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A new class of cost functionals for optimal control of quantum systems which produces controls which are sparse in frequency and smooth in time is proposed. This is achieved by penalizing a suitable time-frequency representation of the control field, rather than the control field itself, and by employing norms which are of $L^1$ or measure form with respect to frequency but smooth with respect to time. We prove existence of optimal controls for the resulting nonsmooth optimization problem, derive necessary optimality conditions, and rigorously establish the frequency-sparsity of the optimizers. More precisely, we show that the time-frequency representation of the control field, which a priori admits a continuum of frequencies, is supported on only textit{ finitely many} frequencies. These results cover important systems of physical interest, including (infinite-dimensional) Schrodinger dynamics on multiple potential energy surfaces as arising in laser control of chemical reactions. Numerical simulations confirm that the optimal controls, unlike those obtained with the usual $L^2$ costs, concentrate on just a few frequencies, even in the infinite-dimensional case of laser-controlled chemical reactions.
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We study the synthesis of optimal control policies for large-scale multi-agent systems. The optimal control design induces a parsimonious control intervention by means of l-1, sparsity-promoting control penalizations. We study instantaneous and infinite horizon sparse optimal feedback controllers. In order to circumvent the dimensionality issues associated to the control of large-scale agent-based models, we follow a Boltzmann approach. We generate (sub)optimal controls signals for the kinetic limit of the multi-agent dynamics, by sampling of the optimal solution of the associated two-agent dynamics. Numerical experiments assess the performance of the proposed sparse design.
In this paper, we discuss the compatibility between the rotating-wave and the adiabatic approximations for controlled quantum systems. Although the paper focuses on applications to two-level quantum systems, the main results apply in higher dimension. Under some suitable hypotheses on the time scales, the two approximations can be combined. As a natural consequence of this, it is possible to design control laws achieving transitions of states between two energy levels of the Hamiltonian that are robust with respect to inhomogeneities of the amplitude of the control input.
We study one-parametric perturbations of finite dimensional real Hamiltonians depending on two controls, and we show that generically in the space of Hamiltonians, conical intersections of eigenvalues can degenerate into semi-conical intersections of eigenvalues. Then, through the use of normal forms, we study the problem of ensemble controllability between the eigenstates of a generic Hamiltonian.
The robustness of quantum control in the presence of uncertainties is important for practical applications but their quantum nature poses many challenges for traditional robust control. In addition to uncertainties in the system and control Hamiltonians and initial state preparation, there is uncertainty about interactions with the environment leading to decoherence. This paper investigates the robust performance of control schemes for open quantum systems subject to such uncertainties. A general formalism is developed, where performance is measured based on the transmission of a dynamic perturbation or initial state preparation error to a final density operator error. This formulation makes it possible to apply tools from classical robust control, especially structured singular value analysis, to assess robust performance of controlled, open quantum systems. However, there are additional difficulties that must be overcome, especially at low frequency ($sapprox0$). For example, at $s=0$, the Bloch equations for the density operator are singular, and this causes lack of continuity of the structured singular value. We address this issue by analyzing the dynamics on invariant subspaces and defining a pseudo-inverse that enables us to formulate a specialized version of the matrix inversion lemma. The concepts are demonstrated with an example of two qubits in a leaky cavity under laser driving fields and spontaneous emission. In addition, a new performance index is introduced for this system. Instead of the tracking or transfer fidelity error, performance is measured by the steady-steady entanglement generated, which is quantified by a non-linear function of the system state called concurrence. Simulations show that there is no conflict between this performance index, its log-sensitivity and stability margin under decoherence, unlike for conventional control problems [...].
A geometric setup for control theory is presented. The argument is developed through the study of the extremals of action functionals defined on piecewise differentiable curves, in the presence of differentiable non-holonomic constraints. Special emphasis is put on the tensorial aspects of the theory. To start with, the kinematical foundations, culminating in the so called variational equation, are put on geometrical grounds, via the introduction of the concept of infinitesimal control . On the same basis, the usual classification of the extremals of a variational problem into normal and abnormal ones is also rationalized, showing the existence of a purely kinematical algorithm assigning to each admissible curve a corresponding abnormality index, defined in terms of a suitable linear map. The whole machinery is then applied to constrained variational calculus. The argument provides an interesting revisitation of Pontryagin maximum principle and of the Erdmann-Weierstrass corner conditions, as well as a proof of the classical Lagrange multipliers method and a local interpretation of Pontryagins equations as dynamical equations for a free (singular) Hamiltonian system. As a final, highly non-trivial topic, a sufficient condition for the existence of finite deformations with fixed endpoints is explicitly stated and proved.
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