A contact covariant approach to optimal control with applications to subRiemannian geometry
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Abstract
We discuss contact geometry naturally related with optimal control problems (and Pontryagin Maximum Principle). We explore and expand the observations of Ohsawa (Autom J IFAC 55:1–5, 2015), providing simple and elegant characterizations of normal and abnormal subRiemannian extremals.
Keywords
Pontryagin maximum principle Contact geometry Contact vector field SubRiemannian geometry Abnormal extremalMathematics Subject Classification
49K15 53D10 53C17 58A301 Introduction
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 welldefined 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 abovementioned 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 timedependent vector field on \(Q\times \mathbb {R}\)). Finally, using the wellknown 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 abovementioned 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 nonuniqueness of the choice of \(\varvec{\varLambda }_t\)).
 Theorem 5 completely characterizes abnormal SR extremals. It states that an absolutely continuous curve \(q(t)\in Q\) tangent to \(\mathcal {D}\) is an abnormal extremal if and only if the minimal distribution along q(t) which contains \(\mathcal {D}_{q(t)}\) and is invariant along q(t) under the flow of the extremal vector field is of rank smaller than \(\dim Q\). As a special case (for smooth vector fields) we obtain, in Corollary 1, the following result: if the distribution spanned by the iterated Lie brackets of a given \(\mathcal {D}\)valued vector field \(X\in \varGamma (\mathcal {D})\) with all possible \(\mathcal {D}\)valued vector fields, i.e.,is of constant rank smaller than \(\dim Q\), then the integral curves of X are abnormal SR extremals.$$\begin{aligned} \big \langle {{\text {ad}}_X^k(Z)\ \ Z\in \varGamma (\mathcal {D}),\; k=0,1,2,\ldots }\big \rangle \end{aligned}$$
 Theorem 6 in a similar manner (yet under an additional assumptions that the controls are normalized with respect to the SR metric g) provides a complete characterization of normal SR extremals. It states that an absolutely continuous curve \(q(t)\in Q\), tangent to \(\mathcal {D}\), is a normal extremal if and only if it is of class \(C^1\) with an absolutely continuous derivative and if the minimal distribution along q(t) which contains these elements of \(\mathcal {D}_{q(t)}\) that are gorthogonal to \(\dot{q}(t)\) and is invariant along q(t) under the flow of the extremal vector field does not contain the direction tangent to q(t) at any point. Again in the smooth case we conclude, in Corollary 2, that if for a given normalized vector field \(X\in \varGamma (\mathcal {D})\) the distribution spanned by the iterated Lie brackets of X with all possible \(\mathcal {D}\)valued vector fields gorthogonal to X, i.e.,is of constant rank and does not contain X at any point of q(t), then the integral curves of X are normal SR extremals.$$\begin{aligned} \big \langle {{\text {ad}}_X^k(Z)\ \ Z\in \varGamma (\mathcal {D}),\; g(Z,X)=0,\; k=0,1,2,\ldots }\big \rangle \end{aligned}$$
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 nonsmooth, 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 rediscover some results of [16, 22] concerning rank2 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 timedependent vector fields, distributions, etc.) in the nonsmooth and timedependent 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 subRiemannian 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.
2 Technical preliminaries
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 timedependent vector field (TDVF) with the timedependence 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 abovementioned 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 selfcontained. 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 abovementioned 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 \(A_{tt_0}(x_0):=x(t;t_0,x_0)\).
Lemma 1

For t close enough to \(t_0\) the maps \(A_{tt_0}:M\rightarrow M\) are welldefined local diffeomorphisms.
 Moreover, they satisfy the following propertieswhenever both sides are defined.$$\begin{aligned} A_{t_0t_0}={\text {id}}_M\quad \text { and }\quad A_{t\tau }(A_{\tau t_0})=A_{t t_0}\ , \end{aligned}$$(2.2)
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 \(X_t\).
Definition 1
The family of local diffeomorphisms \(A_{t\tau }:M\rightarrow M\) described in the above lemma will be called the timedependent flow of \(X_t\) (TD flow).
Clearly \(A_{tt_0}\) is a natural timedependent 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.
Lemma 2
Let \(A_{t\tau }:M\rightarrow M\) be a family of local diffeomorphisms satisfying (2.2) and such that for each choice of \(x_0\in M\) and \(t_0\in \mathbb {R}\) the map \(t\mapsto A_{tt_0}(x_0)\) is ACB. Then \(A_{t\tau }\) is a TD flow of some Caratheodory TDVF \(X_t\).
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 \(X_t\).
Definition 2
Let \(x(t)=x(t;t_0,x_0)\) with \(t\in [t_0,t_1]\) be an integral curve of a Caratheodory TDVF \(X_t\). A distribution \(\mathcal {B}\) along x(t) is a family of linear subspaces \(\mathcal {B}_{x(t)}\subset \mathrm {T}_{x(t)} M\) attached at each point of the considered curve. In general, the dimension of \(\mathcal {B}_{x(t)}\) may vary from point to point.
By an ACB section of \(\mathcal {B}\) we will understand a vector field Z along x(t) such that \(Z(x(t))\in \mathcal {B}_{x(t)}\) for every \(t\in [t_0,t_1]\) and that the map \(t\mapsto Z(x(t))\) is ACB. The space of such sections will be denoted by \(\varGamma _{ACB}(\mathcal {B})\). A distribution \(\mathcal {B}\) along x(t) shall be called charming if pointwise it is spanned by a finite set of elements of \(\varGamma _{ACB}(\mathcal {B})\).
