ON OPTIMAL SETS FOR PRACTICAL STABILITY OF LINEAR SYSTEMS OF DIFFERENTIAL EQUATIONS WITH H -CONVEX PHASE CONSTRAINTS

The article considers the construction of the maximal set of practical stability and practical semistability of a linear homogeneous system of differentialequations with H -convex phase constraints. Such problems arise in the mathematical modeling of systems of different nature and in control problemsfor such systems when the parameters describing the system state are bounded. Such problems are complex and require the development of methodsthat allow taking into account constraints on the system state. One of the main directions of the study of practical stability conditions is based on theproperties of the maximum set of initial conditions for which phase constraints are not violated in the dynamics. Such a set is called the optimal set ofpractical stability. Computational construction of such sets is a complex problem, the complexity of which increases significantly with increasingdimensionality of the phase space. For linear systems of differential equations with convex compact phase constraints, the optimal set of practicalstability is a convex compact set. Therefore, in this case, convex analysis methods are used to find the optimal set of practical stability. In particular,this allows us to describe the optimal set using the Minkowski function, the inverse Minkowski function, and the support function. The articleconsiders the case in which the phase constraints satisfy the H -convexity condition. Such a condition makes it possible to reduce the complexity ofthe representation of constraints and, as a result, simplify the methods for finding the maximum set of initial conditions. The article considers the caseof practical stability and practical semistability. The Minkowski function and the inverse Minkowski function of the optimal sets are obtained for thesecases. An example of phase constraints in the form of polygons is analyzed as an important case of H -convex sets. The theorem on the H -convexityof the optimal set of practical stability and the optimal set of practical semistability is established.