The Markov power moment problem in problems of controllability and frequency extinguishing for the wave equation on a half-axis

In this paper necessary and sufficient conditions of null-controllability and approximate null-controllability are obtained for the wave equation on a half-axis. Controls solving these problems are found explicitly. Moreover, bang-bang controls solving the approximate null-controllability problem are constructed with the aid of solutions of a frequency extinguishing problem in the restricted band (−a,a) for this equation and the Markov power moment problem.     

Integral boundary problem in a layer

Translated from Matematicheskie Zametki, Vol. 53, No. 6, pp. 122–129, June, 1993.     

On stabilizability of evolution systems of partial differential equations on ℝ<Superscript><Emphasis Type="Italic">n</Emphasis></Superscript>×[0,+∞) by time-delayed feedback controlsby time-delayed feedback controls

In this work we obtain sufficient conditions for stabilizability by time-delayed feedback controls for the system

$\frac{{\partial w\left( {x,t} \right)}}{{\partial t}} = A(D_x )w(x,t) - A(D_x )u(x,t), x \in \mathbb{R}^n , t > h, $

where D _x =(-i?/?x ₁,...-i?/?x _n ), A(σ) and B(σ) are polynomial matrices (m×m), det B(σ)≡0 on ? ⁿ , w is an unknown function, u(·,t)=P(D _x )w(·,t?h) is a control, h>0. Here P is an infinite differentiable matrix (m×m), and the norm of each of its derivatives does not exceed Γ(1+|σ|²)^γ for some Γ, γ∈? depending on the order of this derivative. Necessary conditions for stabilizability of this system are also obtained. In particular, we study the stabilizability problem for the systems corresponding to the telegraph equation, the wave equation, the heat equation, the Schrödinger equation and another model equation. To obtain these results we use the Fourier transform method, the Lojasiewicz inequality and the Tarski—Seidenberg theorem and its corollaries. To choose an appropriate P and stabilize this system, we also prove some estimates of the real parts of the zeros of the quasipolynomial det {Iλ-A(σ)+B(σ)P(σ)e ^-hλ.

     

Test for propriety in a layer of a boundary problem with integral condition

A criterion for propriety in the class of bounded functions of a boundary problem in a layer IRⁿ × [0,T], consisting of the solution of an evolutional linear differential equation with constant (complex) coefficients under an additional integral condition containing an arbitrary differential operator (in the spatial variables) and weight function exp {at}, a is obtained. The dependence of the propriety of the given problem on the thickness T of the considered layer is studied.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 42, No. 11, pp. 1546–1551, November, 1990.     

Transformation Operators in the Problems of Controllability for the Degenerate Wave Equation with Variable Coefficients

Ukrainian Mathematical Journal - We study a control system $$ {w}_{tt}=\frac{1}{\rho }{\left(k{w}_x\right)}_x+\gamma w,w\left(0,t\right)=u(t),x\in \left(0,l\right),t\in \left(0,T\right), $$ in...     

Controllability problems for the string equation

For the string equation controlled by boundary conditions, we establish necessary and sufficient conditions for 0-and ε-controllability. The controls that solve such problems are found in explicit form. Moreover, using the Markov trigonometric moment problem, we construct bangbang controls that solve the problem of ε-controllability. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 59, No. 7, pp. 939–952, July, 2007.     

Nonlocal well-posed boundary-value problems in a layer

Translated from Matematicheskie Zametki, Vol. 48, No. 1, pp. 20–25, July, 1990.     

Nonlocal two-point boundary-value problems in a layer with differential operators in the boundary condition

We obtain criteria of well-posedness and strong well-posedness (smoothing of solutions as compared with given functions) of boundary-value problems for linear partial differential evolution equations in an infinite layer. The boundary condition is nonlocal and gives a relation between the values of the unknown function and its derivatives with respect to spatial coordinates on shifts of connected components of the boundary of the layer inside the layer.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 47, No. 8, pp. 1122–1128, August, 1995.