ∈ ( − ε, ε ) , x = x o such that V ( x s ) ≤ V ( x t ) for s ≤ t . Remark 3.3.
The proof of the previous statement can be found in [10], see also [11]. When F ∈ C another proof can be found in Proposition 5.18 of [5] or can be deduced from the optimality principlesin optimal control proved in [24], when F is locally Lipschitz continuous. In the case when F islocally Lipschitz, the two differential inequalities in (i) of Corollary 3.2 turn out to be equivalent andof course there is also uniqueness of the trajectory of the dynamical system ˙ x = F ( x ) , x (0) = x o .When (ii) in the Corollary is satisfied by all trajectories of the dynamical system then the system issaid to be strongly monotone. This occurs in particular if there is at most one trajectory, as when F islocally Lipschitz continuous. More general sufficient conditions for strong monotonicity can be foundin [11], see also [13].In view of the above result, we introduce the following definition. Definition 3.4.
Let Ω ⊂ R n be open and let H : Ω × R n → R satisfying (H1). We say that a function U ∈ C (Ω) is a C − supersolution (resp. subsolution) of the Aronsson equation (2.4) in Ω , if setting V ( x ) = H ( x, ∇ U ( x )) and F ( x ) = − H p ( x, ∇ U ( x )) we have that V is a viscosity subsolution (resp.supersolution) of − F ( x ) · ∇ V ( x ) = 0 and a supersolution (resp. a subsolution) of F ( x ) · ∇ V ( x ) = 0 .It is worth pointing out explicitely the consequence we have reached by Proposition 3.1. Corollary 3.5.
Let U ∈ C (Ω) be a C − supersolution (resp, subsolution) of (2.4). For x o ∈ Ω \H ,then there is a trajectory x t of the Hamiltonian dynamics (2.3) such that H ( x t , ∇ U ( x t )) is nonde-creasing (resp. nonincreasing). Remark 3.6. • Notice that if U is a C − solution of (2.4) and the Hamiltonian dynamics (2.3) iseither strongly decreasing and strongly increasing, as for instance if it has a unique solution fora given initial condition, then for all trajectories x t of (2.3), H ( x t , ∇ U ( x t )) is constant. • In order to comment back to Example 2.2, notice that while U ( x, y ) = | x | / − | y | / is a C function, nevertheless, as easily checked, V ( x, y ) = H ( ∇ U ( x, y )) = 16( | x | / + | y | / ) / isonly a viscosity subsolution but not a supersolution of −∇ V ( x ) · ( − H p ( ∇ U ( x ))) = 0 , ∇ V ( x ) · ( − H p ( ∇ U ( x ))) = 0 . Then it turns out that theHamiltonian is weakly increasing on the trajectories of the Hamiltonian dynamics. Indeedthere is another trajectory of the Hamiltonian dynamics such that ( x (3) (0) , y (3) (0)) = (1 ,
0) =( x (1) (0) , y (1) (0)) , namely ( x (3) ( t ) , y (3) ( t )) = (cid:18) − t (cid:19) / , (cid:18) t (cid:19) / ! along which the Hamiltonian is actually constant, until the trajectory is well defined. • It is clear by Example 2.2 that while classical C solutions of (2.4) are C − solutions, contin-uous or even C viscosity solutions in general are not. The definition of C − solution that weintroduced is meant to preserve the monotonicity property of the Hamiltonian on the trajectoriesof the Hamiltonian dynamics. • Observe that if U is a C − solution, then − U is a C − solution as well, since the Hamiltonianis unchanged and the vector field in the Hamiltonian dynamics becomes the opposite.It may look unpleasant that Definition 3.4 of solution of (2.4) refers to a property that is notformulated directly for the function U . Therefore in the next statement we will reformulate the abovedefinition. The property (ED) below will give an equivalent definition of a C − solution. Proposition 3.7.
Let U ∈ C (Ω) and H satisfying (H1), (H2). The following two statements areequivalent:(ED) for all x o ∈ Ω \H , there is a trajectory x t of the Hamiltonian dynamics (2.3), such that if ϕ ∈ C ([0 , ε )) ∪ C ([( − ε, is a test function and U ( x t ) − ϕ ( t ) has a minimum (respectivelymaximum) at and ˙ ϕ (0) = ddt U ( x t ) t =0 , then we have that − ¨ ϕ (0) ≥ resp. ≤ . • U is a C − supersolution (resp. subsolution) of (2.4).In particular, if H is C at { ( x, p ) : H ( x, p ) = 0 } , a C − supersolution (resp. subsolution) is aviscosity supersolution (resp. subsolution) of (2.4). Remark 3.8.
