Exact controllability of nonLipschitz semilinear systems
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Abstract
We present sufficient conditions for exact controllability of a semilinear infinitedimensional 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 pointtype application of the Schmidt existence theorem and illustrated by a nonlinear transport partial differential equation.
Keywords
Exact controllability nonlinear infinitedimensional dynamical system Schmidt existence theorem fixed point theoryMathematics Subject Classification
37L50 93B05 37C251 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–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 pointtype 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 [12, 18]. In the former case, the authors make use of the fact that the system under consideration can be represented as a sum of Lipschitztype 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 integrodifferential 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 28 years separate articles [12] and [18], they both contain the 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 is that with Lipschitz condition comes either computational ’smoothness’, which goes back to existence results for initial value problem such as the Picard–Lindelöf theorem, 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.
2 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 \(\square \) sign.
2.1 On measures of noncompactness, onesided Lipschitz condition and Schmidt Theorem
Lemma 2.1
Let \(\Xi \) be a normed space and \(x,y\in \Xi \). Then the real function \(p:{\mathbb {R}}\rightarrow [0;+\infty )\), \(p(t):=\Vert x+ty\Vert \) is convex. \(\square \)
Corollary 2.2
If for given \(t\in {\mathbb {R}}\) the lefthand side derivative \(p_{}'\) of p given by above Lemma exists at point t and the righthand side derivative \(p_{+}'\) also exists at point t, then the inequality \(p_{}'(t)\le p_{+}'(t)\) holds.\(\square \)
Definition 2.3
Lemma 2.4
 (a)
\([x,y]_{}\le [x,y]_{+}\)
 (b)
\([x,y]_{\pm }\le \Vert y\Vert \)
 (c)
\([0,y]_{\pm }=\pm \Vert y\Vert \)
 (d)
\([x,y+z]_{\pm }\le [x,y]_{\pm }+\Vert z\Vert \).
Proof
 1.Consider the case \(h<0\):and the case \(h>0\):$$\begin{aligned} \Vert x+h(y+z)\Vert \ge \Vert x+hy\Vert \Vert hz\Vert =\Vert x+hy\Vert +h\Vert z\Vert , \end{aligned}$$$$\begin{aligned} \Vert x+h(y+z)\Vert \le \Vert x+hy\Vert +\Vert hz\Vert =\Vert x+hy\Vert +h\Vert z\Vert . \end{aligned}$$
 2.In both estimations in (1) by subtracting \(\Vert x\Vert \) from both sides and dividing, respectively, by \(h<0\) or \(h>0\), one obtains$$\begin{aligned} \frac{\Vert x+h(y+z)\Vert \Vert x\Vert }{h}\le \frac{\Vert x+hy\Vert \Vert x\Vert }{h}+\Vert z\Vert . \end{aligned}$$
 3.
Going to the limit in (2), the result follows. \(\square \)
Lemma 2.5

(\(l_{}\)) \([xy,f(t,x)f(t,y)]_{}\le M\Vert xy\Vert \quad \forall x,y\in \Xi \)

(\(l_{+}\)) \([xy,f(t,x)f(t,y)]_{+}\le M\Vert xy\Vert \quad \forall x,y\in \Xi \)

(l) \(\Vert f(t,x)f(t,y)\Vert \le M\Vert xy\Vert \quad \forall x,y\in \Xi \)
Proof
Lemma 2.6
 (a)
\([x,y]_{\pm }=\frac{1}{x}\langle x,y\rangle \quad \forall x,y\in \Xi ,\ x\ne 0\).
 (b)Suppose \(D\subset {\mathbb {R}}\times \Xi \) and \(f:D\rightarrow \Xi \). Then both conditionsare equivalent to

(\(l_\pm \)) \([xy,f(t,x)f(t,y)]_{\pm }\le M\Vert xy\Vert \quad \forall (t,x),(t,y)\in D\)
$$\begin{aligned} \langle xy,f(t,x)f(t,y)\rangle \le Mxy^2\qquad \forall (t,x),(t,y)\in D. \end{aligned}$$ 
Proof
Proof follows from a straightforward calculation using the Definition 2.3. \(\square \)
Before proceeding further, we recall
Definition 2.7
We can now introduce one of the most commonly used measures of noncompactness (MNC), namely
Definition 2.8
The Kuratowski MNC has properties given by the following:
Theorem 2.9
 (a)
\(\alpha (A)\le {{\mathrm{diam}}}A\)
 (b)
if \(A\subseteq B\) then \(\alpha (A)\le \alpha (B)\)
 (c)
\(\alpha (A\cup B)=\max \{\alpha (A),\alpha (B)\}\)
 (d)
\(\alpha ({{\mathrm{cl}}}A)=\alpha (A) \) where \({{\mathrm{cl}}}\) stands for closure
 (e)
if \(\Xi \) is a normed space and \(\dim \Xi =\infty \) then \(0\le \alpha (B(0,1))\le 2\)
Additionally, if \(\Xi \) is a Banach space, than the following Theorem is true [4]:
Theorem 2.10
 (f)
\(\alpha (A)=0\) if and only if A is relatively compact
 (g)
\(\alpha (A+B)\le \alpha (A)+\alpha (B)\)
 (h)
\(\alpha (mA)=m\alpha (A)\)
 (i)
\(\alpha ({{\mathrm{conv}}}A)=\alpha (A)\) where \({{\mathrm{conv}}}\) stands for convex hull
In the sequel, to analyse the equicontinuity of a set of functions we will make use of the following items [5].
