Controllability of bilinear systems

Bilinear systems can be developed from the point of view of time-varying linear differential equations or from the symmetry of Lie theory, in particular Lie algebras, Lie groups, and Lie semigroups. For bilinear control systems with bounded control range, we analyze when a unique control set (i.e., a maximal set of approximate controllability) with nonvoid interior exists, for the induced system on projective space. We use the system semigroup by considering piecewise constant controls and use spectral properties to extend the result to bilinear systems in Rd. The contribution of this paper highlights the relationship between all the existent control sets. We show that the controllability property of a bilinear system is equivalent to the existence and uniqueness of a control set of the projective system.

During recent years there has been considerable interest in using bilinear systems [1, 2] as mathematical models to represent the dynamic behavior of a wide class of engineering, biological and economic systems. Moreover, there are some methods [3] which may approximate nonlinear control systems by bilinear systems. For the first time Zubov has proposed a method of stabilization control synthesis for a set of programmed motions in linear systems [4]. In papers [5, 6] this method has been developed and used to solve the same problem for bilinear systems.In the present paper the following problems are considered. First, synthesis of nonlinear control as feedback under which the bilinear control system has a given set of programmed and asymptotic stable motions. Because this control is not unique, the second problem concerns optimal stabilization. In this paper a method for the design of nonlinear optimal control is suggested. This control is constructed in the form of a convergent series. The theorem on the sufficient conditions to solve this problem is represented.

Multipoint interpolation of Volterra series and [formula omitted]-model reduction for a family of bilinear descriptor systems

The systems-theoretic concept of controllability is elaborated for quantum-mechanical systems, sufficient conditions being sought under which the state vector ψ can be guided in time to a chosen point in the Hilbert space ℋ of the system. The Schrödinger equation for a quantum object influenced by adjustable external fields provides a state-evolution equation which is linear in ψ and linear in the external controls (thus a bilinear control system). For such systems the existence of a dense analytic domain 𝒟ω in the sense of Nelson, together with the assumption that the Lie algebra associated with the system dynamics gives rise to a tangent space of constant finite dimension, permits the adaptation of the geometric approach developed for finite-dimensional bilinear and nonlinear control systems. Conditions are derived for global controllability on the intersection of 𝒟ω with a suitably defined finite-dimensional submanifold of the unit sphere Sℋ in ℋ. Several soluble examples are presented to illuminate the general theoretical results.

In this paper, we will discuss the problem of optimal model order reduction of bilinear control systems with respect to the generalization of the well-known ${\cal H}_2$-norm for linear systems. We revisit existing first order necessary conditions for ${\cal H}_2$-optimality based on the solutions of generalized Lyapunov equations arising in bilinear system theory and present an iterative algorithm which, upon convergence, will yield a reduced system fulfilling these conditions. While this approach relies on the solution of certain generalized Sylvester equations, we will establish a connection to another method based on generalized rational interpolation. This will lead to another way of computing the ${\cal H}_2$-norm of a bilinear system and will extend the pole-residue optimality conditions for linear systems, also allowing for an adaption of the successful iterative rational Krylov algorithm to bilinear systems. By means of several numerical examples, we will then demonstrate that the new techniques outperform the method of balanced truncation for bilinear systems with regard to the relative ${\cal H}_2$-error.

Stabilization of unbounded bilinear control systems in Hilbert space

Let G be a connected Lie group with Lie algebra g. This review is devoted to studying the fundamental dynamic properties of elements in the normalizer NG of G. Through an algebraic characterization of NG, we analyze the different dynamics inside the normalizer. NG contains the well-known left-invariant vector fields and the linear and affine vector fields on G. In any case, we show the shape of the solutions of these ordinary differential equations on G. We give examples in low-dimensional Lie groups. It is worth saying that these dynamics generate the linear and bilinear control systems on Euclidean spaces and the invariant and linear control systems on Lie groups. Moreover, the Jouan Equivalence Theorem shows how to extend this theory to control systems on manifolds.