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 nonsmooth 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 \(x(t;t_0,x_0)\).
Proposition 1

A restriction of a locally finitely generated smooth distribution on M to an ACB curve \(x(t)=x(t;t_0,x_0)\) is charming.

Let \(A_{t\tau }\) be the TD flow of a Caratheodory TDVF \(X_t\) and let \(\mathcal {B}\) be a distribution along an integral curve \(x(t)=x(t;t_0,x_0)\) of \(X_t\). Then if \(\mathcal {B}\) is \(A_{t\tau }\)invariant along x(t), it is also charming.
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.
Proposition 2
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.
Definition 3
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.
Proposition 3
Assuming that \(t\mapsto Z(x(t))\) is an ACB map and that \(X_t\) is a Caratheodory TDVF, the Lie bracket \([X_t,Z]_{x(t)}\) is defined by formula (2.3) almost everywhere along x(t). In fact, it is well defined at all regular points of \(t\mapsto Z(x(t))\). Moreover, \(t\mapsto [X_t,Z]_{x(t)}\) is a measurable and locally bounded map.
The Lie bracket \([X_t,Z]_{x(t)}\) is completely determined by the values of Z along x(t) and by the values of \(X_t\) in a neighborhood of x(t).
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
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).
Theorem 1
 (a)
\(\mathcal {B}\) is respected by the TD flow \(A_{t\tau }\) of \(X_t\) along x(t).
 (b)
\(\mathcal {B}\) is a charming distribution \(X_t\)invariant and of constant rank 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.
Lemma 3
Proof
The justification of the above result is quite simple. By construction, \({\text {ad}}^\infty _X(\mathcal {B})_{x(t)}\) is the smallest distribution along x(t) containing \(\mathcal {B}_{x(t)}\) and closed under the operation \({\text {ad}}_X=[X,\cdot ]\). It is clear that \({\text {ad}}^\infty _X(\mathcal {B})\) is spanned by a finite number of smooth vector fields of the form \({\text {ad}}_X^k(Z)\), where \(Z\in \varGamma (\mathcal {B})\), and thus it is charming. Since it is also of constant rank along x(t) we can use Theorem 1 (for a timeindependent vector field X) to prove that \({\text {ad}}^\infty _X(\mathcal {B})_{x(t)}\) is invariant along x(t) under the flow \(A_{t}\). We conclude that \(A_{\bullet }(\mathcal {B})_{x(t)}\subset {\text {ad}}^\infty _X(\mathcal {B})_{x(t)}\). On the other hand, since \(A_{\bullet }(\mathcal {B})_{x(t)}\) is \(A_{t}\)invariant along x(t), again by Theorem 1, it must be closed with respect to the operation \([X,\cdot ]\). In particular, it must contain the smallest distribution along x(t) containing \(\mathcal {B}_{x(t)}\) and closed under the operation \([X,\cdot ]\). Thus, \(A_{\bullet }(\mathcal {B})_{x(t)}\supset {\text {ad}}^\infty _X(\mathcal {B})_{x(t)}\). This ends the proof. \(\square \)
Remark 1
Let us remark that the construction provided by (2.4) would be, in general, not possible in all nonsmooth cases. The basic reason is that the Lie bracket defined by (2.3) is of regularity lower than the initial vector fields, i.e., \([X_t,Z]\) may not be ACB along x(t) even if so were \(X_t\) and Z. Thus, by adding the iterated Lie brackets to the initial distribution \(\mathcal {B}\), we may loose the property that it is charming (cf. also a remark following Definition 4) which is essential for Theorem 1 to hold.
Also the constant rank condition is important, as otherwise the correspondence between \(X_t\) and \(A_{t\tau }\)invariance provided by Theorem 1 does not hold. If (2.4) is not of constant rank along x(t) we may only say that \({\text {ad}}^\infty _X(\mathcal {B})_{x(t)}\subset A_{\bullet }(\mathcal {B})_{x(t)}\) (see also Remark 10).
It is worth noticing that this situation resembles the wellknown results of Sussmann [19] concerning the integrability of distributions: being closed under the Lie bracket is not sufficient for integrability, as the invariance with respect to the flows of distributionvalued vector fields is also needed. After adding an extra assumption that the rank of the distribution is constant, the latter condition can be relaxed.
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.
Lemma 4
Let \(\mathcal {D}\) be a smooth distribution of constant rank on \(M, X_t\) a Caratheodory \(\mathcal {D}\)valued TDVF and \(x(t)=x(t;t_0,x_0)\) (with \(t\in [t_0,t_1]\)) an integral curve of \(X_t\). Let \(\mathcal {B}\) be a charming distribution along x(t), such that \(\mathcal {D}_{x(t)}\subset \mathcal {B}_{x(t)}\) for every t. Then the property of \(\mathcal {B}\) being \(X_t\)invariant along x(t) depends only on the values of \(X_t\) along x(t).
The proof is given in the Appendix.
3 The basics of contact geometry
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 corank one distribution \(\mathcal {C}\subset \mathrm {T}\mathcal {M}\) satisfying a certain maximum nondegeneracy 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 corank one distribution \(\mathcal {C}\) on \(\mathcal {M}\). Sometimes it will be convenient to treat \(\mathcal {C}\) as a vector subbundle of \(\mathrm {T}\mathcal {M}\).
Definition 5
A pair \((\mathcal {M},\mathcal {C})\) consisting of a smooth manifold \(\mathcal {M}\) and a smooth corank one distribution \(\mathcal {C}\subset \mathrm {T}\mathcal {M}\) is called a contact manifold if the associated \(\mathrm {N}\mathcal {C}\)valued 2form \(\beta \) is nondegenerate, i.e., if \(\beta (X,\cdot )\equiv 0\) implies \(X\equiv 0\).
