  This paper studies the parameter tuning problem of positive linear systems
for optimizing their stability properties. We specifically show that, under
certain regularity assumptions on the parametrization, the problem of finding
the minimum-cost parameters that achieve a given requirement on a system norm
reduces to a \emph{geometric program}, which in turn can be exactly and
efficiently solved by convex optimization. The flexibility of geometric
programming allows the state, input, and output matrices of the system to
simultaneously depend on the parameters to be tuned. The class of system norms
under consideration includes the $H^2$ norm, $H^\infty$ norm, Hankel norm, and
Schatten $p$-norm. Also, the parameter tuning problem for ensuring the robust
stability of the system under structural uncertainties is shown to be solved by
geometric programming. The proposed optimization framework is further extended
to delayed positive linear systems, where it is shown that the parameter
tunning problem jointly constrained by the exponential decay rate, the
$\mathcal L^1$-gain, and the $\mathcal L^\infty$-gain can be solved by convex
optimization. The assumption on the system parametrization is stated in terms
of posynomial functions, which form a broad class of functions and thus allow
us to deal with various interesting positive linear systems arising from, for
example, dynamical buffer networks and epidemic spreading processes. We present
numerical examples to illustrate the effectiveness of the proposed optimization
framework.
