Research Article | Open Access
The Aronsson Equation, Lyapunov Functions, and Local Lipschitz Regularity of the Minimum Time Function
We define and study -solutions of the Aronsson equation (AE), a second order quasi linear equation. We show that such super/subsolutions make the Hamiltonian monotone on the trajectories of the closed loop Hamiltonian dynamics. We give a short, general proof that -solutions are absolutely minimizing functions. We discuss how -supersolutions of (AE) become special Lyapunov functions of symmetric control systems, and allow to find continuous feedbacks driving the system to a target in finite time, except on a singular manifold. A consequence is a simple proof that the corresponding minimum time function is locally Lipschitz continuous away from the singular manifold, despite classical results showing that it should only be Hölder continuous unless appropriate conditions hold. We provide two examples for Hörmander and Grushin families of vector fields where we construct -solutions (even classical) explicitly.
In this note we want to describe a possible new, nonstandard way of using the Aronsson equation, a second order partial differential equation, to obtain controllability properties of deterministic control systems. We investigate a symmetric control system
where , is a nonempty and compact subset of a metric space. We define the Hamiltonian
which is therefore nonnegative and positively one homogeneous in the adjoint variable, and we want to drive the system to a target, temporarily we say the origin. We are interested in the relationship of (1) with the Aronsson equation (AE)
which is a quasilinear degenerate elliptic equation. Ideally, if everything is smooth, when we are given a classical solution of (AE) and we consider a trajectory of the Hamiltonian dynamics
which is a closed loop dynamics for the original control system, we find out that (AE) can be rewritten as
Therefore is constant. This is a very desirable property on the control system since it allows to use as a control Lyapunov function, despite the presence of a possibly nonempty singular set
which possibly contains the origin. Indeed if is outside the singular set and has a unique global minimum at the origin, then the trajectory of the Hamiltonian dynamics will reach the origin in finite time.
In general, however, several steps of this path break down. From one side, (AE) does not have classical solutions in general. Even in the case where , is the closed unit ball, and (AE) becomes the well known infinity Laplace equation
solutions are not classical, although known regularity results, see Evans and Savin , show that they are sometimes. Therefore, solutions of (AE) have to be meant in some weak sense, as viscosity solutions. For generic viscosity solutions, we can find counterexamples to the fact that the Hamiltonian is constant along trajectories of the Hamiltonian dynamics, as we show later. For an introduction to the theory of viscosity solutions in optimal control, we refer the reader to the book by Bardi and Capuzzo-Dolcetta .
In this paper, we will first characterize when, for a given super or subsolution of (AE) the Hamiltonian is monotone on the trajectories of the Hamiltonian dynamics (e.g. satisfies the monotonicity property). To this end we introduce the notion of -super/subsolution and prove for them that they satisfy the monotonicity property of the Hamiltonian. We emphasize the fact that not all viscosity solutions that are functions, are -solutions according to our definition. Moreover, as a side result, we also show that our -solutions are absolutely minimizing functions, i.e. local minimizers of the functional that computes the norm of the Hamiltonian. It is a well known equivalent property to being a viscosity solution of (AE) at least when is coercive or possibly in some Carnot Caratheodory spaces, but this fact is not completely understood in general. Therefore -solution appears to be an appropriate notion.
We then prove that if (AE) admits a -supersolution having a unique minimum at the origin, then our control system can be driven to the origin in finite time with a continuous feedback, starting at every initial point outside the singular set . If moreover satisfies appropriate decay in a neighborhood of the origin only at points where the Hamiltonian stays away from zero, then we show that the corresponding minimum 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 function can only be proven to be Hölder 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 singular set. Finally for two explicit well known examples, where the system has an Hörmander type, or a Grushin family of vector fields, we exhibit two explicit not yet known classical solutions of (AE), their gauge functions, providing examples of smooth absolute minimizers for such systems and the proof that their minimum time function is locally Lipschitz continuous outside the singular set. We remark the fact that neither in the general statement nor in the examples, the family of vector fields is ever supposed to span the whole space at the origin; therefore, the classical sufficient attainability condition ensuring that the minimum time function is locally Lipschitz continuous will not be satisfied in general. Indeed in the explicit examples that we illustrate in Section 4, the minimum time function is known to be locally only -Hölder continuous in its domain.
