I'm having a trouble with this optimal control problem: $$\min_u J(x,u) = \min_u\{\int_0^\infty -u(\frac{ct}{ct+cf}Q+\alpha x+\beta u) dt$$ $$\dot x=(\alpha x+\beta u)(\frac{1}{ct}+\frac{1}{cf})$$ where:$\,\,\,\,ct, cf ,Q\gt0,\,\,\,\alpha,\beta\lt0$ and constant
after I constructed the state, costate and the stationery equations i got: $$\begin{pmatrix} \dot x\\ \dot p\\ \end{pmatrix} =\begin{pmatrix} \frac{\alpha}{2}(\frac{1}{ct}+\frac{1}{cf})) & \beta(\frac{1}{ct}+\frac{1}{cf})^2 \\ -\frac{\alpha^2}{4\beta} & -\frac{\alpha}{2}(\frac{1}{ct}+\frac{1}{cf})) \\ \end{pmatrix} \begin{pmatrix} x\\ p\\ \end{pmatrix}+\begin{pmatrix} -\frac{Q}{2cf}\\ -\frac{\alpha Qct}{4\beta(cf + ct)}\\ \end{pmatrix} $$ $p$ is the lagrange multiplier. I stucked here. this matrix is singular. How can i solve a singular (2x2) hamiltanion matrix? can i solve it using the exponential matrix? I got really tired of searching. l'll be grateful if any body give me a push? Thanks