Exact and Positive Controllability of Boundary Control Systems

Using the semigroup approach to abstract boundary control problems we characterize the space of all exactly reachable states. Moreover, we study the situation when the controls of the system are required to be positive. The abstract results are applied to flows in networks with static as well as dynamic boundary conditions.


Introduction
This paper is a continuation of [EKNS08,EKKNS10] where we introduced a semigroup approach to boundary control problems and applied it to the control of flows in networks. While in these previous works we concentrated on maximal approximate controllability, we now focus on the exact-and positive controllability spaces. In particular, this will generalize and refine results given in [BBEAM14,EKNS08,EKKNS10] where further references to the related literature can be found. As a simple motivation, we consider as in [EKNS08] a transport process along the edges of a finite network. This system is governed through the transmission conditions in the vertices of the network which represent the "boundary space" for our problem. We then like to control the behavior of this system by acting upon a single node only. In this context it is reasonable to ask the following questions.
• Can we reach all possible states in final time?
• If not, can we describe the maximal possible set of reachable states?
• Is the choice of a particular control node important?
• Which states can be reached if only positive controls are allowed? In Section 5 we will address all these questions. To this end we first recall in Section 2 our abstract framework from [EKKNS10] as well as some basic results concerning boundary control systems. In Section 3 we then characterize boundary admissible control operators and describe the corresponding exact reachability space. In Section 4 we turn our attention to positive boundary control systems on Banach lattices. Finally, in Section 5 we apply our results and explicitly compute the exact (positive) reachability spaces for three different examples of a transport equation controlled at the boundary: in R m , in a network, and in a network with dynamic boundary conditions.
(i) three Banach spaces X, ∂X and U, called the state, boundary and control space, resp.; (ii) a closed, densely defined system operator A m : D(A m ) ⊆ X → X; (iii) a boundary operator Q ∈ L([D(A m )], ∂X); (iv) a control operator B ∈ L(U, ∂X).
For these operators and spaces and a control function u ∈ L 1 loc (R + , U) we then consider the abstract Cauchy problem with boundary control 1 (2.1) A function x(·) = x(·, x 0 , u) ∈ C 1 (R + , X) with x(t) ∈ D(A m ) for all t ≥ 0 satisfying (2.1) is called a classical solution. Moreover, we denote the abstract boundary control system associated to (2.1) by Σ BC (A m , B, Q). In order to investigate (2.1) we make the following standing assumptions which in particular ensure that the uncontrolled abstract Cauchy problem, i.e., (2.1) with B = 0, is well-posed. (ii) Q| ker(λ−Am) is invertible and the operator The following operators are essential to obtain explicit representations of the solutions of the boundary control problem (2.1).
Definition 2.4. For λ ∈ ρ(A) we call the operator Q λ , introduced in Lemma 2.3.(ii), abstract Dirichlet operator and define By [EKKNS10,Prop. 2.7] the solutions of (2.1) can be represented by the following extrapolated version of the variation of parameters formula.
Proposition 2.5. Let x 0 ∈ X, u ∈ L 1 loc (R + , U) and λ ∈ ρ(A). If x(·) = x(·, x 0 , u) is a classical solution (2.1), then it is given by the variation of parameters formula 1 We denote byẋ(t) the derivative of x with respect to the "time" variable t.
Our aim in the sequel is to investigate which states in X can be exactly reached from x 0 = 0 by solutions of (2.1). To this end we have to impose an additional assumption which, by (2.2), ensures that solutions for L p -controls have values in X.
Definition 2.6. Let 1 ≤ p ≤ +∞. Then the control operator B ∈ L(U, ∂X) is called p-boundary admissible if there exist t > 0 and λ ∈ ρ(A) such that , U) it follows that 1-boundary admissibility implies p-boundary admissibility for all p > 1. Now assume that B ∈ L(U, ∂X) is p-boundary admissible. Then for fixed λ ∈ ρ(A) and are called the controllability maps of the system Σ BC (A m , B, Q), where the second integral initially is taken in the extrapolation space X −1 . Note that by the closed graph theorem B BC t ∈ L(L p [0, t], U , X). Hence, this definition is independent of the particular choice of λ ∈ ρ(A) and gives the (unique) classical solution of (2.1) for given u ∈ W 2,1 ([0, t], U) and x 0 = 0. This motivates the following definition.  Proposition 2.9. Assume that B ∈ L(U, ∂X) is p-boundary admissible. Then the following holds.
Part (iii) shows that there is an upper bound for the reachability space depending on the eigenvectors of A m only, independent of the control operator B. This justifies the following notion.
Definition 2.10. The maximal reachability space of Σ BC (A m , B, Q) is defined by The system Σ BC (A m , B, Q) is called maximally controllable if eR BC = R BC max . We stress that R BC max = X may happen (some basic examples are provided in [EKNS08, Sec. 5]), hence the relevant question about exact or approximate controllability is indeed to compare eR BC or aR BC to the space R BC max and not to the whole space X, as it is usually done in the classical situation. After this short summary on boundary control systems Σ BC (A m , B, Q) taken mainly from [EKKNS10] in the context of approximate controllability, we now turn our attention to the case of exact controllability.

