A contact covariant approach to optimal control with applications to sub-Riemannian geometry

A contact interpretation of the Pontryagin Maximum Principle In a recent paper, Ohsawa [17] observed that for normal solutions of the optimal control problem on a manifold Q, the Hamiltonian evolution of the covector \(\varvec{\varLambda }_t\) in \(\mathrm {T}^*(Q\times \mathbb {R})\) considered in the Pontryagin maximum principle (PMP), projects to a well-defined contact evolution in the projectivization \(\mathbb {P}(\mathrm {T}^*(Q\times \mathbb {R}))\). Here, \(Q\times \mathbb {R}\) is the extended configuration space (consisting of both the configurations Q and the costs \(\mathbb {R}\)) and \(\mathbb {P}(\mathrm {T}^*(Q\times \mathbb {R}))\) is equipped with a natural contact structure. Moreover, Ohsawa observed that the maximized Hamiltonian of the PMP is precisely the generating function of this contact evolution.

The above result was our basic inspiration to undertake this study. Our goal was to understand, from a geometric viewpoint, the role and origins of the above-mentioned contact structure in the PMP and to study possible limitations of the contact approach (does it work alike for abnormal solutions, etc.).

As a result we prove Theorem 3, a version of the PMP, in which the standard Hamiltonian evolution of a covector curve \(\varvec{\varLambda }_t\) in \(\mathrm {T}^*(Q\times \mathbb {R})\) along an optimal solution \(\varvec{q}(t)\in Q\times \mathbb {R}\) is substituted by a contact evolution of a curve of hyperplanes \(\varvec{\mathcal {H}}_t\) in \(\mathrm {T}(Q\times \mathbb {R})\) along this solution. (Note that the space of all hyperplanes in \(\mathrm {T}(Q\times \mathbb {R})\) is actually the manifold of contact elements of \(Q\times \mathbb {R}\) and can be naturally identified with \(\mathbb {P}(\mathrm {T}^*(Q\times \mathbb {R}))\).) It is worth mentioning that this result is valid regardless of the fact whether the solution is normal or abnormal and, moreover, the contact evolution is given by a natural contact lift of the extremal vector field (regarded as a time-dependent vector field on \(Q\times \mathbb {R}\)). Finally, using the well-known relation between contact vector fields and smooth functions we were able to interpret the Pontryagin maximized Hamiltonian as a generating function of the contact evolution of \(\varvec{\mathcal {H}}_t\).

It seems to us that, apart from the very recent paper of Ohsawa [17], the relation between optimal control and contact geometry has not been explored in the literature. This fact is not difficult to explain as the PMP in its Hamiltonian formulation has been very successful and as symplectic geometry is much better developed and understood than contact geometry. In our opinion, the contact approach to the PMP seems to be a promising direction of studies for at least two reasons. First of all it allows for a unified treatment of normal and abnormal solutions and, second, it seems to be closer to the actual geometric meaning of the PMP (we shall justify this statement below).

About the proof The justification of Theorem 3 is rather trivial. In fact, it is just a matter of interpretation of the classical proof of the PMP [18] (see also [13, 15]). Recall that geometrically the PMP says that at each point of the optimal trajectory \(\varvec{q}(t)\), the cone \(\varvec{\mathcal {K}}_t\subset \mathrm {T}_{\varvec{q}(t)}(Q\times \mathbb {R})\) approximating the reachable set can be separated, by a hyperplane \(\varvec{\mathcal {H}}_t\subset \mathrm {T}_{\varvec{q}(t)}(Q\times \mathbb {R})\), from the direction of the decreasing cost (cf. Fig. 2). Thus, in its original sense the PMP describes the evolution of a family of hyperplanes \(\varvec{\mathcal {H}}_t\) (i.e., a curve in the manifold of contact elements of \(Q\times \mathbb {R}\), identified with \(\mathbb {P}(\mathrm {T}^*(Q\times \mathbb {R}))\)) along the optimal solution. This evolution is induced by the flow of the optimal control on \(Q\times \mathbb {R}\). From this perspective, the only ingredient one needs to prove Theorem 3 is to show that this flow induces a contact evolution (with respect to the natural contact structure) on \(\mathbb {P}(\mathrm {T}^*(Q\times \mathbb {R}))\). It is worth mentioning that the covector curve \(\varvec{\varLambda }_t\in \mathrm {T}^*(Q\times \mathbb {R})\) from the standard formulation of the PMP is nothing else than just an alternative description of the above-mentioned curve of hyperplanes, i.e., \(\varvec{\mathcal {H}}_t=\ker \varvec{\varLambda }_t\) for each time t. Obviously, there is an ambiguity in choosing such a \(\varvec{\varLambda }_t\), which is defined up to a rescaling.

Applications From the above perspective, it is obvious that the description of the necessary conditions for optimality of the PMP in terms of \(\varvec{\mathcal {H}}_t\)s (the contact approach) is closer to the actual geometric meaning of the PMP as it contains the direct information about the separating hyperplanes. On the contrary, in the Hamiltonian approach this information is translated into the language of covectors (not to forget the non-uniqueness of the choice of \(\varvec{\varLambda }_t\)).

It should be stressed that the language of flows used throughout is much more effective, and in fact simpler, than the language of Lie brackets usually applied in the study of SR extremals. Indeed, the assertions of Theorems 5 and 6 are valid for non-smooth, i.e., absolutely continuous curves and bounded measurable controls do not require any regularity assumptions (contrary to the characterization in terms of Lie brackets) and work for single trajectories (not necessary families of trajectories).

As an illustration of the above results we give a few examples. In particular, in Examples 1 and 8 we were able to provide a surprisingly easy derivation of the Riemannian geodesic equation (obtaining the equation \(\nabla _{\dot{\gamma }}\dot{\gamma }=0\) from the standard Hamiltonian approach is explained in [1, 20]). In Examples 3, 7, and 9, we re-discover some results of [16, 22] concerning rank-2 distributions.

Organization of the paper We begin our considerations by a technical introduction in Sect. 2. Our main goal in this part is to introduce, in a rigorous way, natural differential geometric tools (Lie brackets, flows of time-dependent vector fields, distributions, etc.) in the non-smooth and time-dependent setting suitable for control theory (in general, we consider controls which are only locally bounded and measurable). Most of the results presented in this section are natural generalizations of the results well known in the smooth case. They are essentially based on the local existence and uniqueness of solutions of ODE in the sense of Caratheodory (Theorem 7). To avoid being too technical, we moved various parts of the exposition of this section (including some proofs and definitions) to the Appendix.

In Sect. 3, we briefly recall basic definitions and constructions of contact geometry. In particular, we show an elegant construction of contact vector fields (infinitesimal symmetries of contact distributions) in terms of equivalence classes of vector fields modulo the contact distribution. This construction is more fundamental than the standard one in terms of generating functions (which requires a particular choice of a contact form). It seems to us that so far it has not been presented in the literature.

In Sect. 4, we discuss in detail a natural contact structure on the projectivization of the cotangent bundle \(\mathbb {P}(\mathrm {T}^*M)\). In particular, we construct a natural contact transformation \(\mathbb {P}(F)\) of \(\mathbb {P}(\mathrm {T}^*M)\) induced by a diffeomorphism F of M. Later we study an infinitesimal counterpart of this construction, i.e., a natural lift of a vector field X on M to a contact vector field \(\mathbf {C}_{X}\) on \(\mathbb {P}(\mathrm {T}^*M)\).

In Sect. 5, we introduce the optimal control problem for a control system on a manifold Q and formulate the PMP in its standard version (Theorem 2). Later we sketch the standard proof of the PMP introducing the cones \(\varvec{\mathcal {K}}_t\) and the separating hyperplanes \(\varvec{\mathcal {H}}_t\). A proper interpretation of these objects, together with our previous considerations about the geometry of \(\mathbb {P}(\mathrm {T}^*M)\) from Sect. 4, allows us to conclude Theorems 3 and 4 which are the contact and the covariant versions of the PMP, respectively.

Finally, in the last Sect. 6, we concentrate our attention on the geometry of the cones \(\varvec{\mathcal {K}}_t\) and hyperplanes \(\varvec{\mathcal {H}}_t\) for the Riemannian and sub-Riemannian geodesic problems. The main results of that section, which characterize normal and abnormal SR extremals, were already discussed in detail in the paragraph “Applications” above.

