Gunther Dirr University of Würzburg Würzburg, Germany [email protected] Jens Jordan University of Würzburg Würzburg, Germany [email protected] Indra Kurniawan ITK Engineering AG Stuttgart, Germany [email protected]
Abstract. This short note deals with the issue of generic accessibility of bilinear control systems. We investigate (right-)invariant control systems evolving on a matrix Lie group G with Lie algebra g. Thereby, both the drift term and the control terms may vary in possibly different analytic subsets of g. Based on standard arguments on analytic functions, we derive a necessary and sufficient condition for generic accessibility within this class of bilinear systems. In combination with previous results in the literature, we obtain a particular simple genericity criterion if g is semisimple. As an application, we demonstrate that almost all finite dimensional open quantum control systems (modelled by a Lindblad-Kossakowski master equation with controls entering only its Hamiltonian part) are accessible.
1
Introduction
Bilinear control systems constitute a class of nonlinear control systems which find numerous applications in many different areas such as physics, engineering, ecology and medicine [8, 18]. In most of these applications, the underlying dynamical models depend on partially unknown parameters. Therefore, one is interested in control properties which are valid not only for a particular bilinear control system but for all or at least a large subclasses of systems.
Probably, accessibility and controllability are the most fundamental properties of control system. Since the work of Lobry, Stefan and Sussmann (see [22] and the references therein) it is known that both properties are robust against small perturba- tions and accessibility is even a generic property for non-linear control systems (with respect to the fine Ck-topology). Furthermore, for linear systems a classical result says that also controllability in generic [21].
If it comes to bilinear systems less is known. There are only a few results mainly concerned with semisimple Lie groups. One result by Jurdjevic and Kupka [11, 12] is essentially that the set of all pairs which generate the whole Lie algebra sln(R)
is open and dense and therefore bilinear control systems on sln(R) are generically
accessible. The aim of this note is to extend this result in two directions: First, we derive a necessary and sufficient condition for generic accessibility (controllability)
Festschrift in Honor of Uwe Helmke G. Dirr et al. which is applicable to any bilinear system. Secondly, we focus on “structured” bilinear systems on semisimple Lie groups. Here, structured means that the drift term and the control terms are not allowed to vary in the entire Lie algebra but only in a prespecified “thin” subset. Such scenarios often arise in systems whose dynamics is related to some underlying weighted graph structure, where the weights may vary but not the graph structure itself.
For deriving the first result, we slightly modify the standard proof of generic con- trollability from linear systems theory. More precisely, the well-known fact that the set
{(A,b) ∈ gln(R)×Rn∣ span{b,Ab,...,An−1b} = Rn}
is open and dense in gln(R)×Rnis usually based on a simple argument about the
zero set of polynomials. The same idea leads in the bilinear case to an if-and-only-if statement on generic accessibility (controllability). The second result, similar to the work by Jurdjevic and Kupka [11, 12] exploits heavily the structure theory of semisimple Lie algebras.
Finally, in the last section, we present an application of our results to quantum control. Most quantum processes (which satisfy the assumption of Markovian dynamics) can be modelled as bilinear control systems, e.g. [5, 6]. The controlled Lindblad- Kossakowski master equation [10, 17], which describes an open quantum system, i.e. a non-isolated quantum systems interacting with the environment, constitutes for instance a bilinear control system on the space of all density operators. It is known that the Lindblad-Kossakowski master equation with controls entering only its Hamiltonian part is never controllable [2, 6]. Nevertheless, accessibility, which guarantees that the reachable sets have at least non-empty interior, may apply. Our goal is to prove that accessibility is actually a generic property of the Lindblad- Kossakowski master equation even in the single control case. Similar statements dealing with the generic accessibility of open quantum systems also appeared in the work by C. Altafini [3].
