In this section, we deal with the approximate controllability for the semilinear equation in

*H* as follows.

$\{\begin{array}{c}{x}^{\prime}(t)+Ax(t)+f(t,x(t),u(t))+Bu(t)=0,\phantom{\rule{1em}{0ex}}0<t\le T,\hfill \\ x(0)={x}_{0}.\hfill \end{array}$

(4.1)

In (4.1), the principal operator −

*A* generates an analytic semigroup

$S(t)$ as stated in Section 2. Let

*U* be a Hilbert space of control variables, and let

*B* be a bounded linear operator from

*U* to

*H*, which is called a controller. The mild solution of initial value problem (4.1) is the following form:

$x(t;f,u)=S(t){x}_{0}-{\int}_{0}^{t}S(t-s)\{f(s,x(s),u(s))+Bu(s)\}\phantom{\rule{0.2em}{0ex}}ds.$

Let *f* be a nonlinear mapping satisfying the following.

**Assumption (F1)** The mapping *f* is demicontinuous bounded from $[0,T]\times H\times U$ into ${V}^{\ast}$. Assume that $f(t,\cdot ,u)$ for each $(t,u)\in [0,T]\times U$ is monotone as a mapping from *V* into ${V}^{\ast}$ with $f(t,\cdot ,0)=0$, and $f(t,x,\cdot )$ for each $(t,x)\in [0,T]\times V$ is monotone as a mapping from *U* into ${V}^{\ast}$.

For each

$u\in {L}^{2}(0,T;U)$, let us define

$F(t,x(t))=f(t,x(t),u(t))$. Then from Theorem 3.1 it follows that solution (4.1) exists and is unique in

${L}^{2}(0,T;V)\cap {W}^{1,2}(0,T;{V}^{\ast})$. Let

$x(T;f,u)$ be a state value of system (4.1) at time

*T* corresponding to the nonlinear term

*f* and the control

*u*. We define the reachable sets for system (4.1) as follows:

$\begin{array}{r}{R}_{T}(f)=\{x(T;f,u):u\in {L}^{2}(0,T;U)\},\\ {R}_{T}(0)=\{x(T;0,u):u\in {L}^{2}(0,T;U)\}.\end{array}$

**Definition 4.1** System (4.1) is said to be approximately controllable at time *T* if for every desired final state ${x}_{1}\in H$ and $\u03f5>0$, there exists a control function $u\in {L}^{2}(0,T;U)$ such that the solution $x(T;f,u)$ of (4.1) satisfies $|x(T;f,u)-{x}_{1}|<\u03f5$, that is, $\overline{{R}_{T}(f)}=H$, where $\overline{{R}_{T}(f)}$ is the closure of ${R}_{T}(f)$ in *H*.

**Definition 4.2** Let

*L* be a mapping from a Banach space

*X* into its conjugate space

${X}^{\ast}$.

*T* is called coercive if there exists

${u}_{0}\in D(L)$ such that

$\underset{u\in D(L),\parallel u\parallel \to \mathrm{\infty}}{lim}\frac{(Lu,u-{u}_{0})}{\parallel u\parallel}=\mathrm{\infty}.$

**Remark 4.1** [[23], Theorem 1.3]

It is well known that if *X* is a reflexive Banach space and *L* is monotone, everywhere defined and hemicontinuous from $D(L)=X$ into ${X}^{\ast}$, then *L* is maximal monotone. If in addition *L* is coercive monotone, then $R(L)={X}^{\ast}$.

First, we consider the approximate controllability of system (4.1) in the case where the controller

*B* is the identity operator on

*H* under Assumption (F1) on the nonlinear operator

*f*. So,

$H=U$ obviously. Consider the linear system given by

$\{\begin{array}{c}{y}^{\prime}(t)+Ay(t)+u(t)=0,\hfill \\ y(0)={x}_{0}\hfill \end{array}$

(4.2)

and the following semilinear control system:

$\{\begin{array}{c}{x}^{\prime}(t)+Ax(t)+f(t,x(t),v(t))+v(t)=0,\hfill \\ x(0)={x}_{0}.\hfill \end{array}$

(4.3)

**Lemma 4.1** *Let Assumption *(F1)

*be satisfied*,

*and let* $y(t)$ *be the solution of* (4.2)

*corresponding to a control* *u*.

