Sufficient conditions for the existence of an extremum in a system of maximal degree of stability. I (Q1284308)

A sufficient condition is established for the optimality of control systems described by the equation \[ x^{(n)}(t)+ \sum_{i=1}^n a_ix^{(i-1)}(t)= k_0u(t). \] The optimized control has the form \[ u(t)=- \sum_{i=1}^m b_ix^{(i-1)}(t) \qquad (m<n-1). \] For optimal parameters \(b_i\) the stability degree \[ I_{0p}= -\min_{b_i} \max_j \text{Re }\lambda_i (b_i) \] is maximal, where \(\lambda_j\) are the zeros of the polynomial \[ \lambda^n+ \sum_{i=1}^n a_i \lambda^{i-1}+ k_0 \sum_{i=1}^m b_i \lambda^{i-1}. \]