Global solvability of singular semilinear differential equations and applications to nonlinear radio engineering
Abstract: The system of differential-algebraic equations which in a vector form has the representation as the semilinear differential-algebraic equation (DAE) d/dt [Ax(t)]+Bx(t)=f(t,x) with a singular characteristic operator pencil is considered. In particular, underdetermined and overdetermined systems of the differential-algebraic equations correspond to this system. The existence theorem of a global solution of the semilinear differential-algebraic equation is proved. In the case of the overdetermined system, the global solution is unique. The global solution of the underdetermined system is not unique and it depends on the functional parameter determining one of its components. The nonlinear right side of the DAE may not satisfy the constraints of the global Lipschitz condition type. This allows to use the theorem efficiently in real practical applications. Two problems for the models of radio engineering filters with nonlinear elements are researched and the restrictions which ensure the smooth evolution of states throughout an arbitrary large period of time are given. Analysis of the problems have shown that the requirements of the proved theorem are physically feasible. The block representations of the singular characteristic pencil and its components necessary to obtain the main results are considered. The extending solution method in terms of differential inequalities with Lyapunov and La Salle functions and the method of spectral projectors are used in the paper.
Keywords: singular pencil, regular block, differential-algebraic equation, global solution, electric circuit, nonlinear element.
Area: Applied Mathematics
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