Definition 2.11
 (a)
\(x\le x\)
 (b)
\(x\le y\) and \(y\le z\) imply \(x\le z\)
 (c)
\(x\le y\) implies \(x+z\le y+z\)
 (d)
\(x\le y\) implies \(ax\le ay\) for all real numbers \(a>0\).
Definition 2.12
Definition 2.13
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 \({\mathcal {F}}\) be a pointwise bounded family of continuous convex operators \(f:M\rightarrow Y\). Then \({\mathcal {F}}\) is equicontinuous.
A generalization of Theorem 2.14 to the class of sconvex 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
We also make use of the following:
Definition 2.16
The main tool we will use to prove our results is given by [21, 25]
Theorem 2.17
 (a)
\([x_1x_2,g(t,x_1)g(t,x_2)]\_\le M_{g}\Vert x_1x_2\Vert \quad \forall t\in [0,T],\ \forall x_1,x_2\in X\)
 (b)
\(\alpha (k([0,T],D))\le M_{k}\alpha (D)\quad \forall D\subseteq X\), D bounded.
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 \(M_k\).
2.2 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.
Basic properties of a generator of a strongly continuous semigroup are gathered in the proposition below [17, Theorem 1.2.4]:
Proposition 2.18
 (a)there exist constants \(\omega \ge 0\) and \(M\ge 1\) such that for every \(t\ge 0\) there is$$\begin{aligned} Q(t)\le Me^{\omega t}, \end{aligned}$$
 (b)
for every \(x\in X\) the function \(t\mapsto Q(t)x\) is continuous from \([0,\infty )\) into X,
 (c)for every \(x\in X\)$$\begin{aligned} \lim _{h\rightarrow 0}\frac{1}{h}\int _{t}^{t+h}Q(s)x\mathrm{d}s=Q(t)x, \end{aligned}$$
 (d)for every \(x\in X\) there is \(\int _{0}^{t}Q(s)x\mathrm{d}s\in D(A)\) and$$\begin{aligned} A\int _{0}^{t}Q(s)x\mathrm{d}s=Q(t)xx, \end{aligned}$$
 (d)for every \(x\in D(A)\) there is \(Q(t)x\in D(A)\) and$$\begin{aligned} \frac{\mathrm{d}}{\mathrm{d}t}Q(t)x=AQ(t)x=Q(t)Ax, \end{aligned}$$
 (e)for every \(x\in D(A)\)$$\begin{aligned} Q(t)xQ(s)x=\int _{s}^{t}Q(\tau )Ax\mathrm{d}\tau =\int _{s}^{t}AQ(\tau )x\mathrm{d}\tau . \end{aligned}$$
The operator \(A^*:D(A^*)\rightarrow X\) is the adjoint of A. Important properties of the adjoint are summarized in the following remark [24, Chapter 2.8].
Remark
 1.
If A is closed (as \(\rho (A)\) is not empty) we conclude that \(A^*\) is also closed, densely defined on X and \(A^{**}=A\).
 2.
There is \({\bar{s}}\in \rho (A^*)\) and \([(sIA)^{1}]^*=({\bar{s}}IA^*)^{1}\).
 3.
Let \((Q(t))_{t\ge 0}\) be a strongly continuous semigroup on X. Then \((Q^*(t))_{t\ge 0}\) 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 \(A:D(A)\rightarrow X\) with \(D(A)\subset X\) densely, we want to establish a setting where it behaves like a bounded one. 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.
Proposition 2.19
Proposition 2.20
The semigroup \((Q(t))_{t\ge 0}\) generated by A has a unique extension \(({Q}_{1}(t))_{t\ge 0}\) such that \({Q}_{1}(t)\in {\mathcal {L}}(X_{1})\) for every \(t\in [0,\infty )\).
Proposition 2.21
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 \((Q(t))_{t\ge 0}\).
