This section briefly summarize some well-known definitions and facts concerning controlled invariant subspaces. Then invariance entropy for controlled invariant
subspaces of linear control systems onX is defined and related to the subspace
entropy of linear flows as defined in the previous section.
The notion of controlled invariant subspaces (also called(A,B)–invariant subspaces) was introduced by Basile and Marro [1]; see the monographs Wonham [12] and Trentelman, Stoorvogel and Hautus [10] for expositions of the theory.
Consider linear control systems in state space form ˙
x(t) = Ax(t)+Bu(t) (5)
with linear maps A∶ X → X and B ∶ Rm→ X , where X is an n-dimensional normed
vector space. The solutions ϕ(t,x,u),t ≥ 0, of (5) with initial condition ϕ(0,x,u) = x are given by the variation-of-constants formula
ϕ(t,x,u) = etAx+∫
t 0
eA(t−s)Bu(s)ds.
Recall that a subspace V is called controlled invariant, if for all x∈ V there is u ∈ Rm with Ax+Bu ∈V, i.e., if AV ⊂V +ImB. Equivalently, there is a linear map F ∶ X → Rm, called a friend of V , such that
(A+BF)V ⊂ V.
This also shows that V is controlled invariant iff for every x∈ V there is an (open loop) continuous control function u∶ [0,∞) → Rmwith ϕ(t,x,u) ∈ V for all t ≥ 0. In fact, differentiating the solution one finds
V∋ d
dtϕ(0,x,u) = Ax+Bu(0).
For the converse, define for x∈ V a control by u(t) = Fe(A+BF)tx,t≥ 0.
We now introduce the central notion of this paper, invariance entropy for controlled invariant subspaces of linear control system (5) and relate it to the subspace entropy defined in the previous section.
Festschrift in Honor of Uwe Helmke F. Colonius
In the following, we consider a fixed controlled invariant subspace V ofX with
dimV= k. Furthermore, we admit arbitrary controls in the space C([0,∞),Rm) of
continuous functions u∶ [0,∞) → Rm.
Definition 6. For a compact subset K⊂ V and for given T,ε > 0 we call a set R ⊂
C([0,∞),Rm) of control functions (T,ε)-spanning if for all x0∈ K there is u ∈ R
with
dist(ϕ(t,x0,u),V) < ε for all t ∈ [0,T]. (6)
By rinv(T,ε,K,V) we denote the minimal cardinality of such a (T,ε)-spanning set.
If no finite(T,ε)-spanning set exists, we set rinv(T,ε,K,V) = ∞.
In other words, we require for a(T,ε)-spanning set R that, for every initial value
in K, there is a control inR such that up to time T the trajectory remains in the
ε -neighborhood of V . Note that, in contrast to the definitions of topological entropy and subspace entropy for flows, Definition 4, here a number of control functions is counted, not a number of initial values. Hence this notion is intrinsic for control systems.
The following elementary observation shows that one cannot require that there are finitely many control functions u such that instead of (6) one has ϕ(t,x0,u) ∈ V for
all t∈ [0,T]. Hence the invariance condition has to be relaxed as indicated above
using ε> 0.
Proposition 7. Let V be a controlled invariant subspace. Furthermore, consider a neighborhood K of the origin in V , let T> 0, and suppose that there is v ∈ V with eATv/∈ V. Then there is no finite set R of controls such that for every x0∈ K there is
u∈ R with ϕ(t,x0,u) ∈ V for all t ∈ [0,T].
Proof. We may assume that γv∈ K for all γ ∈ (0,1˙). The proof is by contradiction.
Suppose thatR = {u1,...,ur} is a finite set of controls such that for every x0∈V there
is a control ujinR with ϕ(T,x0,uj) ∈ V. There is a control in R, say u1, with
ϕ(T,v,u1) = eTAv+∫ T 0
e(T−s)ABu1(s)ds ∈ V.
Since eTAv/∈ V, it follows that ϕ(T,0,u1) = ∫ T 0 e (T−s)ABu 1(s)ds /∈ V. We find for γ∈ (0,1) ϕ(T,γv,u1) = γ [eTAv+∫ T 0 e (T−s)ABu 1(s)ds]+(1−γ)∫ T 0 e (T−s)ABu 1(s)ds = γϕ(T,v,u1)+(1−γ)ϕ(T,0,u1).
This implies ϕ(T,γv,u1) /∈ V for all γ ∈ (0,1). Choose γ1∈ (0,1) and let v1∶= γ1v.
There is a control inR, say u2/= u1, such that ϕ(T,v1,u2) ∈V. Iterating the arguments
On the other hand, there are always finite(T,ε)-spanning sets of controls as shown by the following remark.
Remark8. Let K⊂ V be compact and ε,T > 0. By controlled invariance of V there is for every x∈ K ⊂ V a control function u with ϕ(t,x,u) ∈ V for all t ≥ 0. Hence, using continuous dependence on initial values and compactness of K, one finds finitely many controls u1,...,ursuch that for every x∈ K there is ujwith dist(ϕ(t,x,uj),V) < ε
for all t∈ [0,T]. Hence rinv(T,ε,K,V) < ∞.
