Our main areas of research are:*differential geometry*. Even if you are unfamiliar with the basics of differential geometry, we hope to at least convey the gist of what our research entails. For those interested in obtaining a better understanding of the differential-geometric prerequisites for a more detailed understanding of our research, we recommend the following two excellent introductory textbooks:

- Invariant control systems and invariant optimal control
- Riemannian, sub-Riemannian and nonholonomic Riemannian geometry
- The geometry of homogeneous spaces.

- John M. Lee,
**Introduction to Smooth Manifolds**. Springer, New York, 2013. - Jeffrey M. Lee,
**Manifolds and Differential Geometry**. American Mathematical Society, Providence, RI, 2009.

From a geometric perspective, a *control system* is a family \((\Xi_u)_{u \in \mathbb{R}^{\ell}}\), where each \(\Xi_u\) is a vector field on a manifold \(\mathsf{M}\), and \(u \in \mathbb{R}^{\ell}\) is the control parameter. The control parameters are allowed to vary over time; a choice of such controls over some duration—i.e., a curve \(u : I \subseteq \mathbb{R} \to \mathbb{R}^{\ell}\)—is called a control function. For a fixed control function, the control system reduces to a single, time-dependent vector field; the integral curves of this vector field are called the trajectories of the system.

*Optimal control theory* deals with the problem of steering a control system from one state to another while satisfying some optimality condition; in other words, finding a control function such that the associated trajectory joins the two states while minimizing some cost. Optimal control problems are typically written in the abbreviated form \[\begin{gather*} \dot{\gamma}(t) = \Xi_{u(t)}(\gamma(t)),\quad \gamma : [0,T] \to \mathsf{M}, \quad u : [0,T] \to \mathbb{R}^{\ell}\\ \gamma(0) = q_0,\quad \gamma(T) = q_T\\ \mathcal{J}[u] = \int_0^T \chi(u(t))\,dt \to \text{min}. \end{gather*} \] Explicitly, the above problem is to find a control function \(u : [0,T] \to \mathbb{R}^{\ell}\) and a trajectory \(\gamma : [0,T] \to \mathsf{M}\) such that \(\gamma\) joins the two points \(q_0,q_T \in \mathsf{M}\) and the cost functional \(\mathcal{J}\) is minimised. (The function \(\chi : \mathbb{R}^{\ell} \to \mathbb{R}\) specifies the cost; for instance, a common choice is \(\chi(u) = \sum_{i=1}^{\ell} u_i^2\), where \(u = (u_1,\ldots,u_{\ell}) \in \mathbb{R}^{\ell}\).)

The prototypical control systems (and optimal control problems) are the invariant control systems (invariant optimal control problems) on Lie groups; our research into control theory has been primarily focused on such systems, particularly in three and four dimensions. Particularly, we are concerned with:

For more information on (geometric) control theory and optimal control, we suggest the following references:

The prototypical control systems (and optimal control problems) are the invariant control systems (invariant optimal control problems) on Lie groups; our research into control theory has been primarily focused on such systems, particularly in three and four dimensions. Particularly, we are concerned with:

*Equivalence and classification*. Briefly, the equivalence problem is to determine when two control systems (or two optimal control problems) are essentially the same system (problem) in disguise. The classification problem involves obtaining an exhaustive list of all distinct (i.e., inequivalent) control systems (or optimal control problems), either on a particular manifold (usually, a Lie group), or on a class of manifolds (e.g., low-dimensional Lie groups).*Controllability*(of control systems). The controllability problem is easy to state: given an initial and a final state, does there exist a control function steering the system from the initial to the final state? However, solving the controllability problem (or finding effective characterizations) is often quite difficult.*Calculation of extremal**controls and extremal trajectories*. The problem is lifted to the cotangent bundle of the manifold, where a Hamiltonian approach (using the canonical symplectic structure, as well as Pontryagin's Maximum Principle) is used to find the extremal controls and the extremal trajectories. Whenever feasible, we try to find explicit (analytical) expressions for the controls and trajectories. In the invariant case, the problem of finding the extremal controls reduces to finding the integral curves of a Hamilton–Poisson system on the dual of the Lie algebra (and hence we often study Hamilton–Poisson systems directly).*Optimality analysis of extremal trajectories*. Having calculated the extremal trajectories, the question remains as to which trajectories (between various initial and final states) are optimal (this is referred to as optimal synthesis). This step is often the most difficult (but is in many ways the final product of studying optimal control on a specific manifold).