As indicated in the Introduction, in this paper we shall apply the language of differential geometry to optimal control theory. This requires some attention as differential geometry uses tools such as vector fields, their flows, distributions and Lie brackets which are a priori smooth, while in control theory it is natural to work with objects of lower regularity. The main technical difficulty is a rigorous introduction of the notion of the flow of a time-dependent vector field (TDVF) with the time-dependence being, in general, only measurable. A solution of this problem, provided within the framework of chronological calculus, can be found in [2]. The recent monograph [11] with a detailed discussion of regularity aspects is another exhaustive source of information about this topic.

Despite the existence of the above-mentioned excellent references, we decided to present our own explication of the notion of the flow of a TDVF. The reasons for that decision are threefold. First of all, this makes our paper self-contained. Second, we actually do not need the full machinery of [2] or [11], so we can present a simplified approach. Finally, for future purposes we need to concentrate our attention on some specific aspects (such as the transport of a distribution along an integral curve of a TDVF and the relation of this transport with the Lie bracket) which are present in neither [2] nor [11]. Our goal in this section is to give a minimal yet sufficient introduction to the above-mentioned concepts. We move technical details and rigorous proofs to the Appendix.

A solution of (2.1) with the initial condition \(x(t_0)=x_0\) will be denoted by \(x(t;t_0,x_0)\) and called an integral curve of \(X_t\). When speaking about families of such solutions with different initial conditions it will be convenient to introduce (local) maps \(A_{tt_0}:M\rightarrow M\) defined by \(A_{tt_0}(x_0):=x(t;t_0,x_0)\).

Since \(X_t\) is Caratheodory, it satisfies locally the assumptions of Theorem 7. Now the justification of Lemma 1 follows directly from the latter result. Properties (2.2) are merely a consequence of the fact that \(t\mapsto A_{tt_0}(x_0)\) is an integral curve of \(X_t\).

Clearly \(A_{tt_0}\) is a natural time-dependent analog of the notion of the flow of a vector field. This justifies the name “TD flow”. It is worth noticing that, alike for the standard notion of the flow, there is a natural correspondence between TD flows and Caratheodory TDVFs.

The natural candidate for such a TDVF is simply \(X_t(x):=\frac{\partial }{\partial \tau }\big |_{\tau =t}A_{\tau t}(x)\). The remaining details are left to the reader.

Distributions along integral curves of TDVFs In this paragraph we shall introduce basic definitions and basic properties related with distributions defined along a single ACB integral curve \(x(t)=x(t;t_0,x_0)\) (with \(t\in [t_0,t_1]\)) of a Caratheodory TDVF \(X_t\). In particular, for future purposes it will be crucial to understand the behavior of such distributions under the TD flow \(A_{t\tau }\) of \(X_t\).

Let us remark that the idea behind the notion of a charming distribution is to provide a natural substitution of the notion of smoothness in the situation where a distribution is considered along a non-smooth curve. Observe namely that a restriction of a smooth vector field on M to an ACB curve \(x(t;t_0,x_0)\) is a priori only an ACB vector field along \(x(t;t_0,x_0)\).

The justification of the above result is straightforward. Regarding the first situation it was already observed that a restriction of a smooth vector field to an ACB curve is an ACB vector field. In the second situation, the distribution \(\mathcal {B}\) is spanned by vector fields \(\mathrm {T}A_{tt_0}(X^i)\) with \(i=1,\ldots ,k\), where \(\{X^1,\ldots ,X^k\}\) is any basis of \(\mathcal {B}_{x_0}\). By the results of Lemma 12 these fields are ACB.

Given a distribution \(\mathcal {B}\) along x(t) we can always extend it to the smallest (with respect to inclusion) distribution along x(t) containing \(\mathcal {B}\) and respected by the TD flow \(A_{t\tau }\) along x(t). This construction will play a crucial role in geometric characterization of normal and abnormal SR extremals in Sect. 6.

Obviously, any distribution \(A_{t\tau }\)-invariant along x(t) and containing \(\mathcal {B}_{x(t)}\) must contain \(A_{\bullet }(\mathcal {B})_{x(t)}\). The fact that the latter is indeed \(A_{t\tau }\)-invariant along x(t) follows easily from property (2.2).

Lie brackets and distributions Constructing distributions \(A_{t\tau }\)-invariant along x(t) introduced in Proposition 2, although conceptually very simple, is not very useful from the practical point of view, as it requires calculating the TD flow \(A_{t\tau }\). This difficulty can be overcome by passing to an infinitesimal description in terms of the Lie brackets, however, for a price of loosing some generality. In this paragraph, we shall discuss this and some related problems in detail.

For future purposes, we would like to extend Definition 3 to be able to calculate the bracket \([X_t,Z]_{x(t)}\) also for fields Z of lower regularity. That can be done, but at a price that the bracket \([X_t,Z]_{x(t)}\) would be defined only for almost every (a.e.) \(t\in [t_0,t_1]\). The details of this construction are provided below.

In other words, formula (2.3) is an extension of Definition 3 which allows to calculate the Lie bracket \([X_t,Z]_{x(t)}\) at almost every point of a given integral curve x(t) of \(X_t\), for vector fields Z defined only along x(t) and such that \(t\mapsto Z(x(t))\) is ACB. The latter generalization is necessary in control theory, since, as \(t\mapsto x(t)\) is in general ACB only, even if Z is a smooth vector field, we cannot expect the map \(t\mapsto Z(x(t))\) to be of regularity higher than ACB.

The above construction of the Lie bracket \([X_t,Z]_{x(t)}\) allows to introduce the following natural construction.

Definition 4

Let \(X_t\) be a Caratheodory TDVF, \(x(t)=x(t;t_0,x_0)\) (with \(t\in [t_0,t_1]\)) its integral curve and let \(\mathcal {B}\) be a distribution along x(t). By \([X_t,\mathcal {B}]\) we shall understand the distribution along x(t) generated by the Lie brackets of \(X_t\) and all ACB sections of \(\mathcal {B}\):

$$\begin{aligned}{}[X_t,\mathcal {B}]_{x(t)}:={\text {vect}}_{\mathbb {R}}\{[X_t,Y]_{x(t)}\ |\ Y\in \varGamma _{ACB}(\mathcal {B})\}, \end{aligned}$$

where we consider \([X_t,Y]_{x(t)}\) at all points where it makes sense, i.e., at which the bracket \([X_t,Y]_{x(t)}\) is well defined.

A charming distribution \(\mathcal {B}\) along x(t) will be called \(X_t\)-invariant along x(t) if

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Note that neither \([X_t,\mathcal {B}]\) nor \(\mathcal {B}+[X_t,\mathcal {B}]\) need be charming distributions along x(t) even if so was \(\mathcal {B}\) as, in general, there is no guarantee that these distributions will be spanned by ACB sections (we can loose regularity when calculating the Lie bracket).

The following result explains the relation between the \(A_{t\tau }\)- and \(X_t\)-invariance of distributions along x(t).

The proof is given in the Appendix. Note that the equivalence between \(X_t\)- and \(A_{t\tau }\)-invariance is valid only if the considered distribution \(\mathcal {B}\) along x(t) satisfies regularity conditions: it has to be charming and of constant rank along x(t).

Given a charming distribution \(\mathcal {B}\) along x(t), it is clear in the light of the above result, that \(A_{\bullet }(\mathcal {B})_{x(t)}\), the smallest distribution \(A_{t\tau }\)-invariant along x(t) and containing \(\mathcal {B}\), should be closed under the operation \([X_t,\cdot ]\). Thus, in the smooth case, it is natural to try to construct \(A_{\bullet }(\mathcal {B})\) in the following way.

By the results of Proposition 3, the property that a distribution \(\mathcal {B}\) is \(X_t\)-invariant along x(t) depends not only on \(\mathcal {B}\) and the values of a Caratheodory TDVF \(X_t\) along x(t), but also on the values of \(X_t\) in a neighborhood of that integral curve. It turns out, however, that in a class of natural situations the knowledge of \(X_t\) along x(t) suffices for checking the \(X_t\)-invariance.