Exact controllability
We start this section by giving two characterizations of p-boundary admissibility for a control operator B which frequently simplifies the explicit computation of the associated controllability map B BC t . Here for λ ∈ C we introduce the function ε λ : R → C by ε λ (s) := e λs . Moreover, for f ∈ L p [0, t] and u ∈ U we define (c) There exist t > 0, λ 0 > ω 0 (A) and M ∈ L L p [0, t], U , X such that for all λ ≥ λ 0 and v ∈ U Moreover, in this case the controllability map is given by We start by proving (2.3). The idea is to show this first for functions of the Then by linearity it also holds for linear combinations of such functions and a density argument implies (2.3) for arbitrary u ∈ L p [0, t], U . To this end

Recall that B
follows by similar arguments replacing the total set We note that by linearity it would suffice that Part (b) of Proposition 3.1 is satisfied for α = 0 and all 0 ≤ β ≤ t (or for all 0 ≤ α ≤ t and β = t).
Corollary 3.2. Let 3 n ∈ N 1 and assume that B is p-boundary admissible. Then for all and M ∈ L L p [0, t], U , X is the operator from Proposition 3.1.
Proof. Let u ∈ L p [0, nt], U . Then by (2.4) In Section 5 we will see that (3.1), (3.2), and (3.5) allow us to easily compute the controllability map in the situations studied in [EKNS08, Sect. 4] and [EKKNS10, Sect. 3] dealing with the control of flows in networks.
Corollary 3.3. If B is p-boundary admissible, then the exact reachability space in time nt for n ∈ N 1 is given by where M ∈ L L p [0, t], U , X is the operator from Proposition 3.1.