Sometimes we call \(\mathcal {C}\) a contact structure or a contact distribution on \(\mathcal {M}\).
Observe that \(\mathcal {C}\) is necessarily of even rank (\(\mathcal {M}\) is odddimensional). This follows from a simple fact from linear algebra that every skewsymmetric 2form on an odddimensional space has a nontrivial kernel.
Definition 6
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.
Proposition 4
Let \(X_t\) be a Caratheodory TDVF on a contact manifold \((\mathcal {M},\mathcal {C})\) and let \(A_{t\tau }\) be the TD flow of \(X_t\). Then \(X_t\) is a contact vector field for every \(t\in \mathbb {R}\) (i.e., \([X_t,\mathcal {C}]\subset \mathcal {C}\)) if and only if the TD flow \(A_{t\tau }\) consists of contact transformations.
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 onetoone correspondence between CVFs on \(\mathcal {M}\) and sections of the normal bundle \(\mathrm {N}\mathcal {C}\).
Lemma 5
Remark 2
Throughout we will denote by \(X\in \mathfrak {X}(\mathcal {M})\) vector fields on \(\mathcal {M}\), by \(Y\in \varGamma (\mathcal {C})\) vector fields valued in \(\mathcal {C}\) and by C (also with variants, like \(C_{\phi }, C_{[X]}\) or \(\mathbf {C}_{\phi }\)) contact vector fields.
Proof
Finally, we need to check that every CVF is of the form \(C_{[X]}\). By construction the class of \(C_{[X]}\) in \(\mathrm {N}\mathcal {C}\) is equal to the class of X in \(\mathrm {N}\mathcal {C}\) (these two vector fields differ by a \(\mathcal {C}\)valued vector field h(X)). Thus, the classes of CVFs of the form \(C_{[X]}\) realize every possible section of \(\mathrm {N}\mathcal {C}\). Now it is enough to observe that the \(\mathrm {N}\mathcal {C}\)class uniquely determines a CVF. Indeed, if C and \(C'\) are two CVFs belonging to the same class in \(\mathrm {N}\mathcal {C}\), then their difference \(XX'\) is a \(\mathcal {C}\)valued CVF, i.e., \([CC',Y]\equiv 0\mod \mathcal {C}\) for any \(\mathcal {C}\)valued vector field Y. That is, \(\beta (CC',\cdot )\equiv 0\) and from the nondegeneracy of \(\beta \) we conclude that \(CC'\equiv 0\). This ends the proof. \(\square \)
Remark 3
It is natural to call a vector field \(X\in \mathfrak {X}(\mathcal {M})\) (or its \(\mathrm {N}\mathcal {C}\)class [X]) a generator of the CVF \(C_{[X]}\). Observe that the \(\mathrm {N}\mathcal {C}\)class of the CVF \(C_{[X]}\) is the same as the class of its generator X (they differ by a \(\mathcal {C}\)valued vector field h(X)).
In the literature, see e.g., [14], a contact distribution \(\mathcal {C}\) on \(\mathcal {M}\) is often presented as the kernel of a certain 1form \(\omega \in \varLambda ^1(\mathcal {M})\) (such an \(\omega \) is then called a contact form). In the language of \(\omega \), the maximum nondegeneracy condition can be expressed as the nondegeneracy of the 2form \({\text {d}}\omega \) 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 onetoone 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, noncanonical (as every rescaling of \(\omega \) by a nowherevanishing 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 1forms). 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.
Remark 4
Proposition 5
Let \((\mathcal {M},\mathcal {C})\) be a contact manifold. There is a onetoone correspondence between CVFs on \(\mathcal {M}\) and controlaffine systems (equivalently, affine distributions) on \(\mathcal {M}\) of the form \(\mathcal {A}=X+\mathcal {C}\subset \mathrm {T}\mathcal {M}\), where \(X\in \varGamma (\mathcal {M})\).
Indeed, to every CVF \(C_{}\), we attach the affine distribution (controlaffine system) \(\mathcal {A}=C_{}+\mathcal {C}\). Conversely, given an affine distribution (controlaffine 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 controlaffine systems \(\mathcal {A}=X+\mathcal {C}\), the correspondence being established by the map \(\mathcal {A}=X+\mathcal {C}\mapsto C_{[X]}\).
4 Contact geometry of \(\mathbb {P}(\mathrm {T}^*M)\)
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 nonzero 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 \(\mathcal {H}_{[\theta ]}:=\ker \theta \subset \mathrm {T}_{\pi _M(\theta )}M\).
Lemma 6
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 1form \(\Omega _M\) 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.
