Nonlinear Delay Evolution Inclusions on Graphs

Abstract

We prove a necessary and a sufficient condition for a time-dependent closed set to be viable with respect to a delay evolution inclusion. An application to a null controllability problem is also included.

Supported by a grant of the Romanian National Authority for Scientific Research, CNCS–UEFISCDI, project number PN-II-ID-PCE-2011-3-0052.

1 Introduction

Let $X$ be a real Banach space, $I=[a,b)\subseteq \mathbb {R}$ and let $A:D(A)\subseteq X\leadsto X$ be the infinitesimal generator of a nonlinear semigroup of nonexpansive mappings $\{S(t):\overline{D(A)}\rightarrow \overline{D(A)};\ t\ge 0\}$. Let $\sigma \ge 0$ and let $C_{\sigma }=C([\,-\sigma ,0\,];X)$ be endowed with the usual sup-norm $\Vert \varphi \Vert _\sigma =\sup \{\Vert \varphi (t)\Vert ;\ t\in [\,-\sigma ,0\,]\}$.

If $u\in C([\,\tau -\sigma ,T\,],X)$ and $t\in [\,\tau ,T\,]$, we denote by $u_t\in C_\sigma$ the function defined by $u_t(s)=u(t+s)$ for $s\in [\,-\sigma ,0\,]$. It should be noticed that for $\sigma =0$, i.e. when de delay is absent, $C_{\sigma }$ reduces to $X$. Let $K:I\leadsto X$ and $F:\mathcal {K}\leadsto X$ be nonempty-valued multi-functions, where $\mathcal {K}=\{(t,\varphi )\in I\times C_{\sigma };\ \varphi (0)\in K(t)\}$.

In this paper prove a necessary and a sufficient condition in order that $\mathcal {K}$ be viable with respect to $A+F$. Let $(\tau ,\varphi )\in \mathcal {K}$ and let us consider

$$\begin{aligned} \left\{ \begin{array}{l} u^\prime (t)\in Au(t)+F(t,u_t)\\ u_\tau =\varphi . \end{array} \right. \end{aligned}$$

(1)

Definition 1

A function $u\in C([\,\tau -\sigma ,T\,];X)$ is said to be a $C^0$ -solution of (1) on $[\,\tau ,T\,]\subseteq I$, if $(t,u_t)\in \mathcal {K}$ for $t\in [\,\tau ,T\,]$, $u(t)=\varphi (t-\tau )$ for $t\in [\,\tau -\sigma ,\tau \,]$ and there exists $f\in L^1(\tau ,T;X)$ with $f(t)\in F(t,u_t)$ a.e. for $t\in [\,\tau ,T\,]$ and such that $u$ is a $C^0$-solution of the Cauchy problem

$$ \left\{ \begin{array}{ll}u'(t)\in Au(t)+f(t),\quad t\in [\,\tau ,T\,]\\ u(\tau )=\varphi (0)\end{array}\right. $$

in the usual sense. See Cârjă, Necula, Vrabie [2], Definition $1.6.2$, p. $17$.

We say that the function $u:[\,\tau -\sigma ,T)\rightarrow X$ is a $C^0$ -solution of (1) on $[\,\tau -\sigma ,T)$, if u is a $C^0$-solution on $[\,\tau -\sigma ,\tilde{T}\,]$ for every $\tilde{T}<T$.

Definition 2

We say that $\mathcal {K}$ is $C^0$ -viable with respect to $A+F$, if for each $(\tau ,\varphi )\in \mathcal {K}$, there exists $T>\tau$, such that $[\,\tau ,T\,]\subseteq I$ and (1) has at least one $C^0$-solution $u:[\,\tau -\sigma ,T\,]\rightarrow X$. If $T=\sup I$, we say that $\mathcal {K}$ is globally $C^0$ -viable with respect to $A+F$.

Viability results concerning evolution inclusions without delay, i.e., when $\sigma =0$, using the concepts of tangent set and quasi-tangent set – introduced and studied by Cârjă, Necula and Vrabie [2–4] and [5] –, were obtained by Necula, Popescu and Vrabie [16, 17]. For viability results referring to delay evolution equations and inclusions, we mention the pioneering papers of Pavel and Iacob [18] and Haddad [9]. For related results see Gavioli and Malaguti [8], Lakshmikantham, Leela and Moauro [12], Leela and Moauro [13], Lupulescu and Necula [14]. The semilinear case was very recently considered by Necula and Popescu [15] and the present paper extends to the fully nonlinear case the results there obtained.

The paper is divided into five sections, the second one being concerned with the definitions of the basic concepts used in that follows. In Sect. 3 we state and prove a necessary condition for $C^0$-viability, while Sect. 4 contains the main result of the paper: a sufficient condition for $C^0$-viability. In Sect. 5, we include an application to a control problem.

2 Preliminaries

Let $f\in L^1(\tau ,T;X)$ and $\xi \in \overline{D(A)}$. We denote by $u(\cdot ,\tau ,\xi ,f):[\,\tau ,T\,]\rightarrow \overline{D(A)}$ the unique $C^0$-solution, i.e. integral solution of the Cauchy problem

$$\left\{ \begin{array}{ll}u'(t)\in Au(t)+f(t),\quad t\in [\,\tau ,T\,]\\ u(\tau )=\xi .\end{array}\right. $$

Clearly, $u(\cdot ,\tau ,\xi ,0)=S(\cdot -\tau )\xi$, where $\{S(t):\overline{D(A)}\rightarrow \overline{D(A)};\ t\ge 0\}$ is the semigroup of nonexpansive mappings generated by $A$ on $\overline{D(A)}$ by the Crandall and Liggett Exponential Formula. See Crandall and Liggett [7].

We assume familiarity with the basic concepts and results in nonlinear evolution equations, delay equations and inclusions and we refer the reader to Barbu [1], Cârjă, Necula and Vrabie [2], Lakshmikantham and Leela [11], Hale [10] and Vrabie [19] for details.