In the statement of (ED), when the hamiltonian vector field F ( x ) = − H p ( x, ∇ U ( x )) is locally Lipschitz continuous, we may restrict the test functions to ϕ ∈ C ( − ε, ε ) . Proof.
We only prove the statement for supersolutions, the other case being similar. Let U ∈ C (Ω) .Suppose first that (ED) holds true. Let V ( x ) = H ( x, ∇ U ( x )) and Φ ∈ C (Ω) such that V − Φ has a maximum at x o , V ( x o ) = Φ( x o ) . Therefore if x t is a solution of the hamiltonian dynamics (2.3)that satisfies (ED), we have that, by homogeneity of H ( x, · ) and for F ( x ) = − H p ( x, ∇ U ( x )) , r Φ( x t ) ≥ rV ( x t ) = rH ( x t , ∇ U ( x t )) = −∇ U ( x t ) · F ( x t ) = − ddt U ( x t ) . Thus integrating for small t > we get ϕ ( t ) := U ( x o ) − r Z t Φ( x s ) ds ≤ U ( x t ) , (3.1)7nd thus U ( x t ) − ϕ ( t ) has a minimum at t = 0 on [0 , ε ) for ε small and ˙ ϕ (0) = − r Φ( x t ) = ddt U ( x t ) | t =0 . If instead V − Φ had a minimum at x o , then integrating on ( t, for t < small enough,we would still obtain the same as in (3.1). By (ED), from (3.1) we get in both cases ≥ ¨ ϕ (0) = − r ddt Φ( x t ) | t =0 = − r ∇ Φ( x o ) · F ( x o ) , where F ( x ) = − H p ( x, ∇ U ( x )) . Therefore we conclude that V is a viscosity subsolution of −∇ V · F ≤ (or a supersolution of ∇ V · F ≥ when V − φ has a minimum st x o ). Finally by definition, U is a C − supersolution of (2.4).Suppose now that U is a C − supersolution of (2.4). Then by Proposition 3.1, for all x o ∈ Ω \H ,we can find a trajectory x t of the dynamics (2.3) such that rV ( x t ) = − ddt U ( x t ) is nondecreasing.Therefore U ( x t ) is a concave function of t . Let ϕ ∈ C (( − ε, ∪ C ([0 , ε )) be such that U ( x t ) − ϕ ( t ) has a minimum at t = 0 , U ( x o ) = ϕ (0) and ddt U ( x t ) | t =0 = ˙ ϕ (0) . If we had ¨ ϕ (0) > then ϕ wouldbe strictly convex in its domain. Therefore for t = 0 small enough, and in the domain of ϕ , U ( x t ) ≥ ϕ ( t ) > ϕ (0) + ˙ ϕ (0) t = U ( x o ) + ddt U ( x t ) | t =0 t ≥ U ( x t ) , by concavity of U ( x t ) . This is a contradiction.We prove the last statement on the fact that a C − solution is a viscosity solution. Therefore for a C − supersolution U of (2.4) let now Φ ∈ C (Ω) be such that U − Φ has a minimum at x o . By (ED),for a suitable solution x t of (2.3) we have that U ( x t ) − ϕ ( t ) has a minimum at t = 0 if ϕ ( t ) = Φ( x t ) ,in particular ˙ ϕ (0) = ddt U ( x t ) t =0 . By (ED) and homogeneity of H ( x, · ) , ≤ − ¨ ϕ (0) = ddt ∇ Φ( x t ) · H p ( x t , Φ( x t )) | t =0 = r ddt H ( x t , ∇ Φ( x t )) | t =0 = − r ∇ ( H ( x o , ∇ Φ( x o ))) · H p ( x o , ∇ Φ( x o )) . Therefore U is a viscosity supersolution of (2.4). The case of subsolutions is similar and we skipit. We end this section by proving another important property of C − solutions of (2.4) that in theliterature was the main motivation to the study of (AE). Theorem 3.9.
