Vol 54, No 6 (2018)

We study the stability and asymptotic stability of the zero solution of autonomous stochastic delay differential equations with discontinuous coefficients by the Lyapunov second method. For the equations under study, we obtain analogs of the Lyapunov stability theorem and the Barbashin–Krasovskii and Zubov asymptotic stability theorems for the zero solution.

We consider the Dirac operator on the interval [0, π] with an integrable potential P = (p_ij (x))_i,j=1² and strongly regular boundary conditions U. It is well known that for any integrable potential P the system {y_n}_n∈Z of root functions of the strongly regular operator L_P,U is a Riesz basis in the space H = L₂[0, π] × L₂[0, π]. We obtain estimates, uniform on every compact set of potentials, of the Riesz constants ||T||||T⁻¹||, where T is the operator of transition to an orthonormal basis.

We prove the existence, uniqueness, and continuous dependence on the initial data of the solutions of the Cauchy problem for stochastic evolution functional equations with random coefficients in Hilbert spaces. We propose a method for constructing an approximating sequence for the solution of the Cauchy problem and obtain an estimate for the rate of convergence to the exact solution.

We consider a mixed problem for the one-dimensional biwave equation with boundary conditions and a nonlocal integral condition. We prove the existence and uniqueness of the classical solution of the problem and obtain an analytic representation of the solution.

For linear autonomous completely regular differential-algebraic systems with commensurable delays in the state and control, we study the problem of constructing a state feedback that ensures a finite spectrum for the closed-loop system. We propose criteria for spectral reducibility and weak spectral reducibility whose proofs contain the synthesis schemes of appropriate controllers. Several illustrative examples are given.