The proof is given in the Appendix.

Contact manifolds and contact transformations In this section, we shall recall basic facts from contact geometry. A contact structure on a manifold \(\mathcal {M}\) is a smooth co-rank one distribution \(\mathcal {C}\subset \mathrm {T}\mathcal {M}\) satisfying a certain maximum non-degeneracy condition. To formalize that condition we introduce the following geometric construction. From now on we shall assume that the pair \((\mathcal {M},\mathcal {C})\) consists of a smooth manifold \(\mathcal {M}\) and a smooth co-rank one distribution \(\mathcal {C}\) on \(\mathcal {M}\). Sometimes it will be convenient to treat \(\mathcal {C}\) as a vector subbundle of \(\mathrm {T}\mathcal {M}\).

Observe that \(\mathcal {C}\) is necessarily of even rank (\(\mathcal {M}\) is odd-dimensional). This follows from a simple fact from linear algebra that every skew-symmetric 2-form on an odd-dimensional space has a non-trivial kernel.

It is worth mentioning that the above relation between contact vector fields and flows consisting of contact transformations can be generalized to the context of TDVFs and TD flows (cf. Sect. 2). We will need this generalized relation in Sect. 5 after introducing control systems.

The proof follows directly from Theorem 1 by taking \(\mathcal {B}=\mathcal {C}\) (which is charming, see—cf. Proposition 1).

Characterization of CVFs It turns out that there is a one-to-one correspondence between CVFs on \(\mathcal {M}\) and sections of the normal bundle \(\mathrm {N}\mathcal {C}\).

In the literature, see e.g., [14], a contact distribution \(\mathcal {C}\) on \(\mathcal {M}\) is often presented as the kernel of a certain 1-form \(\omega \in \varLambda ^1(\mathcal {M})\) (such an \(\omega \) is then called a contact form). In the language of \(\omega \), the maximum non-degeneracy condition can be expressed as the non-degeneracy of the 2-form \({\text {d}}\omega \) on \(\mathcal {C}\). The latter is equivalent to the condition that \(\omega \wedge ({\text {d}}\omega )^{\wedge n}\), where \(n=\frac{1}{2}{\text {rank}}\mathcal {C}\), is a volume form on \(\mathcal {M}\) (i.e., \(\omega \wedge ({\text {d}}\omega )^{\wedge n}\ne 0\)).

Also CVFs have an elegant characterization in terms of contact forms. One can show that CVFs are in one-to-one correspondence with smooth functions on \(\mathcal {M}\). Choose a contact form \(\omega \) such that \(\mathcal {C}=\ker \omega \), then this correspondence is given by an assignment \(\phi \mapsto C_{\phi }\), where \(C_{\phi }\) is the unique vector field on M such that \(\omega (C_{\phi })=\phi \) and \((C_{\phi }\lrcorner {\text {d}}\omega )|_{\mathcal {C}}=-{\text {d}}\phi |_{\mathcal {C}}\). A function \(\phi \) is usually called the generating function of the corresponding CVF \(C_{\phi }\) associated with the contact form \(\omega \). Notice that given a contact vector field \(C_{}=C_{\phi }\) and a contact form \(\omega \) one can recover the generating function simply by evaluating \(\omega \) on \(C_{}\), i.e., \(\phi =\omega (C)\).

It is interesting to relate the construction \(\phi \mapsto C_{\phi }\) with the construction \([X]\mapsto C_{[X]}\) given above. Namely, the choice of a contact form \(\omega \) allows to introduce a vector field \(R\in \mathfrak {X}(\mathcal {M})\) (known as the Reeb vector field) defined uniquely by the conditions \(\omega (R)=1\) and \(R\lrcorner {\text {d}}\omega =0\). Since R is not contained in \(\mathcal {C}=\ker \omega \), its class [R] establishes a basis of the normal bundle \(\mathrm {N}\mathcal {C}\). Consequently, we can identify smooth functions on \(\mathcal {M}\) with sections of \(\mathrm {N}\mathcal {C}\), via \(\phi \mapsto [\phi R]\). Now it is not difficult to prove that \(C_{\phi }=C_{[\phi R]}\) and conversely that \(C_{[X]}=C_{\phi }\) for \(\phi =\omega (C_{[X]})=\omega (X)\). The details are left to the reader.

Note, however, that the description of the contact distribution \(\mathcal {C}\) in terms of a contact form \(\omega \) is, in general, non-canonical (as every rescaling of \(\omega \) by a nowhere-vanishing function gives the same kernel \(\mathcal {C}\)) and valid only locally (as there clearly exist contact distributions which cannot be globally presented as kernels of single 1-forms). For this reason, the description of a contact manifold \((\mathcal {M},\mathcal {C})\) in terms of \(\mathcal {C}\) and related objects (e.g., \(\mathrm {N}\mathcal {C}, \beta \)) is more fundamental and often conceptually simpler (for example, in the description of CVFs) than the one in terms of \(\omega \). Not to mention that, for instance, the construction of the CVF \(C_{\phi }\) does depend on the particular choice of \(\omega \), whereas the construction of \(C_{[X]}\) is universal.

Indeed, to every CVF \(C_{}\), we attach the affine distribution (control-affine system) \(\mathcal {A}=C_{}+\mathcal {C}\). Conversely, given an affine distribution (control-affine system) \(\mathcal {A}=X+\mathcal {C}\), there exists a unique CVF \(C_{}\in \varGamma ( \mathcal {A})\), namely \(C_{}=C_{[X]}\), such that \(\mathcal {A}=C_{}+\mathcal {C}=C_{[X]}+\mathcal {C}\). In other words, on every contact manifold \((\mathcal {M},\mathcal {C})\), there are as many CVF’s C as control-affine systems \(\mathcal {A}=X+\mathcal {C}\), the correspondence being established by the map \(\mathcal {A}=X+\mathcal {C}\mapsto C_{[X]}\).

In this section, we shall describe the natural contact structure on \(\mathbb {P}(\mathrm {T}^*M)\) and its relation with the canonical symplectic structure on \(\mathrm {T}^*M\) (see, e.g., [5] or [14]). Later it will turn out that this structure for \(M=Q\times \mathbb {R}\) plays the crucial role in optimal control theory.

The canonical contact structure on \(\mathbb {P}(\mathrm {T}^*M)\)

Let us denote the cotangent bundle of a manifold M by \(\pi _M:\mathrm {T}^*M\rightarrow M\). The projectivized cotangent bundle \(\mathbb {P}(\mathrm {T}^*M)\) is defined as the space of equivalence classes of non-zero covectors from \(\mathrm {T}^*M\) with \([\theta ]=[\theta ']\) if \(\pi _M(\theta )=\pi _M(\theta ')\) and \(\theta =a\cdot \theta '\) for some scalar \(a\in \mathbb {R}{\setminus }\{0\}\). Clearly, \(\mathbb {P}(\mathrm {T}^*M)\) is naturally a smooth manifold and also a fiber bundle over M with the projection \(\pi :\mathbb {P}(\mathrm {T}^*M)\rightarrow M\) given by \(\pi :[\theta ]\mapsto \pi _M(\theta )\). The fiber of \(\pi \) over \(p\in M\) is simply the projective space \(\mathbb {P}(\mathrm {T}^*_p M)\). It is worth noticing that \(\mathbb {P}(\mathrm {T}^*M)\) can be also understood as the space of hyperplanes in \(\mathrm {T}M\) (a manifold of contact elements), where we can identify each point \([\theta ]\in \mathbb {P}(\mathrm {T}^*M)\) with the hyperplane \(\mathcal {H}_{[\theta ]}:=\ker \theta \subset \mathrm {T}_{\pi _M(\theta )}M\).

In other words, \(\mathcal {C}_{[\theta ]}\) consists of all vectors in \(\mathrm {T}_{[\theta ]}\mathbb {P}(\mathrm {T}^*M)\) which project, under \(\mathrm {T}\pi \), to the hyperplane \(\mathcal {H}_{[\theta ]}=\ker \theta \), see Fig. 1. We shall refer to this contact structure as the canonical contact structure on \(\mathbb {P}(\mathrm {T}^*M)\).