The metric $d$ on $\mathcal {K}$ is defined by $d((\tau ,\varphi ),(\theta ,\psi ))=\max \{|\tau -\theta |,\Vert \varphi -\psi \Vert _{\sigma }\}$, for all $(\tau ,\varphi ),(\theta ,\psi )\in \mathcal {K}$. Furthermore, whenever we use the term strongly-weakly u.s.c. multi-function we mean that the domain of the multi-function in question is equipped with the strong topology, while the range is equipped with the weak topology. The term u.s.c. refers to the case in which both domain and range are endowed with the strong, i.e. norm, topology.

Thereafter, $D(\xi ,r)$ denotes the closed ball with center $\xi$ and radius $r$.

Definition 3

The multi-function $F:\mathcal {K}\leadsto X$ is called locally bounded if, for each $(\tau ,\varphi )$ in $\mathcal {K}$, there exist $\delta >0$, $\rho >0$ and $M>0$ such that for all $(t,\psi )$ in $([\,\tau -\delta ,\tau +\delta \,]\times D(\varphi ,\rho ))\cap \mathcal {K}$, we have $\Vert F(t,\psi )\Vert \le M$.

Let $(\tau ,\varphi )\in \mathcal {K}$, let $\eta \in X$ and let $E\subset X$ be a nonempty, bounded subset, let $h>0$ and let $\mathcal {F}_E=\left\{ f\in L^1_\mathrm{loc}(\mathbb {R};X);\ f(s)\in E\,\ \text{ a.e. } \text{ for }\ s\in \mathbb {R}\right\} .$ We denote by $u(\tau +h,\tau ,\varphi (0),\mathcal {F}_E)=\{u(\tau +h,\tau ,\varphi (0),f);\ f\in \mathcal {F}_E\}.$

Definition 4

We say that $E$ is $A$ -right-quasi-tangent to $\mathcal {K}$ at $(\tau ,\varphi )$ if

$$\liminf _{h\downarrow 0}h^{-1} \text{ d }\left( u(\tau +h,\tau ,\varphi (0),\mathcal {F}_E), K(\tau +h)\right) =0.$$

We denote by $\mathcal {QTS}_{\mathcal {K}}^A(\tau ,\varphi )$ the set of all $A$-right-quasi-tangent sets to $\mathcal {K}$ at $(\tau ,\varphi )$.

If $K$ is constant, $E$ is right-quasi-tangent to $\mathcal {K}$ at $(\tau ,\varphi )$ if and only if it is $A$-quasi-tangent to $K$ at $\xi =\varphi (0)$ in the sense of Cârjă, Necula, Vrabie [2].

3 Necessary Conditions for Viability

The following lemma was proved in Necula and Popescu [15].

Lemma 1

Let $f:[\,\tau ,T\,]\rightarrow X$ be a measurable function and $B,C\subset X$ two nonempty sets such that $f(t)\in B+C$ a.e. for $t\in [\,\tau ,T\,]$. Then, for every $\varepsilon >0$ there exist $b:[\,\tau ,T\,]\rightarrow B$, $c:[\,\tau ,T\,]\rightarrow C$ and $r:[\,\tau ,T\,]\rightarrow S(0,\varepsilon )$, all measurable, such that $f(t)=b(t)+c(t)+r(t)\ \text{ a.e. } \text{ for }\ t\in [\,\tau ,T\,].$

Theorem 1

If $F:\mathcal {K}\leadsto X$ is u.s.c. and $\mathcal {K}$ is $C^0$-viable with respect to $A+F$ then, for all $(\tau ,\varphi )\in \mathcal {K}$, $\displaystyle \lim _{h\downarrow 0}h^{-1}\text{ d }\left( u(\tau +h,\tau ,\varphi (0),\mathcal {F}_{F(\tau ,\varphi )}), K(\tau +h)\right) =0$.

Proof

Let $(\tau ,\varphi )\in \mathcal {K}$ and $u:[\,\tau -\sigma ,T\,]\rightarrow X$ be a $C^0$-solution of 1. Hence there exists $f\in L^1 (\tau ,T;X)$ such that $f(s)\in F(s,u_s)$ a.e. for $s\in [\,\tau ,T\,]$ and $u(t)= u(t,\tau ,\varphi (0),f)$ for all $t\in [\,\tau ,T\,]$. Let $\varepsilon >0$ be arbitrary but fixed.

Since $F$ is u.s.c. at $(\tau ,\varphi )$ and $\lim _{t\rightarrow \tau }u_t=u_\tau =\varphi$ in $C_\sigma$, we may find $\delta >0$ such that $f(s)\in F(s,u_s)\subseteq F(\tau ,\varphi )+S(0,\varepsilon )\ \text{ a.e. } \text{ for }\ s\in [\,\tau ,\tau +\delta \,].$

Taking $B=F(\tau ,\varphi )$ and $C=S(0,\varepsilon )$, from Lemma 1, we deduce that there exist two integrable functions $g:[\,\tau ,\tau +\delta \,]\rightarrow F(\tau ,\varphi )$ and $r:[\,\tau ,\tau +\delta \,]\rightarrow S(0,2\varepsilon )$ such that $f(s)=g(s)+r(s)$ a.e. for $s\in [\,\tau ,\tau +\delta \,]$. Since $u(\tau +h)\in K(\tau +h)$, we deduce that, for each $0<h<\delta$, $\displaystyle \text{ d }\left( u(\tau +h,\tau ,\varphi (0),\mathcal {F}_{F(\tau ,\varphi )}), K(\tau +h)\right)$

$\displaystyle \le \text{ d }\left( u(\tau +h,\tau ,\varphi (0),g),u(\tau +h,\tau ,\varphi (0),f\right) \le \int _{\tau }^{\tau +h}\Vert g(s)-f(s)\Vert ds\le 2\varepsilon h.$ So, $\displaystyle \limsup _{h\downarrow 0}h^{-1}\text{ d }\left( u(\tau +h,\tau ,\varphi (0),\mathcal {F}_{F(\tau ,\varphi )}), K(\tau +h)\right) \le 2\varepsilon$. The proof is complete.