The following Definition [24, Definition 4.1.5] is suitable in the context above, namely
Definition 2.22
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 \(z_{0},x_{T}\in X\) and any \(\varepsilon >0\) there exists a control u such that \(\Vert z(T)x_{T}\Vert \le \varepsilon \), 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 \(\varepsilon =0\) 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. First, according 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 \(A\in {\mathcal {L}}(X,X_{1})\) 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]:
Definition 2.25
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 \(\Phi (\tau )\) is such that \(\mathrm {Im}\Phi (\tau )\subset X\) for some \(\tau >0\) then for every \(t\ge 0\) there is \(\Phi (t)\in {\mathcal {L}}(L^2([0,\infty ),U),X)\) [24, Proposition 4.2.2]. Obviously, every \(B\in {\mathcal {L}}(U,X)\) is an admissible operator.
The following Proposition [24, Proposition 4.2.5] shows that if B is admissible and \(u\in L_{loc}^2([0,\infty ),U)\) then the initial value problem (7) has a wellbehaved unique solution in \(X_{1}\).
Proposition 2.26
3 Controllability by Schmidt Theorem
In this section, we present our main findings.
3.1 Problem statement
3.2 Step 1
We make use of the following:
Definition 3.1
 (a)the Pickardtype [2] operator \(L\in {\mathcal {L}}(Z)\),$$\begin{aligned} Lz:=\int _{0}^{\cdot }Q(\cdot s)z(s)\mathrm{d}s, \end{aligned}$$
 (b)the Pickard composition operator \(L(t)\in {\mathcal {L}}(Z,X_{1})\),$$\begin{aligned} L(t)z:=(Lz)(t),\quad t\in J, \end{aligned}$$
 (c)
the nonlinear continuous composition \(fz\in Z\), \((fz)(t):=f(z(t))\)
3.3 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
 (H1)\(f:X\rightarrow X\) is continuous on X and there exists \(M_f\in [0,\infty )\) such that f fulfils a onesided Lipschitz condition, i.e.$$\begin{aligned} \langle x_1  x_2,f(x_1)f(x_2)\rangle \le M_fx_1x_2^2\qquad \forall x_1,x_2\in X,\ \forall t\in J, \end{aligned}$$
 (H2)there exists \(M_g\in [0,\infty )\) such that for every \(t\in J\) the map \(g_t\in {\mathcal {G}}\) (i.e. \(g_t:[0,t]\times X\rightarrow X\), \(g_t(s,x):=Q(ts)f(x)\)) fulfils a onesided Lipschitz condition$$\begin{aligned} \langle x_1  x_2,g_t(s,x_1)g_t(s,x_2)\rangle \le M_gx_1x_2^2\quad \forall x_1,x_2\in X,\ \forall s\in [0,t]. \end{aligned}$$
Proof
 1.
Fix \(x\in X\) and define \(G_{x}:J\rightarrow X\), \(G_{x}(t):=\int _{0}^{t}g_t(s,x)ds=\int _{0}^{t}Q(ts)f(x)ds\).
 2.With the definition in 1. it follows thatwhere the last equality is true provided that both limits on the righthand side exist. We show it below.$$\begin{aligned} \begin{aligned} g(t,x)&=\frac{\mathrm{d}}{\mathrm{d}t}G_{x}(t)=\lim _{h\rightarrow 0}\frac{1}{h}\big (G_{x}(t+h)G_{x}(t)\big )\\&=\lim _{h\rightarrow 0}\frac{1}{h}\Big ( \int _{0}^{t+h}Q(t+hs)f(x)\mathrm{d}s \int _{0}^{t}Q(ts)f(x)\mathrm{d}s \Big )\\&=\lim _{h\rightarrow 0}\frac{1}{h}\Big ( Q(h)\int _{0}^{t+h}Q(ts)f(x)\mathrm{d}s \int _{0}^{t}Q(ts)f(x)\mathrm{d}s \Big )\\&=\lim _{h\rightarrow 0}\frac{1}{h}\Big ((Q(h)I)\int _{0}^{t}Q(ts)f(x)\mathrm{d}s \\&\quad +Q(h)\int _{t}^{t+h}Q(ts)f(x)\mathrm{d}s \Big )\\&=\lim _{h\rightarrow 0}\Big (\frac{1}{h}(Q(h)I)\int _{0}^{t}Q(ts)f(x)\mathrm{d}s \\&\quad +\frac{1}{h}\int _{t}^{t+h}Q(t+hs)f(x)\mathrm{d}s\Big )\\&=AG_{x}(t)+f(x), \end{aligned} \end{aligned}$$