The paper is organized as follows. Section 2 provides the basic facts on accessibility and controllability of bilinear control systems on Lie groups. Section 3 contains the main results: the general case is treated in Subsection 3.1; the real semisimple one in Subsection 3.2. In Section 4, we give an application of our results to open quantum systems. Most proofs are only sketched, more comprehensive details will be provided in a forthcoming full paper.
. . . and now for something completely different: HAPPY BIRTHDAY, UWE!
2
Preliminaries
To fix notation, let gln(R) and gln(C) be the Lie algebra of all real and, respectively, complex n×n matrices. Moreover, let son(R) ⊂ gln(R) and su(n) ⊂ gln(C) denote
the Lie subalgebras of all skew-symmetric and, respectively, all skew-Hermitian
matrices with trace zero. For arbitrary n×n matrices, the trace and the commutator
are given by Tr(A) ∶= ∑nk=1akkand[A,B] ∶= AB−BA, respectively. The identity matrix
Now, let g be a Lie subalgebra of gln(R), i.e. g is a subspace of gln(R) which is closed under taking commutators. Then there exists a unique Lie subgroup G of GLn(R) which corresponds to g in the sense that the tangent space of G at the identity
coincides with g. A bilinear or, equivalently, a (right)-invariant control systems on G is given by
(Σ) X˙= (A0+∑m
k=1
uk(t)Ak)X, X(0) = X0∈ G, (1)
where A0,A1,...Am∈ g and u(t) ∶= (u1(t),...,um(t)) ∈ U ⊂ Rmis an admissible real-
valued control input. For our purposes, the class of piecewise constant controls u(⋅)
assuming values inRm(i.e. U= Rm) is convenient. However, in many cases the
assumption U= Rmcan be considerably relaxed by requiring that only the convex
hull of the control set U contains the origin as an interior point [11].
Next, we define the terms accessibility and controllability for (bilinear) control systems. To this end, we need the concept of reachability. LetRT(X0) be the set of
all X∈ G which can be reached from X0in time T≥ 0, i.e.
RT(X0) ∶= {Xu(T) ∣ u ∶ [0,T] → Rmadmissible control}, (2)
where Xu(⋅) denotes the corresponding solution of Σ. Thus the entire reachable set
of X0and Σ is given by
R(X0) ∶= ⋃
T≥0R
T(X0). (3)
Then, Σ is called accessible if for all X0∈ G the reachable set R(X0) has non-empty
interior in G, and controllable if for all X0∈ G the reachable set R(X0) is equal to
G. As Σ is right-invariant one hasR(X0) = R(I)X0and therefore accessibility and
controllability of Σ is equivalent to accessibility and, respectively, controllability at the identity I. Moreover, the so-called Lie algebra rank condition (LARC) which is in general only sufficient for accessibility yields the following necessary and sufficient accessibility criterion for right-invariant control systems.
Proposition 1. Let Σ be defined as in (1). Then Σ is accessible if and only if Σ satisfies the LARC-condition at the identity, i.e.⟨A0,A1,...,An⟩L= g.
Here and henceforth, ⟨A0,A1,...,An⟩L or, more generally, ⟨A⟩L denotes the Lie
subalgebra generated byA ⊂ gln(R), i.e. ⟨A⟩L is the smallest Lie subalgebra of
gln(R) which contains A or, equivalently, the smallest subspace of gln(R) which
containsA and all iterated commutators of the form
[A1,A2],[A1,[A2,A3]],[[A1,A2],A3]],[A1,[A2,[A3,A4]]],[[A1,A2],[A3,A4]],...
with Ak∈ A.
For controllability there is in general no such simple condition as for accessibility. Yet, for some special cases one has the following results.
Proposition 2. Let Σ be defined as in (1). Then one has:
(a) If Σ is additionally driftless (i.e. A0= 0) then controllability of Σ is equivalent
Festschrift in Honor of Uwe Helmke G. Dirr et al. (b) If G is compact then controllability of Σ is equivalent to the Lie algebra
condition⟨A0,A1,...,An⟩L= g.