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The fact that (4.1) defines a contact structure is well known in the literature. The proof is given, for instance, in Appendix 4 of the book of Arnold [5], where the reasoning is based on the properties of the Liouville 1-form \(\Omega _M\) on the cotangent manifold \(\mathrm {T}^*M\). For convenience of our future considerations in Sect. 5 we shall, however, present a separate proof quite similar to the one of Arnold.

Contact transformations of \(\mathbb {P}(\mathrm {T}^*M)\) induced by diffeomorphisms

In this paragraph, we will define contact transformations of \(\mathbb {P}(\mathrm {T}^*M)\) which are natural lifts of diffeomorphisms of the base M.

Observe that if \(\mathcal {H}=\ker \theta \), then \(\mathrm {T}F(\mathcal {H})=\ker ((F^{-1})^*(\theta ))\). In other words, \(\mathbb {P}(F)\) is the projection of \((F^{-1})^*:\mathrm {T}^*M\rightarrow \mathrm {T}^*M\) to \(\mathbb {P}(\mathrm {T}^*M)\) (note that \((F^{-1})^*\) is linear on fibers of \(\mathrm {T}^*M\), so this projection is well defined)

$$\begin{aligned} \mathbb {P}(F)([\theta ])=[(F^{-1})^*\theta ]. \end{aligned}$$

It is worth noticing that \(\mathbb {P}(F)\) respects the fiber bundle structure of \(\pi :\mathbb {P}(\mathrm {T}^*M)\rightarrow M\):

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(4.4)

We claim that

Let us remark that an alternative way to prove the above result is to show that \((F^{-1})^*\) maps the contact form \(\omega _R\) to \(\omega _{\mathrm {T}F(R)}\). To prove that, one uses the fact that the pullback \((F^{-1})^*\) preserves the Liouville form.

CVFs on \(\mathbb {P}(\mathrm {T}^*M)\) induced by base vector fields The results of the previous paragraph have their natural infinitesimal version.

The correctness of the above definition is a consequence of a simple observation that the contact lift preserves the composition of maps, i.e., \(\mathbb {P}(F\circ G)=\mathbb {P}(F)\circ \mathbb {P}(G)\) for any pair of maps \(F,G:M\rightarrow M\). It follows that the contact lift of the flow \(A_{t}\) is a flow of contact transformations of \(\mathbb {P}(\mathrm {T}^*M)\) and as such it must correspond to some contact vector field (cf. Definition 6).

An analogous reasoning shows that given a Caratheodory TDVF \(X_t\in \mathfrak {X}(M)\) and the related TD flow \(A_{t\tau }:M\rightarrow M\), the contact lift of the latter, i.e., \(\mathbb {P}(A_{t\tau })\), will consist of contact transformations and will satisfy all the properties of the TD flow. By the results of Proposition 4, \(\mathbb {P}(A_{t\tau })\) is a TD flow associated with some contact TDVF (see also Lemma 2). Obviously, this field is just \(\mathbf {C}_{X_t}\). The justification of this fact is left for the reader.

Let now \(q(t)\in Q\) be the trajectory of the (CS) associated with a given control \(u(t)\in U\). It is convenient to regard the related trajectory \(\varvec{q}(t)=(q(t),q_0(t))\) in the extended configuration space \(\varvec{Q}:=Q\times \mathbb {R}\), where \(q_0(t):=\int _0^tL(q(s),u(s)){\text {d}}s\) is the cost of the trajectory at time t.² In fact, \(\varvec{q}(t)\) is a trajectory (associated with the same control u(t)) of the following extension of (CS):

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with \(\varvec{f}:=(f,L\cdot \partial _{q_0}): \varvec{Q}\times U\rightarrow \mathrm {T}\varvec{Q}=\mathrm {T}Q\times \mathrm {T}\mathbb {R}\). Here, we treat both f and L as maps from \(\varvec{Q}\times U\) invariant in the \(\mathbb {R}\)-direction in \(\varvec{Q}=Q\times \mathbb {R}\). In other words, we extended (CS) by incorporating the costs \(q_0(t)\) as additional configurations of the system. The evolution of these additional configurations is governed by the cost function L. Note that the total cost of the trajectory q(t) with \(t\in [0,T]\) is precisely \(q_0(T)\). Since the latter is fully determined by the pair (q(t), u(t)), it is natural to regard the extended pair \((\varvec{q}(t),u(t))\) rather than (q(t), u(t)) as a solution of (OCP).

Note that the extended configuration space \(\varvec{Q}=Q\times \mathbb {R}\ni \varvec{q}=(q,q_0)\) is equipped with the canonical vector field \(\varvec{\partial }_{q_0}:=(0,\partial _{q_0})\in \mathrm {T}\varvec{Q}=\mathrm {T}Q\times \mathrm {T}\mathbb {R}\). We shall denote the distribution spanned by this field by \(\varvec{\mathcal {R}}\subset \mathrm {T}\varvec{Q}\). The ray \(\varvec{\mathcal {R}}^{-}_{\varvec{q}}:=\{-r\varvec{\partial }_{q_0}\ |\ r\in \mathbb {R}_{+}\}\subset \varvec{\mathcal {R}}_{\varvec{q}}\subset \mathrm {T}_{\varvec{q}}\varvec{Q}\) contained in this distribution will be called the direction of the decreasing cost at \(\varvec{q}\in \varvec{Q}\).

Regarding technical assumptions, following [18], we shall assume that U is a subset of an Euclidean space, f(q, u) and L(q, u) are differentiable with respect to the first variable and, moreover, \(f(q,u), L(q,u), \frac{\partial f}{\partial q}(q,u)\) and \(\frac{\partial L}{\partial q}(q,u)\) are continuous as functions of (q, u). In the light of Theorem 7 below it is clear that these conditions guarantee that, for any choice of a bounded measurable control u(t) and any initial condition \(\varvec{q}(0)\), equation (CS) has a unique (Caratheodory) solution defined in a neighborhood of 0. It will be convenient to denote the TDVFs \(q\mapsto f(q,u(t))\) and \(\varvec{q}\mapsto \varvec{f}(\varvec{q},u(t))\) related with such a control u(t) by \(f_{u(t)}\) and \(\varvec{f}_{u(t)}\), respectively. In the language of Sect. 2, technical assumptions considered above guarantee that \(f_{u(t)}\) and \(\varvec{f}_{u(t)}\) are Caratheodory TDVFs. In particular, their TD flows \(F_{t\tau }:Q\rightarrow Q\) and \(\varvec{F}_{t\tau }:\varvec{Q}\rightarrow \varvec{Q}\), respectively, are well-defined families of (local) diffeomorphisms.³ Note that if \(\varvec{q}(t)\) with \(t\in [0,T]\) is a solution of (CS), then for every \(t,\tau \in [0,T]\) the map \(\varvec{F}_{t\tau }\) is well defined in a neighborhood of \(\varvec{q}(\tau )\).

In the above setting, necessary conditions for the optimality of \((\varvec{q}(t),u(t))\) are formulated in the following PMP

Let \((\varvec{q}(t), \widehat{u}(t))\) be a trajectory of (CS). By \(\varvec{F}_{t\tau }: \varvec{Q}\rightarrow \varvec{Q}\), where \(0\le \tau \le t\le T\), denote the TD flow on \(\varvec{Q}\) of the Caratheodory TDVF \(\varvec{f}_{\widehat{u}(t)}\) defined by the control \(\widehat{u}(t)\) (cf. Definition 1). In other words, given a point \(\varvec{q}\in \varvec{Q}\), the curve \(t\mapsto \varvec{F}_{t 0}(\varvec{q})\) is the a trajectory of (CS) associated with the control \(\widehat{u}(t)\) with the initial condition \(\varvec{q}(0)=\varvec{q}\).

The importance of the construction of the cone \(\varvec{\mathcal {K}}_t\) lies in the fact that it approximates the reachable set of the control system (CS) at the point \(\varvec{q}(t)\). In particular, it was proved in [18] that if at any point \(t\in [0,T]\), the interior of the cone \(\varvec{\mathcal {K}}_t\) contains the direction of the decreasing cost \(\varvec{\mathcal {R}}^-_{\varvec{q}(t)}\), then the trajectory \(t\mapsto \varvec{q}(t), t\in [0,T]\), cannot be optimal.

