Differential Equations最新文献

Abstract

The paper discusses the development of a method for constructing asymptotic formulas as(xto infty ) for the fundamental solution system of two-termsingular symmetric differential equations of odd order with coefficients in a broad class offunctions that allow oscillation (with relaxed regularity conditions that do not satisfy the classicalTitchmarsh–Levitan regularity conditions). Using the example of a third-order binomial equation(({i}/{2})bigl [(p(x)y^{prime })^{prime prime }+(p(x)y^{prime prime })^{prime }bigr ] +q(x)y =lambda y), the asymptotics of solutions inthe case of various behavior of the coefficients (q(x)) and(h(x)=-1+{1}big /{sqrt {p(x)}}) is studied. New asymptoticformulas are obtained for the case in which (h(x) notin L_1[1,infty ) ).

Abstract

For the inhomogeneous string vibration equation in a half-strip, we consider a problemperiodic in the spatial variable and a mixed problem. The solutions of these problems in the formof finite sums are obtained by quadratures. When solving these problems, we use thecharacteristic rectangle identity, Riemann invariants, and the method of characteristics.

Abstract

The paper considers the problem of controlling processes whose mathematical model is an initial–boundary value problem for a pseudohyperbolic linear differential equation of high order in the spatial variable and second order in the time variable. The pseudohyperbolic equation is a generalization of the ordinary hyperbolic equation typical in vibration theory. As examples, we consider models of vibrations of moving elastic materials. For the model problems, an energy identity is established and conditions for the uniqueness of a solution are formulated. As an optimization problem, we consider the problem of controlling the right-hand side so as to minimize a quadratic integral functional that evaluates the proximity of the solution to the objective function. From the original functional, a transition is made to a majorant functional, for which the corresponding upper bound is established. An explicit expression for the gradient of this functional is obtained, and adjoint initial–boundary value problems are derived.

Abstract

We study the local dynamics of the delay logistic equation with an additional feedbackcontaining a large delay. Critical cases in the problem of stability of the zero equilibrium state areidentified, and it is shown that they are infinite-dimensional. The well-known methods forstudying local dynamics based on the theory of invariant integral manifolds and normal forms donot apply here. The methods of infinite-dimensional normalization proposed by the author areused and developed. As the main results, special nonlinear boundary value problems of parabolictype are constructed, which play the role of normal forms. They determine the leading terms ofthe asymptotic expansions of solutions of the original equation and are called quasinormal forms.

Abstract

The paper addresses the consensus problem (i.e., the agreement of state vectors) for amultiagent system consisting of identical linear agents. The study focuses on the case where thereis no communication between agents, meaning there is no exchange of information, and agentcontrol is achieved through the agents’ own sensors, providing incomplete information about thestate vector of the agent and its neighbors, with the information possibly being noisy. To solvethis problem, a linear protocol based on observer data for systems under uncertainty is proposed.Cascade observers based on the “super-twisting” method are suggested as such observers. Sufficientconditions are obtained for the existence of a controller where the observation error converges tozero under bounded disturbances. An example illustrating the proposed approach is provided.

Abstract

The paper deals with proving the weak solvability of an initial–boundary value problem forthe modified Kelvin–Voigt model taking into account memory along the trajectories of motion offluid particles. To this end, we consider an approximation problem whose solvability is establishedwith the use of the Leray–Schauder fixed point theorem. Then, based on a priori estimates, weshow that the sequence of solutions of the approximation problem has a subsequence that weaklyconverges to the solution of the original problem as the approximation parameter tends to zero.

Abstract

The paper considers two weakly singular Fredholm boundary integral equations of the firstkind to each of which the three-dimensional Helmholtz transmission problem can be reduced. Theproperties of these equations are studied on the spectra, where they are ill posed. For the firstequation, it is shown that its solution, if it exists on the spectrum, allows finding a solution of thetransmission problem. The second equation in this case always has infinitely many solutions, withonly one of them giving a solution of the transmission problem. The interpolation method forfinding approximate solutions of the integral equations and the transmission problem in questionis discussed.

Abstract

We consider the spectral problem for the Dirac operator with arbitrary two-point boundaryconditions and any square integrable potential (V). Necessary andsufficient conditions for an entire function to be the characteristic determinant of such an operatorare established. In the case of irregular boundary conditions, conditions are found under which aset of complex numbers is the spectrum of the problem under consideration.

Abstract

We consider the regularization of classical optimality conditions—the Lagrangeprinciple and the Pontryagin maximum principle—in a convex optimal control problemwith an operator equality constraint and functional inequality constraints. The controlled systemis specified by a linear functional–operator equation of the second kind of general form in thespace (L^m_2 ), and the main operator on the right-hand side ofthe equation is assumed to be quasinilpotent. The objective functional of the problem is onlyconvex (perhaps not strongly convex). Obtaining regularized classical optimality conditions isbased on the dual regularization method. In this case, two regularization parameters are used, oneof which is “responsible” for the regularization of the dual problem, and the other is contained inthe strongly convex regularizing Tikhonov addition to the objective functional of the originalproblem, thereby ensuring the well-posedness of the problem of minimizing the Lagrange function.The main purpose of the regularized Lagrange principle and Pontryagin maximum principle is thestable generation of minimizing approximate solutions in the sense of J. Warga. The regularizedclassical optimality conditions

1. 1.

   Are formulated as existence theorems for minimizing approximate solutions in the originalproblem with a simultaneous constructive representation of these solutions.

2. 2.

   Are expressed in terms of regular classical Lagrange and Hamilton–Pontryagin functions.

3. 3.

   “Overcome” the properties of the ill-posedness of the classical optimality conditions andprovide regularizing algorithms for solving optimization problems.

Based on the perturbation method, an important property of the regularizedclassical optimality conditions obtained in the work is discussed in sufficient detail; namely, “in thelimit” they lead to their classical counterparts. As an application of the general results obtained inthe paper, a specific example of an optimal control problem associated with an integro-differentialequation of the transport equation type is considered, a special case of which is a certain inversefinal observation problem.

Abstract

For a singularly perturbed reaction–diffusion equation, we study the structure of theinternal transition layer in the case of a balanced reaction with a weak discontinuity. Theexistence of solutions with an internal transition layer (contrast structures) is proved, the questionof their stability is investigated, and asymptotic approximations to solutions of this type areobtained. It is shown that in the case of reaction balance, the presence of even a weak(asymptotically small) reaction discontinuity can lead to the formation of contrast structures offinite size, both stable and unstable.