Proof
To finish the proof it is enough to check that \(\omega _R\) satisfies the maximum nondegeneracy condition. This can be easily seen by introducing local coordinates \((x^0,x^1,\ldots ,x^n)\) on M in which \(R=\partial _{x^0}\) (recall that R is nonvanishing on \(\pi (\mathcal {U}_R)\), so such a choice is locally possible). Let \((x^i,p_i)\) be the induced coordinates on \(\mathrm {T}^*M\). It is clear that in these coordinates the image \(i_R(\mathcal {U}_R)\subset \mathrm {T}^*M\) is characterized by equation \(p_0=1\) and thus the Liouville form \(\Omega _M=\sum _{i=0}^np_i{\text {d}}x^i\) restricted to \(i_R(\mathcal {U}_R)\) is simply \({\text {d}}x^0+\sum _{i=1}^np_i{\text {d}}x^i\). Obviously, the pullback functions \(\widetilde{x}^i:=(i_R)^*x^i\) with \(i=0,\ldots ,n\) and \(\widetilde{p}_j:=(i_R)^*p_j\) with \(j=1,\ldots ,n\) form a coordinate system in \(\mathcal {U}_R\). In these coordinates, the form \(\omega _R=(i_R)^*\Omega _M\) simply reads as \({\text {d}}\widetilde{x}^0+\sum _{i=1}^n\widetilde{p}_i{\text {d}}\widetilde{x}^i\). It is a matter of a simple calculation to check that such a oneform satisfies the maximum nondegeneracy condition. We conclude that \(\omega _R\) is, indeed, a contact form on \(\mathcal {U}_R\) for the canonical contact structure on \(\mathbb {P}(\mathrm {T}^*M)\). \(\square \)
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.
Definition 7
Let \(F:M\rightarrow M\) be a diffeomorphism. Its tangent map \(\mathrm {T}F:\mathrm {T}M\rightarrow \mathrm {T}M\) induces a natural transformation \(\mathbb {P}(F):\mathbb {P}(\mathrm {T}^*M)\rightarrow \mathbb {P}(\mathrm {T}^*M)\) of the space of hyperplanes in \(\mathrm {T}M\), i.e., given a hyperplane \(\mathcal {H}\subset \mathrm {T}_p M\), we define the hyperplane \(\mathbb {P}(F)(\mathcal {H})\subset \mathrm {T}_{F(p)}M\) to be simply the image \(\mathrm {T}F(\mathcal {H})\). The map \(\mathbb {P}(F)\) shall be called the contact lift of F.
Lemma 7
\(\mathbb {P}(F)\) is a contact transformation with respect to the canonical contact structure on \(\mathbb {P}(\mathrm {T}^*M)\).
Proof
Let Y be an element of \(\mathrm {T}_{[\theta ]}\mathbb {P}(\mathrm {T}^*M)\) projecting to \(\mathrm {T}\pi (Y)=:\underline{Y}\) under \(\mathrm {T}\pi \). By diagram (4.4), the tangent map \(\mathrm {T}\mathbb {P}(F)\) sends Y to an element of \(\mathrm {T}_{[(F^{1})^*\theta ]}\mathbb {P}(\mathrm {T}^*M)\) lying over \(\mathrm {T}F(\underline{Y})\).
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.
Definition 8
Let \(X\in \mathfrak {X}(M)\) be a smooth vector field. By the contact lift of X we shall understand the contact vector field \(\mathbf {C}_{X}\) on \(\mathbb {P}(\mathrm {T}^*M)\) whose flow is \(\mathbb {P}(A_{t})\), the contact lift of the flow \(A_{t}\) of X.
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.
Lemma 8
Proof
Since \(\mathbb {P}(A_{t})\), the flow of \(\mathbf {C}_{X}\), projects under \(\pi \) to \(A_{t}\), the flow of X, we conclude that \(X=\mathrm {T}\pi (\mathbf {C}_{X})\). As we already know from the proof of Lemma 5, a CVF is uniquely determined by its class in \(\mathrm {N}\mathcal {C}\). By (4.1), the \(\mathrm {N}\mathcal {C}\)class of a field \(Y\in \mathfrak {X}(\mathbb {P}(\mathrm {T}^*M))\) is completely determined by its \(\mathrm {T}\pi \)projection. In other words, if two fields Y and \(Y'\) have the same \(\mathrm {T}\pi \)projections, then \(YY'\) is a \(\mathcal {C}\)valued vector field. Thus, the field \(\widetilde{X}\) has the same \(\mathrm {N}\mathcal {C}\)class as the CVF \(\mathbf {C}_{X}\) so, by the results of Lemma 5 (see also Remark 3), it follows \(\mathbf {C}_{X}=C_{[\widetilde{X}]}\). \(\square \)
Remark 5
Remark 6
It is worth mentioning the following illustrative picture which was pointed to us by Janusz Grabowski. Every contact structure on a manifold N can be viewed as a homogeneous symplectic structure on some principal \(GL(1,\mathbb {R})\)bundle over N. In the case of the canonical contact structure on \(N=\mathbb {P}(\mathrm {T}^*M)\) the corresponding bundle is simply \(\mathrm {T}^*_0 M\), the cotangent bundle of M with the zero section removed, equipped with the natural action of \(\mathbb {R}{\setminus }\{0\}=GL(1,\mathbb {R})\) being the restriction of the multiplication by reals on \(\mathrm {T}^*M\). The canonical symplectic structure is obviously homogeneous with respect to this action. Now every homogeneous symplectic dynamics on \(\mathrm {T}_0^*M\) reduces to contact dynamics on \(\mathbb {P}(\mathrm {T}^*M)\). For more information on this approach the reader should consult [7, 9].
5 The Pontryagin maximum principle
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 welldefined 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
Theorem 2
Definition 9
A pair \((\varvec{q}(t),\widehat{u}(t))\) satisfying the necessary conditions for optimality provided by Theorem 2 (i.e., the existence of a covector curve \(\varvec{\varLambda }_t\) satisfying (5.1)–(5.3)) is called an extremal.
Proof of the PMP
Although the PMP is a commonly known result, for future purposes it will be convenient to sketch its original proof following [18]. \(\square \)
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.
Lemma 9
([18]) If, for any \(0<t\le T\), the ray \(\varvec{\mathcal {R}}^_{\varvec{q}(t)}\) lies in the interior of \(\varvec{\mathcal {K}}_t\), then \((\varvec{q}(t),\widehat{u}(t))\) cannot be a solution of (OCP).
