Special Right-hand Side Research Articles

On a Linear Differential Game of Pursuit with Integral Constraints in ℓ2

In this paper, we study the stability, controllability, and differential game of pursuit for an infinite system of linear ODEs in ℓ2. The system we consider has a special right-hand side, which is not diagonal and serves as a toy model for controllable system of infinitely many interacting points. We impose integral constraints on the control parameters. We obtain criteria for stability and null controllability of the system. Further, we construct a strategy for the pursuer that guarantees completion of the pursuit problem for the differential game. To prove controllability we use the so called Gramian operators.

The nonlocal solvability conditions for a system of quasilinear equations of the first order special right-hand sides

The Cauchy problem for a system of first-order quasilinear equations with special right-hand sides is considered. The study of solvability of this system in the original coordinates is based on the method of additional argument. It is proved that the local solution of such system exists and that its smoothness is not lower than the smoothness of the initial conditions. For system of two equations non-local solutions are considered that are continued by finite number of steps from the local solution. Sufficient conditions for the existence of such non-local solution are derived. The proof of the non-local resolvability of the system relies on original global estimates.

Traveling-Wave Solutions of the Kolmogorov–Petrovskii–Piskunov Equation

We consider quasi-stationary solutions of a problem without initial conditions for the Kolmogorov–Petrovskii–Piskunov (KPP) equation, which is a quasilinear parabolic one arising in the modeling of certain reaction–diffusion processes in the theory of combustion, mathematical biology, and other areas of natural sciences. A new efficiently numerically implementable analytical representation is constructed for self-similar plane traveling-wave solutions of the KPP equation with a special right-hand side. Sufficient conditions for an auxiliary function involved in this representation to be analytical for all values of its argument, including the endpoints, are obtained. Numerical results are obtained for model examples.

Условия нелокальной разрешимости задачи Коши для системы дифференциальных уравнений в частных производных первого порядка с правыми частями специального вида

Условия нелокальной разрешимости задачи Коши для системы дифференциальных уравнений в частных производных первого порядка с правыми частями специального вида

On the solvability of the characteristic Dirichlet problem for linear degenerate parabolic equations

We investigate the classical solvability for some classes of linear, degenerate equations in divergence form with prescribed Dirichlet data. Since the boundary value problem is characteristic according to Fichera on a part of the boundary, some typical nonlinear phenomena at these points are observed as boundary gradient blowups of the classical solutions in space directions. The regularity results explain the lack of hypoellipticity for special right-hand sides or boundary data for linear degenerate parabolic equations.

Gramian-Based Model Reduction for Descriptor Systems

Model reduction is of fundamental importance in many control applications. We consider model reduction methods for linear time-invariant continuous-time descriptor systems. These methods are based on the balanced truncation technique and closely related to the controllability and observability Gramians and Hankel singular values of descriptor systems. The Gramians can be computed by solving generalized Lyapunov equations with special right-hand sides. Numerical examples are given.

Stability and inertia theorems for generalized Lyapunov equations

We study generalized Lyapunov equations and present generalizations of Lyapunov stability theorems and some matrix inertia theorems for matrix pencils. We discuss applications of generalized Lyapunov equations with special right-hand sides in stability theory and control problems for descriptor systems.

Numerical solution and perturbation theory for generalized Lyapunov equations

We discuss the numerical solution and perturbation theory for the generalized continuous-time Lyapunov equation E * XA+ A * XE=− G with a singular matrix E. If this equation has a solution, it is not unique. We generalize a Bartels–Stewart method and a Hammarling method to compute a partial solution of the generalized Lyapunov equation with a special right-hand side. A spectral condition number is introduced and perturbation bounds for such an equation are presented. Numerical examples are given.

Asymptotics of solutions of a second-order linear differential system with a special right-hand side

Asymptotics of solutions of a second-order linear differential system with a special right-hand side

On a method of solving integral equations and its application to the problem of the bending of a plate with a cruciform inclusion

A scheme for a method of solving a certain integral equation is proposed consisting of the construction of a system of solutions of the characteristic equation with a complete system of special right-hand sides (polynomials, say) and writing the approximate solution of the initial equation in the form of a linear combination of the functions constructed. The scheme is realized in the case when the characteristic equation is reduced to a Mellin convolution and is solved exactly by the factorization method using the Mellin transform. In particular, such an approach provides an efficient solution of an integral equation with a fixed singularity in the kernel. A system of two integral equations obtained in the problem of the bending of a rectangular plate with a cruciform inclusion is solved as an illustration. In the case of branches of the inclusion of identical length the system is solved by the ordinary factorization method, in the case of branches of the inclusion of different length the factorization method is modified substantially.