Positive controllability
In this section we are interested in positive control functions yielding positive states. To this end we will make the following Additional Assumption 4.1. The spaces X and U are Banach lattices.
Moreover, by Y + := {y ∈ Y : Y ≥ 0} we denote the positive cone in a Banach lattice Y . Note that in the sequel we do not make any positivity assumptions on (T (t)) t≥0 , B or Q λ if not stated otherwise.
Moreover, we define the exact positive reachability space (in arbitrary time) by Moreover, we define the approximate positive reachability space (in arbitrary time) by First we give necessary and sufficient conditions implying that starting from the initial state x 0 = 0 positive controls result in positive states.
Moreover, if (T (t)) t≥0 is positive, then the above assertions are satisfied if and only if Proof. The equivalence of (4.5) and (4.6) follows from the closedness of X + . To show the equivalence of (4.5) and (4.7) recall that by [ABHN01, p.14] the step functions are dense , U is continuous, we conclude that the positive step functions are dense in L p [0, t], U + . The claim then follows from (the proof of) Proposition 3.1 using the boundedness of the controllability map B BC t . Now assume that (T (t)) t≥0 is positive. Then the equivalences of (4.5), (4.6) with (4.8), (4.9) follow from Corollary 3.3 using the fact that the reachability spaces are growing in time. In particular, this implies that if (4.7) holds for some t > 0 it holds for arbitrary t > 0 and choosing β = t and α = 0 we obtain (4.10) for arbitrary t > 0. To show the remaining assertions we fix some λ > ω 0 (A) and define on X := X × X the operator matrix Then by [Eng99,Cor. 3.4] the matrix A generates a C 0 -semigroup (T(t)) t≥0 given by Moreover, by [Eng99,Lem. 3.1] we have (0, +∞) ⊂ ρ(A) and Now, if (4.10) holds then T(s) ≥ 0 for all 0 ≤ s ≤ t which implies that (T(t)) t≥0 is positive which is equivalent to the fact that A is resolvent positive. However, by (4.13) the latter is the case if and only if (4.11) is satisfied which shows the equivalence of (4.10) and (4.11). Finally, if (4.10) holds, then This proves (4.7) and completes the proof.
In the sequel we use the notation co M and co M to indicate the convex hull and the closed convex hull of a set M ⊂ X, respectively.
Proposition 4.4. Assume that B ∈ L(U, ∂X) is p-boundary admissible and that e + R BC t ⊂ X + . Then the following holds.
Proof. (i). Clearly, a + R BC is a closed convex cone. Its invariance under (T (t)) t≥0 and R(λ, A) for λ > ω 0 (A) follows from the representations in (iii) and (iv). To show (ii) we note that by (2.4) and (3.3) the inclusion "⊇" holds. Now recall that the positive step functions are dense in L p [0, t], U + and invariant under positive convey combinations. Hence, the boundedness of the controllability maps implies equality of the spaces in (ii). To obtain (iii) we note that by (2.4) and (3.3) we have for all 0 ≤ α ≤ β ≤ t and v ∈ U + . Multiplying this inclusion by e −λβ > 0 and putting s := t − β and r := t − α implies for all 0 ≤ s ≤ r and v ∈ U + . Since λ > ω 0 (A) we obtain and co {T (s)B µ y : s ≥ 0, µ > w, y ∈ U + } is a convex cone. Combining these facts and (3.3) it follows that (4.14) holds if for all 0 ≤ α ≤ β ≤ t, k ∈ N 0 and v ∈ X + . Since (T (t)) t≥0 is strongly continuous the following integral is the limit of Riemann sums, hence for ν > max{0, w} we obtain using Lemma 2.3.(iii) as ν → +∞. This proves (4.15) and completes the proof of (iii). That the right-hand-sides of the equalities in (iii) and (iv) coincide follows from the integral representation of the resolvent (see [  (c) There exists w > ω 0 (A) such that the following implication holds for all ϕ ∈ X ′ R(λ, A) n B λ v, ϕ ≥ 0 for all v ∈ U + , n ∈ N and λ > w ⇒ ϕ ≥ 0. Remark 4.6. The previous two results generalize [BBEAM14,Prop. 3.3 and Thm. 3.4], respectively, where it is assumed that (T (t)) t≥0 , B and Q λ for all λ > λ 0 are all positive and, in particular, the additional hypothesis (H) There exists γ > 0 and λ 0 ∈ R such that Qx ≥ γλ x for all λ > λ 0 and x ∈ ker(λ − A m ) is made. We note that Hypothesis (H) in reflexive state spaces X implies that A = A m , cf. [ABE16, Lem. A.1]. Hence, the results of [BBEAM14] are, e.g., not applicable to state space like X = L p ([a, b], Y ) for p ∈ (1, +∞) and reflexive Y .
Combining Corollary 3.2 and Proposition 4.3 we finally obtain the following characterization of an exact positive reachability space.
Corollary 4.7. Assume that B is p-boundary admissible, t > 0 and n ∈ N 1 . Then the exact positive reachability space in time nt is given by where M ∈ L L p [0, t], U + , X is the operator from Proposition 3.1. Moreover, the operator M is positive if and only if a + R BC t ⊂ X + .

Examples
In this section we will show how our abstract results can be applied to a transport equation with boundary control and to the vertex control of flows in networks.