As a direct corollary, using basic facts about separation of convex sets, one obtains the following

The geometry of this situation is depicted in Fig. 2.

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Note that, as we have already observed in Remark 7, for abnormal solutions, we have \(\varvec{\partial }_{q_0}\in \varvec{\mathcal {H}}_t=\ker \varvec{\lambda }(t)\), and thus \(\big \langle {\varvec{\lambda }(t),\varvec{\partial }_{q_0}}\big \rangle \equiv 0\). For normal solutions it is possible to choose \(\varvec{\lambda }(t)\) in such a way that \(\big \langle {\varvec{\lambda }(t),-\varvec{\partial }_{q_0}}\big \rangle \equiv 1\) along the optimal solution.

The contact formulation of the PMP Expressing the essential geometric information of the PMP (see Fig. 2) in terms of hyperplanes \(\varvec{\mathcal {H}}_t\), instead of covectors \(\varvec{\lambda }(t)\), combined with our considerations about the canonical contact structure on \(\mathbb {P}(\mathrm {T}^*M)\) (see Sect. 4) allows to formulate the following contact version of the PMP.

Let us remark that the contact dynamics (5.7) are valid regardless of the fact whether the considered solution is normal or abnormal. We have a unique contact TDVF \(\mathbf {C}_{\varvec{f}_{\widehat{u}(t)}}\) on \(\mathbb {P}(\mathrm {T}^*\varvec{Q})\) governing the dynamics of the separating hyperplanes \(\varvec{\mathcal {H}}_t\). The difference between normal and abnormal solutions lies in the relative position of the hyperplanes \(\varvec{\mathcal {H}}_t\) with respect to the canonical vector field \(-\varvec{\partial }_{q_0}\) on \(\varvec{Q}\).

Although Eq. (5.7) has a very clear geometric interpretation it is more convenient to avoid, in applications, calculating the contact lift. Combining (5.6) with Theorem 1 allows to substitute equation (5.7) by a simple condition involving the Lie bracket.

In particular, by choosing \(\varvec{R}=\varvec{\partial }_{q_0}\) we can easily recover the results of [17]. Note that \(\varvec{R}=\varvec{\partial }_{q_0}\) is the canonical choice of a vector field transversal to all hyperplanes \(\varvec{\mathcal {H}}_t\)’s in the normal case (note that additionally \(\varvec{R}=\varvec{\partial }_{q_0}\) is \(\varvec{F}_{t\tau }\)-invariant). For such a choice of \(\varvec{R}\), the corresponding embedding \(i_{\varvec{R}}:\mathcal {U}_{\varvec{R}}\hookrightarrow \mathrm {T}^*\varvec{Q}\) is constructed simply by setting \(\big \langle {\varvec{\lambda },\varvec{\partial }_{q_0}}\big \rangle =1\), which is just the standard normalization of the normal solution. The associated contact form is \(\omega _{\varvec{R}}=\pi _2^*{\text {d}}q_0+\pi _1^*\Omega _Q\), where \(\Omega _Q\) is the Liouville form on \(\mathrm {T}^*Q\) and \(\pi _1:Q\times \mathbb {R}\rightarrow Q\) and \(\pi _2:Q\times \mathbb {R}\rightarrow \mathbb {R}\) are natural projections. As observed above, the generating function of the contact dynamics associated with \(\omega _{\varvec{R}}\) is the linear Hamiltonian (5.2). This stays in a perfect agreement with the results of Sect. 2 in [17].

For the abnormal case there is no canonical choice of the field \(\varvec{R}\) transversal to the separating planes. Yet locally such a choice (but not canonical) is possible. The resulting generating function of the contact dynamics (5.7) is again the linear Hamiltonian (5.2).

In this section, we shall apply our covariant approach to the PMP (cf. Remark 8) to concrete problems of optimal control. We shall concentrate our attention on the SR geodesic problem on a manifold Q. Our main idea is to extract, from the geometry of the cone \(\varvec{\mathcal {K}}_t\), as much information as possible about the separating hyperplane \(\varvec{\mathcal {H}}_t\) and then use the contact evolution (in the form (5.6) or (5.8)) to determine the actual extremals of the system.

A sub-Riemannian geodesic problem To be more precise we are considering a control system constituted by choosing in the tangent space \(\mathrm {T}Q\) a smooth constant rank distribution \(\mathcal {D}\subset \mathrm {T}Q\). Clearly (locally and non-canonically), by taking \(f(q,u)=\sum _{i=1}^d u^if_i(q)\), where \(u=(u^1,u^2,\ldots ,u^d)\) and \(\mathcal {D}=\big \langle {f_1,\ldots ,f_d}\big \rangle \), we may present \(\mathcal {D}\) as the image of a map \(f:Q\times U\rightarrow \mathrm {T}Q\) where \(U=\mathbb {R}^d\), with \(d:={\text {rank}}\mathcal {D}\), is an Euclidean space, i.e., a control system of type (CS). We shall refer to it as to the SR control system. In agreement with our notation from the previous Sect. 5, we will write also \(f_u(q)\) instead of \(f(q,u)\in \mathcal {D}_q\).

The geometry of cones and separating hyperplanes Observe that the image \(\varvec{f}(\varvec{q},U)\subset \mathrm {T}_{\varvec{q}}\varvec{Q}=\mathrm {T}_qQ\times \mathrm {T}_{q_0}\mathbb {R}\) is a paraboloid (see Fig. 4). The following fact is a simple consequence of (5.4).

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Here is an easy corollary from the above lemma and our previous considerations.

It turns out that in some cases the above basic information, suffices to find SR extremals. Let us study the following two examples.

In the following two subsections we shall discuss normal and abnormal SR extremals in full generality.

6.2 Normal SR extremals

Observe first that the extremal vector field \(f_{\widehat{u}(t)}\) is normalized by \(g(f_{\widehat{u}(t)},f_{\widehat{u}(t)})\equiv 1\) along every solution of the SR geodesic problem. Indeed, this follows easily from the standard argument involving the Cauchy–Schwartz inequality. From now on we shall thus assume that the extremal vector field \(f_{\widehat{u}(t)}\) is normalized in a neighborhood of a considered trajectory q(t). This assumption allows for an elegant geometric characterization of normal SR extremals in terms of the distribution

$$\begin{aligned} \mathcal {D}^\perp _{q(t)}:=\{Y\in \mathcal {D}_{q(t)}\ |\ g(Y,f_{\widehat{u}(t)})=0\} \end{aligned}$$

consisting of those elements of \(\mathcal {D}\) which are g-orthogonal to \(f_{\widehat{u}(t)}\) along q(t). Note that \(\mathcal {D}^\perp _{q(t)}\) is a subdistribution of \(\mathcal {D}\) along q(t).

Theorem 6

([4]) Assume that the field \(f_{\widehat{u}(t)}\) is normalized, i.e., \(g(f_{\widehat{u}(t)},f_{\widehat{u}(t)})\equiv 1\). Then, for the SR geodesic problem introduced above, the following are equivalent:

1. (a)

   The pair \((\varvec{q}(t),\widehat{u}(t))\) is a normal SR extremal.

    
2. (b)
   The velocity \(f_{u(t)}(q(t))\) is of class ACB with respect to t, and the smallest distribution \(F_{t\tau }\)-invariant along q(t) and containing \(\mathcal {D}^\perp _{q(t)}\), i.e.,

   $$\begin{aligned} F_{\bullet }(\mathcal {D}^\perp )_{q(t)}={\text {vect}}_{\mathbb {R}}\{\mathrm {T}F_{t\tau }(Y)\ |\ Y\in \mathcal {D}_{q(\tau )}, \; g(Y,f_{\widehat{u}(t)})=0, \; 0\le \tau \le T\} \end{aligned}$$

   does not contain \(f_{\widehat{u}(t)}(q(t))\) for any \(t\in [0,T]\). Here \(F_{t\tau }\) denotes the TD flow (in Q) of the Caratheodory TDVF \(f_{\widehat{u}(t)}\).
    

