The stability of linear Volterra integral equations are discussed from the Lyapunov
Direct Method point of view. The characteristic feature of these equations is the fact that their solutions are not continuous functions. We consider the case when the solution class is the Bochner space Lp loc(0;1;X) of locally p-integrable functions.
The first boundary-value problem for the telegraph equation on an interval which
one end is mobile is considered. The method of the solution of such problem is developed and its exact solution is obtained. This method is based on integrated representation of solutions of the telegraph equation and generalization of a method of re°ections with reference to areas with variable border. Variants of movement of the mobile end with subsonic, sound and supersonic speeds, and also with arbitrary speed are considered.
The classification of the weak solutions to Dirichlet initial boundary value problem
associated with a linear degenerate parabolic equation has been studied. Some applications to associated optimal control problems in coeffcients are discussed.
The third boundary-value problem for the telegraph equation in semi-bounded domain is considered. The solution of this problem in quadratures is obtained. Construction of the exact solution to this problem is based on application of the method of extensions and on development of the method of integral representation for rather wide class of solutions to the telegraph equation.
The spatial model of cinematical interaction of two spatial whirlwinds of one direction, but different with respect to the intensity and rotation is considered.
In this paper we study an optimal control problem for a nonlinear elliptic variational inequality with generalized solenoidal coefficients which we adopt as controls in L°°(fi). We prove the existence of optimal solution of the stated problem.
The sufficient optimality conditions of an optimal control by the vibrosystem which is a more compact machine of rolling type are obtained. Using the ideas of penalty method and method of local variations, the numeral solution of optimal control problem is presented for the vibrosystem with two debalances.
We consider the traffic flow models in vector-valued optimization statement, where the flow is controlled on the edges of network. We study the topological properties of the set of all admissible pairs to the problem. The existence of efficient solutions of vector optimization problem for traffic flow on network are proved.
We study the inverse problems which can not be solved in the classical framework: Krylov inverse problem, early diagnostics of a rotor unbalance, the most probable solution. For obtaining the steady solutions of these problems some algorithms based on the method of Tikhonov regularization are offered. Krylov inverse problem in various statements has been considered and numerical calculation on real measurements has been executed. Non-standard statements of inverse problems extend of regularization method possibilities.
On the basis of known potential symmetry the finite nonlocal integro-differential transformation leaving invariant the nonlinear telegraph equation is constructed. The algorithms generating its solutions are obtained. New solutions are present there among generated. Equations connected with the given telegraph equation by means of potential system are received. Lie symmetries for them are investigated and exact solutions are constructed. It is shown, that potential symmetries are a special case of nonlocal symmetries - the invariance under finite Lie-Baklund transformations depending on integral variable. The characteristic equations corresponding to the potential symmetry of the telegraph equation are deduced. They define the nonlocal symmetries of equations connected by means of potential system. They also are used for searching of exact solutions of the specified equations.
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