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Том 53, № 3 (2017)
- Год: 2017
- Статей: 13
- URL: https://journal-vniispk.ru/0012-2661/issue/view/9304
Ordinary Differential Equations
Hyperbolic annulus principle
Аннотация
We study a special class of diffeomorphisms of an annulus (the direct product of a ball in ℝk, k ≥ 2, by an m-dimensional torus). We prove the so-called annulus principle; i.e., we suggest a set of sufficient conditions under which each diffeomorphism in a given class has an m-dimensional expanding hyperbolic attractor.
281-301
Estimate for the amplitude of the limit cycle of the Liénard equation
Аннотация
We consider the nonlinear Liénard equation \(\ddot x\left( t \right) + f\left( x \right)\dot x\left( t \right) + g\left( x \right) = 0\). Liénard obtained sufficient conditions on the functions f(x) and g(x) under which this equation has a unique stable limit cycle. Under additional conditions, we prove a theorem that permits one to estimate the amplitude (the maximum value of x) of this limit cycle from above. The theorem is used to estimate the amplitude of the limit cycle of the van der Pol equation \(\ddot x\left( t \right) + \mu \left[ {{x^2}\left( t \right) - 1} \right]\dot x\left( t \right) + x\left( t \right) = 0\).
302-310
On the geometry of the reachability set of vector fields
Аннотация
We study the geometry of the reachability set of a family of vector fields on a C∞ manifold. We show that, for each real number T, the T-reachability set is a smooth submanifold of an orbit of codimension zero or one and that, on an arbitrary connected C∞ manifold of dimension greater than one, there exists a system of three vector fields such that each 0-reachability set coincides with the manifold itself.
311-316
Partial Differential Equations
Singular boundary integral equations of boundary value problems of the elasticity theory under supersonic transport loads
Аннотация
We consider a transport boundary value problem for an isotropic elastic medium bounded by a cylindrical surface of arbitrary cross-section and subjected to supersonic transport loads. We pose the corresponding hyperbolic boundary value problem and prove the uniqueness of the solution with regard to shock waves. To solve the problem, we use the method of generalized functions. In the space of generalized functions, we obtain the solution, perform its regularization, and construct a dynamic analog of the Somigliana formula and singular boundary equations solving the boundary value problem.
317-332
On the solvability of some boundary value problems for the inhomogeneous polyharmonic equation with boundary operators of the Hadamard type
Аннотация
We study the properties of fractional integro-differential operators. As an application, we analyze the solvability of some boundary value problems for the inhomogeneous polyharmonic equation in the unit ball. These problems generalize the Dirichlet and Neumann problems to the case of fractional boundary operators.
333-344
Control Theory
Boundary control of a hyperbolic system with one space variable
Аннотация
We consider a mixed problem in a half-strip for a hyperbolic system with one space variable and with constant coefficients. The control problem is to find boundary conditions ensuring that the system has a given state vector at a given instant of time. We study whether the problem is asymptotically solvable, i.e., whether there exists a sequence of boundary conditions such that the corresponding sequence of final state vectors uniformly converges to the given vector. We reduce the construction of a family of such sequences of boundary conditions with a function parameter to the solution of a Fredholm integral equation of the second kind and prove a sufficient condition for its unique solvability in terms of the problem data.
345-351
Degenerate abnormal trajectories in a sub-Riemannian problem with growth vector (2, 3, 5, 8)
Аннотация
We consider the nilpotent sub-Riemannian problem with growth vector (2, 3, 5, 8). We describe and study abnormal extremals orthogonal to the cube of the distribution. We analyze the geometric properties of a two-dimensional surface in the state space on which the corresponding abnormal trajectories define optimal synthesis.
352-365
366-381
Multiplicative stochastic systems: Optimization and analysis
Аннотация
We consider the H2/H∞-optimal control problem for a dynamical system defined by a linear stochastic Itô equation whose drift and diffusion coefficients linearly depend on the state vector, the control signal, and the external disturbance. The optimization is carried out under the a priori requirement of maximum possible damping of the harmful influence of external disturbances on the system operation. We present theorems on the solvability of matrix Riccati differential equations to which the original optimization problem is reduced.
382-397
Numerical Methods
Approximate solution of a parabolic equation with the use of a rational approximation to the operator exponential
Аннотация
For the abstract parabolic equation \(\dot x = Bx + bv\left( t \right)\) with an unbounded self-adjoint operator B, where b is a vector and v(t) is a scalar function, we suggest a solution method based on the evaluation of some rational function of the operator B. We obtain a priori estimates of the approximation error for the output function y(t) = <x(t), l>, where l is a given vector. The results of a numerical experiment for the inhomogeneous heat equation are presented.
398-408
Eigenvibrations of a bar with elastically attached load
Аннотация
We study the problem on the eigenvibrations of a bar with an elastically attached load. The problem is reduced to finding the eigenvalues and eigenfunctions of an ordinary secondorder differential problem with a spectral parameter nonlinearly occurring in the boundary condition at the load attachment point. We prove the existence of countably many simple positive eigenvalues of the differential problem. The problem is approximated by a grid scheme of the finite element method. We study the convergence and accuracy of the approximate solutions.
409-423
Short Communications
Remarks on the solvability of a convolution integral equation on a finite interval
Аннотация
We present some results on the solvability of an integral equation of the second kind with a difference kernel on a finite interval, construct a counterexample to an assertion, earlier believed to have been proved, on the solvability of this equation, and pose an open problem.
424-428
An estimate of the norm of the Cauchy operator for linear differential equations
Аннотация
We obtain a necessary and sufficient condition for the norm of the exponential of a linear operator on a Banach space not to exceed the exponential of its spectral radius. The condition is stated in terms of the logarithmic norm of the original operator.
429-432