This paper examines the controllability for quantum control systems with SU(1,1) dynamical symmetry, namely, the ability to use some electromagnetic field to redirect the quantum system toward a desired evolution. The problem is formalized as the control of a right invariant bilinear system evolving on the Lie group SU(1,1) of two dimensional special pseudo-unitary matrices. It is proved that the elliptic condition of the total Hamiltonian is both sufficient and necessary for the controllability. Conditions are also given for small time local controllability and strong controllability. The results obtained are also valid for the control systems on the Lie groups SO(2,1) and SL(2,R).
In this paper we extend the results on controllability of linear systems obtained in "Controllability of linear systems on solvable Lie groups", from solvable Lie groups to Lie groups with finite semisimple center.
We study possibilities to control an ensemble (a parameterized family) of nonlinear control systems by a single parameter-independent control. Proceeding by Lie algebraic methods we establish genericity of exact controllability property for finite ensembles, prove sufficient approximate controllability condition for a model problem in $\mathbb{R}^3$, and provide a variant of Rashevsky-Chow theorem for approximate controllability of control-linear ensembles.
Linear systems on Lie groups are a natural generalization of linear system on Euclidian spaces. For such systems, this paper studies controllability by taking in consideration the eigenvalues of an associated derivation D. When the state space is a solvable connected Lie group, controllability of the systems is guaranteed if the reachable set of the neutral element is open and the derivation D has only pure imaginary eigenvalues. For bounded systems on nilpotent Lie groups such conditions are also necessary.
In this paper we show that a complete characterization of the controllability property for linear control system on three-dimensional solvable nonnilpotent Lie groups is possible by the LARC and the knowledge of the eigenvalues of the derivation associated with the drift of the system.
Bilinear systems emerge in a wide variety of fields as natural models for dynamical systems ranging from robotics to quantum dots. Analyzing controllability of such systems is of fundamental and practical importance, for example, for the design of optimal control laws, stabilization of unstable systems, and minimal realization of input-output relations. Tools from Lie theory have been adopted to establish controllability conditions for bilinear systems, and the most notable development was the Lie algebra rank condition (LARC). However, the application of the LARC may be computationally expens…
This paper presents graph theoretic conditions for the controllability and accessibility of bilinear systems over the special orthogonal group, the special linear group and the general linear group, respectively, in the presence of drift terms. Such bilinear systems naturally induce two interaction graphs: one graph from the drift, and another from the controlled dynamics. As a result, the system controllability or accessibility becomes a property of the two graphs in view of the classical Lie algebra rank condition. We establish a systemic way of transforming the Lie bracket operations in the…
Problems involving control of large ensmebles of structurally identical dynamical systems, called \emph{ensemble control}, arise in numerous scientific areas from quantum control and robotics to brain medicine. In many of such applications, control can only be implemented at the population level, i.e., through broadcasting an input signal to all the systems in the population, and this new control paradigm challenges the classical systems theory. In recent years, considerable efforts have been made to investigate controllability properties of ensemble systems, and most works emphasized on linea…
In this paper, we study graphical conditions for structural controllability and accessibility of drifted bilinear systems over Lie groups. We consider a bilinear control system with drift and controlled terms that evolves over the special orthogonal group, the general linear group, and the special unitary group. Zero patterns are prescribed for the drift and controlled dynamics with respect to a set of base elements in the corresponding Lie algebra. The drift dynamics must respect a rigid zero-pattern in the sense that the drift takes values as a linear combination of base elements with strict…
Let $S$ be subsemigroup with nonempty interior of a complex simple Lie group $G$. It is proved that $S=G$ if $S$ contains a subgroup $G(α) \approx \mathrm{Sl}(2,\mathbb{C}) $ generated by the $\exp \mathfrak{g}_{\pm α}$, where $\mathfrak{g}%_α$ is the root space of the root $α$. The proof uses the fact, proved before, that the invariant control set of $S$ is contractible in some flag manifold if $S$ is proper, and exploits the fact that several orbits of $G(α)$ are 2-spheres not null homotopic. The result is applied to revisit a controllability theorem and get some improvements.
This paper is devoted to the study of controllability of linear systems on generalized Heisenberg groups. Some general necessary controllability conditions and some sufficient ones are provided. We introduce the notion of decoupled systems, and more precise controllability criteria are stated for them.
In this paper we study the controllability problem for a symmetric-top molecule, both for its classical and quantum rotational dynamics. The molecule is controlled through three orthogonal electric fields interacting with its electric dipole. We characterize the controllability in terms of the dipole position: when it lies along the symmetry axis of the molecule neither the classical nor the quantum dynamics are controllable, due to the presence of a conserved quantity, the third component of the total angular momentum; when it lies in the orthogonal plane to the symmetry axis, a quantum symme…
We study the controllability of a closed control-affine quantum system driven by two or more external fields. We provide a sufficient condition for controllability in terms of existence of conical intersections between eigenvalues of the Hamiltonian in dependence of the controls seen as parameters. Such spectral condition is structurally stable in the case of three controls or in the case of two controls when the Hamiltonian is real. The spectral condition appears naturally in the adiabatic control framework and yields approximate controllability in the infinite-dimensional case. In the finite…
We study local controllability and optimal control problems for invertible discrete-time control systems. We present second order necessary conditions for optimality and sufficient conditions for local controllability. The conditions are stated in geometric terms, using vector fields naturally associated to the system. The Hessian of the optimal problem is computed in terms of Lie brackets of vector fields of the system.
It is first shown that a smooth controllable system on a compact manifold is finite time controllable. The technique of proof is close to the one of Sussmann's orbit theorem, and no rank condition is required. This technique is also used to give a new and elementary proof of the equivalence between controllability for essentially bounded inputs and for piecewise constant ones. Two sufficient conditions for controllability at exact time on a compact manifold are then stated. Some applications, in particular to linear systems on Lie groups, are provided.