The condition that the velocity curve \(t\mapsto f_{u(t)}(q(t))\) is of class ACB is equivalent to the fact that the velocities are preserved by the flow \(F_{t \tau }\), i.e.,

$$\begin{aligned} \mathrm {T}F_{t\tau }\left[ f_{u(\tau )}(q(\tau ))\right] =f_{u(t)}(q(t)) \quad \text { for every } \; t,\tau \in [0,T]. \end{aligned}$$

(6.5)

Theorem 3.1 of [4] contains a formulation of the above result equivalent to ours.

Again if q(t) is sufficiently regular we can use the method introduced in Lemma 3 to check condition (b) in the above theorem. The result stated below can be easily derived from Theorem 6 using similar arguments as in the proof of Corollary 1. For the case \({\text {rank}}\mathcal {D}=2\) it was proved as Theorem 6 in [16].

Corollary 2

Let X be a \(C^\infty \)-smooth \(\mathcal {D}\)-valued vector field and let q(t) with \(t\in [0,T]\) be an integral curve of X. Then q(t) is an SR normal extremal in the following two (non-exhaustive) situations:

• The distribution spanned by the iterated Lie brackets of X and all possible smooth \(\mathcal {D}\)-valued vector fields g-orthogonal to X, i.e.,

  $$\begin{aligned} {\text {ad}}^\infty _X(\mathcal {D}^\perp )=\big \langle {{\text {ad}}_X^k(Y)\ |\ Y\in \varGamma (\mathcal {D}), \; g(X,Y)=0, \; k=0,1,2,\ldots }\big \rangle \end{aligned}$$

  is of constant rank r along q(t) and it does not contain X(q(t)) for any \(t\in [0,T]\).
• There exists a smooth distribution \(\mathcal {B}\) on Q, such that

  $$\begin{aligned}{}[X,\mathcal {B}]_{q(t)}\subset \mathcal {B}_{q(t)}, \quad X(q(t))\notin \mathcal {B}_{q(t)} \quad \text { and } \quad \mathcal {D}^\perp _{q(t)}\subset \mathcal {B}_{q(t)} \quad \text { for any } \; t\in [0,T]. \end{aligned}$$

Proof of Theorem 6

The fact that property (6.5) is equivalent to \(t\mapsto f_{u(t)}(q(t))\) being of class ACB follows directly from Lemma 12. Indeed, the field \(f_{u(t)}\) satisfies \([f_{u(t)},f_{u(t)}]_{q(t)}=0\) a.e. Thus, it is ACB along q(t) if and only if it is respected by the flow \(F_{t\tau }\).

Assume first that \((\varvec{q}(t),\widehat{u}(t))\) is a normal SR extremal. Let \(\varvec{\mathcal {H}}_t\subset \mathrm {T}_{\varvec{q}(t)}\varvec{Q}\) be the related curve of separating hyperplanes given by the PMP. Note that, since \(\varvec{\mathcal {H}}_t\) for each t is a hyperplane transversal to the line \(\mathcal {R}_{\varvec{q}(t)}\subset \mathrm {T}_{\varvec{q}(t)} \varvec{Q}\), it must be of the form

$$\begin{aligned} \varvec{\mathcal {H}}_t=\{Y+\alpha _t(Y)\partial _{q_0}\ |\ Y\in \mathrm {T}_{q(t)}Q\}, \end{aligned}$$

where \(\alpha _t:\mathrm {T}_{q(t)}Q\rightarrow \mathbb {R}\) is a linear map. Using the results of Lemma 11 we know that Open image in new window , i.e., \(\alpha _t(f_{u(t)}(q(t)))=1\) and \(\mathcal {D}^\perp _{q(t)}\subset \ker \alpha _t\). In particular, \(f_{u(t)}(q(t))\) is transversal to \(\ker \alpha _t\supset D^\perp _{q(t)}\).

Now, since \(\varvec{f}_{\widehat{u}(t)}=f_{\widehat{u}(t)}+\frac{1}{2}\partial _{q_0}\) (here we use the normalization of \(f_{\widehat{u}(t)}\)), it is clear that

$$\begin{aligned} \varvec{F}_{t\tau }(q,q_0)=(F_{t\tau }(q),q_0+\frac{1}{2}(t-\tau )). \end{aligned}$$

It follows that \(\mathrm {T}\varvec{F}_{t\tau }\left[ Y+\alpha _{\tau }(Y)\partial _{q_0}\right] =\mathrm {T}F_{t\tau }(Y)+\alpha _{\tau }(Y)\partial _{q_0}\), for every \(t,\tau \in [0,T]\) and \(Y\in \mathrm {T}_{q(t)}Q\). Since \(\mathrm {T}\varvec{F}_{t\tau }\left( \varvec{\mathcal {H}}_{\tau }\right) =\varvec{\mathcal {H}}_t\), the above vector must be of the form \(X+\alpha _t(X)\partial _{q_0}\), where \(X=\mathrm {T}F_{t\tau }(Y)\). That is, \(\alpha _t(\mathrm {T}F_{t\tau }(X))=\alpha _\tau (X)\). In particular, \(t\mapsto \alpha _t\) is continuous and, moreover, \(\mathrm {T}F_{t\tau }\left( \ker \alpha _\tau \right) =\ker \alpha _t\) for every \(t,\tau \in [0,T]\). We conclude that \(\ker \alpha _t\) is a distribution along q(t) which is \(F_{t\tau }\)-invariant, contains \(D^\perp _{q(t)}\) and is transversal to \(f_{u(t)}(q(t))\). Clearly, \(F_{\bullet }(D^\perp )_{q(t)}\subset \ker \alpha _t\) and thus it is also transversal to \(f_{u(t)}(q(t))\).

To prove that \(t\mapsto f_{u(t)}(q(t))\) is ACB, observe first that \(D^\perp _{q(t)}=\ker \alpha _t\cap D_{q(t)}\) admits locally a g-orthonormal basis of ACB sections. Indeed, \(\ker \alpha _t\) is charming since it is \(F_{t\tau }\)-invariant (cf. Proposition 1). Let now \(\{X_1,\ldots ,X_{n-1}\}\) be a local basis of ACB sections of \(\ker \alpha _t\) along q(t). Choose a minimal subset of this basis, say \(\{X_1,\ldots ,X_s\}\), such that \(\big \langle {X_1,\ldots ,X_s}\big \rangle _{q(t)}\oplus \mathcal {D}^\perp _{q(t)}=\ker \alpha _t\) for every t in a relatively compact neighborhood of a given point \(t_0\in [0,T]\). Extend locally the SR metric g to a metric \(\widetilde{g}\) on \(\ker \alpha _t\) by taking \(\widetilde{g}\big |_{\mathcal {D}^\perp _{q(t)}}=g\big |_{\mathcal {D}^\perp _{q(t)}}\) and by setting vectors \(X_1,\ldots ,X_s\) to be \(\widetilde{g}\)-orthonormal and \(\widetilde{g}\)-orthogonal to \(\mathcal {D}^\perp _{q(t)}\). Clearly, this new metric is ACB in the considered neighborhood of \(t_0\). Now we can apply Lemma 13 to the ACB basis \(\{X_1,\ldots ,X_{n-1}\}\) and obtain an ACB \(\widetilde{g}\)-orthonormal basis \(\{X_1,\ldots ,X_s,Y_{s+1},\ldots ,Y_{n-1}\}\) of \(\ker \alpha _t\). Clearly, by the construction of the Gram–Schmidt algorithm, \(\{Y_{s+1},\ldots ,Y_{n-1}\}\) is a \(\widetilde{g}\)-, and thus also a g-orthonormal basis of \(\mathcal {D}^\perp _{q(t)}\) (the relative compactness of the neighborhood is used to assure that the \(\widetilde{g}\)-lengths of sections \(X_i\) are separated from zero).