Theorem 2

If $F:\mathcal {K}\leadsto X$ is u.s.c. and $\mathcal {K}$ is $C^0$-viable with respect to $A+F$ then $F(\tau ,\varphi )\in \mathcal {QTS}_{\mathcal {K}}^{A}(\tau ,\varphi )$ for all $(\tau ,\varphi )\in \mathcal {K}.$

4 Sufficient Conditions for Viability

Definition 5

We say that the multi-function $K:I\leadsto X$ is$\,:$

• (i) closed from the left on $I$ if for any sequence $((t_n,x_n))_{n\ge 1}$ from $I\times X$, with $x_n\in K(t_n)$ and $(t_n)_n$ nondecreasing, $\lim _nt_n=t\in I$ and $\lim _nx_n=x$, we have $x\in K( t).$

• (ii) locally closed from the left if for each $(\tau ,\xi )\in I\times X$ with $\xi \in K(\tau )$ there exist $T>\tau$ and $\rho >0$ such that the multi-function $t\leadsto K(t)\cap D(\xi ,\rho )$ is closed from the left on $[\,\tau ,T\,]$.

Definition 6

An $m$-dissipative operator $A:D(A)\subseteq X\leadsto X$ is of complete continuous type if for each sequence $(f_n,u_n)_n$ in $L^1(\tau ,T;X)\times C([\,\tau ,T\,];X)$ with $u_n$ a $C^0$-solution of the problem $u'_n(t)\in Au_n(t)+f_n(t)$ on $[\,\tau , T\,]$ for $n=1,2,\dots$, $\lim _nf_n=f$ weakly in $L^1(\tau ,T;X)$ and $\lim _nu_n=u$ strongly in $C([\,\tau ,T\,];X)$, it follows that $u$ is a $C^0$-solution of the problem $u'(t)\in Au(t)+f(t)$ on $[\,\tau , T\,]$.

If the dual of $X$ is uniformly convex and $A$ generates a compact semigroup, then $A$ is of complete continuous type. See Vrabie [19, Corollary 2.3.1, p. 49].

Theorem 3

Let $K$ be locally closed from the left and let $F:\mathcal {K} \leadsto X$ be nonempty, convex and weakly compact valued. If $F$ is strongly-weakly u.s.c., locally bounded and $A:D(A)\leadsto X$ is of complete continuous type and generates a compact semigroup, then a sufficient condition in order that $\mathcal {K}$ be $C^0$-viable with respect to $A+F$ is the tangency condition $F(\tau ,\varphi )\in \mathcal {QTS}_{\mathcal {K}}^A(\tau ,\varphi )$ for all $(\tau ,\varphi )\in \mathcal {K}.$ If, in addition, $F$ is u.s.c., then the tangency condition is also necessary in order that $\mathcal {K}$ be $C^0$-viable with respect to $A+F$.

The next lemma is inspired from Cârjă and Vrabie [6].

Lemma 2

Let $K:I\leadsto X$ be locally closed from the left, $F:\mathcal {K}\leadsto X$ be locally bounded and let $(\tau ,\varphi )\in \mathcal {K}$. Let us assume that the tangency condition is satisfied. Let $\rho >0$, $T>\tau$ and $M>0$ be such that:

• (1) the multi-function $t\leadsto K(t)\cap D(\varphi (0),\rho )$ is closed from the left on $[\,\tau ,T)\,;$

• (2) $\Vert F(t,\psi )\Vert \le M$ for all $t\in [\,\tau ,T\,]$ and all $\psi \in D_\sigma (\varphi ,\rho )$ with $(t,\psi )\in \mathcal {K}\,;$

• (3) $\displaystyle \sup _{t\in [\,\tau ,T\,]}\Vert S(t-\tau )\varphi (0)-\varphi (0)\Vert +\sup _{|t-s|\le T-\tau }\Vert \varphi (t)-\varphi (s)\Vert +(T-\tau )(M+1)<\rho .$

Then, for each $\varepsilon \in (0,1)$, there exist a family $\mathcal {P}_T=\{[\,t_m,s_m);m\in \varGamma \}$ of disjoint intervals, with $\varGamma$ finite or at most countable, and two functions: $f\in L^1(\tau ,T;X)$, and $u\in C([\,\tau -\sigma ,T\,];X)$ such that$\,:$

• (i) $\cup [\,t_m,s_m)=[\,\tau ,T)$ and $s_m-t_m\le \varepsilon$, for all $m\in \varGamma \,;$

• (ii) $u(t_m)\in K(t_m)$, for all $m\in \varGamma$ and $u(T)\in K(T)\,;$

• (iii) $f(s)\in F(t_m,u_{t_m})$ a.e. for $s\in [\,t_m,s_m)$ and $\Vert f(s)\Vert \le M$ a.e. for $s\in [\,\tau ,T\,]\,;$

• (iv) $u(t)=\varphi (t-\tau )$ for $t\in [\,\tau -\sigma ,\tau \,]$ and $\Vert u(t)-u(t,t_m,u(t_m),f)\Vert \le (t-t_m)\varepsilon$ for $t\in [\,t_m,T\,]$ and $m\in \varGamma \,;$

• (v) $\Vert u_t-\varphi \Vert _\sigma <\rho$ for all $t\in [\,\tau ,T\,]\,;$

• (vi) $\Vert u(t)-u(t_m)\Vert \le \varepsilon$ for all $t\in [\,t_m,s_m)$ and all $m\in \varGamma .$