As a direct corollary, using basic facts about separation of convex sets, one obtains the following
Proposition 6
([18]) Assume that \((\varvec{q}(t),\widehat{u}(t))\) is a solution of (OCP). Then for each \(t\in (0,T]\) there exists a hyperplane \(\varvec{\mathcal {H}}_t\subset \mathrm {T}_{\varvec{q}(t)}\varvec{Q}\) separating the convex cone \(\varvec{\mathcal {K}}_t\) from the ray \(\varvec{\mathcal {R}}^_{\varvec{q}(t)}\).
Remark 7

normal if Open image in new window for any \(t\in [0,T]\). Note that, in consequence, the ray \(\varvec{\mathcal {R}}^{}_{\varvec{q}(t)}\) can be strictly separated from the cone \(\varvec{\mathcal {K}}_t\) for each \(t\in [0,T]\);

abnormal if \(\varvec{\mathcal {R}}_{\varvec{q}(t)}\subset \varvec{\mathcal {H}}_t\) for each \(t\in [0,T]\);

strictly abnormal if for some \(t\in [0,T]\) the ray \(\varvec{\mathcal {R}}^_{\varvec{q}(t)}\) cannot be strictly separated from the cone \(\varvec{\mathcal {K}}_t\) (and thus \(\varvec{\mathcal {R}}_{\varvec{q}(t)}\subset \varvec{\mathcal {H}}_t\) for each \(t\in [0,T]\)).
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.
Theorem 3
Moreover, each \(\varvec{\mathcal {H}}_t\) separates the convex cone \(\varvec{\mathcal {K}}_t\) defined by (5.4) from the ray \(\varvec{\mathcal {R}}^{}_{\varvec{q}(t)}\).
Proof
The family of hyperplanes \(\varvec{\mathcal {H}}_t\) separating the cone \(\varvec{\mathcal {K}}_t\) from the ray \(\varvec{\mathcal {R}}^{}_{\varvec{q}(t)}\) and satisfying (5.6) was constructed in the course of the proof of Theorem 2 sketched in the previous paragraph. To end the proof it is enough to check that \(\varvec{\mathcal {H}}_t\) evolves according to (5.7). From (5.6) and Definition 7 of the contact lift we know that \(\varvec{\mathcal {H}}_t\) evolves according to \(\mathbb {P}(\varvec{F}_{\tau t})\). By the remark following Definition 8 this is precisely the TD flow induced by the TDVF \(\mathbf {C}_{\varvec{f}_{\widehat{u}(t)}}\). \(\square \)
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 \(\varvec{Q}\).
Remark 8
Actually, the fact that the evolution of \(\varvec{\mathcal {H}}_t\) is contact (and at the same time that the evolution of \(\varvec{\varLambda }_t\) is Hamiltonian) is in a sense “accidental”. Namely, it is merely a natural contact (Hamiltonian) evolution induced on \(\mathbb {P}(\mathrm {T}^*\varvec{Q})\) (on \(\mathrm {T}^*\varvec{Q}\)) by the TD flow on \(\varvec{Q}\) defined by means of the extremal vector field. In the Hamiltonian case this was, of course, already observed—see, e.g., Chapter 12 in [2]. Thus, it is perhaps more proper to speak rather about covariant (in terms of hyperplanes) and contravariant (in terms of covectors) formulations of the PMP, than about its contact and Hamiltonian versions. It may seem that the choice between one of these two approaches is a matter of a personal taste, yet obviously the covariant formulation is closer to the original geometric meaning of the PMP, as it contains a direct information about the separating hyperplanes, contrary to the contravariant version where this information is translated to the language of covectors (not to forget the nonuniqueness of the choice of \(\varvec{\varLambda }_t\)). In the next Sect. 6 we shall show a few applications of the covariant approach to the subRiemannian geometry. Expressing the optimality in the language of hyperplanes allows to see a direct relation between abnormal extremals and special directions in the constraint distribution. It also provides an elegant geometric characterization of normal extremals.
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.
Theorem 4
Proof
The proof is immediate. The existence of separating hyperplanes \(\varvec{\mathcal {H}}_t\) satisfying (5.6) was already proved in the course of this section. The only part that needs some attention is the justification of equation (5.8). It follows directly from the \(\varvec{F}_{t\tau }\)invariance along \(\varvec{q}(t)\) of \(\varvec{\mathcal {H}}_t\) and Theorem 1. (Note that \(\varvec{\mathcal {H}}_t\) is charming along \(\varvec{q}(t)\) by Proposition 1.) \(\square \)
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).
6 Applications to the subRiemannian geodesic problem
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 subRiemannian 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 noncanonically), 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 \(f(q,u)\in \mathcal {D}_q\).
Definition 10
By a SR extremal we shall understand a trajectory \((\varvec{q}(t),\widehat{u}(t))\) of the SR control system satisfying the necessary conditions for optimality given by the PMP (in the form provided by Theorem 2 or, equivalently, Theorems 3 or 4).
