Optimizing problem on controllable abundances for promoting the performance of the control systems

According to the difinition of the controllable abundances and the theorem on the relation among the controllable abundance, the solution set of the control laws, and the performance of the control systems, in my peper arXiv1705.08064(On Controllable Abundance Of Saturated-input Linear Discrete Systems), the following optimiztion problem on controllable abundances for promoting the performance of the control systems for the linear discrete-time systems $\varSigma(A(\alpha),B(\alpha))$ with the tenic parameters $\alpha$ is presented

              $\max_{\alpha\in\Phi}\:v_{c,N}(\alpha)$

              $s.t.\;\min\left\{ \left|\lambda_{i}(\alpha)\right|,i=\overline{1,n}\right\} \geq\lambda_{\textrm{min}}$

where $v_{c,N}(\alpha)$ is the control abundance of the systems the systems $\varSigma(A(\alpha),B(\alpha))$ ,   $\alpha\in\Phi$ is the technic parameter set to be determined and $\Phi$ is the corresponding parameter space, $\lambda_{i}(\alpha)(i=\overline{1,n})$ is the poles of the systems, $\lambda_{\textrm{min}}$ is the expected minimum modulus of poles.

In the above optimizing problem, the reason why the modulus of poles is restricted with the lower bound is that $\left|\lambda_{i}(\alpha)\right|\rightarrow0$ will lead to $v_{c,N}(\alpha)\rightarrow\infty$ and $\left|\lambda_{i}(\alpha)\right|<1$ will lead to $v_{c,\infty}(\alpha)\rightarrow\infty$ . Therefor, for the optimizing problem with $N\rightarrow\infty$ , $\lambda_{\textrm{min}}$ must be greater than 1.