*Then there exists* $v\in {L}^{2}(0,T;H)$ *such that* $\{\begin{array}{c}v(t)=u(t)-f(t,y(t),v(t)),\phantom{\rule{1em}{0ex}}0<t\le T,\hfill \\ v(0)=u(0).\hfill \end{array}$

(4.4)

*Proof* Set

$w(t)=v(t)-u(t),\phantom{\rule{2em}{0ex}}g(t,w(t))=f(t,y(t),w(t)+u(t)).$

Let

$(Gw)(t)=g(t,w(t))$. Then equation (

4.4) is equivalent to

It is easy to see that

*G* is monotone as an operator from

${L}^{2}(0,T;H)$ to

${L}^{2}(0,T;{V}^{\ast})$, and is a demicontinuous bounded mapping as an operator from

${L}^{2}(0,T;H)$ into

${L}^{2}(0,T;{V}^{\ast})$. Let the collection of all finite dimensional subspaces of

*H* be denoted by

$\mathcal{Y}$, and when

$Y\in \mathcal{Y}$, let the orthogonal projection on

*Y* be denoted by

${P}_{Y}$. For

$u\in {L}^{2}(0,T;H)$, let us define

$({P}_{Y}u)(t)={P}_{Y}u(t)$; thus

${P}_{Y}$ also denotes the orthogonal projection in

${L}^{2}(0,T;H)$. According to Assumption (F1), we have that the operator

$I+{P}_{Y}g$ is a coercive monotone operator from

*Y* into itself. In general, any demicontinuous operator is hemicontinuous. Therefore, by Remark 4.1, we have

$R(I+{P}_{Y}g)=Y$, which (4.5) implies the existence of a solution to

${w}_{Y}(t)+{P}_{Y}g(t,{w}_{Y}(t))=0,\phantom{\rule{2em}{0ex}}{w}_{Y}(0)={u}_{Y}(0).$

(4.6)

Since

$\begin{array}{rl}{|{w}_{Y}(t)|}^{2}& =-({P}_{Y}g(t,{w}_{Y}(t)),{w}_{Y}(t))\\ =-({P}_{Y}g(t,{w}_{Y}(t))-{P}_{Y}g(t,0),{w}_{Y}(t))-({P}_{Y}g(t,0),{w}_{Y}(t))\\ \le |g(t,0)||{w}_{Y}(t)|,\end{array}$

we have

${\parallel {w}_{Y}\parallel}_{C([0,T];H)}\le C.$

Hence, the solution of (4.6) is bounded on

$C([0,T];H)$. Let

*w* be an arbitrary element of

${L}^{2}(0,T;H)$. Then we can take

${w}_{n}\in {Y}_{n}\in \mathcal{Y}$ satisfying (4.6) such that

${w}_{n}\to w$ in

${L}^{2}(0,T;H)$. Since

*G* is monotone as an operator from

${L}^{2}(0,T;H)$ to

${L}^{2}(0,T;{V}^{\ast})$, and is a demicontinuous bounded mapping, we have

$G{w}_{n}\rightharpoonup Gw$ in

${L}^{2}(0,T;{V}^{\ast})$. Hence, we obtain that for

$v\in {L}^{2}(0,T;V)$,

$0\le ((I+{P}_{Y}G)v-(I+{P}_{Y}G){w}_{n},v-{w}_{n})=((I+{P}_{Y}G)v,v-{w}_{n})$

as

$n\to \mathrm{\infty}$, so that

$((I+{P}_{Y}G)v,v-w)\ge 0.$

(4.7)

If

*v* is replaced by

$w+{n}^{-1}v$ in (4.7), we have

$0\le (w+{n}^{-1}v+{P}_{Y}G(w+{n}^{-1}v),{n}^{-1}v),$

which, by the demicontinuity of

*G*, leads to

$0\le (w+Gw,v),\phantom{\rule{1em}{0ex}}\mathrm{\forall}v\in {L}^{2}(0,T;V)$

in the limit as $n\to \mathrm{\infty}$. Since *v* is arbitrary, we obtain $w+Gw=0$. □

**Remark 4.2** As seen in [24], we know that if *X* is a Hilbert space and $G\subset X\times X$ is maximal monotone, then $R(I+G)=X$. So, (4.5) is easily obtained if the operator *G* in Theorem 4.1 is maximal monotone.