Small time local attainability and regularity of the minimum time function is an important subject in optimal control. Classical results by Petrov  show sufficient conditions for attainability at a single point by requiring that the convex hull of the vector fields at the point contains the origin in its interior. Such result was later improved by Liverovskii  augmenting the vector fields with the family of their Lie brackets, see also the paper by author . More recently such results had several extensions in the work by Krastanov and Quincampoix  and Marigonda et al. [7–9]. Our regularity results rather go in the direction of those contained in two recent papers by Albano et al. [10, 11], where they show, by completely different methods, that if a family of smooth vector fields satisfies the Hörmander condition, then the set where the local Lipschitz continuity of the minimum time function fails is the union of singular trajectories, and that it is analytic except on a subset of null measure. Our approach is instead more direct and comes as a consequence of constructing Lyapunov functions as -supersolutions of the Aronsson equation. We finally mention the paper by Motta and Rampazzo  where the authors study higher order Hamiltonians obtained by adding iterated Lie brackets as additional vector fields, in order to prove global asymptotic controllability to a target. While we do not study asymptotic controllability in this paper, 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 , as the Euler Lagrange equation for absolute minimizers, i.e. local minima of functionals, typically the norm of the gradient. There has been a lot of work in more recent years to develop that theory using viscosity solutions by authors like Jensen , Barron–Jensen–Wang , Juutinen , Crandall . For the main results on the infinity Laplace equation, we refer the reader to the paper  and the references therein. For results for equation (AE) especially in the dependent case, we also refer to the paper by the author  and the references therein, see also [20–22]. In particular, we mention that equation (AE) has been studied in Carnot groups by Bieske–Capogna , by Bieske  in the Grushin space, and by Wang  in the case of 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 motivating example. In Section 3, we introduce -solutions of (AE) and show for them some important properties: monotonicity of the Hamiltonian on the Hamiltonian dynamics, an equivalent definition and the fact that they are absolutely minimizing functions. In Section 4, we use -solutions of (AE) as Lyapunov functions for nonlinear control systems and obtain local Lipschitz regularity of the minimum time function away from the singular set. In Section 5, we provide two new examples of explicit classical solutions of (AE) in two important cases of nonlinear control systems where the results of Section 4 apply.
This paper appeared as preprint on ArXiv with number 1907.07436.
2. Control Theory and the Aronsson Equation
As we mentioned in the introduction, throughout the paper we consider the controlled dynamical system (1) where is open, is a nonempty, compact subset of some metric space, and is a continuous function, continuously differentiable and uniformly Lipschitz continuous in the first group of variables, i.e.
We suppose moreover that is convex for every and that the system is symmetric, i.e. for all and define the Hamiltonian
so that and by symmetry. Notice that is at least locally Lipschitz continuous, and is positively homogeneous of degree one by compactness of . We will also assume that is continuously differentiable on .
The case we are mostly interested in the following sections is when
where is set of matrices and is the closed unit ball. In this case .
Given a smooth function and , where is the singular set as in (6), we consider the Hamiltonian dynamics
where indicates the gradient of the Hamiltonian with respect to the group of adjoint variables .
Remark 2.1. When the Hamiltonian is differentiable, notice that for such that we have that
Therefore, under appropriate regularity, trajectories of (11) are indeed trajectories of the system (1) and moreover (11) is a closed loop system of (1) with feedback . If in particular is as in (10), then, for ,
Therefore, in this case the feedback control is at least continuous on and the closed loop system always has a well defined local solution starting out on that set.
We want to discuss when is monotone on a trajectory of (11). If we can compute derivatives, then we need to discuss the sign of
Therefore, a sufficient condition is that is a super or subsolution of the following pde
which is named Aronsson equation in the literature. Notice that is actually constant if is a classical solution of (15). The above computation is correct only under the supposed regularity on and unfortunately if such regularity is not satisfied and we interpret super/subsolutions of (15) as viscosity solutions this is no longer true in general, as the following example shows. Notice that if is not differentiable at a point where , then is multivalued, precisely the closed convex subgradient of the Lipschitz function computed at and contains the origin by the symmetry of the system. Therefore, the dynamics (11) has at least the constant solution also in this case. In some statements below it will be sometimes more convenient to look at (AE) for in order to gain regularity at points where vanishes.
Example 2.2. In the plane, suppose that hence it is smooth and independent of the state variables. In this case (AE) becomes the well known infinity Laplace equation
It is easy to check that a viscosity solution of the equation is . The function . Among solutions of the Hamiltonian dynamics , we can find the following two trajectories
defined in a neighborhood of . Clearly the Hamiltonian along the two trajectories is
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 are going to understand the reason.