Let Ω ⊂ R n open and bounded, H satisfying (H1), and having the structure (2.1). Let U ∈ C (Ω) ∩ C (Ω) be a C − solution of (2.4). For any function W ∈ C (Ω) such that : (cid:26) H ( x, ∇ W ( x )) ≤ k ∈ R , x ∈ Ω ,W ( x ) = U ( x ) , x ∈ ∂ Ω (3.2)in the viscosity sense, then H ( x, ∇ U ( x )) ≤ k in Ω . Remark 3.10.
When D ⊂ R n is an open set and the property of a function U ∈ C ( D ) in Theorem(3.9) holds for all open subsets Ω ⊂ D then we say that U is an Absolutely minimizing function in D for the Hamiltonian H . This means that U is a local minimizer of k H ( · , ∇ U ( · )) k L ∞ . It is well knownthat for the infinity Laplace equation, where we minimize the Lipschitz constant of U , it is equivalentto be a viscosity solution and an absolutely minimizing function. Such equivalence is also knownfor coercive Hamiltonians and for the norm of the horizontal gradient in some Carnot Caratheodoryspaces. For more general Hamiltonians this equivalence is not known. Here we prove one implicationat least for C − solutions of (2.4). 8 roof. Let
U, W be as in the statement and suppose for convenience that H ( x, · ) is positively 1-homogeneous. We define V ( x ) = H ( x, ∇ U ( x )) ≥ and look at solutions x t of the Hamiltoniandynamics (2.3). If V ( x o ) = 0 , then clearly V ( x o ) ≤ k and we have nothing left to show. If otherwise V ( x o ) > since U is a C − solution of (2.4), we already know that we can construct a solution of(2.3) starting out at x o ∈ Ω such that V ( x t ) is nondecreasing for t ≥ and nonincreasing for t ≤ (by a concatenation of two trajectories of (2.3) with monotone Hamiltonian). Since Ω is bounded,then the curve x t will not stay indefinitely in Ω because as we already observed U ( x t ) − U ( x o ) ≤ − Z t V ( x s ) ds ≤ − tV ( x o ) , for t ≥ , and U ( x t ) − U ( x o ) ≥ − tV ( x o ) , for t ≤ . Hence x t will hit ∂ Ω forward and backward in finite time. Let t < < t be such that x t , x t ∈ ∂ Ω and x t ∈ Ω for t ∈ ( t , t ) . Therefore U ( x t ) + t V ( x o ) ≤ U ( x o ) ≤ U ( x t ) + t V ( x o ) (3.3)and then W ( x t ) − W ( x t ) = U ( x t ) − U ( x t ) ≥ ( t − t ) V ( x o ) . (3.4)Now we use the differential inequality (3.2) in the viscosity sense and the lower optimality principlein control theory as in [24] for subsolutions of the Hamilton-Jacobi equation. Therefore since x t is atrajectory of the control system (1.1) we have that for all ε > and t + ε < t < t , as x s ∈ Ω for s ∈ [ t + ε, t ] , W ( x t + ε ) ≤ k ( t − t − ε ) + W ( x t ) . By letting t → t − and ε → we conclude, by continuity of W at the boundary of Ω and (3.4), V ( x o )( t − t ) ≤ W ( x t ) − W ( x t ) ≤ k ( t − t ) which is what we want. Remark 3.11.
Notice that in (3.3) equalities hold if V is constant on a given trajectory of (2.3) andwe obtain that U ( x o ) − U ( x t ) t = U ( x o ) − U ( x t ) t and then U ( x o ) = t t − t U ( x t ) − t t − t U ( x t ) , which is an implicit representation formula for U through its boundary values, since the points x t , x t depend on the Hamiltonian dynamics (2.3) and U itself. In this section, we go back to the stucture (2.1) for H and want to discuss the classical idea of controlLyapunov function. Let T ⊂ R n be a closed target set, we want to find U : R n → [0 , + ∞ ) at leastlower semicontinuous and such that: U ( x ) = 0 if and only if x ∈ T and such that for all x ∈ R n \T a · ∈ L ∞ (0 , + ∞ ) and t x ≤ + ∞ such that the corresponding trajectory of (1.1)satisfies: U ( x t ) is nonincreasing and U ( x t ) → , as t → t x . Classical necessary and sufficient conditions lead to look for strict supersolutions of the HamiltonJacobi equation, namely to find U such that H ( x, ∇ U ( x )) ≥ l ( x ) , (4.1)with l : R n → [0 , + ∞ ) continuous and such that l ( x ) = 0 if and only if x ∈ T . The case T = { } isalready quite interesting for the theory.Here we will apply the results of the previous section and plan consider Lyapunov functionsbuilt as follows. We analyse the existence of U ∈ C (Ω \ ( T ∩ H )) ∩ C (Ω \T ) such that U is a C − supersolution of (AE), i.e. satisfies − ∇ ( H ( x, ∇ U ( x )) · H p ( x, ∇ U ( x ))) ≥ x ∈ Ω \ ( T ∩ H ) . (4.2) Remark 4.1.
