Interdisciplinary Application of Lie Groups Thesis

The study of Lie groups has long captivated researchers in the realms of differential geometry, control theory, and quantum mechanics. In pursuit of a deeper understanding of these interconnected fields, my undergraduate thesis in Mathematics focused on exploring the applications of Lie groups within each discipline. The thesis provided a succinct yet comprehensive overview of the role Lie groups play in these fields, while also introducing the fundamental principles of Lie group theory.

For brevity, I will present one of the applications I analyzed, Affine Nonlinear Control Systems. To delve into additional examples and access the complete thesis, you can download it here.

Application for Affine Nonlinear Control Systems

Lie algebra rank condition

In control theory, the Lie algebra rank condition is necessary for the controllability of a nonlinear system. It relates the rank of the Lie bracket of vector fields associated with the system to the dimension of the state space.

Controllability

In control theory, a system is controllable if, starting from any initial state, it is possible to steer it to any desired state using a suitable control input. Controllability is a vital system property because it determines the extent to which a system can be controlled.

The Lie algebra rank condition is a condition that must be satisfied for a nonlinear system to be controllable. It states that the system is controllable if and only if the rank of the Lie bracket of the vector fields associated with the system is equal to the dimension of the state space.

Example

The Lie bracket of two vector fields is another vector field that represents the commutator of the two vector fields. In other words, it measures how much the two vector fields "fail to commute." The Lie bracket is defined as:

where $L_u$ and $L_v$ are the vector fields associated with $u$ and $v$, respectively, and $[u,v]$ is the vector commutator of $u$ and $v$.

The rank of the Lie bracket of the vector fields associated with a system is then defined as the rank of the matrix whose entries are the Lie brackets of all pairs of vector fields associated with the system.

The Lie algebra rank condition states that if this rank is equal to the dimension of the state space, then the system is controllable. Intuitively, this means that the vector fields associated with the system are "independent enough" to span the entire state space, which is necessary for controllability.

To illustrate this with an example, consider a nonlinear system given by the following set of differential equations:

where $x$ and $y$ are the state variables, and $u$ and $v$ are the control inputs. The vector fields associated with this system are:

The Lie bracket of these vector fields is:

The rank of this Lie bracket is 1, which is less than the dimension of the state space (2). Therefore, according to the Lie algebra rank condition, the system is not controllable.

In summary, the Lie algebra rank condition is necessary for the controllability of a nonlinear system. It relates the rank of the Lie bracket of vector fields associated with the system to the dimension of the state space.