Consider the following bilinear system on a open and bounded domain $\Omega$
            \begin{equation}
            \left\{\begin{array}{r c l}
            \displaystyle\frac{dy(t)}{dt} &=& Ay(t)+u(t)By(t)\\
            y(0) &=& y_0
            \end{array}
            \right.
            \end{equation}
            where the state space is $ H^1(\Omega)$  endowed with
            its usual inner product, denoted by $\langle ., . \rangle$ and  $\|.\|$ the associated norm, the operator $A$ generates   a semigroup of contractions $S(t)$ on $H^1(\Omega)$,   $u(t)$ is a scalar valued control and  $B$ is a linear and bounded  operator mapping  $H^1(\Omega)$ into itself.

Suppose that this system admits a unique mild solution $y$.

Since $A$ generates a semigroup of contractions then from Lumer-Phillips Theorem $A$ is dissipative (\langle Ay,y\rangle \leq 0), moreover we have $G = \nabla^*\nabla$ is positive (\langle By,y\rangle = \langle \nabla y,\nabla y\rangle = |\nabla y|^2_{(L^2(\Omega))^n} \geq 0).

So in this case, are there some assumptions to add in order to get $\langle GAy,y\rangle \leq 0$?

Consider the following bilinear system on a open and bounded domain $\Omega$ \begin{equation} \left\{\begin{array}{r c l} \displaystyle\frac{dy(t)}{dt} &=& Ay(t)+u(t)By(t)\\ y(0) &=& y_0 \end{array} \right. \end{equation} where the state space is $ H^1(\Omega)$ endowed with its usual inner product, denoted by $\langle ., . \rangle$ and $\|.\|$ the associated norm, the operator $A$ generates a semigroup of contractions $S(t)$ on $H^1(\Omega)$, $u(t)$ is a scalar valued control and $B$ is a linear and bounded operator mapping $H^1(\Omega)$ into itself.

Suppose that this system admits a unique mild solution $y$.

Since $A$ generates a semigroup of contractions then from Lumer-Phillips Theorem $A$ is dissipative ($\langle Ay,y\rangle \leq 0$), moreover we have $G = \nabla^*\nabla$ is positive ($\langle Gy,y\rangle = \langle \nabla y,\nabla y\rangle = \|\nabla y\|^2_{(L^2(\Omega))^n} \geq 0$).

So in this case, are there some sufficient assumptions on $A$ that guarantees $\langle GAy,y\rangle \leq 0$?