Lemma 10
Proof
It follows from (5.4) (after taking \(k=1, t_1=t\), and thus \(\varvec{F}_{tt_1}={\text {id}}_{\varvec{Q}}\)) that \(\varvec{\mathcal {K}}_t\) contains every secant ray \(\mathbb {R}_+\cdot \{\varvec{f}_v(\varvec{q}(t)) \varvec{f}_{{\widehat{u}}(t)}(\varvec{q}(t))\}\) of the paraboloid \(\varvec{f}(\varvec{q}(t),U)=\{\varvec{f}_v(\varvec{q}(t))\ \ f_v(q(t))\in \mathcal {D}_{q(t)}\}\) passing through the point \(\varvec{f}_{{\widehat{u}}(t)}(\varvec{q}(t))\). Using these secant rays we may approximate every tangent ray of the paraboloid \(\varvec{f}(\varvec{q}(t),U)\) passing through \(\varvec{f}_{{\widehat{u}}(t)}(\varvec{q}(t))\) with an arbitrary accuracy. Since \(\varvec{\mathcal {K}}_t\) is closed, it has to contain this tangent ray and, consequently, the whole tangent space of \(\varvec{f}(\varvec{q}(t),U)\) at \(\varvec{f}_{{\widehat{u}}(t)}(\varvec{q}(t))\) (see Fig. 5). The fact that this tangent space is described by equality (6.1) is an easy exercise. \(\square \)
Remark 9
In general, for an arbitrary control system and an arbitrary cost function, the cone \(\varvec{\mathcal {K}}_t\) contains all secant rays of the image \(\varvec{f}(\varvec{q}(t),U)\) passing through \(\varvec{f}_{\widehat{u}(t)}(\varvec{q}(t))\). Thus, after passing to the limit, the whole tangent cone to \(\varvec{f}(\varvec{q}(t),U)\) at \(\varvec{f}_{\widehat{u}(t)}(\varvec{q}(t))\) is contained in \(\varvec{\mathcal {K}}_t\). If \(\varvec{f}(\varvec{q}(t),U)\) is a submanifold, as it is the case in the SR geodesic problem, this tangent cone is simply the tangent space at \(\varvec{f}_{\widehat{u}(t)}(\varvec{q}(t))\).
Here is an easy corollary from the above lemma and our previous considerations.
Lemma 11
Proof
To justify the first part of the assertion, observe that if, in a linear space V, a hyperplane \(\mathcal {H}\subset V\) supports a cone \(\mathcal {K}\subset V\) which contains a line \(l\subset \mathcal {K}\) (and all these sets contain the zero vector), then necessarily \(l\subset \mathcal {H}\) (each line containing 0 either intersects the hyperplane or is tangent to it). Since, by Lemma 10, \(\varvec{\mathcal {K}}_t\) contains the subspace \(\{Y+g(f_{\widehat{u}(t)},Y)\partial _{q_0}\ \ Y\in \mathcal {D}_{q(t)}\}\), we conclude that this subspace must lie in \(\varvec{\mathcal {H}}_t\).
It turns out that in some cases the above basic information, suffices to find SR extremals. Let us study the following two examples.
Example 1
(Riemannian extremals) In the Riemannian case \(\mathcal {D}=\mathrm {T}Q\) is the full tangent space and g is a Riemannian metric on Q. Let us introduce any connection \(\nabla \) on Q compatible with the metric. By \(T_\nabla (X,Y):=\nabla _XY\nabla _YX[X,Y]\) denote the torsion of \(\nabla \) (in particular, if we take the Levi–Civita connection \(\nabla =\nabla ^{LC}\), then \(T_{\nabla ^{LC}}\equiv 0\)).
Example 2
In the following two subsections we shall discuss normal and abnormal SR extremals in full generality.
6.1 Abnormal SR extremals
Our previous considerations allow us to give the following characterization of SR abnormal extremals.
Theorem 5
 (a)
The pair \((\varvec{q}(t),\widehat{u}(t))\) is an abnormal SR extremal.
 (b)The smallest distribution \(F_{t\tau }\)invariant along q(t) and containing \(\mathcal {D}_{q(t)}\), i.e.,is of rank smaller than \(\dim Q\). Here \(F_{t\tau }\) denotes the TD flow (in Q) of the Caratheodory TDVF \(f_{\widehat{u}(t)}\).$$\begin{aligned} F_{\bullet }(\mathcal {D})_{q(t)}={\text {vect}}_{\mathbb {R}}\{\mathrm {T}F_{t\tau }(Y)\ \ Y\in \mathcal {D}_{q(\tau )},\, 0\le \tau \le T\} \end{aligned}$$
Note that Theorem 5 reduces the problem of finding abnormal SR extremals to the study of the minimal distribution \(F_{t\tau }\)invariant along q(t) and containing \(\mathcal {D}_{q(t)}\). Often, if q(t) is sufficiently regular, this problem can be solved by the methods introduced in Lemma 3, which are more practical from the computational viewpoint.
Corollary 1
 The distribution spanned by the iterated Lie brackets of X with all possible smooth \(\mathcal {D}\)valued vector fields, i.e.,is of constant rank r along q(t) and \(r<\dim Q\).$$\begin{aligned} {\text {ad}}^\infty _X(\mathcal {D})=\big \langle {{\text {ad}}_X^k(Y)\ \ Y\in \varGamma (\mathcal {D}),\; k=0,1,2,\ldots }\big \rangle \end{aligned}$$
 There exists a smooth distribution \(\mathcal {B}\supset \mathcal {D}\) on Q of constant corank at least one, such that$$\begin{aligned}{}[X,\mathcal {B}]_{q(t)}\subset \mathcal {B}_{q(t)} \quad \text { for any } \; t\in [0,T]. \end{aligned}$$
The above fact follows directly from Theorem 5, Lemma 3 and Theorem 1. In each of the two cases along q(t) we have a constant rank smooth (and thus charming) distribution which contains \(\mathcal {D}\), is Xinvariant along q(t) (and thus by Theorem 1 also \(F_{t\tau }\)invariant along q(t)) and of corank at least one. Clearly such a distribution must contain \(F_{\bullet }(\mathcal {D})_{q(t)}\), which in consequence also is of corank at least one.
