0, We; shall investigate the stability of an equilibrium state of such electrical systems which may be considered as a connection of a passive linear n-port containing constant lumped elements, with a nonlinear purely resistive n-port.For this purpose we shall first establish certain relations between the properties of the admittance (impedance) matrix of a linear passive n-port and the behavior of its free transients.On applying the extended Popov's criteria (see [1]), there result certain assertions on the dependence of regimes accross common terminals on the initial state of the linear system.These results yield the desired conditions for stability of an equilibrium of the entire connection.1. Let us begin with some basic definitions.If G is an oriented graph with branches h u ..., h r which contains at least one loop, and L, R, S are constant symmetric positive semidefinite r x r matrices such that each element L ik , R ik , S ik is assigned to a pair (h i9 h k ), i, k = 1, ..., r of branches, then the quadruple N, = (G, L, R, 3) will be called a passive network on a current basis, or simply J-network.Let X be an r x m matrix, whose columns constitute a complete set of linearly independent real solutions of the equation d'x = 0, where a is the branch-node incidence matrix of G (see [2]); if X\L+ R + S) X is a positive definite matrix, Nj will be called J-regular.Let Nj be J-regular; if E(t) is a real r-vector function locally integrable on <0, oo) (or, more generally, a distribution vanishing on ( -oo, 0)), and J 0 , q 0 are real constant r-vectors such that the component E^t), J oi , q 0i of E(t), J 0 , q 0 , respectively, is as signed to the branch h i9 i = 1, ..., r, then the r-vector J(t) given by (1.1)where H 0 = 1 for t = 0, H 0 = 0 for t < 0, S 0 = H 0 and(1.2) Z(D) = ID + R + SD~l , will be called the solution of N, on a current basis corresponding to E(t), J 0 , q 0 .
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