Optimal control in systems with distributed parameters of hyperbolic type

On an example of transmission of a constant current, problems of optimal control in systems with distributed parameters of hyperbolic type are considered. A solution of corresponding problems is brought up to double extremal relations. Examples of application of the general theory to concrete control problems are given.Translated from Dinamicheskie Sistemy, No. 7, pp. 71–78, 1988.     

Necessary and sufficient conditions for discrete and differential inclusions of elliptic type

This paper deals for the first time with the Dirichlet problem for discrete (PD), discrete approximation problem on a uniform grid and differential (PC) inclusions of elliptic type. In the form of Euler-Lagrange inclusion necessary and sufficient conditions for optimality are derived for the problems under consideration on the basis of new concepts of locally adjoint mappings. The results obtained are generalized to the multidimensional case with a second order elliptic operator.     

Existence results for impulsive hyperbolic differential inclusions

In this article we investigate the existence of solutions for second order impulsive hyperbolic differential inclusions in separable Banach spaces. By using suitable fixed point theorems, we study the case when the multi-valued map has convex and non-convex values.     

Existence theory for perturbed impulsive hyperbolic differential inclusions with variable times

In this paper, the existence of solutions for a perturbed impulsive hyperbolic differential inclusion with variable times is proved under the mixed generalized Lipschitz and Carathéodory's conditions.     

Existence theorems for hyperbolic differential inclusions in Banach algebras

In this paper, an existence theorem for a certain hyperbolic differential inclusions in Banach algebras is proved under the mixed Lipschitz and Carathéodory conditions. The existence of extremal solutions for Carathéodory as well as discontinuous hyperbolic differential inclusions is also proved under certain monotonicity conditions.     

On solutions of hyperbolic differential inclusions with nonconvex right-hand side

The existence of a generalized solution with continuous derivativesu _x ,u _y is proved for the differential inclusionu _xy F(x, y, u) with a nonconvex right-hand side satisfying the Lipschitz conditioninx, y, andu.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 47, No. 4, pp. 531–534, April, 1995.     

Two-dimensional state-space discrete models for hyperbolic partial differential equations

It is shown that the finite difference technique can be used to transform some important linear distributed processes described by partial differential equations into so-called Roesser discrete state-space models. Two illustrative examples of numerical solutions are given and compared with exact solutions.     

Partial differential equations with discrete and distributed state-dependent delays

This work is an attempt to treat partial differential equations with discrete (concentrated) state-dependent delay. The main idea is to approximate the discrete delay term by a sequence of distributed delay terms (all with state-dependent delays). We study local existence and long-time asymptotic behavior of solutions and prove that the model with distributed delay has a global attractor while the one with discrete delay possesses the trajectory attractor.     

Extremal solutions of Cauchy problems for abstract fractional differential equations

In this paper, we study the extremal solutions of Cauchy problems for abstract fractional differential equations. Some definitions such as L ¹-Lipschitz-like, L ¹-Carathéodory-like and L ¹-Chandrabhan-like are introduced. By virtue of the singular integral inequalities with several nonlinearities due to Medved’, the properties of solutions are given. By using a hybrid fixed point theorem due to Dhage, existence results for extremal solutions are established. Finally, we present an example to illustrate our main results.     

Some problems in the theory of differential pursuit games in systems with distributed parameters

A game with dynamics described by partial differential equations is considered. The equations of the players are additively represented on the right-hand side and are subject to integral or pointwise restrictions. The goal of the first player, who is informed of the instantaneous value of the control of his partner, is to bring the system into an unperturbed state. To solve the problem the decomposition method developed in [1] for a controlled system (with one player) is used. Three combinations of restrictions on the players are considered. In all cases the control of the first player is presented explicitly. The main complication, compared with the problem considered previously [1] is that this control consists of two terms estimated in different norms.     

Existence and general decay for nondissipative hyperbolic differential inclusions with acoustic/memory boundary conditions

In this paper, we prove the existence and general energy decay rate of global solution to the mixed problem for nondissipative multi‐valued hyperbolic differential inclusions with memory boundary conditions on a portion of the boundary and acoustic boundary conditions on the rest of it. For the existence of solutions, we prove the global existence of weak solution by using Galerkin's method and compactness arguments. For the energy decay rates, we first consider the general nonlinear case of h satisfying a smallness condition, and prove the general energy decay rate by using perturbed modified energy method. Then, we consider the linear case of h: and prove the general decay estimates of equivalent energy.     

The optimality principle for discrete and first order partial differential inclusions

Necessary and sufficient conditions for optimality are derived for the problems under consideration on the basis of the apparatus of locally conjugate mappings and local tents.     

Approximation and optimization of higher order discrete and differential inclusions

This paper is mainly concerned with the necessary and sufficient conditions of optimality for Cauchy problem of higher order discrete and differential inclusions. Applying optimality conditions of problems with geometric constraints, for arbitrary higher order (say s-order) discrete inclusions optimality conditions are formulated. Also some special transversality conditions, which are peculiar to problems including third order derivatives are formulated. Formulation of sufficient conditions both for convex and non-convex discrete and differential inclusions are based on the apparatus of locally adjoint mappings. Furthermore, an application of these results is demonstrated by solving the problems with third order linear discrete and differential inclusions.     

Duality and optimality for convex extremal problems described by discrete inclusions

This paper is devoted to the Hamiltonian approach for extremal problems concerning convex (multi-valued) mapping. The approach exploits the concept of a Hamiltonian function permitting simplified proofs and useful mathematical insights. Moreover it provides in a duality framework a common point ox view upon the methods used. by Rockafellar, (the theory of convex processes), Pshenichnyi (the conjugate transformation method) and CASS (the symmetric duality scheme) to construct optimality conditions. The theory is used to develop a complete characterization of optimal solutions for multi-period convex programming problems.     

Boundary-value problems for systems of first-order partial differential inclusions

((Without abstract.))