 3.Fix \(t\in J\). Thenfor some \(s\in [t,t+h]\). As \(h\rightarrow 0\) there is also \(s\rightarrow t\) and by strong continuity of the semigroup Q(t), we have$$\begin{aligned} \begin{aligned}&\Big \Vert \frac{1}{h}\int _{t}^{t+h}Q(t+hs)f(x)\mathrm{d}sf(x) \Big \Vert \\&\quad =\Big \Vert \frac{1}{h}\int _{t}^{t+h}\big (Q(t+hs)f(x)f(x)\big )\mathrm{d}s \Big \Vert \\&\quad \le Q(t+hs)f(x)f(x) \end{aligned}, \end{aligned}$$$$\begin{aligned} \lim _{h\rightarrow 0}\frac{1}{h}\int _{t}^{t+h}Q(t+hs)f(x)ds=f(x),\quad \forall t\in J. \end{aligned}$$
 4.For any fixed \(x\in X\), hence fixed \(f(x)\in X\), by Proposition 2.18 there isand we have$$\begin{aligned} \int _{0}^{t}Q(ts)f(x)\mathrm{d}s=\int _{0}^{t}Q(\tau )f(x)d\tau \in D(A), \end{aligned}$$In particular, although the integration is formally carried out in \(X_{1}\), the result is in X.$$\begin{aligned} A\int _{0}^{t}Q(ts)f(x)\mathrm{d}s=A\int _{0}^{t}Q(\tau )f(x)\mathrm{d}\tau =Q(t)f(x)f(x). \end{aligned}$$
 5.From points 3 and 4 it follows thatis continuous and bounded.$$\begin{aligned} g(t,x)=Q(t)f(x) \end{aligned}$$(16)
 6.Fix \(t\in J\) and \(x_1,x_2\in X\). We may write the following estimation:$$\begin{aligned} \langle x_1x_2,g(t,x_1)g(t,x_2)\rangle&=\langle x_1x_2,Q(t)f(x_1)Q(t)f(x_2)\rangle \\&=\langle x_1x_2,g_{t}(0,x_1)g_{t}(0,x_2)\rangle \le M_gx_1x_2^2. \end{aligned}$$
 7.
As X is a Hilbert space over \({\mathbb {R}}\), the result of point 6 is equivalent, due to Lemma 2.6, to condition a) of Theorem 2.17. \(\square \)
Before proceeding further, we state a useful lemma.
Lemma 3.3
Proof
 1.Fix \(u\in C(J,X)\). We havewhere \(\tau =t_0<t_1<t_2<\dots <t_n=t\).$$\begin{aligned} (\Phi u)(t)\approx a+\sum _{k=1}^{n}(t_kt_{k1})f(\tau _k,u(\tau _k)), \end{aligned}$$
 2.Rewriting above, we getwhere \(\Theta \in S\cup \{0\}\).$$\begin{aligned} (\Phi u)(t)\approx a+(T\tau )\left[ \sum _{k=1}^{n}\frac{t_kt_{k1}}{T\tau }f(\tau _k,u(\tau _k))+\frac{Tt}{T\tau }\Theta \right] , \end{aligned}$$
 3.As \(\sum _{k=1}^{n}\frac{t_kt_{k1}}{T\tau }+\frac{Tt}{T\tau }=1\), there is$$\begin{aligned} \left[ \sum _{k=1}^{n}\frac{t_kt_{k1}}{T\tau }f(\tau _k,u(\tau _k))+\frac{Tt}{T\tau }\Theta \right] \in {{\mathrm{conv}}}(S\cup \{0\})\subseteq {{\mathrm{cl}}}{{\mathrm{conv}}}(S\cup \{0\}). \end{aligned}$$
 4.As integral is a limit to the Riemann sums, each of which belongs to \({{\mathrm{conv}}}(S\cup \{0\})\), the integral itself belong to \({{\mathrm{cl}}}{{\mathrm{conv}}}(S\cup \{0\})\), i.e.$$\begin{aligned} (\Phi u)(t)\in a+(T\tau ){{\mathrm{cl}}}{{\mathrm{conv}}}(S\cup \{0\}). \end{aligned}$$
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
 (H3)
the operator \(B\in {\mathcal {L}}(U,X)\) (hence, as bounded from U to X, it is an admissible control operator for \((Q(t))_{t\ge 0}\)) and the linear system (8) is exactly controllable to the space \(X=\big (L(T)B\big )\),
 (H4)
the operator \(W:V/\ker (L(T)B)\rightarrow X\), \(W(u):=L(T)Bu\) has a bounded inverse operator \(W^{1}\),