3. Monotonicity of the Hamiltonian along the Hamiltonian Dynamics
Throughout this section, we consider a Hamiltonian not necessarily with the structure as in (9), but satisfying the following:
We will also refer to the following property:
Given , the monotonicity of the Hamiltonian along trajectories of (11) is the object of this section. It is a consequence of the following known general result.
Proposition 3.1. Let be an open set and be a continuous vector field. The following are equivalent:(i) is a continuous viscosity solution of in .(ii)The system is forward weakly increasing, i.e. for every , there is a solution of the differential equation , for , such that for .Moreover, the following are also equivalent(iii) is a continuous viscosity solution of in .(iv)The system is backward weakly increasing, i.e. for every , there is a solution of the differential equation , for , such that for .
Corollary 3.2. Let be an open set and be a continuous vector field. The following are equivalent:(i) is a continuous viscosity solution of and of in .(ii)The system is weakly increasing, i.e. for every , there is a solution of the differential equation , for , such that for .
Remark 3.3. The proof of the previous statement can be found in , see also . When another proof can be found in Proposition 5.18 of  or can be deduced from the optimality principles in optimal control proved in , when is locally Lipschitz continuous. In the case when is locally Lipschitz, the two differential inequalities in (i) of Corollary 3.2 turn out to be equivalent and of course there is also uniqueness of the trajectory of the dynamical system , . When (ii) in the Corollary is satisfied by all trajectories of the dynamical system then the system is said to be strongly monotone. This occurs in particular if there is at most one trajectory, as when is locally Lipschitz continuous. More general sufficient conditions for strong monotonicity can be found in , see also .
In view of the above result, we introduce the following definition.
Definition 3.4. Let be open and let satisfying (19). We say that a function is a -supersolution (resp. subsolution) of the Aronsson equation (15) in , if setting and we have that is a viscosity subsolution (resp. supersolution) of and a supersolution (resp. a subsolution) of .
It is worth pointing out explicitly the consequence we have reached by Proposition 3.1.
Remark 3.6. (i)Notice that if is a -solution of (15) and the Hamiltonian dynamics (11) is either strongly decreasing and strongly increasing, as for instance if it has a unique solution for a given initial condition, then for all trajectories of (11), is constant.(ii)In order to comment back to Example 2.2, notice that while is a function, nevertheless, as easily checked, is only a viscosity subsolution but not a supersolution ofwhile it is a viscosity solution of . Then it turns out that the Hamiltonian is weakly increasing on the trajectories of the Hamiltonian dynamics. Indeed there is another trajectory of the Hamiltonian dynamics such that , namelyalong which the Hamiltonian is actually constant, until the trajectory is well defined.(iii)It is clear by Example 2.2 that while classical solutions of (15) are -solutions, continuous or even viscosity solutions in general are not. The definition of -solution that we introduced is meant to preserve the monotonicity property of the Hamiltonian on the trajectories of the Hamiltonian dynamics.(iv)Observe that if is a -solution, then is a -solution as well, since the Hamiltonian is unchanged and the vector field in the Hamiltonian dynamics becomes the opposite.
It may look unpleasant that Definition 3.4 of solution of (15) refers to a property that is not formulated directly for the function . Therefore, in the next statement we will reformulate the above definition. The property (ED) below will give an equivalent definition of a -solution.
Proposition 3.7. Let and satisfying (19), (20). The following two statements are equivalent:
(ED) for all , there is a trajectory of the Hamiltonian dynamics (11), such that if is a test function and has a minimum (respectively maximum) at and , then we have that(i) is a -supersolution (resp. subsolution) of (15).In particular, if is at , a -supersolution (resp. subsolution) is a viscosity supersolution (resp. subsolution) of (15).
Remark 3.8. In the statement of (ED), when the Hamiltonian vector field is locally Lipschitz continuous, we may restrict the test functions to .
Proof. We only prove the statement for supersolutions, the other case being similar. Let .
Suppose first that (ED) holds true. Let and such that has a maximum at , . Therefore if is a solution of the Hamiltonian dynamics (11) that satisfies (ED), we have that, by homogeneity of and for ,Thus integrating for small we getand thus has a minimum at on for small and . If instead had a minimum at , then integrating on for small enough, we would still obtain the same as in (25). By (ED), from (25) we get in both caseswhere . Therefore we conclude that is a viscosity subsolution of (or a supersolution of when has a minimum st ). Finally by definition, is a -supersolution of (15).