Proof

Let us observe that, if $(i)$ $\sim$ $(iv)$ are satisfied, then $(v)$ is satisfied too, i.e. $\Vert u(t+s)-\varphi (s)\Vert <\rho$ for all $t\in [\,\tau ,T\,]$ and $s\in [\,-\sigma ,0\,]$. Indeed, if $t+s\le \tau$ then

$$\Vert u(t+s)-\varphi (s)\Vert =\Vert \varphi (t+s-\tau )-\varphi (s)\Vert \le \sup _{|t_1-t_2|\le T-\tau }\Vert \varphi (t_1)-\varphi (t_2)\Vert <\rho .$$

If $t+s>\tau$ then $|s|<T-\tau$ and from $(3)$, $(iii)$ and $(iv)$, we get

$$\Vert u(t+s)-\varphi (s)\Vert \le \Vert u(t+s)-u(t+s,\tau ,\varphi (0),f)\Vert $$

$$+\Vert u(t+s,\tau ,\varphi (0),f)-u(t+s,\tau ,\varphi (0),0)\Vert $$

$$+\Vert u(t+s,\tau ,\varphi (0),0)-\varphi (0)\Vert +\Vert \varphi (0)-\varphi (s)\Vert $$

$$\le (t+s-\tau )\varepsilon +\int _{\tau }^{t+s}\Vert f(\theta )\Vert d\theta +\Vert S(t+s-\tau )\varphi (0)-\varphi (0)\Vert +\Vert \varphi (0)-\varphi (s)\Vert $$

$$\le (T-\tau )(1+M)+\Vert S(t+s-\tau )\varphi (0)-\varphi (0)\Vert +\Vert \varphi (0)-\varphi (s)\Vert <\rho .$$

Let $\varepsilon \in (0,1)$ be arbitrary, but fixed. We will show that there exist $\delta =\delta (\varepsilon )$ in $(\tau ,T)$ and $\mathcal {P}_{\delta }$, $f$, $u$ such that $(i)$ $\sim$ $(vi)$ hold true with $\delta$ instead of $T$.

From the tangency condition, it follows that there exist $h_n\downarrow 0$, $g_n\in \mathcal {F}_{F(\tau ,\varphi )}$ and $p_n\in X$, with $\Vert p_n\Vert \rightarrow 0$ and $u(\tau +h_n,\tau ,\varphi (0),g_n)+p_nh_n\in K(\tau +h_n)$ for every $n\in \mathbb {N}$, $n\ge 1$. Let $n_0\in \mathbb N$ and $\delta =\tau +h_{n_0}$ be such that $\delta \in (\tau ,T)$, $h_{n_0}<\varepsilon$ and $\Vert p_{n_0}\Vert <\varepsilon$.

Let $\mathcal {P}_\delta =\{[\,\tau ,\delta )\}$, $f(t)=g_{n_0}(t)$ and $u(t)=u(t,\tau ,\varphi (0),g_{n_0})+(t-\tau )p_{n_0}$ for $t\in [\,\tau ,\delta \,]$. Obviously, $(i)$ $\sim$ $(v)$ are satisfied. Moreover, we may diminish $\delta >\tau$ (increase $n_0$), if necessary, in order to $(vi)$ be satisfied too.

Let $\mathcal {U}=\{(\mathcal {P}_\delta ,f,u);\ \delta \in (\tau ,T\,]\ \text{ and } (i)\sim (vi) \text{ are } \text{ satisfied } \text{ with } \delta \text{ instead } \text{ of } T\}.$

As we already have shown, $\mathcal {U}\ne \emptyset$. On $\mathcal {U}$ we define a partial order by:

$$(\mathcal {P}_{\delta _1},f_1,u_1)\preceq (\mathcal {P}_{\delta _2},f_2,u_2),$$

if $\delta _1\le \delta _2$, $\mathcal {P}_{\delta _1}\subseteq \mathcal {P}_{\delta _2}$, $f_1(s)=f_2(s)$ a.e. for $s\in [\,\tau ,\delta _1\,]$ and $u_1(s)=u_2(s)$ for all $s\in [\,\tau ,\delta _1\,]$. We will prove that each nondecreasing sequence in $\mathcal {U}$ is bounded from above. Let $((\mathcal {P}_{\delta _j},f_j,u_j))_{j\ge 1}$ be a nondecreasing sequence in $\mathcal {U}$ and let $\delta =\sup _{j\ge 1}\delta _j$. If there exists $j_0\in \mathbb N$ such that $\delta _{j_0}=\delta$, then $(\mathcal {P}_{\delta _{j_0}},f_{j_0},u_{j_0})$ is an upper bound for the sequence. So, let us assume that $\delta _j<\delta$, for all $j\ge 1$. Obviously, $\delta \in (\tau ,T\,]$. We define $\mathcal {P}_\delta =\cup _{j\ge 1}\mathcal {P}_{\delta _j}$, $f(t)=f_j(t)$ and $u(t)=u_j(t)$ for all $j\ge 1$ and $t\in [\,\tau ,\delta _j)$. Clearly, $f\in L^1(\tau ,\delta ;X)$ and $u\in C([\,\tau ,\delta );X)$.

Let us observe that, in view of $(iv)$, we have

$$\Vert u(t)-u(s)\Vert \le \Vert u(t)-u(t,\delta _j,u(\delta _j),f)\Vert $$

$$+\Vert u(t,\delta _j,u(\delta _j),f)-u(s,\delta _j,u(\delta _j),f)\Vert +\Vert u(s,\delta _j,u(\delta _j),f)-u(s)\Vert $$

$$\le (t-\delta _j)\varepsilon +\Vert u(t,\delta _j,u(\delta _j),f)-u(s,\delta _j,u(\delta _j),f)\Vert +(s-\delta _j)\varepsilon $$

$$\le 2(\delta -\delta _j)\varepsilon +\Vert u(t,\delta _j,u(\delta _j),f)-u(s,\delta _j,u(\delta _j),f)\Vert $$