For more information on (geometric) control theory and optimal control, we suggest the following references:

- A.A. Agrachev and Yu.L. Sachkov,
**Control Theory from the Geometric Viewpoint**. Springer, Berlin, 2004. - R. Biggs and C.C. Remsing,
**Invariant control systems on Lie groups**, in: G. Falcone (ed.), Lie Groups, Differential Equations, and Geometry: Advances and Surveys. Springer, 2017, pp. 127–182. - V. Jurdjevic,
**Geometric Control Theory**. Cambridge University Press, Cambridge, 1997. - Yu.L. Sachkov,
**Control theory on Lie groups**, J. Math. Sci. 156 (2009) 381–439.

A Riemannian manifold is a manifold \(\mathsf{M}\), together with a mapping \(\mathbf{g} : \mathsf{M} \ni q \mapsto \mathbf{g}_q\) (the “Riemannian metric”) that assigns an inner product \(\mathbf{g}_q\) to each tangent space \(T_q\mathsf{M}\). (That is, at each point \(q \in \mathsf{M}\), there is an inner product \(\mathbf{g}_q\) defined on the space of tangent vectors to \(\mathsf{M}\) at \(q\).) The metric allows one to define on \(\mathsf{M}\) notions such as angles, the length of curves, volumes, etc. In particular, since the length of curves is defined, one can talk about the *shortest curves* between two points, or, in other words, to generalise the idea of a *straight line* to \(\mathsf{M}\). These shortest curves are called *geodesics*; the study of Riemannian geodesics, and how the geometry of \(\mathsf{M}\) affects the geodesics, is a central theme of Riemannian geometry.

There is also an alternative approach to the geodesics. This approach uses the idea of an affine connection. Briefly, a connection is means to translate geometric objects (for instance, a tangent vector) along a curve in a parallel fashion. The connection also allows one to define:*Levi-Civita connection*. Using this connection, one can define the *straightest curves* in \(\mathsf{M}\) to be the curves with zero acceleration (thereby obtaining another generalisation of straight lines to \(\mathsf{M}\)). Remarkably, it turns out that *the straightest curves are also the shortest curves!* A connection also allows one to measure the curvature of \(\mathsf{M}\). Curvature essentially measures how different a Riemannian manifold is from Euclidean space (which is flat, i.e., it has zero curvature). The study of curvature is another major theme in Riemannian geometry; for instance, studying how curvature effects the geometry of the space (e.g., the geodesics), or using the curvature to compare different Riemannian manifolds.

While we are interested in Riemannian geometry, and have done some research in this area, we have primarily concentrated on two generalisations of Riemannian geometry, namely: sub-Riemannian geometry, and nonholonomic Riemannian geometry. We briefly describe these two fields here, with an emphasis on the type of research topics we are interested in.

**Sub-Riemannian Geometry**

*Sub-Riemannian* (SR) *geometry *starts from the same principles as Riemannian geometry, but introduces *constraints on motion*. More specifically, a SR manifold consists of a manifold \(\mathsf{M}\), a subbundle \(\mathcal{D} \subsetneq T\mathsf{M}\), and a metric \(\mathbf{g}\) on \(\mathcal{D}\). By a “subbundle” we mean that \(\mathcal{D}_q\) is a subspace of \(T_q\mathsf{M}\) for each \(q \in \mathsf{M}\), and that each subspace \(\mathcal{D}_q\) has the same dimension. (Intuitively, one should interpret \(\mathcal{D}_q\) to be the subspace of admissible directions in which one can move, if positioned at \(q\).) We also require \(\mathcal{D}\) to satisfy a technical condition called complete nonholonomy; essentially, this condition allows one to generate all possible directions from vector fields in \(\mathcal{D}\). That \(\mathbf{g}\) is only defined on \(\mathcal{D}\) means that \(\mathbf{g}_q\) is an inner product only on \(\mathcal{D}_q\) (and not on the whole of \(T_q\mathsf{M}\), as is the case on a Riemannian manifold). Notice that, if we were to take \(\mathcal{D} = T\mathsf{M}\), then the SR manifold \((\mathsf{M},\mathcal{D},\mathbf{g})\) is simply a Riemannian manifold \(\mathsf{M},\mathbf{g})\).

