Stability of a set of trajectories of nonlinear dynamics (Q2377477)

The stability problem and the stability conditions for the set of stationary solutions for the system of a set of differential equations \[ D_HX=F(t,X), \quad X(t_0)=X_0\in K_c(\mathbb R^n) \] is stated in terms of the existence of a suitable matrix-valued function, where \(t_0\geq 0\), \(F\in C(\mathbb R_+ \times K_c(\mathbb R^n),K_c(\mathbb R^n))\), \(D_H X\) is the Hukuhara derivative of the set X and \(K_c(\mathbb R^n)\) represents all the compact convex subsets of \(\mathbb R^n\). In the analysis of such systems, the properties of the right-hand side of set differential equations under consideration and some geometric characteristics of X are naturally taken into account in the construction of elements of the matrix-valued Lyapunov function. The stability theory of set differential equations can be useful in the mathematical simulation of phenomena in real-world systems associated with beams of charged particles, wave packets, dissipative control, etc.