On the Chernous'ko Time-Optimal Problem for the Equation of Heat Conductivity in a Rod

Abstract

The time-optimal problem for the controllable equation of heat conductivity in a rod is considered. By means of the Fourier expansion, the problem reduced to a countable system of one-dimensional control systems with a combined constraint joining control parameters in one relation. In order to improve the time of a suboptimal control constructed by F.L. Chernous'ko, a method of grouping coupled terms of the Fourier expansion of a control function is applied, and a synthesis of the improved suboptimal control is obtained in an explicit form.

On the Stability and Null-Controllability of an Infinite System of Linear Differential Equations

“…Control problems in Banach or Hilbert spaces arise naturally in processes described by partial differential equations (see for example [ 1 , 3 , 7 , 8 , 11 , 13 , 15 , 16 , 19 , 22 ] and references therein). Sometimes it is useful to reduce the control problem for partial differential equations to infinite systems of ODEs [ 4 , 5 , 9 , 10 ]. Also, it is of independent interest to consider control systems governed by infinite system as models in Banach spaces.…”

“…Often it is useful to study finite dimensional approximations of the infinite system, such an approach is taken in [ 4 , 5 ]. The main difficulty is then to prove that the approximate solutions converge to a solution of the initial control problem.…”

On the stability and null-controllability of an infinite system of linear differential equations

“…Control problems in Banach or Hilbert spaces arise naturally in processes described by partial differential equations (see for example [2,6,7,10,12,14,15,19,22] and references therein). Sometimes it is useful to reduce the control problem for partial differential equations to infinite systems of ODEs [3,4,8,9]. Also, it is of independent interest to consider control systems governed by infinite system as models in Banach spaces.…”

“…Often it is useful to study finite dimensional approximations of the infinite system, such an approach is taken in [3,4]. The main difficulty is then to prove that the approximate solutions converge to a solution of the initial control problem.…”

On a linear differential game in the Hilbert space $\ell^2$

“…A motivation for the setup comes from control problems for evolutionary PDEs, where using suitable decomposition of the control problem (see for example [4,5,[8][9][10]) x i would be a Fourier coefficients of an unknown function, while u i and v i would be that of control parameters. Also, the setup is of independent interest as a controlled system in a Banach space (for works in this spirit see for example [6,13]).…”