Remark 10
Proof of Theorem 5
If \((\varvec{q}(t),\widehat{u}(t))\) is an abnormal extremal then, by the results of Lemma 11, \(\mathcal {H}_t\), the \(\mathrm {T}Q\)projection of the curve of supporting hyperplanes \(\varvec{\mathcal {H}}_t\subset \mathrm {T}_{\varvec{q}(t)} \varvec{Q}\), is a curve of hyperplanes along q(t) (i.e., a distribution of corank one along q(t)), it contains \(\mathcal {D}_{q(t)}\) and is \(F_{t\tau }\)invariant along q(t). In particular, it must contain the smallest distribution \(F_{t\tau }\)invariant along q(t) and containing \(\mathcal {D}\) (cf. Proposition 2). Thus, \({\text {rank}}F_{\bullet }(\mathcal {D})_{q(t)}\le {\text {rank}}\mathcal {H}_t=\dim Q1\).
Conversely, assume that \({\text {rank}}F_{\bullet }(\mathcal {D})_{q(t)}<\dim Q\). Now by adding (if necessary) to \(F_{\bullet }(\mathcal {D})_{q(t)}\) several vector fields of the form \(F_{tt_0}(X)\) where \(X\in T_{q(0)}Q\), we can extend \(F_{\bullet }(\mathcal {D})_{q(t)}\) to \(\mathcal {H}_t\), a corank one distribution \(F_{t\tau }\)invariant along q(t). Define now the curve of hyperplanes \(\varvec{\mathcal {H}}_t:=\mathcal {H}_t\oplus \mathcal {R}_{q_0(t)}\subset \mathrm {T}_{\varvec{q}(t)}\varvec{Q}\). We claim that \(\varvec{\mathcal {H}}_t\) is a curve of supporting hyperplanes described in the assertion of Theorem 3. Indeed, the \(\varvec{F}_{t\tau }\)invariance of \(\varvec{\mathcal {H}}_t\) should be clear, as on the product \(\varvec{Q}=Q\times \mathbb {R}\) the TD flow \(\varvec{F}_{t\tau }\) takes the form \(\varvec{F}_{t\tau }(q,q_0)=(F_{t\tau }(q), B_{t\tau }(q_0))\), for some TD flow \(B_{t\tau }\) on \(\mathbb {R}\). Clearly, since \(\mathcal {H}_t\) is \(F_{t\tau }\)invariant along q(t), the tangent map of \(\varvec{F}_{t\tau }\) preserves \(\varvec{\mathcal {H}}_t=\mathcal {H}_t\oplus \mathcal {R}_{q_0(t)}\). To prove that \(\varvec{\mathcal {H}}_t\) indeed separates the cone \(\varvec{\mathcal {K}}_t\) from the direction of the decreasing cost \(\varvec{\mathcal {R}}^_{\varvec{q}(t)}\) observe that any vector of the form \(\varvec{f}_{v}(\varvec{q}(t))\varvec{f}_{\widehat{u}(t)}(\varvec{q}(t))\), where \(f_{v}\in \mathcal {D}_{q(t)}\), lies in \(\mathcal {D}_{q(t)}\oplus \mathcal {R}_{q_0(t)}\subset \varvec{\mathcal {H}}_t\). Moreover, any vector of the form \(\mathrm {T}\varvec{F}_{t\tau }\left[ \varvec{f}_v(\varvec{q}(\tau ))\varvec{f}_{\widehat{u}(t)}(\varvec{q}(\tau ))\right] \), where \(f_{v}\in \mathcal {D}_{q(\tau )}\), lies in \(\mathrm {T}\varvec{F}_{t\tau }(\mathcal {D}_{q(\tau )}\oplus \mathcal {R}_{q_0(\tau )})\subset \mathrm {T}\varvec{F}_{t\tau }(\varvec{\mathcal {H}}_\tau )\subset \varvec{\mathcal {H}}_t\). Thus, the whole cone \(\varvec{\mathcal {K}}_t\) is contained in \(\varvec{\mathcal {H}}_t\) (cf. formula (5.4)). Since also \(\varvec{\mathcal {R}}^_{\varvec{q}(t)}\subset \varvec{\mathcal {R}}_{\varvec{q}(t)}\subset \varvec{\mathcal {H}}_t\), we conclude that indeed \(\varvec{\mathcal {H}}_t\) separates \(\varvec{\mathcal {K}}_t\) from \(\varvec{\mathcal {R}}^_{\varvec{q}(t)}\) (in a trivial way).
Examples
Example 3
Example 4
(Zelenko) The following example by Igor Zelenko [21] became known to us thanks to the lecture of Boris Doubrov. The interested reader may consult also [3, 8].
First let us show that the integral curves of \(\partial _t\) are abnormal extremals. Indeed, it is easy to see that \([\partial _t,X_1+t X_2]=X_2\) and that \([\partial _t,X_2]=0\), i.e., the minimal distribution \({\partial _t}\)invariant and containing \(\mathcal {D}\) is precisely \(\big \langle {\partial _t,X_1,X_2}\big \rangle \). This distribution is of constant rank smaller than \(6=\dim Q\), so by Corollary 1, indeed, the integral curves of \(\partial _t\) are abnormal extremals.