 (H5)the space X is ordered by a normal wedge C and the operator \(W^{1}\) is such that for every \(y\in X\) the function \(f:[0,t]\rightarrow X\),is convex,$$\begin{aligned} f(s):=Q(ts)BW^{1}(y)(s) \end{aligned}$$
 (H6)there exists \(M_w\in [0,\infty )\) such that$$\begin{aligned} \alpha (W^{1}(D)(t))\le M_w\alpha (D)\quad \forall t\in J,\ \forall D\subset X,D\text { bounded}, \end{aligned}$$
 (H7)there exists \(M_w'\in [0,\infty )\) such thatwhere \(D':=\{y\in X: W^{1}(y)=\frac{\mathrm{d}}{\mathrm{d}s}W^{1}(x),\ x\in D\}\),$$\begin{aligned} \alpha (W^{1}(D')(t))\le M_w'\alpha (D)\qquad \forall t\in J,\ \forall D\subset X,D\text { bounded}, \end{aligned}$$
Proof
 1.From Definition 2.8 for every \(s\in [0,T]\) and every bounded \(D\subset X\), we havewhere \(\Sigma _{i}\subset V/\ker (L(T)B)=L_{loc}^{2}([0,\infty ),U)\cap C^\infty ([0,\infty ),U)/\ker (L(T)B)\).$$\begin{aligned} \alpha \big (W^{1}(D)(s)\big )=&\inf \big \{\delta (s)\ge 0:W^{1}(D)(s)\subset \bigcup _{i=1}^{n}\Sigma _{i}(s); \\&{{\mathrm{diam}}}\Sigma _{i}(s)\le \delta (s),\ \Sigma _{i}(s)\subset U,\ i\in \{1,\dots ,n\};n\in {\mathbb {N}}\big \}, \end{aligned}$$
 2.Further, for every \(\tau \in [0,T]\), we haveand$$\begin{aligned} \alpha \big (Q(\tau )BW^{1}(D)(s)\big )=&\inf \big \{\delta '(s)\ge 0:Q(\tau )BW^{1}(D)(s){\subset }\bigcup _{i=1}^{n}Q(\tau )B\Sigma _{i}(s);\\&{{\mathrm{diam}}}Q(\tau )B\Sigma _{i}(s)\le \delta '(s)\big \} \end{aligned}$$Let us fix \(u(s),v(s)\in \Sigma _{i}(s)\) and let \(x(s):=Q(\tau )Bu(s)\), \(y(s):=Q(\tau )Bv(s)\). We then have \(x(s),y(s)\in Q(\tau )B\Sigma _{i}(s)\) and$$\begin{aligned} {{\mathrm{diam}}}(\Sigma _{i}(s))=\sup _{u(s),v(s)\in \Sigma _{i}(s)}u(s)v(s). \end{aligned}$$Hence, \({{\mathrm{diam}}}(Q(\tau )B\Sigma _{i}(s))\le M_qB{{\mathrm{diam}}}(\Sigma _{i}(s))\), where \(M_q:=\max _{t\in J}Q(t)\). Using point 1, we may now write$$\begin{aligned} x(s)y(s)=Q(\tau )B(u(s)v(s))\le Q(\tau )Bu(s)v(s). \end{aligned}$$for suitably chosen \(\delta '(s),\delta (s)\). Passing to infimum we get the estimation$$\begin{aligned} {{\mathrm{diam}}}(Q(\tau )B\Sigma _{i}(s))\le \delta '(s)\le M_qB{{\mathrm{diam}}}(\Sigma _{i}(s))\le M_qB\delta (s) \end{aligned}$$for every \(s,\tau \in J\) and every bounded \(D\subset X\).$$\begin{aligned} \alpha \big (Q(\tau )BW^{1}(D)(s)\big )\le M_qB\alpha (W^{1}(D)(s)) \end{aligned}$$
 3.Using now assumption (H6), we obtainfor every \(s,\tau \in J\) and every bounded \(D\subset X\).$$\begin{aligned} \alpha \big (Q(\tau )BW^{1}(D)(s)\big )\le M_qBM_w\alpha (D) \end{aligned}$$
 4.Preparing the ground for the Ambrosetti Theorem 2.15 let \(J_t:=[0,t]\subset [0,T]\) and define a family of operators indexed by the elements of \(D\subset X\), namelyWe will show that for every \(t\in J\) and every bounded \(D\subset X\) the family \({\mathcal {F}}_{t}\subset C(J_t,X)\) is equicontinuous. For that purpose note first that the operator \(W^{1}\) is a bounded and linear operator, hence it is continuous on X.