Suppose now that is a -supersolution of (15). Then by Proposition 3.1, for all , we can find a trajectory of the dynamics (11) such that is nondecreasing. Therefore, is a concave function of . Let be such that has a minimum at , and . If we had then would be strictly convex in its domain. Therefore, for small enough, and in the domain of ,by concavity of . This is a contradiction.
We prove the last statement on the fact that a -solution is a viscosity solution. Therefore for a -supersolution of (15) let now be such that has a minimum at . By (ED), for a suitable solution of (11) we have that has a minimum at if , in particular . By (ED) and homogeneity of ,Therefore is a viscosity supersolution of (15). The case of subsolutions is similar and we skip it.
We end this section by proving another important property of -solutions of (15) that in the literature was the main motivation to the study of (AE).
Remark 3.10. When is an open set and the property of a function in Theorem (3.9) holds for all open subsets then we say that is an Absolutely minimizing function in for the Hamiltonian . This means that is a local minimizer of . It is well known that for the infinity Laplace equation, where we minimize the Lipschitz constant of , it is equivalent to be a viscosity solution and an absolutely minimizing function. Such equivalence is also known for coercive Hamiltonians and for the norm of the horizontal gradient in some Carnot Caratheodory spaces. For more general Hamiltonians this equivalence is not known. Here we prove one implication at least for -solutions of (15).
Proof. Let be as in the statement and we know that is positively 1-homogeneous. We define and look at solutions of the Hamiltonian dynamics (11). If , then clearly and we have nothing left to show. If otherwise since is a -solution of (15), we already know that we can construct a solution of (11) starting out at such that is nondecreasing for and nonincreasing for (by a concatenation of two trajectories of (11) with monotone Hamiltonian). Since is bounded, then the curve will not stay indefinitely in because as we already observedand
Hence will hit forward and backward in finite time. Let be such that and for . Therefore,
Now we use the differential inequality (29) in the viscosity sense and the lower optimality principle in control theory as in  for subsolutions of the Hamilton-Jacobi equation. Therefore, since is a trajectory of the control system (1) we have that for all and , as for ,
By letting and we conclude, by continuity of at the boundary of and (33),
which is what we want.
Remark 3.11. Notice that in (32) equalities hold if is constant on a given trajectory of (11) and we obtain thatand thenwhich is an implicit representation formula for through its boundary values, since the points depend on the Hamiltonian dynamics (11) and itself.
4. Lyapunov Functions and (AE)
In this section, we go back to the structure (9) for and want to discuss the classical idea of control Lyapunov function. Let be a closed target set, we want to find at least lower semicontinuous and such that: if and only if and such that for all there exists a control and such that the corresponding trajectory of (1) satisfies:
Classical necessary and sufficient conditions lead to look for strict supersolutions of the Hamilton-Jacobi equation, namely to find such that
with continuous and such that if and only if . The case is already quite interesting for the theory.
Here we will apply the results of the previous section and consider Lyapunov functions built as follows. We analyse the existence of such that is a -supersolution of (AE), i.e. satisfies
Remark 4.1. To study (40) in the case when is as in (9) and as in (10), it is sometimes more convenient to write it for the Hamiltonian squared, . Thuswhere we indicated, are the columns of , and . Therefore, a special sufficient condition for to satisfy (40) is that is negative semidefinite, which means that is -concave with respect to the family of vector fields , in the sense of Bardi-Dragoni . We recall that the matrix also appears in  to study second order controllability conditions for symmetric control systems.
Define the minimum time function for system (1) aswhere . We prove the following result, recall that is the singular set.
Proposition 4.2. Let be open and a closed target. Let have the structure (9). Assume that is nonnegative and a -solution of (40) in and that for , for and for and some . For any there exists a solution of the closed loop system (11) such that(i) is a nondecreasing function of ;(ii) is a strictly decreasing function of ;(iii)The trajectory reaches the target in finite time and the minimum time function for system (1) satisfies the estimate
Proof. The thesis (i) follows from the results of the previous section since is a supersolution of (AE). Let be a point where . By homogeneity of the Hamiltonian we get, for and (ii) follows. Integrating now the last inequality we obtainand thus the solution of (11) reaches the target before timeTherefore (44) follows by definition.
The estimate (44) can be used to obtain local regularity of the minimum time function. The proof of regularity now follows a more standard path although under weaker assumptions than usual literature and will allow us to obtain a new regularity result. We emphasize that nothing in the next statement is assumed on the structure of the vectogram when . In particular the target need not be even small time locally attainable.
