Computing the State Transition Matrix of a modified system

Question

For the linear system $$ \dot{x} = A(t)x(t) + B(t)u(t),$$ if the transition matrix for $A(t)$ is $$ \phi_A(t,\tau)$$ then for what matrix $F(t)$ is $$ \phi_F(t,\tau) = \phi_A^T(-\tau,-t) ?$$ What I found was for the adjoint state equation $$ \dot{z}(t) = -A^T(t)z(t) \implies \phi_Z(t,\tau) = \phi_A^T(\tau,t).$$

I'm guessing I have to work around this...

Answer 1

Let $\psi(t,\tau) = \phi_A^T(-\tau,-t)$. Note that $\psi(t,\tau) = \phi_A^T(-t,-\tau)^{-1}$

\begin{eqnarray} {\partial \psi(t,\tau) \over \partial t} &=& -\phi_A^T(-t,-\tau)^{-1} (-{ \partial \phi_A^T(-t,-\tau)\over \partial t} ) \phi_A^T(-t,-\tau)^{-1} \\ &=& \phi_A^T(-\tau,-t) \phi_A^T(-t,-\tau) A^T(-t) \phi_A^T(-\tau,-t) \\ &=& A^T(-t) \phi_A^T(-\tau,-t) \\ &=& A^T(-t) \psi(t,\tau) \end{eqnarray}