Synthesis of minimal linear stabilizers

We consider the stabilizer synthesis problem for linear stationary control dynamical systems; attention is mainly focused on the investigation of possibilities to reduce the order of such stabilizers. The problem is considered in two settings. The first setting deals with the construction of stabilizers of given order (in particular, the minimum possible order); no conditions are imposed on the spectrum of the closed system. For scalar (single-input-single-output) control systems, we suggest an approach to the solution of this problem and obtain necessary and sufficient conditions for the existence of a stabilizer of a given order. The second problem is that of synthesizing a stabilizer of minimum order with given dynamic properties (a given spectrum or a given distribution of the spectrum of the closed system, in particular, with a guaranteed convergence rate of the closed system). For this problem, we suggest two approaches that permit one to obtain an upper bound for the dimension of such stabilizers.