Example 5
Example 6
Example 7
(Zhitomirskii) Let \(\mathcal {D}\) be a 2distribution on a manifold Q such that \(\mathcal {D}^2:=\mathcal {D}+[\mathcal {D},\mathcal {D}]\) is of rank 3. In [22] Zhitomirskii introduced the following definition.

\(\mathcal {Z}\) is involutive

for any \(q\in Q\) we have \(\mathcal {D}_q{\not \subseteq } \mathcal {Z}_q\)

\({\text {rank}}(\mathcal {D}^2\cap \mathcal {Z})=2\).
Now it should be clear that the smallest distribution containing \(\mathcal {D}\) and invariant with respect to the TD flow of Y is contained in \(\mathcal {H}\), which is of corank one. Thus, by Theorem 5, the integral curves of X are abnormal SR extremals.
6.2 Normal SR extremals
Theorem 6
 (a)
The pair \((\varvec{q}(t),\widehat{u}(t))\) is a normal SR extremal.
 (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.,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 \(f_{\widehat{u}(t)}\).$$\begin{aligned} F_{\bullet }(\mathcal {D}^\perp )_{q(t)}={\text {vect}}_{\mathbb {R}}\{\mathrm {T}F_{t\tau }(Y)\ \ Y\in \mathcal {D}_{q(\tau )}, \; g(Y,f_{\widehat{u}(t)})=0, \; 0\le \tau \le T\} \end{aligned}$$
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
 The distribution spanned by the iterated Lie brackets of X and all possible smooth \(\mathcal {D}\)valued vector fields gorthogonal to X, i.e.,is of constant rank r along q(t) and it does not contain X(q(t)) for any \(t\in [0,T]\).$$\begin{aligned} {\text {ad}}^\infty _X(\mathcal {D}^\perp )=\big \langle {{\text {ad}}_X^k(Y)\ \ Y\in \varGamma (\mathcal {D}), \; g(X,Y)=0, \; k=0,1,2,\ldots }\big \rangle \end{aligned}$$
 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 \(F_{t\tau }\).
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 gorthonormal basis of ACB sections. Indeed, \(\ker \alpha _t\) is charming since it is \(F_{t\tau }\)invariant (cf. Proposition 1). Let now \(\{X_1,\ldots ,X_{n1}\}\) 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 \(\{X_1,\ldots ,X_{n1}\}\) and obtain an ACB \(\widetilde{g}\)orthonormal basis \(\{X_1,\ldots ,X_s,Y_{s+1},\ldots ,Y_{n1}\}\) of \(\ker \alpha _t\). Clearly, by the construction of the Gram–Schmidt algorithm, \(\{Y_{s+1},\ldots ,Y_{n1}\}\) is a \(\widetilde{g}\), and thus also a gorthonormal 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 \(\{Y_{s+1},\ldots ,Y_{n1},Y_n\}\) of \(\mathcal {D}_{q(t)}\) to a gorthonormal ACB local basis \(\{Y_{s+1},\ldots ,Y_{n1},\widetilde{Y}_n\}\). Obviously, \(\widetilde{Y}_n(q(t))\) is a gnormalized vector gorthogonal to \(\mathcal {D}^\perp _{q(t)}=\big \langle {Y_{s+1},\ldots ,Y_{n1}}\big \rangle \), 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 corank 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 corank 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 \(\mathcal {B}_{q(t)}\oplus \big \langle {f_{\widehat{u}(t)}(q(t))}\big \rangle \).
In a similar way one deals with a cusptype 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 cornertype or cusptype 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.
Example 9
Footnotes
 1.
Sometimes it is convenient to identify a TDVF \(X_t\) on M with the vector field \(\widetilde{X}(x,t)=X_t(x)+\partial _t\) on \(M\times \mathbb {R}\). Within this identification Eq. (2.1) is an Mprojection of the autonomous ODE \((\dot{x},\dot{t})=\widetilde{X}(x,t)\) defined on \(M\times \mathbb {R}\).
 2.
From now on, geometric objects and constructions associated with the extended configuration space \(\varvec{Q}\) will be denoted by bold symbols, e.g., \(\varvec{f}, \varvec{q}, \varvec{F}_{tt_0}, \varvec{\mathcal {H}}_t\) etc. Normalfont symbols, e.g., \(f, q, F_{tt_0}, \mathcal {H}_t\), will denote analogous objects in Q being, in general, projections of the corresponding objects from \(\varvec{Q}\).
 3.
From now on we will use symbols \(\varvec{F}_{t\tau }\) and \(F_{t\tau }\) to denote the TD flows of TDVFs \(\varvec{f}_{u(t)}\) and \(f_{u(t)}\), respectively, for a particular control u(t). Note that \(\varvec{F}_{t\tau }\) projects to \(F_{t\tau }\) under \(\pi _1:\varvec{Q}=Q\times \mathbb {R}\rightarrow Q\).
 4.
In the original proof in [18], the optimal control problem with a free time interval [0, T] is considered. In this case, the sets \(\varvec{\mathcal {K}}_t\) contain additional elements.
 5.
By choosing \(\varvec{\mathcal {K}}_0:=\{0\}\) we can easily extend \(\varvec{\mathcal {H}}_t\) to the whole interval [0, T].
 6.
We leave the proof of the fact that the Lie bracket (2.3) satisfies the Leibniz rule as an exercise.
 7.
The existence of measurable functions \(\phi _i^{\ j}\) can be justified in a similar manner to the existence of ACB functions \(\phi _i\) above.
Notes
Acknowledgments
This research was supported by the National Science Center under the Grant DEC2011/02/A/ST1/00208 “Solvability, chaos and control in quantum systems”.
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