5.a. Exact Boundary Controllability for a Transport Equation.
In this subsection we study a transport equation in R m given by . With these choices the controlled transport equation (5.1) can be reformulated as an abstract Cauchy problem with boundary control of the form (2.1). Clearly, the boundary operator Q is surjective. By [BKR17,Cor. 18.4] we know that for λ ∈ C and A = A m | ker(Q) as above we have λ ∈ ρ(A) ⇐⇒ e λ ∈ ρ(B).
Moreover, by [BKR17,Prop. 18.7] the operator A generates a strongly continuous semigroup given by where B 0 := Id. This shows that the Assumptions 2.2 are satisfied. To proceed we have to compute the associated Dirichlet operator.
Lemma 5.1. For λ ∈ ρ(A) the Dirichlet operator Q λ ∈ L C m , L p [0, 1], C m is given by Proof. By Lemma 2.3.(ii) we know that Q : Next we verify that in this context (3.1) holds.
Thus B is p-boundary admissible. Next we compute the appropriate reachability space. Remark 5.6. The previous result characterizes the exact maximal boundary controllability by a one-dimensional control in terms of a Kalman-type condition which is well-known in control theory.
Combining Remark 5.4 and Corollary 5.5 we furthermore obtain the following Corollary 5.7. Let l ∈ N be the degree of the minimal polynomial of B. If l < m, the transport equation (5.1) is not maximally controllable, i.e., eR BC R BC max . Finally, we investigate positive controllability and consider • the positive cone X + := L p [0, 1], R m + in the state space X, • the positive cone U + := R + in the control space U, • a positive matrix B ∈ M m (R + ), • a positive control operator B := b ∈ R m + . Then by (5.2)-(5.3) the operators T (t) ∈ L(X) for t ≥ 0 and B λ ∈ L(U, X) for λ > ω 0 (A) are positive. Thus arguing as above using Proposition 4.3 and Corollary 4.7 we obtain the following.
Corollary 5.8. The exact positive reachability space of the controlled transport equation (5.1) is given by Hence, the problem is exactly positive controllable if and only if Vertex control of flows in networks. The previous example can be easily adapted to cover a transport problem on a network controlled in a single vertex. More precisely, consider a network consisting of n vertices {v 1 , . . . , v n } and m edges {e 1 , . . . , e m }. As shown in [BKR17,Sec. 18.1], its structure can be described by either the transposed weighted adjacency matrix A ∈ M n (C) given by otherwise, or by the transposed weighted adjacency matrix of the line graph B ∈ M m (C) where To proceed we also need the transposed weighted outgoing incidence matrix (Φ − w ) ⊤ =: Ψ ∈ M m×n (C) defined by and the corresponding unweighted outgoing incidence matrix denoted by Φ − ∈ M n×m (C).
For the weights we assume 0 ≤ w ij ≤ 1, thus all these matrices are positive. Moreover, we assume that Ψ is column stochastic (i.e., the weights on all the outgoing edges from a given vertex sum up to 1). For a detailed account of the various graph matrices we refer to [BKR17,Sec. 18.1]. Here we only mention the following relations (5.5) ΨA = BΨ, ΨR(λ, A) = R(λ, B)Ψ, and Φ − Ψ = Id C n which we will need in the sequel.
We then consider a transport equation on the m edges imposing n boundary conditions in the vertices, controlled in a single vertex v i , i.e., Note that in big connected networks one usually has n ≤ m, hence the latter space is the more important one for applications. Positive control for this problem was already treated in [BBEAM14] and the approximate positive reachability spaces was computed. However, our approach even yields the exact reachability space.
Corollary 5.10. The exact positive reachability space of the controlled transport in network problem (5.6) is given by

