Exact controllability of non-Lipschitz semilinear systems

We present sufficient conditions for exact controllability of a semilinear infinite dimensional dynamical system. The system mild solution is formed by a noncompact semigroup and a nonlinear disturbance that does not need to be Lipschitz continuous. Our main result is based on a fixed point type application of the Schmidt existence theorem and illustrated by a nonlinear transport partial differential equation.


Introduction
Controllability of nonlinear systems is a mature subject of research -see [3,16,18] and references therein. In recent years various applications of fixed point theorems are particularly popular among researchers tackling this problem. These range from classical Banach or Schauder Fixed Point Theorems (FPT) to more specific, such as Nussbaum [15] FPT in [18], Schaefer [22] FPT in [20] or Mönch [13] FPT in [12]. A short survey on fixed point approaches is given in [26].
In most cases the nonlinearities present in (otherwise linear) systems are regarded as disturbances, in some way influencing the normal operation of a system. The choice of the fixed-point-type approach depends on the nature of nonlinearity and the structure of the system itself. The examples of such approach we particularly focus on, are given in [18] and [12]. In the former case the authors make use of the fact that the system under consideration can be represented as a sum of Lipschitz-type and compact operators, what allows them to apply the Nussbaum [15] fixed point theorem. In the latter case the authors examine the controllability conditions of a semilinear impulsive mixed Volterra-Fredholm functional integro-differential evolution differential system with finite delay and nonlocal conditions by means of measures of noncompactness and Mönch fixed point theorem [13].
Interesting, however, is that although almost 30 years separate these articles they both contain assumption of the Lipschitz type of nonlinearity. The author of this article is actually not aware of any example of fixed point theorem application to the problem of exact controllability where the nonlinearity is not Lipschitz in some way. The reason for that is that with Lipschitz condition comes either computational 'smoothness', which goes back to existence results for initial value problem such as e.g. Picard-Lindelöf, or equicontinuity needed by the Ambrosetti Theorem to express the measure of noncompactness.
For this reason we intentionally drop the Lipschitz condition. In this way this article combines and expands above results by means of the Schmidt existence theorem, originally developed for the Cauchy problem in Banach spaces [21]. This requires a reformulation of the theorem into the fixed point form. The set of assumptions is discussed and the results follow. In particular, we do not impose any compactness condition. The article is finished with an illustrative example.

Preliminaries
This section gives the basic definitions and background material. It also defines the notation. If for lemmas or theorems given without reference to a particular source the proof is short and simple, they are immediately followed by the sign.