(c) For U= Rm, controllability of Σ is guaranteed by the Lie algebra condition
⟨A1,...,An⟩L= g.
Note that in the compact case accessibility and controllability of Σ are equivalent. A proof of both propositions can be found e.g. in [8, 11, 13]
Now, assume that the drift A0and the control terms A1,...,Ammay vary in some
non-empty subsets D,C1,...,Cm⊂ g, respectively. Then Σ(D;C1,...,Cm) denotes
the family of all bilinear systems which can be obtained while(A0,A1,...,Am) runs
through D×C1×⋯×Cm. For Σ(D;C,...,C) we also write Σ(D;Cm). To specify a
particular system in Σ(D;C1,...,Cm) we use the notation Σ(A0; A1,...,Am). Thus
we are prepared to state precisely what generic accessibility, controllability or, more general, genericity of any property of the family Σ(D;C1,...,Cm) means.
• A property P is called (topologically) generic for Σ(D;C1,...,Cm) if the set
{(A0,A1,...,Am) ∈ D×C1×⋯×Cm∣ Σ(A0; A1,...,Am) satisfies P} (4)
contains an open and dense subset of D×C1×⋯×Cm, where D×C1×⋯×Cmis
equipped with the topology induced by gln(R)m+1.
• If D and C1,...,Cm are smooth submanifolds of gln(R), then P is called
generic(with respect to the Lebesgue measure) for Σ(D;C1,...,Cm) if the
complement of the set defined by (4) has measure zero in D×C1× ⋯ ×Cm
(cf. Remark 3 below).
Remark3.
(a) If D×C1×⋯×Cmis a Baire space, for instance, if D,C1,...,Cmare smooth
submanifolds, then topological genericity implies that the set defined by (4) is of second category.
(b) In general, topological genericity does not imply genericity with respect to the Lebesgue measure nor vice versa. Counterexamples can be obtained by Cantor-like sets.
(c) Sets of measure zero in D×C1× ⋯ ×Cm can simply be defined locally in
coordinate charts and “globalized” via the partition of the unity. General assumption and convention:
(a) From now on we assume U= Rnand that D and C1,...,Cmare real analytic
connected submanifolds of gln(R).
(b) Whenever we do not specify the type of genericity (topological or with respect to the Lebesgue measure), the corresponding result holds for both types.
The trivial, but useful observation that Σ(D,C1,...,Cm′) is generically accessible
(controllable) for all m′≥ m if Σ(D;C1,...,Cm) is generically accessible (controllable)
allows us to put emphasis on the case m= 1.
We complete this preliminary section with an auxiliary results that is well known in Lie theory but maybe not in control theory. It yields an upper bound for the maximal Lie word length which has to be considered in “constructing”⟨A0,...,Am⟩L. To this
end, we define recursively the following sets. LetA be an arbitrary subset of g. Then,
L1(A) ∶= A, Ln(A) ∶=
n−1
⋃
k=1
[Lk(A),Ln−k(A)] for n ≥ 2 (5)
and
L1′(A) ∶= A, L′n(A) ∶= [L′1(A),L′n−1(A)] for n ≥ 2. (6)
Clearly, while Lnincludes all Lie words (over the alphabetA) of length n, the set L′n
contains only Lie words of length n of the particular type
[An,[An−1,[...[A1,A0]]]] with Ak∈ A. (7)
Lemma 4. ForA ⊂ g let Ln(A) and L′n(A) be defined as above. Then
(a) span Ln(A) = span Ln′(A) for all n ∈ N.
(b) If span L′n
∗+1(A) ⊂ ∑
n∗
k=1span L′k(A) for some n∗∈ N then span L′n′(A) ⊂
∑n∗
k=1span L′k(A) for all n′≥ n∗and thus
⟨A⟩L= n∗
∑
k=1
span L′k(A). (8)
Proof. Part (a) follows by induction and the Jacobi identity; part (b) is a straightfor- ward consequence of (a). A complete proof can be found in [4].