**Theorem 4.1** *Under Assumption *(F1)

*and* $B=I$,

*we have* ${R}_{T}(0)\subset {R}_{T}(f).$

*Therefore*, *if linear system* (4.2) *with* $f=0$ *is approximately controllable at time* *T*, *then so is semilinear system* (4.3).

*Proof* Let

*y*,

*x* be the solutions of (4.2) and (4.3), respectively. Let

$v(t)=u(t)-f(t,y(t),v(t))$ in the sense of Lemma 4.1. Then, since

$\begin{array}{r}{x}^{\prime}(t)+Ax(t)+f(t,x(t),v(t))+v(t)\\ \phantom{\rule{1em}{0ex}}={x}^{\prime}(t)+Ax(t)+f(t,x(t),v(t))+u(t)-f(t,y(t),v(t)),\end{array}$

we have

$\{\begin{array}{c}{x}^{\prime}(t)-{y}^{\prime}(t)+A(x(t)-y(t))+f(t,x(t),v(t))-f(t,y(t),v(t))=0,\hfill \\ x(0)-y(0)=0.\hfill \end{array}$

Acting on both sides of the above equation, by

$x(t)-y(t)$, from the monotonicity of

*f*, it follows

$\frac{1}{2}\frac{d}{dt}{|x(t)-y(t)|}^{2}+{\omega}_{1}{\parallel x(t)-y(t)\parallel}^{2}\le {\omega}_{2}{|x(t)-y(t)|}^{2},$

which is

${|x(t)-y(t)|}^{2}+2{\omega}_{1}{\int}_{0}^{t}{\parallel x(s)-y(s)\parallel}^{2}\phantom{\rule{0.2em}{0ex}}ds\le 2{\omega}_{2}{\int}_{0}^{t}{|x(s)-y(s)|}^{2}\phantom{\rule{0.2em}{0ex}}ds.$

By using Gronwall’s inequality, we get $x=y$ in $C([0,T];H)$. Noting that $x(\cdot ),y(\cdot )\in C([0,T];H)$, every solution of the linear system with control *u* is also a solution of the semilinear system with control *v*, that is, we have that ${R}_{T}(0)\subset {R}_{T}(f)$. □

From now on, we consider the initial value problem for semilinear parabolic equation (4.1). Let *U* be a Hilbert space, and let the controller operator *B* be a nonlinear operator from *U* to *H*.

**Theorem 4.2** *Let Assumption *(F1)

*and* $R(f)\subset R(B)$ *be satisfied*.

*Assume that the inverse mapping* ${B}^{-1}$ *of the controller* *B* *exists and is monotone*.

*Then the linear system* $\{\begin{array}{c}{y}^{\prime}(t)+Ay(t)+Bu(t)=0,\hfill \\ y(0)={x}_{0},\hfill \end{array}$

(4.8)

*is approximately controllable at time* *T*, *so is nonlinear system* (4.1).

*Proof* Let

*y* be a solution of (4.8) corresponding to a control

*u*. Consider the following semilinear system:

$\{\begin{array}{c}{x}^{\prime}(t)+Ax(t)+f(t,x(t),v(t))+Bu(t)-f(t,y(t),v(t))=0,\hfill \\ x(0)={x}_{0}.\hfill \end{array}$

(4.9)

Set

$v(t)=u(t)-{B}^{-1}f(t,y(t),v(t)).$

(4.10)

We put

$w(t)=v(t)-u(t),\phantom{\rule{2em}{0ex}}({B}_{1}u)(t)=Bu(t),\phantom{\rule{2em}{0ex}}(Gw)(t)=g(t,w(t))=f(t,y(t),w(t)+u(t)).$

Equation (4.10) is equivalent to

$(I+{B}_{1}^{-1}G)w=0.$

(4.11)

Here, similarly to the proof of Lemma 4.1, we have that there exists an element $v\in {L}^{2}(0,T;U)$ satisfying (4.10), that is, $Bv(t)=Bu(t)-f(t,y(t),v(t))$. In a similar way to the proof of Theorem 4.1, we get $x=y$. Since system (4.1) is equivalent to (4.9), we conclude that ${R}_{T}(0)\subset {R}_{T}(f)$. □

Now we consider the control problem of (4.1) when the controller *B* is a nonlinear mapping in the case where $U=V$. In this case, we suppose that Assumption (F) and the next additional assumption are satisfied.

