Sensitivities of stability constraints and their applications

Let the real polynomial
(a(s) = a_{0} + a_{1}s + ... + a_{n}s^{n}
with the coefficients being known differentiable functions
a_{k}(x)
be given and let the constraints
g_{i}(x) > 0
determine the strictly Hurwitz property of the polynomial
a(s)
. A simple and efficient method to calculate the derivatives
\partial g_{i}(x)/\partial x_{j}
is proposed. Then, the application of the method to the problem of stability of polynomials under coefficient perturbation by gradient optimization is discussed. Also, a theorem characterizing the stability region and the newly introduced regions of nondestabilizing perturbations is given.