To study (4.2) in the case when H is as in (2.1) and f as in (2.2), it is sometimes moreconvenient to write it for the Hamiltonian squared H ( x, ∇ U ( x )) = |∇ U ( x ) σ ( x ) | . Thus −∇ ( H ( x, ∇ U ( x )) · ( H ) p ( x, ∇ U ( x )) = − t D ( ∇ U σ ( x )) t ( ∇ U ( x ) σ ( x )) · (cid:0) σ ( x ) t ( ∇ U ( x ) σ ( x )) (cid:1) = − S ∗ t ( ∇ U ( x ) σ ( x )) · t ( ∇ U ( x ) σ ( x )) , where we indicated S = t σ ( x ) t D ( ∇ U σ ( x )) = t σ ( x ) D U ( x ) σ ( x ) + ( Dσ j σ i ( x ) · ∇ U ( x )) i,j =1 ,...,m ,σ j , j = 1 , . . . , k are the columns of σ , and S ∗ = ( S + t S ) / . Therefore a special sufficient conditionfor U to satisfy (4.2) is that S ∗ is negative semidefinite, which means that U is σ − concave withrespect to the family of vector fields σ j , in the sense of Bardi-Dragoni [6]. We recall that the matrix S also appears in [29] to study second order controllability conditions for symmetric control systems.Define the minimum time function for system (1.1) as T ( x ) = inf a ∈ L ∞ (0 , + ∞ ) t x ( a ) , where t x ( a ) = inf { t ≥ x t ∈ T , x t solution of (1 . } ≤ + ∞ . We prove the following result,recall that H = { x : H ( x, ∇ U ( x )) = 0 } is the singular set. Proposition 4.2.
Let Ω ⊂ R n be open and T ⊂ Ω a closed target. Let H have the structure (2.1).Assume that U ∈ C (Ω \T ) ∩ C (Ω \ ( T ∩H )) is nonnegative and a C − solution of (4.2) in Ω \ ( T ∩H ) and that U ( x ) = 0 for x ∈ T , U ( x ) = M for x ∈ ∂ Ω and U ( x ) ∈ (0 , M ) for x ∈ Ω \T and some M > . For any x o ∈ Ω \ ( T ∪ H ) there exists a solution of the closed loop system (2.3) such that(i) H ( x t , ∇ U ( x t )) is a nondecreasing function of t ;(ii) U ( x t ) is a strictly decreasing function of t (iii) The trajectory ( x t ) t ≥ reaches the target in finite time and the minimum time function for sys-tem (1.1) satisfies the estimate T ( x o ) ≤ U ( x o ) H ( x o , ∇ U ( x o )) . (4.3)10 roof. The thesis (i) follows from the results of the previous section since U is a supersolution of(AE). Let x o be a point where H ( x o , ∇ U ( x o )) > . By homogeneity of the Hamiltonian we get, for t ≥ < H ( x o , ∇ U ( x o )) ≤ H ( x t , ∇ U ( x t )) = ∇ U ( x t ) · H p ( x t , ∇ U ( x t )) = − ddt U ( x t ) and (ii) follows. Integrating now the last inequality we obtain ≤ U ( x t ) ≤ U ( x o ) − H ( x o , ∇ U ( x o )) t and thus the solution of (2.3) reaches the target before time ¯ t = U ( x o ) H ( x o , ∇ U ( x o )) . (4.4)Therefore (4.3) follows by definition.The estimate (4.3) can be used to obtain local regularity of the minimum time function. Theproof of regularity now follows a more standard path although under weaker assumptions than usualliterature and will allow us to obtain a new regularity result. We emphasize that nothing in the nextstatement is assumed on the structure of the vectogram f ( x, A ) when x ∈ T . In particular the targetneed not be even small time locally attainable. Theorem 4.3.