For a symmetric system, we want to study the problem of crossing an hypersurface in the neighborhood of a given point, when we suppose that all of the available vector fields are tangent to the hypersurface at the point. Classically one requires transversality of at least one Lie bracket generated by two available vector fields. However such condition does not take into account neither the geometry of the hypersurface nor the practical fact that in order to realize the direction of a Lie bracket one needs three switches among the vector fields in a short time. We find a new sufficient conditio…
Controllability of chemical reactions is an important problem in chemical engineering science. In control theory, analysis of the controllability of linear systems is well-founded, however the dynamics of chemical reactions is usually nonlinear. Global controllability properties of chemical reactions are analyzed here based on the Lie-algebra of the vector fields associated to elementary reactions. A chemical reaction is controllable almost everywhere if all the reaction rate coefficients can be used as control inputs. The problem where one can not control all the reaction rate coefficients is…
For a control system two major issues can be considered: the stabilizability with respect to a given target, and the minimization of an integral functional (while the trajectories reach this target). Here we consider a problem where stabilizability or controllability are investigated together with the further aim of a "cost regulation", namely a state-dependent upper bounding of the functional. This paper is devoted to a crucial step in the program of establishing a chain of equivalences among degree-k stabilizability with regulated cost, asymptotic controllability with regulated cost, and the…
In this article, we study the controllability issues of the Landau-Lifshitz-Gilbert Equations (LLGEs), accompanied with non-zero exchange energy only, in an interval in one spatial dimension with Neumann boundary conditions. The paper is of twofold. In the first part of the paper, we study the controllability issues of the LLGEs. The control force acting here is degenerate i.e., it acts through a few numbers of low mode frequencies. We exploit the Fourier series expansion of the solution. We borrow methods of differential geometric control theory (Lie bracket generating property) to establish …
We consider a network of n spin 1/2 systems which are pairwise interacting via Ising interaction and are controlled by the same electro-magnetic control field. Such a system presents symmetries since the Hamiltonian is unchanged if we permute two spins. This prevents full (operator) controllability in that not every unitary evolution can be obtained. We prove however that controllability is verified if we restrict ourselves to unitary evolutions which preserve the above permutation invariance. For low dimensional cases, n=2 and n=3, we provide an analysis of the Lie group of available evolutio…
The purpose of this paper is to present explicitly the solution curve for affine control systems on Lie groups under the assumption that automorphisms associated to the linear vector fields commutes. If we assume that the derivations associated to linear vector fields are inner, we obtain a simpler solution and we show some results of controllability. To end, we work with conjugation by homomorphism of Lie groups between affine systems.
The controllability of a fully three-dimensional $N$-link swimmer is studied. After deriving the equations of motion in a low Reynolds number fluid by means of Resistive Force Theory, the controllability of the minimal $2$-link swimmer is tackled using techniques from Geometric Control Theory. The shape of the $2$-link swimmer is described by two angle parameters. It is shown that the associated vector fields that govern the dynamics generate, via taking their Lie brackets, all six linearly independent directions in the configuration space; every direction and orientation can be achieved by op…
The ergodicity and the output-controllability of stochastic reaction networks have been shown to be essential properties to fulfill to enable their control using, for instance, antithetic integral control. We propose here to extend those properties to the case of uncertain networks. To this aim, the notions of interval, robust, sign, and structural ergodicity/output-controllability are introduced. The obtained results lie in the same spirit as those obtained in [Briat, Gupta & Khammash, Cell Systems, 2016] where those properties are characterized in terms of control theoretic concepts, linear …
Let $\mathrm{Sl}\left( n,\mathbb{H}\right)$ be the Lie group of $n\times n$ quaternionic matrices $g$ with $\left\vert \det g\right\vert =1$. We prove that a subsemigroup $S \subset \mathrm{Sl}\left( n,\mathbb{H}\right)$ with nonempty interior is equal to $\mathrm{Sl}\left( n,\mathbb{H}\right)$ if $S$ contains a subgroup isomorphic to $\mathrm{Sl}\left( 2,\mathbb{H}\right)$. As application we give sufficient conditions on $A,B\in \mathfrak{sl}\left( n,\mathbb{H}\right)$ to ensuring that the invariant control system $\dot{g}=Ag+uBg$ is controllable on $\mathrm{Sl}\left( n,\mathbb{H}\right)$. We…
The purpose of this paper is to describe explicitly the solution for linear control systems on Lie groups. In case of linear control systems with inner derivations, the solution is given basically by the product of the exponential of the associated invariant system and the exponential of the associated invariant drift field. We present the solutions in low dimensional cases and apply the results to obtain some controllability results.
We consider scalar-input control systems in the vicinity of an equilibrium, at which the linearized systems are not controllable. For finite dimensional control systems, the authors recently classified the possible quadratic behaviors. Quadratic terms introduce coercive drifts in the dynamics, quantified by integer negative Sobolev norms, which are linked to Lie brackets and which prevent smooth small-time local controllability for the full nonlinear system. In the context of nonlinear parabolic equations, we prove that the same obstructions persist. More importantly, we prove that two new beh…
We consider nonlinear scalar-input differential control systems in the vicinity of an equilibrium. When the linearized system at the equilibrium is controllable, the nonlinear system is smoothly small-time locally controllable, i.e., whatever $m>0$ and $T>0$, the state can reach a whole neighborhood of the equilibrium at time $T$ with controls arbitrary small in $C^m$-norm. When the linearized system is not controllable, we prove that small-time local controllability cannot be recovered from the quadratic expansion and that the following quadratic alternative holds. Either the state is constra…