Theorem 4.3. Let be open and a closed target. Assume that is nonnegative and -solution of (40) in and that for , for and for and some . Let be the distance function from the target. Suppose that satisfies the following: for all there are such that
Then the minimum time function for system (1) to reach the target is finite and locally Lipschitz continuous in .
Proof. Let , and be such that , for all . The parameter will be small enough to be decided later. We apply the assumption (48) and find accordingly. The fact that is finite in , for sufficiently small, follows from Proposition 4.2
Take and suppose that are the trajectories solutions of (1) corresponding to the initial conditions respectively. To fix the ideas we may suppose that and for any we choose a control and time such that . Note that by (44), , for all . Moreover, by the Gronwall inequality for system (1) and since ,and the right hand side is smaller than if is small enough. Now we can estimate, by the dynamic programming principle and by (44), (48),As , the result follows.
The extra estimate (48) is crucial in the sought regularity of the minimum time function, but contrary to the existing literature is only asked in a possibly proper subset of a neighborhood of the target. We will show in the examples of the next section how it may follow from (AE) as well. In order to achieve small time local attainability of the target, one needs in addition that the system can evade from .
Corollary 4.4. In addition to the assumptions of Theorem 4.3 suppose that is a manifold of codimension at least one and that for all we have , the tangent space of at . Then for any we can reach the target in finite time.
Proof. By following the vector field , we immediately exit the singular set and then reach the target in finite time by Proposition 4.2.
5. Some Smooth Explicit Solutions of the Aronsson Equation
In this section, we show two examples of well known nonlinear systems where we can find an explicit smooth solution of (AE) and then apply Theorem 4.3 to obtain local Lipschitz regularity of the minimum time function. Our system will be in the form (9), (10) and .
5.1. Hörmander-Like Vector Fields
We consider the case where and
where is the identity matrix and is not singular, is also . In particular is an even number and . It is known that the corresponding symmetric control system is globally controllable to the origin and that its minimum time function is locally -Hölder continuous. We want to prove higher regularity except on its singular set.
We consider the two functions
and want to show that is a solution of (AE) for in . is a so called gauge function for the family of vector fields. We easily check that, after denoting ,
Notice in particular that if and only if and thus the singular set contains the target and is a smooth manifold, being the axis. As a consequence of the last displayed equation we have
which is an information that we need to apply Theorem 4.3. Finally, if ,
Therefore, is even a classical solution of (AE) for Hamiltonian in and then is constant along the trajectories of the closed loop system (11). Hence, by Theorem 4.3, the system (1) is controllable in finite time to the origin from
and the corresponding minimum time function is locally Lipschitz continuous on that set. Notice that, for , . Also the last Corollary applies.
Proposition 5.1. Consider the symmetric control systemwhere is given in (51). Then the gauge function (52) is a solution of the Aronsson equation (15) for in , it is an absolutely minimizing function for the corresponding norm of the subelliptic gradient and the minimum time function to reach the origin is locally Lipschitz continuous in . The system is small time locally controllable and there is a continuous feedback leading the system to the target outside the singular set.
5.2. Grushin Vector Fields
We consider the system where and
where is matrix. Also in this case it is known that the corresponding symmetric control system is globally controllable to the origin and that its minimum time function is locally -Hölder continuous. We consider as before in (52) want to show that is a solution of (AE) in . In this case we can check that,
and again we have, for ,
Finally, if ,
Therefore is a solution of (AE) for Hamiltonian and hence the system (1) is controllable in finite time to the origin from and we prove the following result.
Proposition 5.2. Consider the symmetric control system (57) where is given in (58). Then the gauge function (52) is a solution of (AE) for in , it is an absolutely minimizing function for the corresponding norm of the subelliptic gradient and the minimum time function to reach the origin is locally Lipschitz continuous in .
We introduced -solutions of the Aronsson equation in order to preserve the monotonicity of the Hamiltonian on the trajectories of the Hamiltonian dynamics. We proved that -solutions are absolute minimisers of the Hamiltonian and a subclass of viscosity solutions of (AE). We also discussed the fact that -solutions of (AE) are a good class of Lyapunov functions to prove that a nonlinear symmetric system is controllable to the target with a continuous feedback outside a singular set and that the minimum time function is locally Lipschitz continuous on that domain. We have provided new examples of smooth explicit solutions of (AE) for Heisenberg and Grushin systems to implement the results.
The arguments and references data used to support the findings of this study are included within the article.
Conflicts of Interest
The author declares that they have no conflicts of interest.
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