5.c. Exact Boundary Controllability for Flows in Networks with Dynamical
Boundary Conditions. In this subsection we investigate exact controllability in the situation of [EKKNS10,Sect. 3]. Without going much into details we only introduce the necessary facts to state the problem and to compute the exact reachability space eR BC t . We start from the transport problem in the network introduced in the previous example, but now change the transmission process in the vertices allowing for dynamical boundary conditions. To encode the structure of the underlying network and the imposed boundary conditions we use the incidence matrices introduced above as well as the weighted incoming incidence matrix Φ + w given by otherwise, for some 0 ≤ w + ij ≤ 1. Defining (5.7) A := Φ + w Ψ and B := ΨΦ + w we obtain the adjacency matrices as above (with different nonzero weights). We mention that the relations (5.5) remain valid also in this case.
We are then interested in the network transport problem with dynamical boundary conditions in s = 1 considered already in [Sik05] and [EKKNS10, Sect. 3], i.e., As is shown in [EKKNS10,Prop. 3.4] these spaces and operators satisfy all assumptions of Section 2. To proceed we first need to compute the associated Dirichlet operator Q λ and an explicit representation of the semigroup operators T (t) for t ∈ [0, 1].
(i) For each 0 = λ ∈ ρ(A), the Dirichlet operator Q λ ∈ L(C n , X) is given by (ii) The semigroup (T (t)) t≥0 generated by A is given by 6 Next we apply Proposition 3.1 to the present situation.
Lemma 5.12. Let λ ∈ ρ(A). Then for all 0 ≤ α ≤ 1 Hence the equality in (3.1) is satisfied with Proof. Using the explicit representations of Q λ and T (t) given in Lemma 5.11 and the relations (5.5) we obtain Similarly, for the second coordinate we have where we used (5.7).
We note that by [EKKNS10,Prop. 3.5] the states of the controlled flow at time t ≥ 0 are given by the first coordinate of the states in our "extended" state space X = L p [0, 1], C m × C n . For this reason we also need to compute the first coordinate of T (1) k Ψg 0 . Lemma 5.13. We have where the operator V s ∈ L L p [0, 1], C m , W 1,p [0, 1], C m is defined in (5.12). Moreover, for k ∈ N 1 we have The formula for T (1) follows immediately from Lemma 5.11.(ii). Since ΨA = BΨ, it suffices to show the second equality in (5.15). Obviously this equation holds for k = 1. To verify it for k > 1 we note that by (5.5) the matrix Ψ is left invertible with left inverse Φ − . Hence, we obtain = (BV s + δ 1 ) · (BV s + δ 1 ) k−1 BΨ V s g = (BV s + δ 1 ) k BΨ V s g.
The previous two lemmas together with Corollary 3.2 imply the following result.
Corollary 5.14. For l ∈ N 2 and u ∈ L p [0, l] we have where u k ∈ L p [0, 1] is defined as in (3.6).
Using this explicit representation of the controllability map we now compute the exact reachability space for the control problem given in (5.8).
Corollary 5.15. If t ≥ min{m, n} =: l then the exact reachability space of the controlled flow with dynamic boundary conditions (5.8) is given by 7 Proof. The equality of the two sets on the right-hand-side follows immediately from (5.5). To show the inclusion in the second set we combine Corollaries 3.3 and 5.14. First observe, that for the operators B, V s , and δ 1 we have for every f ∈ L p [0, 1], C m while δ k 1 f = δ 1 f = f (1) for k ≥ 1. So, when expanding (BV s + δ 1 ) k−1 V s we can rearrange the terms to obtain expressions of the form α i B i V s 1 · · · V s i+1 , 0 ≤ i ≤ k − 1, where α i are scalar coefficients and s j ∈ {s, 1}, 1 ≤ j ≤ i+1. Next, for arbitrary u ∈ L p [0, 1] and 0 ≤ k ≤ l we have V s 1 · · · V s k u ∈ W k,p [0, 1], s j ∈ {s, 1}, 1 ≤ j ≤ k.
Combining these facts we obtain the desired result by considering (5.16) for all u ∈ L p [0, l].
From the previous Corollary we immediately obtain the following result which improves [EKKNS10,Thm. 3.10] and shows that aR BC In the same manner as before we also obtain the following result on positive controllability.

Conclusion
Using a new characterization of admissible boundary control operators (see Proposition 3.1) we are able to describe explicitly the exact reachability space of the abstract boundary control system Σ BC (A m , B, Q), cf. (2.1). Moreover, this approach allows us also to determine the positive reachability space obtained allowing only positive control functions. Our results generalize and improve the ones obtained in the former works [BBEAM14,EKNS08,EKKNS10] where only approximate controllability or positive controllability under quite restrictive assumptions are studied.