How to solve the integral of controllability Gramian? If $A$ and $P$ are square matrices, how to solve the following integral?
$$\int_{t_1}^{t} e^{A^T\tau}Pe^{A\tau}d\tau$$
I know it is similar to the controllability Gramian where it can be solved using the Lyapunov equation. But in my case the integration interval is $[t_1,t]$ instead of $[0,\infty]$ which makes me confused.
Edited:
Matrix $P$ can be decomposed as $P=R^TR$ and $A$ is a symmetric matrix.
Is there any closed-form analytical solution to the problem with the condition mentioned above?
 A: There is no closed-form expression in general for this integral. So, you will have to solve it numerically. There are different ways.
One way to that is with a software (like Matlab) or a programming language (like Python) that can do integration symbolically where A and P are given real matrices (non symbolic).
Another possibility would be to define the matrix-valued differential equation
$$\dot{X}(t)=-A^TX(t)-X(t)A+P$$ with $X(t_1)=0$.
An explicit solution for that differential equation is given by
$$X(t)=e^{-A^T(t-t_1)}X(t_1)e^{-A(t-t_1)}+\int_{t_1}^te^{-A^T(t-s)}Pe^{-A(t-s)}ds.$$
Since $X(t_1)=0$, we get that
$$X(t)=\int_{t_1}^te^{-A^T(t-s)}Pe^{-A(t-s)}ds.$$
Then, pre- and post-multiplying $X(t)$ by $e^{A^Tt}$ and $e^{At}$, respectively, yields
$$\int_{t_1}^te^{A^Ts}Pe^{As}ds=e^{A^Tt}X(t)e^{At}.$$
Therefore, you can solve for $X$ numerically over the interval $[t_1,t]$ from which you can compute the desired integral from the above expression. For more information about Lyapunov differential equations, see the book by Gajic & Qureshi, "Lyapunov Matrix Equation in System Stability and Control".