The metric \(\mathbf{g}\) on \(\mathcal{D}\) allows us to define angles, length, etc. of admissible tangent vectors (i.e., tangent vectors in \(\mathcal{D}\)). In particular, we can define the length of curves in \(\mathsf{M}\) tangent to \(\mathcal{D}\) (i.e., curves \(\gamma : [0,T] \to \mathsf{M}\) such that \(\dot{\gamma}(t) \in \mathcal{D}_{\gamma(t)}\) for every \(t \in [0,T]\)), and thus can talk about the*shortest* such curves; these are called the *sub-Riemannian geodesics*. Having a measure of the length of curves tangent to \(\mathcal{D}\) also gives rise to a distance function on \(\mathsf{M}\) (hence \(\mathsf{M}\) is actually a metric space); this distance (called the *Carnot**–Carathé**odory distance*) is a central feature of SR geometry. (Riemannian geometry also possesses such a distance function, and it is of likewise importance).

Some of the problems in SR geometry that we are concerned with are:*are* similar. Optimal control theory provides many useful tools for the study of sub-Riemannian manifolds; in particular, the problem of finding the sub-Riemannian geodesics may be phrased as an optimal control problem. Accordingly, there is a lot of overlap in control theory and sub-Riemannian geometry.

**Nonholonomic Riemannian Geometry**

*Nonholonomic Riemannian* (NHR) *geometry* may be viewed as another generalisation of Riemannian geometry, distinct from SR geometry. While SR geometry may be thought of as generalising the “distance” aspects of Riemannian geometry (the central object in a SR manifold is a distance function, namely the Carnot–Carathéodory), NHR geometry generalises the “connection” aspects (the central object in a NHR manifold is a connection called the *nonholonomic connection*). The ingredients for a NHR manifold are: a manifold \(\mathsf{M}\), a subbundle \(\mathcal{D} \subsetneq T\mathsf{M}\) (satisfying the same conditions as on a SR manifold), a complement \(\mathcal{D}^{\perp}\) to \(\mathcal{D}\) (so that \(T\mathsf{M} = \mathcal{D} \oplus \mathcal{D}^{\perp}\)), and a metric \(\mathbf{g}\) defined on \(\mathcal{D}\). (Essentially, a NHR structure is a SR structure + the complement \(\mathcal{D}^{\perp}\). Once again, if \(\mathcal{D} = T\mathsf{M}\), then we recover the case of a Riemannian manifold.)

The additional structure on a NHR manifold ensures the existence of the*nonholonomic connection*. This connection generalises the Levi-Civita connection on a Riemannian manifold. Not only does the connection give us a means to parallel transport geometric objects along curves tangent to \(\mathcal{D}\), it also allows us to define the *straightest curves* (called the *nonholonomic geodesics*). Moreover, unlike in Riemannian geometry, shortest ≠ straightest. Hence, one can view SR geometry as the geometry of shortest curves, and NHR geometry as the geometry of straightest curves. The nonholonomic connection also allows one to measure the curvature of the NHR manifold. Having said that, the curvature situation is considerably more complicated than that in Riemannian geometry, and there is still very little known about the curvature of NHR manifolds.

Some of the problems in NHR geometry that we are interested in are:

For more information on Riemannian, sub-Riemannian, and nonholonomic Riemannian geometry, we suggest the following references:

There is also an alternative approach to the geodesics. This approach uses the idea of an affine connection. Briefly, a connection is means to translate geometric objects (for instance, a tangent vector) along a curve in a parallel fashion. The connection also allows one to define:

- the acceleration of a curve (which is not possible without a connection);
- the derivative of a vector field along another vector field.

While we are interested in Riemannian geometry, and have done some research in this area, we have primarily concentrated on two generalisations of Riemannian geometry, namely: sub-Riemannian geometry, and nonholonomic Riemannian geometry. We briefly describe these two fields here, with an emphasis on the type of research topics we are interested in.