$$\begin{aligned} {\mathcal {F}}_{t}:=\{Q(t\cdot )BW^{1}(y)(\cdot )\}_{y\in D}\subset C(J_t,X). \end{aligned}$$(17)Now fix \(t\in J\) and define a function \(\varphi _{t}:J_{t}\times X\rightarrow X\),We will show that \(\varphi _t\) is continuous on the product \(J_t\times X\). Fix \(y\in X\) and let \(s_1,s_2\in J_t\) be such that \(s_1+\tau =s_2\), \(\tau >0\). We then have$$\begin{aligned} \varphi _{t}(s,y):=Q(ts)BW^{1}(y)(s). \end{aligned}$$(18)where the last part tends to 0 with \(s_1\rightarrow s_2\), that is with \(\tau \rightarrow 0\). This follows from the continuity of \(W^{1}(y)\) on \(J_t\) and strong continuity of the semigroup Q. Now joint continuity of \(\varphi _t\) follows from linearity and continuity of \(W^{1}\) on X and the decomposition$$\begin{aligned}&Q(ts_1)BW^{1}(y)(s_1)Q(ts_2)BW^{1}(y)(s_2)\\&\quad =Q(ts_2+\tau ))BW^{1}(y)(s_2\tau )Q(ts_2)BW^{1}(y)(s_2)\\&\quad =Q(ts_2)Q(\tau )BW^{1}(y)(s_2\tau )Q(ts_2)BW^{1}(y)(s_2)\\&\quad =\Big \Vert Q(ts_2)\Big (Q(\tau )BW^{1}(y)(s_2\tau )Q(\tau )BW^{1}(y)(s_2)\\&\qquad +Q(\tau )BW^{1}(y)(s_2)BW^{1}(y)(s_2)\Big )\Big \Vert \\&\quad \le Q(ts_2)Q(\tau )BW^{1}(y)(s_2\tau )BW^{1}(y)(s_2)\\&\qquad +Q(ts_2)Q(\tau )BW^{1}(y)(s_2)BW^{1}(y)(s_2), \end{aligned}$$where \((\tau ,z)\rightarrow (s,y)\).$$\begin{aligned} \varphi _t(s,y)\varphi _t(\tau ,z)=\varphi _t(s,y)\varphi _t(s,z)+\varphi _t(s,z)\varphi _t(\tau ,z), \end{aligned}$$From continuity of \(\varphi _t\) it follows that for every bounded \(D\subset X\) the set \(\varphi _t(J_t,D)\subset X\) remains bounded, i.e. there exists such \(r<\infty \) that \(\varphi _t\subset B(0,r)\), a zerocentred ball with a finite radius r. Defining, for a given bounded \(D\subset X\), the setwe have$$\begin{aligned} {\mathcal {F}}_{t}(s):=\{\theta (s):\theta \in {\mathcal {F}}_t\}, \end{aligned}$$(19)for suitable \(r<\infty \). By (H5) and Theorem 2.14 it follows that the collection of continuous mappings \({\mathcal {F}}_t\) is equicontinuous for every \(t\in J\) and every bounded \(D\subset X\).$$\begin{aligned} {\mathcal {F}}_{t}(s)=\bigcup _{y\in D}\varphi _t(s,y)\subset B(0,r) \end{aligned}$$
 5.Fix bounded \(D\subset X\). From the Ambrosetti Theorem 2.15 and point 4 we havewhere \({\mathcal {F}}_t(J_t)=\bigcup _{s\in J_t}{\mathcal {F}}_t(s)\). Point 3 gives$$\begin{aligned} \alpha ({\mathcal {F}}_{t})=\sup _{s\in J_t}\alpha ({\mathcal {F}}_t(s))=\alpha ({\mathcal {F}}_t(J_t))\quad \forall t\in J, \end{aligned}$$In consequence, we have$$\begin{aligned} \alpha ({\mathcal {F}}_t(s))=\alpha \big (Q(ts)BW^{1}(D)(s)\big )\le M_qBM_w\alpha (D)\quad \forall t\in J\ \forall s\in J_t. \end{aligned}$$$$\begin{aligned} \sup _{s\in J_t}\alpha ({\mathcal {F}}_t(s))=\alpha ({\mathcal {F}}_t(J_t))\le M_qBM_w\alpha (D)\quad \forall t\in J. \end{aligned}$$(20)Note that for every \(t\in J\) and every \(y\in D\) there is \(\varphi _t(s,y)\in {\mathcal {F}}_t(s)\). Hence, \(\varphi _t(J_t,D)=\{\varphi (s,y):s\in J_t,y\in D\}={\mathcal {F}}_t(J_t)\) and for every \(t\in J\) and every bounded \(D\subset X\), we gethence, for every \(t\in J\) the mapping \(\varphi _t:J_t\times X\) is condensing with constant \(M_k:=M_qBM_w\).$$\begin{aligned} \alpha (\varphi _t(J_t,D))\le M_k\alpha (D), \end{aligned}$$
