# convexity

Consider
\begin{align}
\dot{x}=A(p)x+B(p)u
\end{align}
where $A(p)$ and $B(p)$ are linear in the fixed-parameter(but unknown)vector $p$.
The solution of the above problem is
\begin{align}
x(t,p)=e^{A(p)t}x_{0}+\int_0^te^{A(p)(t-\tau)}B(p)u(\tau)d\tau
\end{align}
Assuming that experimental data $x_e$ is available at time points $t_1,t_2,\cdots, t_N$,
the objective is to minimize
\begin{align}
f(p)=\sum_{i=1}^N\left|\left|e^{A(p)t_i}x_{0}+\int_0^{t_i}e^{A(p)(t_i-\tau)}B(p)u(\tau)d\tau-x_e(t_i)\right|\right|^2
\end{align}
Let
\begin{align}
f_i(p):=\left|\left|e^{A(p)t_i}x_{0}+\int_0^{t_i}e^{A(p)(t_i-\tau)}B(p)u(\tau)d\tau-x_e(t_i)\right|\right|^2
\end{align}
Then,
\begin{align}
f(p)=\sum_{i=1}^N f_i(p)
\end{align}
Since sum of convex functions is convex, to show that $f(p)$ is convex we have to show that
$f_i(p)$ are convex.
\\\\
\textbf{Question}: are $f_i(p)$ convex?