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Research

 

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OVERVIEW

 

My research is at the intersection of control and systems theory and mathematical physics, in particular quantum physics and quantum information. I study differential geometry, Lie theory and nonlinear analysis and their impact in problems in physics and control. Some of my papers can be found here For a full list see the current CV. 

 

 

 

MOST RECENT  PROJECTS (as of 12/10/2020)

 

 

Optimal Synthesis for quantum systems in the KP configuration

 

The dynamics of quantum control systems which are controlled in open loop via a semiclassical field and are not interacting with the environment can be described by the Schrodinger equation for the evolution operator. In some cases such an equation has a structure called KP. That is, the Hamiltonian generators belong to the K and P spaces of an underlying Cartan decomposition of su(n). Such a structure can be used to solve explicitly several optimal control problems for these systems. These include steering in minimum time or with minimum energy between two two given points. Explicit expressions for the optimal controls and a complete optimal synthesis have been found for low dimensional examples (2 and 3 level quantum  systems). References: [1],[2],[3],[4],[5] 

 

 

 

Symmetry Reduction in sub-Riemannian Geometry and the interplay between Riemannian and sub-Riemannian geometry

 

Certain sub-Riemannian problems on a manifold M and equivalent optimal control problems admit a group of symmetries G which allows us to reduced the problem to the  quotient space M/G of the orbits of M under G. It is in fact often possible to endow M/G with a Riemannian metric so that the Riemannian length of curves on M/G coincides with the sub-Riemannian length on M. Care must be taken because M/G is not in general a manifold but it has the more general structure of a stratified space. It has however a subset which is open and dense in M/G, called the `regular part' which can be given the structure of a manifold. This research aims at understanding how much information on the sub-Riemannian structure on M can be inferred from the corresponding Riemannian structure on M/G. These include for example the characterization of geometric objects such as the sub-Riemannian cut locus  and the practical design of control laws for systems with non-holonomic constraints.

References: [6],[7], [8]

 

Controllability and Dynamical Decomposition of Quantum Control Systems

 

Quantum control systems are generically controllable which means that every unitary operation can be obtained using an appropriate control. However in physical systems, due to the inherent symmetries of the model, controllability is often lost and the dynamics splits into a parallel of different systems on a vector space with smaller dimension. Several questions therefore arise: How do you mathematically obtain such a splitting and identify to smaller invariant subspaces?  Is the system controllable on each subspace (subspace controllability)? How do you control in parallel? Do quantum states of any significance fall in any of these subspaces? For example highly  entangled states. Methods of Lie algebra structure theory and representation theory have been applied to obtain model decomposition. Results have been obtained so far for networks of spins where the topology of the network itself suggests the set of symmetries of the system. [

References: [9], [10], [11], [12].  

 

 

 

 

 

REFERENCES

 

[1] D. D'Alessandro, B. Sheller and Z. Zhu, Time-optimal control of quantum Lambda systems in the KP configuration, Journal of Mathematical Physics, 61, 052107 (2020) .

[2] F. Albertini, D. D'Alessandro and B. Sheller, Sub-Riemannian geodesics on SU(n)/S(U(n-1) X U(1)) and optimal control of three level quantum systems, IEEE Transactions on Automatic Control, 65, Vol. 3, 1176-1191 (2020).

[3]  Y. Ji, J. Bian, M. Jiang, D. D'Alessandro and X. Peng, Time-optimal control of independent spin-1/2 systems under simultaneous control,  Physical Review A 98, 062108 December 2018.

[4]  F. Albertini and D. D'Alessandro, The K-P problem on tensor products of Lie groups and time-optimal control of n quantum bits with a bounded field,  IEEE Transactions on Automatic Control, Vol 63, No. 2, February 2018, pp. 518-524. 

[5]  F. Albertini and D. D'Alessandro, Time optimal simultaneous control of two level quantum systems,  Automatica, Volume 74, December 2016, Pages 55-62

[6]  D. D'Alessandro and B. Sheller, Algorithms for Quantum Control without Discontinuities; Application to the Simultaneous Control of two Qubits, J. Math. Phys.,  60 (2019), no. 9, 092101. 

[7]  F. Albertini and D. D'Alessandro, On symmetries in time optimal control, sub-Riemannian geometries and the K-P problem, J. Dyn. Control Syst.,  24 (2018), no. 1, pp. 13-38

[8] D. D'Alessandro and Z. Zhu, A different look at the optimal control of the Brockett integrator, Preprint Department of Mathematics Iowa State University,  Nov. 2020, Submitted. 

[9] D. D'Alessandro and J. Hartwig, Dynamical Decomposition of Bilinear Control Systems subject to Symmetries,  Journal of Dynamical and Control Systems,  (2020)

[10]  F. Albertini and D. D'Alessandro, Subspace controllability of bipartite symmetric spin networks,  Linear Algebra Appl,  585 (2020), 1-23. Corrigendum in Linear Algebra Appl., Volume 603, 15 October 2020, Pages 508-510.

[11]  F. Albertini and D. D'Alessandro, Controllability of Symmetric Spin Networks,  Journal of Mathematical Physics,  59, 052102, (2018)

[12]   F. Albertini and D. D'Alessandro, Subspace controllability of bipartite spin networks, Preprint, xxx.arXiv:2006.11402, Submitted.