Properties of Optimal Sets of External Practical Weak Stability of Differential Inclusions

On practical stability of differential inclusions using Lyapunov functions

“…It means that for arbitrary x 0 ∈ G * any suitable solution of differential inclusion (8) belongs to Φ(t) for all t ∈ [t 0 , T ]. Under the conditions of this section G * is the convex compact set [6,12]. Now we prove the following result.…”

“…where x ∈ R n is an n-dimensional vector of phase coordinate, A(t) is a continuous n × n-matrix, H(t) is a continuous symmetric positive definite n × n-matrix. We denote by Θ(t, s) a fundamental matrix of the linear system This inclusion implies 0 ∈ intG * [6,12]. Here G * is the maximum set of initial conditions under the phase constraints Φ(t), t ∈ [t 0 , T ].…”

“…Sufficient conditions of practical stability have been obtained for different types of practical stability, in some cases proved the necessary conditions. In [6,7,8,12], the concept that studies the properties of maximum sets of initial conditions has been proposed and the effective numerical methods for different practical problems have been developed (for instance, for the charge beams optimization problem). This approach was effective in studying the practical stability problems of differential inclusions solutions.…”

“…This approach was effective in studying the practical stability problems of differential inclusions solutions. In [6,12] the topological properties of optimal set of initial conditions (compactness, boundary and interior properties) have been studied both for strong and weak practical stability. In the case of linear differential inclusions and convex phase…”