Now let us choose any ACB section \(Y_n\) of \(\mathcal {D}_{q(t)}\) which is transversal to \(\mathcal {D}^\perp _{q(t)}\). Again using Lemma 13 we modify the ACB local basis \(\{Y_{s+1},\ldots ,Y_{n-1},Y_n\}\) of \(\mathcal {D}_{q(t)}\) to a g-orthonormal ACB local basis \(\{Y_{s+1},\ldots ,Y_{n-1},\widetilde{Y}_n\}\). Obviously, \(\widetilde{Y}_n(q(t))\) is a g-normalized vector g-orthogonal to \(\mathcal {D}^\perp _{q(t)}=\big \langle {Y_{s+1},\ldots ,Y_{n-1}}\big \rangle \), thus \(\widetilde{Y}_n(q(t))=\pm f_{u(t)}(q(t))\). Now \(\alpha _t(\widetilde{Y}_n(q(t)))=\pm \alpha _t(f_{u(t)}(q(t)))=\pm 1\). And since both \(\alpha _t\) and \(\widetilde{Y}_n(q(t))\) are continuous with respect to t the sign ± must be constant along [0, T]. We conclude that \(t\mapsto f_{u(t)}(q(t))\) is ACB alike \(t\mapsto \widetilde{Y}_n(q(t))\).

Conversely, assume that (b) holds. The crucial step is to build, along the projected trajectory \(q(t)\in Q\), a splitting \(\mathrm {T}_{q(t)} Q=\mathcal {B}_{q(t)}\oplus \big \langle {f_{\widehat{u}(t)}(q(t))}\big \rangle \), where \(\mathcal {B}_{q(t)}\) is a co-rank one distribution along q(t), which is \(F_{t\tau }\)-invariant along q(t) and contains \(\mathcal {D}^\perp _{q(t)}\). Such a \(\mathcal {B}_{q(t)}\) can be constructed by adding, if necessary, to \(F_{\bullet }(\mathcal {D}^\perp )_{q(t)}\) several vector fields of the form \(F_{t0}(X_{i})\), where \(X_{i}\in \mathrm {T}_{q(0)}Q\) together with \(F_{\widehat{u}(0)}(q(0))\) are independent. Clearly, in this way we can build \(\mathcal {B}_{q(t)}\) which is \(F_{t\tau }\)-invariant along q(t), of co-rank one and contains \(\mathcal {D}^\perp _{q(t)}\). What is more, since \(f_{u(t)}(q(t))\) is \(F_{t\tau }\)-invariant, the flow \(F_{t\tau }\) respects the splitting \(\mathcal {B}_{q(t)}\oplus \big \langle {f_{\widehat{u}(t)}(q(t))}\big \rangle \).

Now we can construct the curve of separating hyperplanes \(\varvec{\mathcal {H}}_t\) for \(\varvec{q}(t)\) by the formula

$$\begin{aligned} \varvec{\mathcal {H}}_t:=(\mathcal B_{q(t)}\oplus 0\cdot \partial _{q_0})\oplus \big \langle {f_{\widehat{u}(t)}(q(t))+1\cdot \partial _{q_0}}\big \rangle . \end{aligned}$$

By construction it is clear that \(\varvec{\mathcal {H}}_t\) is a hyperplane in \(\mathrm {T}_{\varvec{q}(t)}\varvec{Q}\) which contains the tangent space to the paraboloid (6.1) and does not contain the line \(\varvec{\mathcal {R}}_{\varvec{q}(t)}\). From these properties we conclude that \(\varvec{\mathcal {H}}_t\) separates strictly the ray \(\varvec{\mathcal {R}}^-_{\varvec{q}(t)}\) from the elements of \(\varvec{\mathcal {K}}_t\) of the form \(\varvec{f}_{v}(q(t))-\varvec{f}_{\widehat{u}(t)}(q(t))\). It is also clear (here we use the normalization of \(f_{\widehat{u}(t)}\), cf. the first part of this proof) that \(\varvec{\mathcal {H}}_t\) is \(\varvec{F}_{t\tau }\)-invariant along \(\varvec{q}(t)\). Thus, it also separates strictly the ray \(\varvec{\mathcal {R}}^-_{\varvec{q}(t)}=\mathrm {T}\varvec{F}_{t\tau }(\varvec{\mathcal {R}}^-_{q(\tau )})\) from the elements of \(\varvec{\mathcal {K}}_t\) of the form \(\mathrm {T}\varvec{F}_{t\tau }[\varvec{f}_{v}(\varvec{q}(\tau ))-\varvec{f}_{\widehat{u}(t)} (\varvec{q}(\tau ))]\). Consequently, using the fact that \(\varvec{\mathcal {H}}_t\) and \(\varvec{\mathcal {R}}^-_{\varvec{q}(t)}\) are convex, we can use \(\varvec{\mathcal {H}}_t\) to separate strictly \(\varvec{\mathcal {R}}^-_{\varvec{q}(t)}\) from any finite convex combination of the above-mentioned elements of \(\varvec{\mathcal {K}}_t\). Since \(\varvec{\mathcal {K}}_t\) is by definition the closure of the set of such finite convex combinations, \(\varvec{\mathcal {H}}_t\) is indeed the separating hyperplane described by the PMP. \(\square \)

A remark on smoothness of normal SR geodesics As was proved above normal SR extremals are \(C^1\)-smooth (and even more: their derivatives are ACB maps). It is worth discussing geometric reasons for this regularity in a less technical manner than in the proof of Theorem 6. Let \(\varvec{q}(t)\) be such an extremal and let \(\varvec{\mathcal {H}}_t\) be the corresponding curve of supporting hyperplanes. As we know from Lemma 11

$$\begin{aligned} \mathcal {D}^\perp _{q(t)}\oplus 0\cdot \partial _{q_0}\subset \varvec{\mathcal {H}}_t \quad \text { and } \quad f_{u(t)}(q(t))+\partial _{q_0}\in \varvec{\mathcal {H}}_t \quad \text { for every } \; t\in [0,T]. \end{aligned}$$

These two facts are enough to exclude, at least in a heuristic way, the existence of singularities of corner-type and of cusp-type along q(t). Indeed, since \(\varvec{\mathcal {H}}_t\) is \(\varvec{F}_{t\tau }\)-invariant it must be continuous. Note that by the continuity of \(\varvec{\mathcal {H}}_t\), the limit subspaces \(\mathcal {D}^\perp _{q(t_0)\pm }\oplus 0\cdot \partial _{q_0}\) coming from both sides of a given point \(t_0\in [0,T]\) must belong to \(\varvec{\mathcal {H}}_{t_0}\). Now if \(\varvec{q}(t)\) had a corner-type singularity at \(t_0\), these limit subspaces would be different and thus they would span together the whole space \(\mathcal {D}_{q(t_0)}\oplus 0\cdot \partial _{q_0}\) (cf. Fig. 6). In particular, \(f_{u(t_0)}(q(t_0))+0\cdot \partial _{q_0}\in \mathcal {D}_{q(t_0)}\oplus 0\cdot \partial _{q_0}\) would belong to \(\varvec{\mathcal {H}}_{t_0}\). Yet, since \(f_{u(t_0)}(q(t_0))+\partial _{q_0}\in \varvec{\mathcal {H}}_{t_0}\), this would mean that also the difference of the latter vectors, \(0+\partial _{q_0}\), lies in \(\varvec{\mathcal {H}}_{t_0}\), which is impossible since \(\varvec{q}(t)\) is normal.

Open image in new window

In a similar way one deals with a cusp-type singularity. At a cusp we would have limit vectors \(\pm f_{u(t_0)}(q(t_0))+\partial _{q_0}\) in \(\varvec{\mathcal {H}}_{t_0}\) (see Fig. 6). Now \(0+2\partial _{q_0}\), the sum of these two vectors, would belong to \(\varvec{\mathcal {H}}_{t_0}\) which contradicts the normality of the extremal. Roughly speaking, the existence of singularities of corner-type or cusp-type implies \(\partial _{q_0}\in \mathcal {H}_{t_0}\), i.e., either a trajectory is not an extremal or it is abnormal.

Examples

Example 8

(Geodesic equation revisited) Theorem 6 provides an alternative way to derive the geodesic equation in the Riemannian case (i.e., when \(\mathcal {D}=\mathrm {T}Q\)). Let \((\varvec{q}(t),\widehat{u}(t))\) be a trajectory of the SR control system (we shall assume that \(f_{\widehat{u}(t)}\) is normalized). Since \(\mathcal {D}=\mathrm {T}Q\), by the assertion of Theorem 5, in the Riemannian case there are no abnormal extremals.