for all $j\ge 1$ and all $t,s\in [\delta _j,\delta )$. Since $\lim _j\delta _j=\delta$ and $u(\cdot ,\delta _j,u(\delta _j),f)$ is continuous at $t=\delta$, we conclude that $u$ satisfies the Cauchy condition for the existence of the limit at $t=\delta$. So, $u$ can be extended by continuity to the whole interval $[\,\tau ,\delta \,].$ By observing that $u(\delta )=\displaystyle \lim _{t\uparrow \delta }u(t) =\lim _{j\rightarrow \infty }u(\delta _j)=\lim _{j\rightarrow \infty }u_j(\delta _j)$, $u_j(\delta _j)\in D(\varphi (0),\rho )\cap K(\delta _j)$ and the latter is closed from the left, we deduce that $u(\delta )\in D(\varphi (0),\rho )\cap K(\delta )$. The rest of conditions in lemma being obviously satisfied, it follows that $(\mathcal {P}_\delta ,f,u)$ is an upper bound for the sequence. Consequently, $(\mathcal {U},\preceq )$ and $\mathcal {N}:(\mathcal {U},\preceq )\rightarrow R$, defined by $\mathcal {N}(\mathcal {P}_\delta ,f,u)=\delta$, for each $(\mathcal {P}_\delta ,f,u)\in \mathcal {U}$, satisfy the hypotheses of the Brezis-Browder Ordering Principle – see Cârjă, Necula and Vrabie [2, Theorem 2.1.1, p. 30]. Accordingly, there exists an $\mathcal {N}$-maximal element in $\mathcal {U}$. This means that there exists $(\mathcal {P}_{\delta ^*},f^*,u^*)\in \mathcal {U}$ such that, whenever $(\mathcal {P}_{\delta ^*},f^*,u^*)\preceq (\mathcal {P}_{\overline{\delta }},\overline{f},\overline{u})$, we necessarily have $\mathcal {N}(\mathcal {P}_{\delta ^*},f^*,u^*)= \mathcal {N}(\mathcal {P}_{\overline{\delta }},\overline{f},\overline{u})$. We will show that $\delta ^*=T$. To this aim, let us assume by contradiction that $\delta ^*<T$.

Since $(\delta ^*,u^*_{\delta ^*})\in \mathcal {K}$, using the tangency condition, we deduce that there exist the sequences $h_n\downarrow 0$, $g_n\in \mathcal {F}_{F(\delta ^*,u^*_{\delta ^*})}$ and $p_n\in X$, with $\Vert p_n\Vert \rightarrow 0$, such that $u(\delta ^*+h_n,\delta ^*,u^*(\delta ^*),g_n)+p_nh_n\in K(\delta ^*+h_n)$ for all $n\in \mathbb {N}$, $n\ge 1$. Let $n_0\in \mathbb {N}$ and $\overline{\delta }=\delta ^*+h_{n_0}$ with $\overline{\delta }\in (\delta ^*,T)$, $h_{n_0}<\varepsilon$ and $\Vert p_{n_0}\Vert <\varepsilon$. Let $\mathcal {P}_{\overline{\delta }}=\mathcal {P}_{\delta ^*}\cup \{[\,\delta ^*,\overline{\delta }\,]\}$,

$$\overline{f}(t)=\left\{ \begin{array}{ll} f^*(t),\ t\in [\,\tau ,\delta ^*\,]\\ f_{n_0}(t),\ t\in (\delta ^*,\overline{\delta }\,] \end{array} \right. , $$

$$ \overline{u}(t)=\left\{ \begin{array}{ll} u^*(t),\ t\in [\,\tau ,\delta ^*\,]\\ u(t,\delta ^*,u^*(\delta ^*),f_{n_0})+(t-\delta ^*)p_{n_0},\ t\in (\delta ^*,\overline{\delta }\,]. \end{array} \right. $$

By $(v)$, we have $u^*_{\delta ^*}\in S_\sigma (\varphi ,\rho )$. So, $(2)$ implies that $\Vert \overline{f}(s)\Vert \le M$ a.e. for $s\in (\tau ,\overline{\delta })$.

Clearly $(i)$ $\sim$ $(iii)$ are satisfied. In order to prove $(iv)$ we will consider only the case $t_m\le \delta ^*\le t$, the other cases being obvious. Using the evolution property, i.e. $u(t,a,\xi ,f)=u(t,b,u(b,a,\xi ,f),f)$ for $\tau \le a\le b \le t\le T$, we get

$$\Vert \overline{u}(t)-u(t,t_m,u^*(t_m),\overline{f})\Vert $$

$$\le \Vert u(t,\delta ^*,u^*(\delta ^*),\overline{f})-u(t,t_m,u^*(t_m),\overline{f})\Vert +(t-\delta ^*)\varepsilon $$

$$=\Vert u(t,\delta ^*,u^*(\delta ^*),\overline{f})-u(t,\delta ^*,u(\delta ^*,t_m,u^*(t_m),\overline{f}),\overline{f})\Vert +(t-\delta ^*)\varepsilon $$

$$\le \Vert u^*(\delta ^*)-u(\delta ^*,t_m,u^*(t_m),\overline{f})\Vert +(t-\delta ^*)\varepsilon $$

$$\le (\delta ^*-t_m)\varepsilon +(t-\delta ^*)\varepsilon =(t-t_m)\varepsilon ,$$

which proves $(iv)$.

Similarly, we can diminish $\overline{\delta }$ (increase $n_0$) in order that $(vi)$ be satisfied too.

So, $(\mathcal {P}_{\overline{\delta }},\overline{f},\overline{u})\in \mathcal {U}$, $(\mathcal {P}_{\delta ^*},f^*,u^*)\preceq (\mathcal {P}_{\overline{\delta }},\overline{f},\overline{u})$, but $\delta ^*<\overline{\delta }$ which contradicts the maximality of $(\mathcal {P}_{\delta ^*},f^*,u^*)$. Hence $\delta ^*=T$, and $\mathcal {P}_{\delta ^*}$, $f^*$ and $u^*$ satisfy all the conditions $(i)$ $\sim$ $(vi)$. The proof is complete.

