convexity

## Primary tabs

# 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?

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