Differential Equations最新文献

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.

Abstract

The relationships between the notion of generalized integral and the notions of reflectingfunction and Poincaré map (period map) for periodic differential systems are traced.The notion of generalized first integral is used to study questions of the existence and stability ofperiodic solutions of periodic differential systems and analyze the center–focus problem.

Abstract

New approaches are proposed in the problem of constructing equivalent scalar differentialequations for multidimensional nonlinear systems of control theory, as well as in the problem ofconstructing equivalent Hamiltonian systems for nonlinear Lurie equations (scalar differentialequations containing derivatives of only even orders). The conditions for the solvability of thecorresponding problems are studied, and new formulas for the transition to equivalent equationsand systems are proposed. For the Lurie equations, the proposed approaches are based on thetransition from the linear part to the normal forms of the corresponding Hamiltonian systems witha subsequent transformation of the resulting system. Calculation formulas and algorithms areobtained, and their efficiency is illustrated by examples.

Abstract

We consider the symmetries of partial differential equations based on the use ofdifferential-geometric and algebraic methods of the theory of dynamical control systems.

Abstract

We consider the problem of stabilization to zero under disturbance in terms ofa differential pursuit game. The dynamics is described by a nonlinear autonomous system ofdifferential equations. The set of control values of the pursuer is finite, and that of the evader(disturbance) is a compact set. The control objective, i.e., the pursuer’s goal, is to bring thetrajectory to any predetermined neighborhood of zero in finite time regardless of the disturbance.To construct the control, the pursuer knows only the state coordinates at some discrete times, andthe choice of the disturbance’s control is unknown. In the paper, we obtain conditions for theexistence of a neighborhood of zero from each point of which a capture occurs in the indicatedsense. A winning control is constructed constructively and has an additional property specified ina theorem. In addition, an estimate of the capture time sharp in some sense is produced.

Abstract

For a linear autonomous system of neutral type with commensurable delays, an algorithmis given for solving the modal controllability problem (in particular, the finite spectrumassignment problem), which provides a closed-loop system with a given characteristicquasipolynomial. A procedure for editing the finite part of the spectrum is proposed. A criterionfor exponential stabilization of the system under study is constructively justified. When thecriterion is met, the closed-loop system can be made exponentially stable according to theproposed spectral reduction algorithm. The obtained statements and spectrum assignmentalgorithms are illustrated with examples.

Abstract

Sufficient conditions are established for the boundedness of all solutions and their first twoderivatives of a third-order linear integro-differential equation of the Volterra type on the half-line.To this end, using a method proposed by the first author in 2006, first, we reduce the equationunder consideration to an equivalent system consisting of one first-order differential equation andone second-order Volterra integro-differential equation. Then a new generalized Lyapunovfunctional is proposed for this system, the nonnegativity of this functional on solutions of thissystem is proved, and an upper bound is given for the derivative of this functional via the originalfunctional. The resulting estimate is an integro-differential inequality whose solution gives anestimate of the functional.

Abstract

Direct and inverse problems for a model equation of mixed parabolic-hyperbolic type arestudied. In the direct problem, we consider a Tricomi-type problem for this equation with anoncharacteristic line of type change. The unknowns of the inverse problem are the variablecoefficients of the lower-order terms in the equation. To determine these coefficients, an integraloverdetermination condition is specified relative to the solution defined in the parabolic part of thedomain, and in the hyperbolic part, conditions are specified on the characteristics: on onecharacteristic it is the value of the normal derivative and on the other, the value of the functionitself. Theorems for the unique solvability of the posed problems in the sense of classical solutionare proved.

Abstract

This paper continues a series of papers in which simultaneous (2times 2 ) matrix solutions of two scalar evolution equations,which are analogs of time-dependent Schrödinger equations, were constructed. In theconstructions in the present paper, these equations correspond to the Hamiltonian system(H^{2+2+1} )—one of the representatives of the hierarchyof degenerations of the isomonodromic Garnier system. The mentioned hierarchy was described byH. Kimura in 1986. In terms of solutions of linear systems of differential equations in the methodof isomonodromic deformations, the consistency condition for which is the Hamiltonian equationsof the (H^{2+2+1} ) system, the constructed simultaneous matrixsolutions of analogs of time-dependent Schrödinger equations are written out explicitly inthis paper.