On measures of noncompactness, one-sided Lipschitz condition and Schmidt Theorem
Lemma 2.1. Let Ξ be a normed space and x, y ∈ Ξ. Then the real function p : R → [0; +∞), p(t) := x + ty is convex.
Corollary 2.2. If for given t ∈ R the left hand side derivative p ′ − of p given by above Lemma exists at point t and the right hand side derivative p ′ + also exists at point t, then the inequality p ′ − (t) ≤ p ′ + (t) holds. Definition 2.3 (One-sided Lipschitz condition). Let Ξ be a Banach space and x, y ∈ Ξ. We define the symbol and say that a function f : Ξ → Ξ fulfills one-sided Lipschitz condition (left "-" or right "+", respectively) if there exists a non-negative constant M such that for any x, y ∈ Ξ.
Proof. Proofs of points a) to c) follow directly from the bracket definition above. We will show only the last one.
1. Consider the case h < 0: x + h(y + z) ≥ x + hy − hz = x + hy + h z and the case h > 0: x + h(y + z) ≤ x + hy + hz = x + hy + h z 2. In both estimations in 1. by subtracting x from both sides and dividing, respectively, by h < 0 or h > 0 one obtains Going to the limit in 2. the result follows.
Lemma 2.5. Let Ξ be a Banach space, f : R × Ξ → Ξ and M be a nonnegative constant. Introducing the notation Proof. Based on the previous lemma the following estimation holds Lemma 2.6. Let Ξ be a real inner product space. Then Proof. Proof follows from a straightforward calculation using the Definition 2.3.
Before proceeding further we recall Definition 2.7 (Diameter of a set). Let Ξ be a metric space with a metric ρ.
The diameter of a set A ⊆ Ξ is defined as For the case of an empty set we take diam ∅ = 0.
We can now introduce one of the most commonly used measures of noncompactness (MNC), namely Definition 2.8 (Kuratowski measure of noncompactness [11]). For a bounded subset A of a metric space Ξ we call The Kuratowski MNC has properties given by the following Theorem 2.9 (Properties of the Kuratowski MNC [7,2]). For bounded A, B ⊆ Ξ and α MNC we have Additionally, if Ξ is a Banach space, than the following Theorem is true [4].
In the sequel we will need [5] the following Definitions and the Theorem on equicontinuity of a set of functions.
Definition 2.11. An ordered linear space Y is a space Y on which there is defined a binary relation ≤ such that for all x, y, z ∈ Y the following conditions are satisfied x ≤ y implies ax ≤ ay for all real numbers a > 0.
A positive wedge of an ordered linear space Y is the set Y + of all elemets x ∈ Y such that 0 ≤ x, where 0 denotes the zero element of Y . We see that Y + is a wedge. Conversely, if C is a wedge in a real linear space Y , then the binary relation ≤ given by satisfies all conditions in Definition 2.11 for all x, y, z ∈ Y , and in consequence makes Y into an ordered linear space whose positive wedge is exactly C. The relation ≤ defined by (1) is called the ordering induced by C. Let Y be a topological linear space. Then C is said to be a normal wedge if for each neighbourhood Definition 2.13. Let M be a convex subset of a linear space X and Y be an ordered linear space. Then f : M → Y is called a convex function when for all a ∈ [0, 1] and x, y ∈ M the inequality holds. When the order on Y is induced by a wedge C, the above can be written as The following Theorem, taken from [10], is a generalization of the well known Banach-Steinhaus Theorem [19, Theorem 2.5].
Theorem 2.14. Let M be an open convex subset of a topological vector space X of the second category, let Y be a topological vector space ordered by a normal wedge C and let F be a pointwise bounded family of continuous convex operators f : M → Y . Then F is equicontinuous.
A generalization of Theorem 2.14 to the class of s-convex functions, containing a necessary and sufficient condition of equicontinuity, can be found in [5], and is further developed in [6].
In expressing MNC in function space a key role is played by the following Theorem 2.15 (Ambrosetti Theorem [1]). Suppose that J is a compact interval, F ⊂ C(J, E), E is a Banach space, F is bounded and equicontinuous. Then We also make use of the following Definition 2.16. Let Ξ 1 and Ξ 2 be metric spaces, Φ : Ξ 1 → Ξ 2 be continuous and mapping bounded sets A ⊆ Ξ 1 onto bounded sets Φ(A) ⊆ Ξ 2 . If there exists a constant M ≥ 0 such that for every bounded F ⊂ Ξ 1 the inequality holds, then Φ is called an α-condensing operator with constant M .
The main tool we will use to prove our results is given by Theorem 2.17 (Schmidt Theorem for the Initial Value Problem [21,25]). Let X be a Banach space and T, M g , M k be reals. Suppose g, k : [0, T ] × X → X are continuous, bounded and Then the initial value problem (IVP) has a solution x : [0, T ] → E.
A function g with properties as above will be called dissipative with constant M g , or dissipative with M g for short, and a function k with properties as above will be called condensing with constant M k or condensing with M k .
We will use an integral form of (2), as it better suits our needs, that is as every solution of (2) is also a solution of (3).

On dynamical systems
From this point onward we drop the general Banach space setting. Although some of the definitions make sense and the results are true, the Hilbert space setting allows us to obtain more concrete results. Hence, throughout the rest of this paper, X and U are Hilbert spaces which are identified with their duals. For the whole remaining part J := [0, T ] is a compact interval.
We will also use the Sobolev space of vector valued functions Let A : D(A) → X be a densly defined, linear, closed and unbounded operator on which the Cauchy problem of interest is based. Before introducing the Cauchy problem formally we describe the setting in which it will be considered. Basic properties of a generator of a strongly continuous semigroup are gathered in the proposition below [17, Theorem 1.

