, supplemented with the adjoint equations , and the transversality conditions $$\label{EQ_trans}
\psi_2(+\infty)=0,\qquad \psi_3(+\infty)=1,$$ it holds that $$\begin{gathered}
\label{EQ_Max-principle}
\max_{(c_1,c_2)} \Re H\big(a_2(t),a_3(t),\psi_2(t),\psi_3(t),t,c_1,c_2\big)\\ = \Re H\big(a_2(t),a_3(t),\psi_2(t),\psi_3(t),t,c^*_1(t),c^*_2(t)\big),\end{gathered}$$ where the maximum is taken over all $(c_1,c_2)\in\C^2$ subject to conditions .

System – can be integrated using integrals to ,: $$\label{EQ_adjoint}
\psi_2(t)=a-2a_2(t),\quad \psi_3(t)=1,$$ where $a:=2a_2(+\infty)$.

To find the maximum of $\Re H$ as a function of $c_1$ and $c_2$, we first fix a $c_1\in\C$ with $|c_1|\le2$ and optimize $\Re H$ in the disk described by the second of the inequalities in . The maximum is achieved for ${c_2=c^*_2:=(\Re c_1)^2+i\,\Re c_1\,\Im c_1-2}$. For this value of $c_2$ and taking into account , we get $$\begin{aligned}
-\frac{e^{2t}}{k}\,\Re H&=e^t\,\Re(ac_1)+\frac{1+k}{2}\,c_1'^{\,2}+\frac{1-k}{2}\,c_2''^{\,2}-2\notag\\                        &=\,\frac{1+k}{2}\left(c_1'+\frac{e^ta'}{1+k}\right)^{\!\!2}\,                            +\,\frac{1-k}{2}\left(c_1''-\frac{e^ta''}{1-k}\right)^{\!\!2}\,+\,C,\label{EQ_quadratic}\end{aligned}$$ where $a=:a'+ia''$, $c_1=:c_1'+ic_1''$, and $C$ is a quantity independent of $c_1$. The absolute minimum of  is achieved at $c_1^\star:=e^t\big(-a'/(1+k)+ia''/(1-k)\big)$. Moreover, even if $|c_1^\star|>2$, the minimum point $c_1^*$ of  over the disk $|c_1|\le 2$ still satisfies $$\label{EQ_sgn}
 {\mathop{\mathrm{sgn}}}\Re c_1^*=-{\mathop{\mathrm{sgn}}}a',\qquad {\mathop{\mathrm{sgn}}}\Im c_1^*={\mathop{\mathrm{sgn}}}a'',$$ where ${\mathop{\mathrm{sgn}}}x:=x/|x|$ for $x\in{\mathbb{R}}\setminus\{0\}$ and ${\mathop{\mathrm{sgn}}}0:=0$. For the optimal trajectory, according to , we have $$\label{EQ_a}
 a=-2k\int_0^{+\infty}\!\!e^{-t}c_1^*(t)\,{\mathrm{d}}t,$$ which would contradict  whenever $a''\neq0$. Therefore, $a$ is real and $$\label{EQ_c1}
c_1^*(t)=
\begin{cases}
-e^ta/(1+k),& \text{if~}\big|e^ta/(1+k)\big|<2,\\
-2{\mathop{\mathrm{sgn}}}a,&      \text{otherwise}.
\end{cases}$$

Consider two cases. First suppose that $a=0$. Then $c_1^*(t)=0$ and $c_2^*(t)=-2$ 