In optimal control theory,  one tries to find the best strategy to drive a system of interest to a desired state.   Such types of problems are easy to formalize in mathematical terms but  difficult to solve explicitly. In fact,  a solution becomes nearly impossible when the dimension of the mathematical model increases. Several problems however admits symmetries. Symmetries  allow us to decrease the dimension of a problem and can facilitate  its solution. This research has developed techniques to use symmetries to analyze the dynamics and solve optimal control problems for various systems relevant in applications.  Particular emphasis has been placed on  models  that appear in the manipulation of quantum mechanical systems. Several problems of practical interest have been given an explicit solution. Some notable outcomes  are as follows:

1) Quantum Lambda Systems are quantum systems which have three energy levels. It is possible to transfer population directly between the lowest two energy levels and the highest level but not  between the lowest two energy levels themselves (cf. Figure 1). The control,  which in physical terms is typically an appropriately shaped electro-magnetic field,  has to be designed  to minimize the time of transfer and-or to minimize the average occupancy in the highest level. Systems in this level of energy  are the one most subject to the de-cohering effect of the environment. This interaction with the external environment destroys the quantum nature of the dynamical evolution which is useful in technological applications. Explicit expressions have been found for the optimal control laws in these cases. 

2) Quantum Bits are quantum systems with two energy levels. They are of paramount importance in quantum computation as they are the elementary physical unit of information. Optimal fields have been calculated for their manipulation. These fields  minimize time of state  transfer and-or the energy used. These results have been also validated with practical experiments which have shown that such theoretical-mathematical results can in fact be physically implemented.

3) Sub-Riemannian geometry problems are ubiquitous in applications not only for quantum systems but also in every area of engineering. They appear when  the dynamics is naturally modeled on a configuration manifold and the controls can only steer at every point in certain directions and not in others. The admissible trajectories are the ones that, at every point are tangent to the given directions (cf. Figure 2). A typical problem of interest is to find admissible  trajectories between two points which minimize an appropriately  defined length. Such trajectories are called sub-Riemannian geodesics. This research has shown how, in the presence of symmetries,  a sub-Riemannian problem can be reduced to a Riemannian problem where the whole, well developed, machinery of Riemannian geometry can be applied. Furthermore the Riemannian problem has  typically lower dimension and the Riemannian geodesics minimizing length  can sometimes be visualized. From them the sub-Riemannian geodesics for the original problem can be calculated. Using this technique several sub-Riemannian problems of interest have been explicitly solved. 

4)  Networks of particles with spin such has atoms or electrons in crystals appear naturally in applications. In the presence of symmetries,  the dynamics of such systems under the action of a common electromagnetic field, happen  in `invariant subspaces'. As a consequence,  there are transitions from a certain set of states to others which are forbidden. The results of this research have provided mathematical methods to obtain such a decomposition in invariant subspaces explicitly. This allows us, once again, to reduce the dimension of a control problem by focusing only on certain sub-systems of interest which are of lower dimension. 

Overall, this research has provided a comprehensive mathematical theory to analyze how the presence of symmetries affects the control and analysis of dynamical systems. The research has  revealed  general principles and, at the same time, has led to the explicit solution of control and analysis problems for practical systems with an emphasis on systems which behave quantum mechanically. 

The ability to effectively manipulate quantum mechanical systems is at the heart of any future application of quantum mechanics. These applications involve fast quantum computing, secure communication, precise quantum measurement of physical quantities (quantum metrology), medical diagnostics through NMR and MRI, laser technology, etc. All these applications indicate that quantum mechanics will have a larger impact on our everyday life in the future. For this reason the knowledge and application of quantum mechanics will have to be part of the standard engineering background. In this respect, this research has also contributed to the development of a curriculum in `Quantum Engineering' by promoting research experiences in quantum mechanics for undergraduate students and new courses and instructional material describing mathematical techniques for the  manipulation of quantum systems. 

Last Modified: 12/20/2022
Modified by: Domenico D'alessandro