Definition 7

Let $\varepsilon >0$. An element $(\mathcal {P}_{T},f,u)$ satisfying $(i)$ $\sim$ $(vi)$ in Lemma 2, is called an $\varepsilon$ -approximate $C^0$ -solution of (1).

We can proceed now to the proof of Theorem 3.

Proof

The necessity follows from Theorem 2. As long as the proof of the sufficiency is concerned, let $\rho >0$, $T>\tau$ and $M>0$ be as in Lemma 2. Let $\varepsilon _n\in (0,1)$, with $\varepsilon _n\downarrow 0$. Let $((\mathcal {P}_T^n,f_n,u_n))_n$ be a sequence of $\varepsilon _n$-approximate $C^0$-solutions of (1) given by Lemma 2. If $\mathcal {P}_T^n=\{[\,t_m^n,s_m^n);\ m\in \varGamma _n\}$ with $\varGamma _n$ finite or at most countable, we denote by $a_n:[\,\tau ,T)\rightarrow [\,\tau ,T)$ the step function, defined by $a_n(s)=t_m^n$ for each $s\in [\,t_m^n,s_m^n)$. Clearly $\displaystyle \lim _na_n(s)=s$ uniformly for $s\in [\,\tau ,T)$, while from $(vi)$, deduce that $\displaystyle \lim _n\Vert u_n(t)-u_n(a_n(t))\Vert =0$, uniformly for $t\in [\,\tau ,T)$. From $(iv)$, we get

$$\begin{aligned} \lim _{n}(u_n(t)-u(t,\tau ,\varphi (0),f_n))=0\end{aligned}$$

(2)

uniformly for $t\in [\,\tau ,T\,]$. Since $\Vert f_n(t)\Vert \le M$ for all $n\in \mathbb {N}$ and a.e. for $t\in [\,\tau ,T\,]$ and the semigroup generated by $A$ is compact, by Vrabie [19, Theorem 2.3.3, p. 47], we deduce that the set $\{u(\cdot ,\tau ,\varphi (0),f_n);\,n\ge 1\}$ is relatively compact in $C([\,\tau ,T\,];X)$. From this remark and (2), we conclude that $(u_n)_n$ has at least one uniformly convergent subsequence to some function $u$, subsequence denoted again by $(u_n)_n.$

Since $a_n(t)\uparrow t$, $\lim _nu_n(a_n(t))=u(t)$, uniformly for $t\in [\,\tau ,T)$ and the mapping $t\rightarrow K(t)\cap D(\varphi (0),\rho )$ is closed from the left, we get that $u(t)\in K(t)$ for all $t\in [\,\tau ,T\,]$. But $\lim _n (u_n)_{a_n(t)}=u_t$ in $C_\sigma$, uniformly for $t\in [\,\tau ,T)$. Hence, the set $C=\overline{\{(a_n(t),(u_n)_{a_n(t)});n\ge 1,\ t\in [\,\tau ,T)\}}$ is compact and $C\subseteq \mathcal {K}$.

At this point, recalling that $F$ is strongly-weakly u.s.c. and has weakly compact values, by Cârjă, Necula and Vrabie [2, Lemma 2.6.1, p. 47], it follows that $B=\overline{\text{ conv }}\left( \bigcup _{n\ge 1}\bigcup _{t\in [\,\tau ,T)}F(a_n(t),(u_n)_{a_n(t)})\right)$ is weakly compact. We notice that $f_n(s)\in B$ for all $n\ge 1$ and a.e. for $s\in [\,\tau ,T\,]$. An appeal to Cârjă, Necula and Vrabie [2, Theorem 1.3.8, p. 10] shows that, at least on a subsequence, $\displaystyle \lim _nf_n=f$ weakly in $L^1(\tau ,T;X)$. As $F$ is strongly-weakly u.s.c. with closed and convex values while, by Lemma 2, for each $n\ge 1$, we have $f_n(s)\in F(a_n(s),(u_n)_{a_n(s)})$ a.e. for $s\in [\,\tau ,T\,]$, from Vrabie [19, Theorem 3.1.2, p. 88], we conclude that $f(s)\in F(s,u_s)$ a.e. for $s\in [\,\tau ,T\,]$.

Finally, by (2) and the fact that $A$ is of complete continuous type, we get $u(t)=u(t,\tau ,\varphi (0),f)$ for each $t\in [\,\tau ,T\,]$ and so, $u$ is a $C^0-$solution of (1).

Theorem 4

Let $K$ be closed from the left and let $F:\mathcal {K} \leadsto X$ be nonempty, convex and weakly compact valued. If there exist $a,b\in C(I)$ such that

$$ \Vert F(t,\varphi )\Vert \le a(t)+b(t)\Vert \varphi (0)\Vert \ \ \text{ for } \text{ all }\ t\in I\ \text{ and } \text{ all }\ \varphi \in C_\sigma , $$

$F$ is strongly-weakly u.s.c. and $A:D(A)\leadsto X$ is of complete continuous type and generates a compact semigroup, then a sufficient condition in order that $\mathcal {K}$ be globally $C^0$-viable with respect to $A+F$ is the tangency condition in Theorem 3. If, in addition, $F$ is u.s.c., then the tangency condition is also necessary in order that $\mathcal {K}$ be mild-viable with respect to $A+F$.