Since \(\mathcal {D}^\perp _{q(t)}=\{f_{\widehat{u}(t)}(q(t))\}^\perp \) is of co-rank one, the only distribution of higher rank along q(t) containing \(\mathcal {D}^\perp _{q(t)}\) is \(\mathrm {T}_{q(t)} Q\) which contains also \(f_{\widehat{u}(t)}(q(t))\). Now, by Theorem 6, \((\varvec{q}(t),\widehat{u}(t))\) is a normal extremal if and only if \(F_{\bullet }(\mathcal {D}^\perp )_{q(t)}=\mathcal {D}^\perp _{q(t)}\), i.e., if

$$\begin{aligned} \mathrm {T}F_{t\tau }(\mathcal {D}^\perp _{q(\tau )})=\mathcal {D}^\perp _{q(t)}, \end{aligned}$$

for every \(t,\tau \in [0,T]\). By the results of Theorem 1 this is equivalent to

$$\begin{aligned}{}[f_{\widehat{u}(t)},\mathcal {D}^\perp ]_{q(t)}=\mathcal {D}^\perp _{q(t)}, \end{aligned}$$

i.e., \(g([f_{\widehat{u}(t)},Y],f_{\widehat{u}(t)})_{q(t)}=0\) whenever \(g(Y,f_{\widehat{u}(t)})_{q(t)}=0\). Now for such a Y, after introducing a metric-compatible connection as in Example 1, we have

$$\begin{aligned} 0&=g([f_{\widehat{u}(t)},Y],f_{\widehat{u}(t)})=g(\nabla _{f_{\widehat{u}(t)}}Y-\nabla _Yf_{\widehat{u}(t)}-T_\nabla (f_{\widehat{u}(t)}, Y),f_{\widehat{u}(t)}) \\&=g(\nabla _{f_{\widehat{u}(t)}}Y,f_{\widehat{u}(t)})-g(\nabla _Yf_{\widehat{u}(t)},f_{\widehat{u}(t)})-g(T_\nabla (f_{\widehat{u}(t)}, Y),f_{\widehat{u}(t)}) \\&=f_{\widehat{u}(t)} g(Y,f_{\widehat{u}(t)})-g(Y,\nabla _{f_{\widehat{u}(t)}}f_{\widehat{u}(t)})-\frac{1}{2} Yg(f_{\widehat{u}(t)},f_{\widehat{u}(t)})-g(T_\nabla (f_{\widehat{u}(t)}, Y),f_{\widehat{u}(t)}). \end{aligned}$$

Using the fact that \(g(Y,f_{\widehat{u}(t)})\equiv 0\) and that \(g(f_{\widehat{u}(t)}, f_{\widehat{u}(t)})\equiv 1\) we get

$$\begin{aligned} g(Y,\nabla _{f_{\widehat{u}(t)}}f_{\widehat{u}(t)})+g(T_\nabla (f_{\widehat{u}(t)}, Y),f_{\widehat{u}(t)})=0 \end{aligned}$$

in agreement with the results of Example 1.

Example 9

(Heisenberg system) Consider an SR system on \(\mathbb {R}^3\ni (x,y,z)\) constituted by a 2-distribution

$$\begin{aligned} \mathcal {D}_{(x,y,z)}=\big \langle {Y:=\partial _x-y\partial _z,Z:=\partial _y+x\partial _z}\big \rangle \end{aligned}$$

and an SR metric such that the fields Y and Z form an orthonormal basis. Such a system is usually called the Heisenberg system. It is easy to check that the system in question is strongly bracket generating (cf. Example 5) and as such does not admit any abnormal SR extremal. Our goal will thus be to determine the normal SR extremals using the results of Theorem 6.

Take now any normalized \(\mathcal {D}\)-valued vector field \(X=X_t:=\phi (t) Y+\psi (t) Z\), where \(\phi ^2+\psi ^2=1\). We have \(\mathcal {D}^\perp =\big \langle {X'={X_t}':=\psi (t) Y-\phi (t) Z}\big \rangle \) and, by the results of Theorem 6, the integral curve q(t) of X is an SR normal extremal if and only if \(F_{\bullet }(\mathcal {D}^\perp )_{q(t)}\) does not contain X at any point q(t). Clearly distribution \(F_{\bullet }(\mathcal {D}^\perp )_{q(t)}\), being \({\text {ad}}_X\)-invariant, contains the fields \(X', [X,X'], [X,[X,X']]\), etc. Skipping some simple calculations one can show that

$$\begin{aligned}{}[X,X']=-2\partial _z+AY+BZ, \end{aligned}$$

where

$$\begin{aligned} \begin{aligned} A= & {} \phi Y(\psi )-\psi Y(\phi )+\frac{1}{2} Z(\phi ^2+\psi ^2)=\phi Y(\psi )-\psi Y(\phi ) \\ B= & {} \phi Z(\psi )-\psi Z(\phi )-\frac{1}{2} Y(\phi ^2+\psi ^2)=\phi Z(\psi )-\psi Z(\phi ). \end{aligned} \end{aligned}$$

(6.6)

Let us now present vector field \([X,X']\) as (note that \(\{X,X'\}\) is a basis of sections of \(\mathcal {D}\))

$$\begin{aligned}{}[X,X']=-2\partial _z+\alpha X+\beta X'. \end{aligned}$$

Then

$$\begin{aligned}{}[X,[X,X']]=X(\alpha )X+\beta [X,X']+X(\beta ) X'. \end{aligned}$$

Now clearly \(\big \langle {X',[X,X'],[X,[X,X']]}\big \rangle \) would contain X if and only if \(X(\alpha )\ne 0\). Thus, a necessary condition for an integral curve q(t) of X to be a normal SR extremal is that \(\alpha =const\) along q(t). Note that if \(X(\alpha )=const\), then the integral curves of X will indeed be normal SR extremals, as then \([X,[X,X']]=\beta [X,X']+X(\beta ) X'\) and, consequently, \(F_{\bullet }(\mathcal {D}^\perp )_{q(t)}\) will be equal to the 2-dimensional distribution \(\big \langle {X',[X,X']}\big \rangle \) which does not contain X (cf. Corollary 2).

By comparing the coefficients of \([X,X']\) expressed in terms of the bases \(\{Y,Z\}\) and \(\{X,X'\}\) we get

$$\begin{aligned} A Y+B Z= & {} \alpha X+\beta X'=\alpha (\phi Y+\psi Z)+\beta (\psi Y-\phi Z) \\= & {} (\alpha \phi +\beta \psi )Y+(\alpha \psi -\beta \phi )Z. \end{aligned}$$

Thus, by (6.6),

$$\begin{aligned} \phi Y(\psi )-\psi Y(\phi )-\alpha \phi&=\beta \psi \\ \phi Z(\psi )-\psi Z(\phi )-\alpha \psi&=-\beta \phi . \end{aligned}$$

Consequently,

$$\begin{aligned} \phi ^2 Y(\psi )-\phi \psi Y(\phi )-\alpha \phi ^2=\beta \phi \psi =-\phi \psi Z(\psi )+\psi ^2 Z(\phi )+\alpha \psi ^2, \end{aligned}$$

which, after substituting \(\phi Y+\psi Z\) by X, leads to

$$\begin{aligned} X(\psi /\phi )=\frac{\phi X(\psi )-\psi X(\phi )}{\phi ^2}=\alpha (1+(\psi /\phi )^2), \end{aligned}$$

i.e., the quotient \(\psi /\phi \) satisfies the equation \(X(x)=\alpha (1+x^2)\), where \(\alpha \) is a constant. For \(\alpha =0\) we get \(x=const\) (i.e., \(\phi \) and \(\psi \) are constant along q(t)), and for \(\alpha \ne 0\) we get \(x={\text {arctan}}(\alpha t+\gamma )\) (i.e., \(\phi =\cos (\alpha t+\gamma )\) and \(\psi =\sin (\alpha t+\gamma )\)). This corresponds to the two well-known families of normal SR extremals of the Heisenberg system (see Sect. 2 of [16]), whose projections to the (x, y)-plane are straight lines and circles, respectively.