**Assumption (F2)** Assume that $f(t,x,\cdot )$ for each $(t,x)\in [0,T]\times V$ is maximal monotone as a mapping from *U* into ${V}^{\ast}$.

The following result is well known from semigroup properties.

**Lemma 4.2**
*If*
$p\in {L}^{1}(0,T;H)$
*and*
${\int}_{0}^{t}S(t-s)p(s)\phantom{\rule{0.2em}{0ex}}ds=0,\phantom{\rule{1em}{0ex}}0\le t\le T,$

*then* $p(t)=0$ *for almost all* $t\in [0,T]$.

**Theorem 4.3** *Let Assumption *(F2) *and* $R(f)\subset R(B)$ *be satisfied*. *Assume that* *B* *is a hemicontinuous monotone mapping from* *V* *into* ${V}^{\ast}$; *moreover*, *if it is coercive*, *then linear system* (4.8) *is approximately controllable at time* *T*, *so is semilinear system* (4.1).

*Proof* Let

${\xi}_{T}\in D(A)$. We define the linear operator

$\stackrel{\u02c6}{S}$ from

${L}^{2}(0,T;H)$ to

*H* by

$\stackrel{\u02c6}{S}p={\int}_{0}^{T}S(T-s)p(s)\phantom{\rule{0.2em}{0ex}}ds$

for

$p\in {L}^{2}(0,T;H)$. As

${\xi}_{T}\in D(A)$, there exists

$p\in {C}^{1}(0,T;H)$ such that

$\stackrel{\u02c6}{S}p={\xi}_{T}-S(T){x}_{0};$

for instance, take

$p(s)=({\xi}_{T}-sA{\xi}_{T})-S(s){x}_{0}/T$. By expressing

$({B}_{1}u)(t)=Bu(t)$ for all

$u\in {L}^{2}(0,T;V)$. By Remark 4.1, since

$R(B)={V}^{\ast}$, there exists

${u}_{1}\in {L}^{2}(0,T;V)$ such that

Since *p* is an arbitrary element of ${L}^{2}(0,T;{V}^{\ast})$, ${p}_{n}\in {C}^{1}(0,T;H)$ and ${p}_{n}\to p$ in ${L}^{2}(0,T;{V}^{\ast})$. This implies that linear system (4.8) is approximately controllable.

To prove the approximate controllability of (4.1), we will show that

$D(A)\subset \overline{{R}_{T}(f)}$,

*i.e.*, for given

$\epsilon >0$ and

${\xi}_{T}\in D(A)$, there exists

$u\in {L}^{2}(0,T;V)$ such that

$\parallel {\xi}_{T}-x(T;f,u)\parallel <\epsilon .$

(4.12)

Let

$x\in {L}^{2}(0,T;V)$. Then we write

$Gu(t)=f(t,x(t),u(t))$ for each

$u\in {L}^{2}(0,T;H)$. Then we rewrite (4.12) as

$\parallel \stackrel{\u02c6}{S}(p+Gu+{B}_{1}u)\parallel <\epsilon .$

Thus, in view of Lemma 4.2, it is enough to verify that there exists an arbitrary element *u* of ${L}^{2}(0,T;V)$ such that $(G+{B}_{1})u=-p$. By (2) of Lemma 2.3, *B* is pseudo-monotone and satisfies the condition (5) of Lemma 2.3. Thus, we have $R(G+{B}_{1})={V}^{\ast}$. Since *p* is an arbitrary element of ${L}^{2}(0,T;{V}^{\ast})$, ${p}_{n}\in {C}^{1}(0,T;H)$ and ${p}_{n}\to p$ in ${L}^{2}(0,T;{V}^{\ast})$. This implies inequality (4.3) and completes the proof of the theorem. □

**Remark 4.3** We know that by Assumption (F1) and (4.8), ${B}_{1}+G$ is monotone, hemicontinuous and coercive from *U* into ${V}^{\ast}$. Therefore, as seen in Remark 4.1, we have $R({B}_{1}+G)={L}^{2}(0,T;{V}^{\ast})$, that is, system (4.1) is approximately controllable.