5 A Sufficient Condition for Null Controllability

Let $X$ be a Banach space, $A:D(A)\subseteq X\leadsto X$ an $m$-dissipative operator, $g:\mathbb {R}_+\times C_{\sigma }\rightarrow X$ a given function and $(\tau ,\varphi )\in \mathbb {R}_+\times C_\sigma$ with $\varphi (0)\in \overline{D(A)}$. The problem is how to find a measurable control $c(\cdot )$ taking values in $D(0,1)$ in order to reach the origin in some time $T$, by $C^0$-solutions of the state equation

$$\begin{aligned} \left\{ \begin{array}{ll} u'(t)\in Au(t)+ g(t,u_t)+c(t)\\ u_{\tau }=\varphi .\end{array}\right. \end{aligned}$$

(3)

With $G:\mathbb {R}_+\times C_{\sigma }\leadsto X$, defined by $G(t,v)=av(0)+ g(t,v)+D(0,1)$, the above problem reformulates: find $T>0$ and a $C^0$-solution of problem

$$\begin{aligned} \left\{ \begin{array}{ll} u'(t)\in (A-aI)u(t)+G(t,u_t)\\ u_\tau =\varphi ,\ \ u(\tau +T)=0.\end{array}\right. \end{aligned}$$

(4)

Theorem 5 and Corollary 1 below are “delay” versions of Cârjă, Necula and Vrabie [3, Theorem 12.1 and Corollary 12.1].

Theorem 5

Let $X$ be a reflexive Banach space and let $A:D(A)\subseteq X\leadsto X$ be such that, for some $a\in \mathbb {R}$, $A-aI$ is an $m$-dissipative operator of complete continuous type and which is the infinitesimal generator of a compact semigroup of contractions, $\{S(t):\overline{D(A)}\rightarrow \overline{D(A)};\ \ t \ge 0\}$. Let $g:\mathbb {R}_+\times C_{\sigma }\rightarrow X$ be a continuous function such that for some $L>0$ we have

$$\begin{aligned} \Vert g(t,v)\Vert \le L\Vert v(0)\Vert ,\ \ \text{ for } \text{ all }\ (t,v)\in \mathbb {R}_+\times C_{\sigma }.\end{aligned}$$

(5)

Assume that $0\in D(A)$ and $0\in A0$. Then, for each $(\tau ,\varphi )\in \mathbb {R}_+\times C_\sigma$ with $\xi =\varphi (0)\in \overline{D(A)}\setminus \{0\}$, there exists a $C^0$-solution $u:[\,\tau ,\infty )\rightarrow X$ of (4) satisfying

$$\begin{aligned} \Vert u(t)\Vert \le \Vert \xi \Vert -(t-\tau ) +(L+a)\int _\tau ^t \Vert u(s)\Vert ds,\ \ \text{ for } \text{ all }\ t\ge \tau \ \text{ with }\ \ u(t)\ne 0. \end{aligned}$$

(6)

Proof

Let $(\tau ,\varphi )\in \mathbb {R}_+\times C_\sigma$ with $\xi =\varphi (0)\in \overline{D(A)}\setminus \{0\}$. We show that there exist $T\in (0,+\infty )$ and a noncontinuable $C^0$-solution $(z,u):[\,\tau ,\tau +T)\rightarrow \mathbb {R}\times X$ of the problem

$$\begin{aligned} \left\{ \begin{array}{ll} z'(t)=(L+a)\Vert u(t)\Vert -1,&{}t\in [\,\tau ,\tau +T)\\ u'(t)\in (A-aI)u(t)+G(t,u_t),&{}t\in [\,\tau ,\tau +T)\\ z_{\tau }=\Vert \varphi \Vert \,\,\,\,\,\,\text{ and } \,\,\,\,\,\,u_{\tau }=\varphi ,\\ \Vert u(t)\Vert \le z(t),&{}t\in [\,\tau ,\tau +T). \end{array}\right. \end{aligned}$$

(7)

On the Banach space $\mathcal {X}=\mathbb {R}\times X$ the operator $\mathcal {A}=(0,A-aI)$ generates a compact semigroup of contractions $\{(1,S(t));\ (1,S(t)): \mathbb {R}\times \overline{D(A)}\rightarrow \mathcal {X}\}$.

We denote by $\mathcal {C}_{\sigma }=C([\,-\sigma ,0\,];\mathcal {X})=C([\,-\sigma ,0\,];\mathbb {R})\times C([\,-\sigma ,0\,];X)$. Let $K$ be the locally closed set $K= \{(x_1,x_2)\in \mathbb {R}_+\times (\overline{D(A)}\setminus \{0\});\ \Vert x_2\Vert \le x_1\}$, with the associate set $\mathcal {K}=\{(t,\psi )\in \mathbb {R}\times \mathcal {C}_{\sigma };\ \psi (0)\in K\}$, i.e.

$$\mathcal {K}=\{(t,\psi _1,\psi _2)\in \mathbb {R}\times C([\,-\sigma ,0\,];\mathbb {R})\times C([\,-\sigma ,0\,];X);\ \Vert \psi _2(0)\Vert \le \psi _1(0)\}$$

and let the multi-function $\mathcal {F}: \mathcal {K}\leadsto \mathbb {R}\times X$ be defined by

$$\mathcal {F}(t,\psi _1,\psi _2)=((L+a)\Vert \psi _2(0)\Vert -1, a\psi _2(0) +g(t,\psi _2)+D(0,1)),\ \text{ for }\ (t,\psi _1,\psi _2)\in \mathcal {K}.$$

To show that $\mathcal {F}(\tau ,\psi _1,\psi _2)\in \mathcal {QTS}_{\mathcal {K}}^\mathcal {A}(\tau ,\psi _1,\psi _2)$, for every $(\tau ,\psi _1,\psi _2)\in \mathcal {K}$, we shall prove the stronger condition: there exists $(\eta _1,\eta _2)\in \mathcal {F}(\tau ,\psi _1,\psi _2)$ such that

$$\begin{aligned} \liminf _{h\downarrow 0}h^{-1}\text{ d } \left( \mathcal {U}(\tau +h,\tau ,(\xi _1,\xi _2),(\eta _1,\eta _2)), K\right) =0,\end{aligned}$$

