Critical Problem Of Optimal Stabilization Without Control Constraints

The article analyzes the linear-quadratic optimal stabilization problem in the so-called "critical case", namely, the situation is considered when the spectrum of the system matrix contains purely imaginary eigenvalues or when the standard conditions of positive definiteness of the weight matrices of the quality functional are violated. Methods for regularizing critical problems by perturbing the system matrices and the functional are investigated, and algorithms for decomposing multidimensional problems into a set of one-dimensional canonical systems are proposed. The results are of practical importance for constructing optimal synthesis in various engineering and economic systems, in particular, the results can be used for stabilizing unmanned aerial vehicles, robotic complexes and intelligent power grids.