2.4]:
Proposition 2.18. Let (Q(t)) t≥0 be a strongly continuous semigroup and let (A, D(A)) be its generator. Then a) there exist constants ω ≥ 0 and M ≥ 1 such that for every t ≥ 0 there is The operator A * : D(A * ) → X is the adjoint of A. Important properties of the adjoint are summarized in the following remark [24, Chapter 2.8].
Remark. Let A : D(A) → X be a densely defined operator with s ∈ ρ(A). The following holds is not empty) we conclude that A * is also closed, densely defined on X and A * * = A.
(iii) Let (Q(t)) t≥0 be a strongly continuous semigroup on X. Then (Q * (t)) t≥0 is also a strongly continuous semigroup on X and its generator is A * .
To overcome certain difficulties with unboundedness of the generator A, we make use of the duality with respect to a pivot space. In general, the idea of (duality with respect to) a pivot space can be described as follows. Having an unbounded closed linear operator A : D(A) → X with D(A) ⊂ X densely, we want to establish a setting where it behaves much like a bounded one but defined on a Banach space. One instance of such situation is when we restrict ourselves to the space made out of its domain, but equipped with a graph (or graphequivalent) norm. It is then reasonable to ask what is the dual of such space. It turns out that it can be represented as a completion of the original space X with a resolvent-induced norm. As the space X is pivotal in the described setting, the name follows. A precise description of such situation can be found in [24,Chapter 2.9] or in [8,Chapter II.5] The following three propositions from [24, Chapter 2.10] introduce duality with respect to a pivot space (sometimes referred to also as a rigged Hilbert space construction) in the context which we will use later.
is a Hilbert space denoted X 1 . The norms generated as above for different β ∈ ρ(A) are equivalent to the graph norm. The embedding X 1 ⊂ X is continuous.
For A as in Proposition 2.19 its adjoint A * has the same properties. Thus, we can define the space where β ∈ ρ(A), and this is also a Hilbert space.
Proposition 2.20. Let A : D(A) → X be a densely defined operator and let β ∈ ρ(A). We denote by X −1 the completion of X with respect to the norm Then the norms generated as above for different β ∈ ρ(A) are equivalent (in particular X −1 is independent of the choice of β). Moreover, X −1 is the dual of X d 1 with respect to the pivot space (X, · X ). The semigroup (Q(t)) t≥0 generated by A has a unique extension (Q −1 (t)) t≥0 such that Q −1 (t) ∈ L(X −1 ) for every t ∈ [0, ∞). Proposition 2.21. Let A : D(A) → X be a densely defined operator with ρ(A) = ∅, β ∈ ρ(A), X 1 be as in Proposition 2.19 and let X −1 be as in Proposition 2.20. Then A ∈ L(X 1 , X) and it has a unique extension A −1 ∈ L(X, X −1 ). Moreover, , and these two operators are unitary.
Remark. In the remaining part we denote the extension A −1 and the generator A by the same symbol A. The same applies to the semigroup (Q(t)) t≥0 .
Consider now a (linear) dynamical system described by the following initial value problem where X (called state space) and U (called control space) are Hilbert spaces; The following Definition [24, Definition 4.1.5] is suitable in the context above, namely is called the mild solution of the corresponding differential equation (7).
The two basic types of controllability are given by the following Definition 2.23 (approximate controllability). The control process described by (8) is said to be approximately controllable when for any given z 0 , x T ∈ X and any ε > 0 there exists a control u such that z(T ) − x T ≤ ε, where z 0 is the initial condition and u is the control.
Definition 2.24 (exact controllability). The control process described by (8) is said to be exactly controllable when ε = 0 in definition 2.23.
In classical literature (see e.g. [23]), when no rigged Hilbert space construction was used, the following problem was of great importance. Namely, when taking equations (7) as a primary model, its solution must lay in D(A), which is only a dense subset of X. That means that the system (7) cannot be exactly controllable. For the same reason, if considering infinite T every approximately controllable system is exactly controllable.
By the use of the rigged Hilbert space construction (called also "a duality with respect to a pivot space") the controllability problem is greatly simplified. Firstly, accodring to [24,Proposition 4.1.4] every solution to (7) in X −1 is a mild solution of (7). Although the converse, in general, still does not have to be true, due to the fact that now A ∈ L(X, X −1 ) greatly simplifies many considerations.
This, however, comes at a price of the operator B mostly being unbounded from U to X. As we would like all the mild solutions (8) to be continuous Xvalued functions, additional constraints must be put on the operator B. This is expressed by the following [24, Definition 4.2.1] The operator B ∈ L(U, X −1 ) is called an admissible control operator for (Q(t)) t≥0 if for some τ > 0 there is ImΦ(τ ) ⊂ X.
Remark. Note that if B is admissible, then in (9) we integrate in X −1 but the integral is in X. Also, if the operator Φ(τ ) is such that ImΦ(τ ) ⊂ X for some Obviously every B ∈ L(U, X) is an admissible operator.
The following Proposition [24,Proposition 4.2.5] shows that if B is admissible and u ∈ L 2 loc ([0, ∞), U ) then the initial value problem (7) has a well-behaved unique solution in X −1 .