Let Ω ⊂ R n be open and T ⊂ Ω a closed target. Assume that U ∈ C (Ω \T ) ∩ C (Ω \ ( T ∩ H )) is nonnegative and C − solution of (4.2) in Ω \ ( T ∩ H ) and that U ( x ) = 0 for x ∈ T , U ( x ) = M for x ∈ ∂ Ω and U ( x ) ∈ (0 , M ) for x ∈ Ω \T and some M > . Let d ( x ) = dist ( x, T ) be the distance function from the target. Suppose that U satisfies the following: for all ε > there are δ, c > such that U ( x ) ≤ c d ( x ) , if H ( x, ∇ U ( x )) ≥ ε, d ( x ) < δ. (4.5)Then the minimum time function T for system (1.1) to reach the target is finite and locally Lipschitzcontinuous in Ω \ ( T ∪ H ) . Proof.
Let x o ∈ Ω , x o / ∈ ( T ∪ H ) and r, ε > be such that H ( x, ∇ U ( x )) ≥ ε , for all x ∈ B r ( x o ) .The parameter r will be small enough to be decided later. We apply the assumption (4.5) and find δ, c > correspondingly. The fact that T is finite in B r ( x o ) , for r sufficiently small, follows fromProposition 4.2.Take x , x ∈ B r ( x o ) and suppose that x t , x t are the trajectories solutions of (1.1) correspondingto the initial conditions x = x , x respectively. To fix the ideas we may suppose that T ( x ) ≤ T ( x ) < + ∞ and for any ρ ∈ (0 , we choose a control a ρ and time t = t x ( a ε ) ≤ T ( x ) + ρ suchthat d ( x t ) = 0 . Note that by (4.3), t ≤ U ( x ) ε + ρ ≤ M ε , for all x ∈ B r ( x o ) . Moreover by theGronwall inequality for system (1.1) and since d ( x t ) = 0 , d ( x t ) ≤ | x t − x t | ≤ | x − x | e Lt ≤ | x − x | e LM ε and the right hand side is smaller than δ if r is small enough. Now we can estimate, by the dynamicprogramming principle and by (4.3), (4.5), ≤ T ( x ) − T ( x ) ≤ ( t + T ( x t )) − t + ρ ≤ U ( x t ) ε + ρ ≤ cε d ( x t ) + ρ ≤ ce LM ε ε | x − x | + ρ. As ρ → , the result follows. 11he extra estimate (4.5) is crucial in the sought regularity of the minimum time function butcontrary to the existing literature is only asked in a possibly proper subset of a neighborhood of thetarget. We will show in the examples of the next section how it may follow from (AE) as well. Inorder to achieve small time local attainability of the target, one needs in addition that the system canevade from H . Corollary 4.4.
In addition to the assumptions of Theorem 4.3 suppose that H is a manifold of codi-mension at least one and that for all x o ∈ H ∩ (Ω \T ) we have f ( x o , A ) T x o ( H ) , the tangent spaceof H at x o . Then for any x o ∈ Ω \T we can reach the target in finite time. Proof.