Krylov subspace methods for model order reduction of bilinear control systems

Suboptimal control for bilinear systems is discussed by use of an extension of the linear‐quadratic optimal control index. The design method of this bilinear suboptimal control system is presented. Its application to the moisture control of a paper‐making process is given as an example. The simulation results show that this suboptimal control system functions very well.

In quantum systems theory one of the fundamental problems boils down to: Given an initial state, which final states can be reached by the dynamic system in question? Formulated in the framework of bilinear control systems, the evolution shall be governed by an inevitable Hamiltonian drift term, finitely many control Hamiltonians allowing for (at least) piecewise constant control amplitudes, plus a (possibly bang-bang switchable) noise term in Kossakowski-Lindblad form. Now assuming switchable coupling of finite-dimensional systems to a thermal bath of arbitrary temperature, the core problem of reachability boils down to studying points in the standard simplex amenable to two types of controls that can be used interleaved: Permutations within the simplex, and contractions by a dissipative one-parameter semigroup. We illustrate how the solutions of the core problem pertain to the reachable set of the original controlled Markovian quantum system. This allows us to show that for global as well as local switchable coupling to a temperature-zero bath one can generate every quantum state from every initial state up to arbitrary precision. Moreover we present an inclusion for non-zero temperatures as a consequence of our results on d-majorization. Then we consider infinite-dimensional open quantum-dynamical systems following a unital Kossakowski-Lindblad master equation extended by controls. Here the drift Hamiltonian can be arbitrary, the finitely many control Hamiltonians are bounded, and the switchable noise term is generated by a single compact normal operator. Via new majorization results of ours, we show that such bilinear quantum control systems allow to approximately reach any target state majorized by the initial one, as up to now only has been known in finite-dimensional analogues.

In this study, the author investigates controllability of discrete‐time bilinear systems by using invariant sets. An invariant set defined in this study is a set on which the system is invariant, namely for any initial state that belongs to the set, the system cannot be steered away from this set. If a system is controllable, then it has no invariant set in its state space other than the state space itself. Invariant sets are thus closely related to controllability. By studying the invariant sets of discrete‐time bilinear systems, the author first presents a necessary and sufficient condition for controllability of the systems in dimension two, which covers a classical result under the same condition. Then the author considers the high‐dimensional systems and improve the recent results on controllability by relaxing a condition concerning invariant sets, where more general controllability criteria are derived. Finally, examples are given to illustrate the obtained results.

We study some model control problems which arise in connection with optimal manipulation of dissipative quantum dynamics. It is shown that the problem of optimal control of quantum mechanical phenomenon in presence of dissipation can be reduced to the study of optimization problems associated with a class of constrained bilinear control systems. These bilinear systems x/spl dot/=(A+/spl Sigma//sub i//sup n/u/sub i/B/sub i/)x are characterized by the fact that the controls can be expressed as polynomial functions of fewer parameters, i.e. u/sub i/=g/sub i/ (/spl upsi//sub 1/, /spl upsi//sub 2/,...,/spl upsi//sub k/) where g/sub i/ are polynomials and k<n. A general study of these systems is expected to find immediate applications in coherent control of quantum mechanical phenomenon.

This paper presents graph theoretic conditions for the controllability and accessibility of bilinear systems over the special orthogonal 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 underlying Lie algebra, into specific operations of removing or creating links over the drift and controlled interaction graphs. As a result, we establish a series of graphical conditions for the controllability and accessibility of such bilinear systems, which rely only on the connectivity of the union of the drift and controlled interaction graphs. We present examples to illustrate the validity of the established results, and show that the proposed conditions are in fact considerably tight.

Reachable set estimation for discrete-time bilinear systems with time-varying delays

We discuss two methods of an exact stochastic representation of the non-Markovian quantum dynamics of open systems. The first method employs a pair of stochastic product vectors in the total system's state space, while the second method uses a pair of state vectors in the open system's state space and a random operator acting on the state space of the environment. Both techniques lead to an exact solution of the von Neumann equation for the density matrix of the total system. Employing a spin star model describing a central spin coupled to the bath of surrounding spins, we perform Monte Carlo simulations for both variants of the stochastic dynamics. In addition, we derive an analytical expression for the expectation values of the stochastic dynamics to obtain the exact solution for the density matrix of the central spin.