The metric \(\mathbf{g}\) on \(\mathcal{D}\) allows us to define angles, length, etc. of admissible tangent vectors (i.e., tangent vectors in \(\mathcal{D}\)). In particular, we can define the length of curves in \(\mathsf{M}\) tangent to \(\mathcal{D}\) (i.e., curves \(\gamma : [0,T] \to \mathsf{M}\) such that \(\dot{\gamma}(t) \in \mathcal{D}_{\gamma(t)}\) for every \(t \in [0,T]\)), and thus can talk about the

Some of the problems in SR geometry that we are concerned with are:

*Equivalence and classification*. The equivalence problem is to decide when two sub-Riemannian structures are essentially the same structure in disguise. The main form of equivalence is with respect to “isometries,” i.e., distance-preserving maps. The classification problem involves obtaining an exhaustive list of all distinct (i.e., inequivalent) SR structures, either on a particular manifold (usually, a Lie group), or on a class of manifolds (e.g., low-dimensional Lie groups).*Calculating the sub-Riemannian geodesics*. The problem of determining the sub-Riemannian geodesics may be phrased as an optimal control problem. Hence, finding explicit expressions for a sub-Riemannian geodesic reduces to the problem of finding the extremal trajectories of an optimal control problem.*Optimality analysis of the geodesics*. Just as one analyses the optimality of extremal trajectories in optimal control, so we are interested in the analogous problem for sub-Riemannian geodesics. For instance, one such problem is to determine at what point a particular geodesic loses optimality (in that, beyond that point, there is a shorter geodesic that does the same job).

The additional structure on a NHR manifold ensures the existence of the

Some of the problems in NHR geometry that we are interested in are:

*Equivalence and classification*. The equivalence problem is to determine when two NHR structures are really the same structure in disguise. There are a few natural equivalence relations one might consider; we have primarily been concerned with equivalence up to isometries, i.e., distance-preserving maps. The classification problem involves obtaining an exhaustive list of all distinct (i.e., inequivalent) NHR structures, either on a particular manifold (usually, a Lie group), or on a class of manifolds (e.g., low-dimensional Lie groups).*Curvature*. Curvature plays a very significant role in Riemannian geometry; the situation is no different in NHR geometry. However, as mentioned previously, there is still much ot be done in this area. We have looked at different notions of curvature for NHR manifolds, studied how the curvature affects the geometry of the space (e.g., the nonholonomic geodesics), and have also considered the*flat structures*(i.e., those with vanishing curvature).*Invariant NHR structures on Lie groups*. The invariant NHR structures on Lie groups exhibit some special properties, and are the prototypes for more general structures. There is also a class of invariant structures whose nonholonomic geodesics are compatible (in a certain sense) with the Lie group structure; this NHR manifolds are of particular interest.

For more information on Riemannian, sub-Riemannian, and nonholonomic Riemannian geometry, we suggest the following references:

- J.M. Lee,
**Riemannian manifolds**, Springer, New York, 1997. - P. Petersen,
**Riemannian geometry**, Springer, New York, 2006. - A.A. Agrachev, D. Barilari, and U. Boscain,
**Introduction to Riemannian and sub-Riemannian geometry**, available from http://webusers.imj-prg.fr/~davide.barilari/Notes.php (accessed July 2018). - R. Montgomery,
**A tour of subriemannian geometries, their geodesics and applications**, American Mathematical Society, Providence, RI, 2002. - A.M. Bloch,
**Nonholonomic mechanics and control**, Springer, New York, 2003. - J. Cortés Monforte,
**Geometric, control and numerical aspects of nonholonomic systems**, Springer, Berlin, 2002. - A.M. Vershik and V.Ya. Gershkovich,
**Nonholonomic problems and the theory of distributions**, Acta Appl. Math. 12 (1988), 181–209. - J. Koiller, P.R. Rodrigues, and P. Pitanga,
**Non-holonomic connections following Élie Cartan**, An. Acad. Bras. Cienc. 73 (2001), 165–190. - V. Dragović and B. Gajić,
**The Wagner curvature tensor in nonholonomic mechanics**, Regul. Chaotic Dyn. 8 (2003), 105–123.

To be completed.