Controllability by Schmidt Theorem
In this section we present our main findings.

Problem statement
Consider the nonlinear dynamical system with zero initial condition stated by the differential equation in X −1 as , A ∈ L(X, X −1 ) and B ∈ L(U, X −1 ) is an admissible control operator for (Q(t)) t≥0 , f : X → X is a given continuous function. The mild solution of the above initial value problem is The main problem we tackle in this article is to find the conditions under which the dynamical system expressed by (10) is exactly controllable.

Step 1
To show the existence of a solution of problem (10) we build an appropriate integral operator Ψ : Z → Z and show that it has a fixed point. Let then In Theorem 2.17 for z to be a unique solution of the Cauchy problem stated there, z has to be also a solution of the integral equation (3). What follows, the integral operator associated with (2) has the form To show the existence of a fixed point of the operator (11) it is enough to show that appropriate parts of (12) fulfil assumptions of Theorem 2.17. Unfortunately, the obvious choice of functions under integrals in (12), namely g(s, z(s)) := Q(t − s)f (z(s)) and k(s, z(s)) := Q(t − s)Bu(s) is not possible. The reason for that is that the semigroup (Q(t)) t≥0 is defined for all t ∈ [0, T ], as well as functions g, k : [0, T ] × X → X in Theorem 2.17. Hence, for given nonlinearity f we introduce formally two parameter-dependent families of functions where the steering trajectory u x is built based on an element x of the state space, as explained below in (15).Taking into account that members g t and k t of both families "work under the integral", the upper limit of which changes in the interval [0, T ], Theorem 2.17 cannot be used directly. Instead, we will work it out from other facts.
We make use of the following With the above definition, the mild solution (10) may be rewritten in the form Let x T ∈ X be the desired final state. Following a canonical procedure [23,18], we assume exact controllability of the linear system without the nonlinear part f , given by Definition 2.22. Then, without loss of generality, we assume that the attainable set A T is equal to the image of the L(T )B operator, that is The reason of such approach is to have a possibility to drive the system with nonlinear disturbance f to every point it could attain without such disturbance.
Define a linear and invertible operator W : V / ker L(T )B → X, which has a bounded inverse operator W −1 : As we are interested in exact controllability, let us fix x T ∈ X. We construct a control signal based on this x T by selecting one element which is explicitly related to a trajectory z (with values in X due to admissibility of B) which system (10) will follow. Substituting control function defined by (15) into operator equation (11) for t = T we obtain By putting the same control function u x into mild solution (10) we get z(T ) = x T , what gives Ψ(z)(T ) = z(T ) and shows that the trajectory end point matches.
The only thing left is to show that with the control function u x defined by (15) the operator Ψ defined by (11) has a fixed point in (note again that the operator B is assumed to be admissible -Proposition 2.26) what means that there exists such trajectory z of the system (10) which leads it to the given final point x T .