 6.Define \(K:J\times X\rightarrow X\), \(K(t,x):=\int _{0}^{t}\varphi _t(s,x)\mathrm{d}s=\int _{0}^{t}Q(ts)BW^{1}(x)(s)\mathrm{d}s\). Fix \(t\in J\) and \(x\in X\), thenprovided that appropriate limits exist. We show it below.$$\begin{aligned} \begin{aligned} k(t,x)&=\frac{\mathrm{d}}{\mathrm{d}t}K(t,x)=\lim _{h\rightarrow 0}\frac{1}{h}\big (K(t+h,x)K(t,x)\big )\\&=\lim _{h\rightarrow 0}\frac{1}{h}\Bigg (\int _{0}^{t}Q(t+hs)BW^{1}(x)(s)\mathrm{d}s \\&\quad \int _{0}^{t}Q(ts)BW^{1}(x)(s)\mathrm{d}s\\&\quad +\int _{t}^{t+h}Q(t+hs)BW^{1}(x)(s)\mathrm{d}s \Bigg )\\&=\lim _{h\rightarrow 0}\Bigg (\frac{1}{h}(Q(h)I)\int _{0}^{t}Q(ts)BW^{1}(x)(s))\mathrm{d}s\\&\quad +\frac{1}{h}\int _{t}^{t+h}Q(t+hs)BW^{1}(x)(s)ds\Bigg )\\&=AK(t,x)+BW^{1}(x)(t)+\int _{0}^{t}Q(ts)B\frac{\mathrm{d}}{\mathrm{d}s}W^{1}(x)(s)\mathrm{d}s\\&=Q(t)BW^{1}(x)(0)+L(t)Bu_{x}', \end{aligned} \end{aligned}$$(21)
 7.Consider again the function \(\varphi _{t}\) defined in (18). Calculating its time derivative at \(s\in [0,t]\) we obtainInitially, 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 righthand 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$$\begin{aligned} \frac{\mathrm{d}}{s}\varphi _{t}(s,x)=AQ(ts)BW^{1}(x)(s)+Q(t)B\frac{\mathrm{d}}{\mathrm{d}s}W^{1}(x)(s). \end{aligned}$$exists for all \(x\in X\) and is continuous.$$\begin{aligned} u_{x}':[0,t]\rightarrow U,\quad u_{x}':=\frac{\mathrm{d}}{\mathrm{d}s}W^{1}(x) \end{aligned}$$(22)We also have the following:$$\begin{aligned}&BW^{1}(x)(t)Q(t)BW^{1}(x)(0)=\varphi _{t}(t,x)\varphi _{t}(0,x)=\int _{0}^{t}\frac{\mathrm{d}}{\mathrm{d}s}\varphi _{t}(s,x)\mathrm{d}s\\&\quad =\int _{0}^{t}A\varphi _{t}(s,x)\mathrm{d}s+L(t)Bu_{x}'=A\int _{0}^{t}\varphi _{t}(s,x)\mathrm{d}s+L(t)Bu_{x}', \end{aligned}$$where due to Proposition 2.18, we can move from the extension to the original generator A. In consequence,$$\begin{aligned} AK(t,x)&=A\int _{0}^{t}\varphi _{t}(s,x)\mathrm{d}s=A\int _{0}^{t}Q(ts)BW^{1}(x)(s)\mathrm{d}s \\&=Q(t)BW^{1}(x)(0)BW^{1}(x)(t)+L(t)Bu_{x}'. \end{aligned}$$
 8.To finish the proof of (21), consider the following estimation:for some \(s\in [t,t+h]\). Taking the limit as \(h\rightarrow 0\) there is also \(s\rightarrow t\) and due the continuity of \(t\mapsto Q(t)\) and continuity of \(W^{1}(x)\) above tends to zero and we obtain$$\begin{aligned}&\left\ \frac{1}{h}\int _{t}^{t+h}Q(t+hs)BW^{1}(x)(s)\mathrm{d}sBW^{1}(x)(t)\right\ \\&\quad =\frac{1}{h}\left\ \int _{t}^{t+h}\Big (Q(t+hs)BW^{1}(x)(s)BW^{1}(x)(t)\Big )\mathrm{d}s\right\ \\&\quad \le \Vert Q(t+hs)BW^{1}(x)(s)Q(t+hs)BW^{1}(x)(t)\Vert \\&\qquad +\Vert Q(t+hs)BW^{1}(x)(t)BW^{1}(x)(t)\Vert , \end{aligned}$$Combining now this result and the one of point 7, we obtain (21).$$\begin{aligned} \lim _{h\rightarrow 0}\frac{1}{h}\int _{t}^{t+h}Q(t+hs)BW^{1}(x)(s)\mathrm{d}s=BW^{1}(x)(t). \end{aligned}$$