Now we consider the exponential growth rate of rinv(T,ε,K,V) as in Definition 6
for T → ∞ and let ε → 0. The resulting invariance entropy is the main subject of the present paper.
Definition 9. Let V be a controlled invariant subspace for a control system of the
form (5). Then, for a compact subset K⊂ V, the invariance entropy hinv(K,V) is
defined by hinv(ε,K,V) ∶= limsup T→∞ 1 Tlog rinv(T,ε,K,V), hinv(K,V) ∶= lim ε↘0 hinv(ε,K,V).
Finally, the invariance entropy of V is defined by
hinv(V;A,B) ∶= supKhinv(K,V),
where the supremum is taken over all compact subsets K⊂ V.
In the sequel, we will use the shorthand notation hinv(V) for hinv(V;A,B), when it
is clear which control system is considered. Note that hinv(ε1,K,V) ≤ hinv(ε2,K,V)
for ε2≤ ε1. Hence the limit for ε→ 0 exists (it might be infinite.) Since all norms
on finite dimensional vector spaces are equivalent, the invariance entropy of V is independent of the chosen norm. We will show later that every controlled invariant subspace has finite invariance entropy. It is clear by inspection, that, as the subspace entropy hsub(V), also the invariance entropy hinv(V) is invariant under state space similarity; i.e. hinv(SV;SAS−1,SB) = hinv(V;A,B) for S ∈ GL(X ).
We are interested in the problem to keep the system in the subspace V for all t≥ 0. Then the exponential growth rate of the required number of control functions will give information on the difficulty of this task. A motivation to consider open-loop controls in this context comes, in particular, from model predictive control (see, e.g., Grüne and Pannek [7]), where optimal open-loop controls are computed and applied on short time intervals.
The following theorem (taken from Colonius and Helmke [3]) shows that the entropy of a controlled invariant subspace V can be characterized by the entropy of V for the corresponding uncontrolled system ˙x= Ax. This result will be useful in order to compute entropy bounds.
Theorem 10. Let V be a controlled invariant subspace for system (5) and consider the invariance entropy hinv(V) of control system (5) and the subspace entropy hsub(V)
of V of the uncontrolled system Φt= etA. Then
Festschrift in Honor of Uwe Helmke F. Colonius
Proof. (i) Let K⊂ V be compact, and fix T,ε > 0. Consider a (T,ε,K,V)-spanning
setR = {u1,...,ur} of controls with minimal cardinality r = rinv(T,ε,K,V). This
means that for every x∈ K there is ujwith
dist(ϕ(t,x,uj),V) < ε for all t ∈ [0,T].
By minimality, we can for every ujpick xj∈ K with dist(ϕ(t,xj,uj),V) < ε for all
t∈ [0,T]. Then, using linearity, one finds for all x ∈ K a control ujand a point xj∈ K
such that for all t∈ [0,T]
dist(etAx−etAxj,V) = dist(ϕ(t,x,uj)−ϕ(t,xj,uj),V) < 2ε.
This shows that the points xjform a(T,2ε)-spanning set for the subspace entropy,
and hence
rinv(T,ε,K,V) ≥ rsub(T,2ε,K,V).
Letting T tend to infinity, then ε→ 0 and, finally, taking the supremum over all
compact subsets K⊂ V, one obtains hinv(V) ≥ hsub(V).
(ii) For the converse inequality, let K be a compact subset of V and T,ε> 0. Let E= {x1,...,xr} ⊂ K be a minimal (T,ε)-spanning set for the subspace entropy which
means that for all x∈ K there is j ∈ {1,...,r},r = rsub(T,ε,K,V), such that for all
t∈ [0,T]
dist(etAx−etAxj,V) = inf z∈V∥e
tAx−etAx
j−z∥ < ε.
Since V is controlled invariant, we can assign to each xj, j∈ {1,...,r}, a control func-
tion uj∈ C([0,∞),Rm) such that ϕ(t,xj,uj) ∈ V for all t ≥ 0. Let R ∶= {u1,...,ur}.
Using linearity we obtain that for every x∈ K there is j such that for all t ∈ [0,T] dist(ϕ(t,x,uj)−ϕ(t,xj,uj),V) = dist(etAx−etAxj,V) < ε.
Since ϕ(t,xj,uj) ∈ V for t ∈ [0,T], it follows that
dist(ϕ(t,x,uj),V) = inf
z∈V∥ϕ(t,x,uj)−z∥
≤ ∥ϕ(t,x,uj)−ϕ(t,xj,uj)∥ < ε.
Thus for every x∈ K there is uj∈ R such that for all t ∈ [0,T] one has dist(ϕ(t,x,uj),
V) < ε. Hence R is (T,ε)-spanning for the invariance entropy and it follows that rinv(T,ε,K,V) ≤ rsub(T,ε,K,V) for all T,ε > 0,
and consequently hinv(K,V) ≤ hsub(V;Φ).