Step 2
The existence of a solution to integral equation (12) is equivalent to the existence of a fixed point of the operator (11). We begin with the following Proposition 3.2. Let X be a real Hilbert space and we assume that (H1) f : X → X is continuous on X and there exists M f ∈ [0, ∞) such that f fulfils a one-sided Lipschitz condition i.e.
) fulfills a one-sided Lipschitz condition Then for a function g : J × X → X given by g(t, where the last equality is true provided that both limits on the right hand side exist. We show it below.

Fix t ∈ J. Then
for some s ∈ [t, t + h]. As h → 0 there is also s → t and by strong continuity of the semigroup Q(t) we have 4. For any fixed x ∈ X, hence fixed f (x) ∈ X, by Proposition 2.18 there is and we have In particular, although the integration is formally carried out in X −1 , the result is in X.

From points 3 and 4 it follows that
is continuous and bounded.
6. Fix t ∈ J and x 1 , x 2 ∈ X. We may write the following estimation 7. As X is a Hilbert space over R, the result of point 6 is equivalent, due to Lemma 2.6, to condtion a) of Theorem 2.17.

As
4. As integral is a limit to the Riemann sums, each of which belongs to conv(S ∪ {0}), the integral itself belong to cl conv(S ∪ {0}), i.e.
Let us now focus on assumption b) of Theorem 2.17. We can relate it to our controllability setting by the following Proposition 3.4. Using previously defined notation, if (H3) the operator B ∈ L(U, X) (hence, as bounded from U to X, it is an admissible control operator for (Q(t)) t≥0 ) and the linear system is convex , then for a function k : J×X → X given by k(t, x) := d Proof.
Using point 1 we may now write for suitably chosen δ ′ (s), δ(s). Passing to infimum we get the estimation for every s, τ ∈ J and every bounded D ⊂ X.
3. Using now assumption (H6) we obtain for every s, τ ∈ J and every bounded D ⊂ X.
We will show that for every t ∈ J and every bounded D ⊂ X the family F t ⊂ C(J t , X) is equicontinuous. For that purpose note firstly that the operator W −1 is a bounded and linear operator, hence it is continuous on X.
Now fix t ∈ J and define a function ϕ t : J t × X → X, We will show that ϕ t is continuous on the product J t × X. Fix y ∈ X and let s 1 , s 2 ∈ J t be such that s 1 + τ = s 2 , τ > 0. We then have where the last part tends to 0 with s 1 → s 2 , that is with τ → 0. This follows from the continuity of W −1 (y) on J t and strong continuity of the semigroup Q. Now joint continuity of ϕ t follows from linearity and continuity of W −1 on X and the decomposition where (τ, z) → (s, y).
From continuity of ϕ t it follows that for every bounded D ⊂ X the set ϕ t (J t , D) ⊂ X remains bounded, i.e. there exists such r < ∞ that ϕ t ⊂ B(0, r), a zero-centred ball with a finite radius r. Defining, for a given bounded D ⊂ X, the set we have F t (s) = y∈D ϕ t (s, y) ⊂ B(0, r) for suitable r < ∞. By (H5) and Theorem 2.14 it follows that the collection of continuous mappings F t is equicontinuous for every t ∈ J and every bounded D ⊂ X.