 9.Fix bounded \(D\subset X\). We haveand$$\begin{aligned} k(t,x)=Q(t)BW^{1}(x)(0)+L(t)Bu_{x}'=k_t(0,x)+L(t)Bu_{x}' \end{aligned}$$Note that for every \(t\in J\) and every \(x\in D\) there is \(Q(t)BW^{1}(x)(0)\in {\mathcal {F}}_t(0)\) with \({\mathcal {F}}_t\) defined for the same index set D.$$\begin{aligned} k(J,D)=\bigcup _{x\in D}\bigcup _{t\in J}k(t,x). \end{aligned}$$Due to the fact that W is an injection, for every \(x\in X\), there exists a unique \(y\in X\) such thatand \(u_{x}'=W^{1}(y)<\infty \). In consequence, for every bounded \(D\subset X\) the set$$\begin{aligned} W^{1}(y)=u_{x}'=\frac{\mathrm{d}}{\mathrm{d}s}W^{1}(x)\in V/\ker L(T)B \end{aligned}$$is unique.$$\begin{aligned} D':=\left\{ y\in X: W^{1}(y)=u_{x}'=\frac{\mathrm{d}}{\mathrm{d}s}W^{1}(x),\ x\in D\right\} \end{aligned}$$Define, similar to point 4, the equicontinuous family of operatorswhich 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), we have$$\begin{aligned} \begin{aligned} {\mathcal {F}}_{t}':=\{Q(t\cdot )BW^{1}(y)(\cdot )\}_{y\in D'} =\{Q(t\cdot )Bu_{x}'\}_{x\in D}\subset C(J_t,X). \end{aligned} \end{aligned}$$(23)As the range of the function \([0,t]\ni s\mapsto Q(ts)Bu_{x}'(s)\in X\) is contined in \({\mathcal {F}}_{t}'(J_t)\), according to Lemma 3.3 there is$$\begin{aligned} \alpha ({\mathcal {F}}_t'(J_t))\le M_qBM_w'\alpha (D) \end{aligned}$$(24)Define now a collection of operators \({\mathcal {P}}:=\{k(\cdot ,x)\}_{x\in D}\), where each member acts from J to X. From point 4 and above considerations, we see that \({\mathcal {P}}\) is in fact a collection of bounded operators, indexed again by elements of \(D\subset X\). With the same reasoning as in point 4 we see that \({\mathcal {P}}\) is an equicontinuous set and, by Ambrosetti Theorem 2.15, we have$$\begin{aligned} L(t)Bu_{x}'\in T{{\mathrm{cl}}}{{\mathrm{conv}}}({\mathcal {F}}_{t}'(J_t)\cup \{0\}). \end{aligned}$$Due to the definition of \({\mathcal {P}}\), we have \(k(J,D)={\mathcal {P}}(J)\) and$$\begin{aligned} \alpha ({\mathcal {P}})=\sup _{t\in J}\alpha ({\mathcal {P}}(t))=\alpha ({\mathcal {P}}(J))\quad \forall t\in J. \end{aligned}$$(25)for every \(t\in J\). From (20), (24) and (25) and Theorems 2.9 and 2.10 it now follows that$$\begin{aligned}&{\mathcal {P}}(t)=\bigcup _{x\in D}Q(t)BW^{1}(x)(0)+L(t)Bu_{x}'\subset {\mathcal {F}}_t(J_t)\cup \\&T{{\mathrm{cl}}}{{\mathrm{conv}}}({\mathcal {F}}_{t}'(J_t)\cup \{0\}) \end{aligned}$$and function k is condensing. \(\square \)$$\begin{aligned} \alpha (k(J,D))&=\alpha (P(J))\le \alpha \big ({\mathcal {F}}_t(J_t)\cup T{{\mathrm{cl}}}{{\mathrm{conv}}}({\mathcal {F}}_{t}'(J_t)\cup \{0\})\big )\\&\le \max \{M_w,M_w'\}TM_qB\alpha (D), \end{aligned}$$
Based on Propositions 3.2 and 3.4, we may state the main Theorem of this article which gives sufficient conditions for the existence of a solution to integral equation (12). This is equivalent to the existence of a fixed point of the operator (11) and results in exact controllability of system (10).
Theorem 3.5
Assume \((H1)(H7)\). Then a dynamical system with mild solution given by (10) is exactly controllable to the space \(\mathrm {Im}\big (L(T)B\big )=X\), with trajectory \(z\in C([0,\infty ),X)\cap {\mathcal {H}}_{loc}^1((0,\infty ),X_{1})\).
4 Example
5 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].
Notes
Acknowledgements
This project has received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie SkłodowskaCurie Grant Agreement No. 700833.
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