By following the vector field f ( x o , a ) / ∈ T x o ( H ) , we immediately exit the singular set. In this section we show two examples of well known nonlinear systems where we can find an ex-plicit smooth solution of (AE) and then apply Theorem 4.3 to obtain local Lipschitz regularity of theminimum time function. Our system will be in the form (2.1), (2.2) and T = { } . We consider the case where x = ( x h , x v ) ∈ R m +1 and σ ( x ) = (cid:18) I mt ( Bx h ) (cid:19) , (5.1)where I m is the m × m identity matrix and B is not singular, t B = − B = B − is also m × m .In particular m is an even number and | Bx h | = | x h | . It is known that the corresponding symmetriccontrol system is globally controllable to the origin and that its minimum time function is locally / − H ¨older continuous. We want to prove higher regularity except on its singular set.We consider the two functions u ( x ) = | x h | + 4 x v , U ( x ) = ( u ( x )) / , (5.2)and want to show that U is a solution of (AE) for H in R m +1 \{ } . U is a so called gauge functionfor the family of vector fields. We easily check that, after denoting A ( x ) = σ ( x ) t σ ( x ) , ∇ u ( x ) = (4 | x h | x h , x v ) , A ( x ) t ∇ u ( x ) = (cid:18) | x h | x h + 8 x v Bx h x v | x h | (cid:19) ,H ( x, ∇ u ( x )) = |∇ U ( x ) σ ( x ) | = A ( x ) t ∇ U ( x ) · t ∇ U ( x ) = 16 | x h | + 64 x v | Bx h | = 16 | x h | u ( x ) ,H ( x, ∇ U ( x )) = | x h | U ( x ) . Notice in particular that H ( x, ∇ U ( x )) = 0 if and only if x h = 0 and thus the singular set { x : H ( x, ∇ U ( x )) = 0 } contains the target and is a smooth manifold, being the x v axis. As a consequenceof the last displayed equation we have U ( x ) ≤ | x h | ε ≤ | x | ε , in H ( x, ∇ U ( x )) ≥ ε, x = 0 , −∇ ( H ( x, ∇ U ( x ))) · ( H ) p ( x, ∇ U ( x )) = − (cid:16) ( x h , U ( x ) − | x h | U ( x ) ∇ U ( x ) (cid:17) · A ( x ) t ∇ U ( x )= − U ( x ) (cid:16) U ( x ) | x h | U ( x ) − | x h | | x h | U ( x ) (cid:17) = 0 . Therefore U is even a classical C solution of (AE) for Hamiltonian H in R m +1 \{ } and then H is constant along the trajectories of the closed loop system (2.3). Hence, by Theorem 4.3, the system(1.1) is controllable in finite time to the origin from { x : H ( x, ∇ U ( x )) > } = R m +1 \{ (0 , x v ) : x v ∈ R } and the corresponding minimum time function is locally Lipschitz continuous on that set. Notice that,for ε < , { x : H ( x, ∇ U ( x )) ≥ ε } = { x : 4 x v ≤ (1 /ε − | x h | } . Also the last Corollary applies. Proposition 5.1.
Consider the symmetric control system (cid:26) ˙ x t = σ ( x t ) a t , t > ,x o ∈ R n , (5.3)where σ is given in (5.1). Then the gauge function (5.2) is a solution of the Aronsson equation (2.4)for H in R m +1 \{ } , it is an absolutely minimizing function for the corresponding L ∞ norm of thesubelliptic gradient and the minimum time function to reach the origin is locally Lipschitz continuousin { x = ( x h , x v ) ∈ R m +1 : x h = 0 } . The system is small time locally controllable and there is acontinuous feedback leading the system to the target outside the singular set. We consider the system where x = ( x h , x v ) ∈ R m +1 and σ ( x ) = (cid:18) I m m t x h (cid:19) , (5.4)where σ ( x ) is ( m + 1) × m matrix. Also in this case it is known that the corresponding symmetriccontrol system is globally controllable to the origin and that its minimum time function is locally / − H ¨older continuous. We consider u, U as before in (5.2) want to show that U is a solution of(AE) in R m +1 \{ } . In this case we can check that, A ( x ) t ∇ u ( x ) = (cid:18) | x h | x h x v | x h | (cid:19) , H ( x, ∇ u ( x )) = 16 | x h | u ( x ) , H ( x, ∇ U ( x )) = | x h | U ( x ) , and again we have, for ε > , U ( x ) ≤ | x h | ε ≤ | x | ε , in H ( x, ∇ U ( x )) ≥ ε. Finally, if x = 0 , −∇ ( H ( x, ∇ U ( x ))) · ( H ) p ( x, ∇ U ( x )) = − U ( x ) (cid:0) U ( x )( x h , − | x h | ∇ U ( x ) (cid:1) · A ( x ) t ∇ U ( x )= − U ( x ) (cid:16) U ( x ) | x h | U ( x ) − | x h | | x h | U ( x ) (cid:17) = 0 . Therefore U is a solution of (AE) for Hamiltonian H and hence the system (1.1) is controllable infinite time to the origin from { x : H ( x, ∇ U ( x )) > } and we prove the following result.13 roposition 5.2. Consider the symmetric control system (5.3) where σ is given in (5.4). Then thegauge function (5.2) is a solution of (AE) for H in R m +1 \{ } , it is an absolutely minimizing functionfor the corresponding L ∞ norm of the subelliptic gradient and the minimum time function to reachthe origin is locally Lipschitz continuous in { x = ( x h , x v ) ∈ R m +1 : x h = 0 } . References [1] A
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