(8)

where $(\xi _1,\xi _2)=(\psi _1(0),\psi _2(0))$ and $\mathcal {U}(\cdot ,\tau ,(\xi _1,\xi _2),(\eta _1,\eta _2))$ is the $C^0$-solution of the corresponding Cauchy problem for the operator $\mathcal {A}$, i.e.

$$\mathcal {U}(t,\tau ,(\xi _1,\xi _2),(\eta _1,\eta _2))=(\xi _1+(t-\tau )\eta _1,u(t,\tau ,\xi _2,\eta _2))\in \mathcal {X},$$

$u(\cdot ,\tau ,\xi _2,\eta _2)$ being the corresponding solution for $A-aI$. To this end, it suffices to prove that there exist $(h_n)_n$ in $\mathbb {R}_+$, with $h_n \downarrow 0$, and $(\theta _n,p_n)$ in $\mathbb {R}\times X$, with $(\theta _n,p_n)\rightarrow (0,0)$, such that, for every $n\in \mathbb {N}$, we have

$$\begin{aligned} \Vert u(\tau +h_n,\tau ,\xi _2,\eta _2)+h_n p_n\Vert \le \xi _1+h_n\eta _1+h_n\theta _n.\end{aligned}$$

(9)

Clearly, $\displaystyle \left\| u\left( \tau +h,\tau , \xi _2,\eta _2\right) \right\| \le \Vert \xi _2\Vert + \int _{\tau }^{\tau +h}[\,u(s,\tau , \xi _2,\eta _2),\eta _2\,]_+ ds$ for all $h>0$. The normalized semi-inner product, $(x,y)\mapsto [\,x,y\,]_+ =\displaystyle \lim _{h\downarrow 0}h^{-1}(\Vert x+hy\Vert -\Vert x\Vert )$, is u.s.c. Hence, setting $\ell (s):=u(s,\tau , \xi _2,\eta _2)$, we get

$$\liminf _{h\downarrow 0}h^{-1}\int _{\tau }^{\tau +h}[\ell (s),\eta _2]_+ds\le \limsup _{h\downarrow 0}h^{-1}\int _{\tau }^{\tau +h}[\,\ell (s),\eta \,]_+ ds\le [\,\xi _2,\eta _2\,]_+.$$

Let $\eta _1=(L+a)\Vert \psi _2(0)\Vert -1=(L+a)\Vert \xi _2\Vert -1$ and $\eta _2=a\xi _2+g(\tau ,\psi _2)-\frac{\xi _2}{\Vert \xi _2\Vert }$. Clearly, $\eta _2\in a\xi _2 +g(\tau ,\psi _2)+D(0,1)$ and so, $(\eta _1,\eta _2)\in \mathcal {F}(\tau ,\psi _1,\psi _2)$. From (5), we get $[\xi _2,\eta _2]_+=a\Vert \xi _2\Vert +[\xi _2,g(\tau ,\psi _2)]_+ -1\le (L+a)\Vert \xi _2\Vert -1=\eta _1$ and hence $\displaystyle \liminf _{h\downarrow 0}h^{-1}\left( \Vert u(\tau +h,\tau ,\xi _2,\eta _2)\Vert -\Vert \xi _2\Vert \right) \le \eta _1$. Keeping in mind that $\Vert \xi _2\Vert =\Vert \psi _2(0)\Vert \le \psi _1(0)=\xi _1$ since $(\tau ,\psi _1,\psi _2)\in \mathcal {K}$, the last inequality proves (9) with $p_n= 0$. Thus we get (8). From Theorem 3, $\mathcal {K}$ is $C^0$-viable with respect to $\mathcal {A}+\mathcal {F}$. As $(\tau , \Vert \varphi \Vert ,\varphi )\in \mathcal {K}$, thanks to Brezis-Browder Ordering Principle [2, Theorem 2.1.1, p. 30] –, we obtain further that there exist $T\in (0,+\infty \,]$ and a noncontinuable $C^0$-solution of $(z,u):[\,\tau ,\tau +T)\rightarrow \mathbb {R}\times X$ of (7) which satisfies $(z(t),u(t))\in K$ for every $t\in [\,\tau ,\tau +T)$. This means that (6) is satisfied for every $t\in [\,\tau ,\tau +T)$. Since $G$ has sublinear growth, $u$, as a solution of (4), can be continued to $\mathbb {R}_+$. So, $u(\tau +T)$ exists, even though the solution $(z,u)$ of (7) is defined merely on $[\,\tau ,\tau +T)$ if $T$ is finite. In this case, $u(\tau +T)=0$ since otherwise $(z,u)$ can be continued to the right of $T$ which is a contradiction.

Corollary 1

Under the hypothesis of Theorem 5, the following properties hold.

• (i) If $L+a \le 0$, for any $(\tau ,\varphi )\in \mathbb {R}_+\times C_\sigma$ with $\xi =\varphi (0)\in \overline{D(A)}\setminus \{0\}$, there exist a control $c(\cdot )$ and a $C^0$-solution of (3) that reaches the origin of $X$ in some time $T\le \Vert \xi \Vert$ and satisfies $\Vert u(t)\Vert \le \Vert x\Vert -(t-\tau )\ \text{ for } \text{ all }\ \tau \le t\le \tau +T.$

• (ii) If $L+a> 0$, for every $(\tau ,\varphi )\in \mathbb {R}_+\times C_\sigma$ with $\xi =\varphi (0)\in \overline{D(A)}\setminus \{0\}$ satisfying $0< \Vert \xi \Vert < 1/(L+a)$, there exist a control $c(\cdot )$ and a $C^0$-solution of (3) that reaches the origin of $X$ in some time $T\le (L+a)^{-1}\log \left\{ \left[ 1-(L+a)\Vert \xi \Vert \right] ^{-1}\right\}$ and $\Vert u(t)\Vert \le e^{(L+a)(t-\tau )}\left[ \Vert \xi \Vert -(L+a)^{-1}\right] +(L+a)^{-1}$ for $t\in [\tau ,\tau +T].$