Fix bounded D ⊂ X. From the Ambrosetti Theorem 2.15 and point 4 we
where F t (J t ) = s∈Jt F t (s). Point 3 gives In consequence we have Note that for every t ∈ J and every y ∈ D there is ϕ t (s, y) ∈ F t (s). Hence, ϕ t (J t , D) = {ϕ(s, y) : s ∈ J t , y ∈ D} = F t (J t ) and for every t ∈ J and every bounded D ⊂ X we get hence for every t ∈ J the mapping ϕ t : J t × X is condensing with constant provided that appropriate limits exist. We show it below.
7. Consider again the function ϕ t defined in (18). Calulating its time derivative at s ∈ [0, t] we obtain Initially the above result can be found either by elementary limit calculation or one can use the result in [8,Lemma B.16]. Here, both parts on the right hand side exist, although for the sake of clarity we skip all the routine limit considerations in the argument of the generator A leading to application of its extension -for more details see Proposition 2.21 and [24,Proposition 2.10.3]. Note also, that by assumption the function exists for all x ∈ X and is continuous.
We also have the following where, due to Proposion 2.18 we can move from the extension to the original generator A. In consequence 8. To finish the proof of (21) consider the following estimaiton for some s ∈ [t, t + h]. Taking the limit as h → 0 there is also s → t and due the continuity of t → Q(t) and continuity of W −1 (x) above tends to zero and we obatin Combining now this result and the one of point 7 we obtain (21).
9. Fix bounded D ⊂ X. We have Note that for every t ∈ J and every x ∈ D there is Q(t)BW −1 (x)(0) ∈ F t (0) with F t defined for the same index set D.
Due to the fact that W is an injection, for every x ∈ X there exists a unique y ∈ X such that and u ′ x = W −1 (y) < ∞. In consequence, for every bounded D ⊂ X the set Define, similarly to point 4, the equicontinuous family of operators which may be regarded as indexed by elements of either D ′ or D. Using the assumption (H7) and following the same procedure which led to (20), As the range of the function Define now a collection of operators P := {k(·, x)} x∈D , where each member acts from J to X. From point 4 and above considerations we see that P is in fact a collection of bounded operators, indexed again by elements of D ⊂ X. With the same reasoning as in point 4 we see that P is an equicontinuous set and, by Ambrosetti Theorem 2.15, we have α(P) = sup t∈J α(P(t)) = α(P(J)) ∀t ∈ J.
Due to the definition of P we have k(J, D) = P(J) and for every t ∈ J. From (20), (24) and (25) and Theorems 2.9 and 2.10 it follows now that α(k(J, D)) =α(P (J)) ≤ α F t (J t ) ∪ T cl conv(F ′ t (J t ) ∪ {0}) ≤ max{M w , M ′ w }T M q B α(D) and function k is condensing.
for every t ∈ J and ξ ∈ [0, 1] where m(ξ) provides a spatial distribution of control.
The dissipativity condition from assumption (H1) in Proposition 3.2, that is ∀x, y ∈ X is equivalent to P x − P y, ρP x − ρP y ≤ M f P x − P y 2 ∀x, y ∈ X.
We will show that f does not fulfil Lipschitz condition which, using the definition of f , is equivalent to ∃ M f <∞ ∀ x,y∈L 2 (0,1) ρP x − ρP y ≤ M f P x − P y .
It remains to show that f fulfils condition (28). Fix P x = α and P y = β. We have because, due to monotonicity of ϕ, the element (α i − β i )(ϕ(α i ) − ϕ(β i )) ≤ 0 for every i ∈ N. Hence, it is enough to take any positive M f in (28) and the condition is met. Note also that with the semigroup Definition (27) assumption (H2) is equivalent to (28).

Conclusions
In this article we showed new results in establishing sufficient conditions for controllability of particular types of dynamical systems. Our results expand the results found in [18], where the authors use Nussbaum fixed point theorem. In particular, we did not impose any compactness condition on the semigroup, instead we used its condensing property. This is a considerably weaker assumption than the one taken in the above mentioned work.
The second improvement in comparison to the present state of literature is that we did not assume that the nonlinearity is Lipschitz. The price paid for that is that we used an existence result initially intended for the initial value problem, not formulated in a fixed point form. The authors are not aware whether there exists a similar fixed point theorem.
Our result can be expanded to incorporate phenomena such as impulsive behaviour or nonlocal conditions in a way similar to [12]. Note, however, that the assumptions of the Schmidt theorem are in a sense weaker than the demands of the Mönch's condition used be the authors of [12].