Kirilich V. M.
Problem without initial conditions for a countable semilinear hyperbolic system of first-order equations
Ukr. Mat. Zh. - 2016. - 68, № 8. - pp. 1043-1055
We derive sufficient conditions for the solvability of the problem without initial conditions for a countable semilinear hyperbolic system of first-order equations and establish conditions for the classical solvability of the initial-boundary value problem for countable hyperbolic systems of semilinear equations of the first-order in a semistrip.
Problem of Optimal Control for a Semilinear Hyperbolic System of Equations of the First Order with Infinite Horizon Planning
Ukr. Mat. Zh. - 2015. - 67, № 2. - pp. 185–201
We establish necessary conditions for the optimality of smooth boundary and initial controls in a semilinear hyperbolic system of the first order. The problem adjoint to the original problem is a semilinear hyperbolic system without initial conditions. The suggested approach is based on the use of special variations of continuously differentiable controls. The existence of global generalized solutions for a semilinear first-order hyperbolic system in a domain unbounded in time is proved. The proof is based on the use of the Banach fixed-point theorem and a space metric with weight functions.
Ukr. Mat. Zh. - 2010. - 62, № 9. - pp. 1173–1199
Using the method of contracting mappings, we prove, for small values of time, the existence and uniqueness of a generalized Lipschitz solution of a mixed problem with unknown boundaries for a hyperbolic quasilinear system of first-order equations represented in terms of Riemann invariants with nonlocal (nonseparated and integral) boundary conditions.
Ukr. Mat. Zh. - 2009. - 61, № 8. - pp. 1148-1153
We investigate the problem of the existence of periodic solutions of the problem of oscillations of a diaphragm with friction and pulse feedback in the case where the times of pulse action are determined by a solution of the system.
Classical solvability of a problem with moving boundaries for a hyperbolic system of quasilinear equations
Ukr. Mat. Zh. - 2009. - 61, № 7. - pp. 867-891
Using the method of characteristics and the method of contracting mappings, we establish the local classical solvability of a problem for a hyperbolic system of quasilinear first-order equations with moving boundaries and nonlinear boundary conditions. Under additional assumptions on the monotonicity and sign constancy of initial data and a restriction on the growth of the right-hand sides of the system, we formulate sufficient conditions for the global classical solvability of the problem.
Ukr. Mat. Zh. - 1997. - 49, № 12. - pp. 1684–1689
The problem with unknown boundaries for a first-order semilinear hyperbolic system is studied in the case where the curve of definition of the initial conditions degenerates to a point. An existence and uniqueness theorem for a classical solution of the problem is proved for small t.
Ukr. Mat. Zh. - 1988. - 40, № 1. - pp. 121-124
Problems without initial conditions with integral restrictions for hyperbolic equations and systems on a line
Ukr. Mat. Zh. - 1983. - 35, № 6. - pp. 722-727