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Abstract In this paper we consider a control problem with pointwise observation for a one-dimensional parabolic equation which arises in a mathematical model of climate control in industrial greenhouses. We study a general equation with variable diffusion coefficient, convection coefficient, and depletion potential. For the extremum problem of minimizing an integral weighted quadratic cost functional, we establish the existence and uniqueness of a minimizing function. We also study exact controllability and dense controllability of the problem. Necessary conditions for an extremum are obtained, and qualitative properties of the minimizing function are studied.

An Erratum to this paper has been published: https://doi.org/10.1134/S106456242207002X

For the third-order equation $$y^{\prime \prime \prime }(x)=p(x)|y(x)|^{k}\mathrm{sgn}\, y(x)$$ with a singular nonlinearity $$0

We consider a temperature control problem based on the heat equation with a convective term and a quadratic quality functional. We examine the structure of the set of attainable functions and establish controllability of the problem on various sets of admissible controls.

We study the behavior of singular solutions of the Emden-Fowler type equation y(n) = p(x, y, y’,..., y(n-1)|y|k sgn y, n > 2, with a regular (k > 1) or singular (0 1) or a solution that vanishes together with derivatives of order < n at some point or has a point of accumulation of zeros (for 0 < k < 1).

We consider a control problem for one-dimensional heat equation with quadratic cost functional. We prove the existence and uniqueness of a control function from a prescribed set, and study the structure of the set of accessible temperature functions. We also prove the dense controllability of the problem for some set of control functions.

We propose an asymptotic classification for solutions of the equation yIV(x) − po|y|k sgn y = 0, 0 0, and establish the existence of its periodic solutions.

We present results on the existence of oscillating solutions of specific form (“quasiperiodic solutions) for a nonlinear differential equation with power nonlinearity. For oscillating solutions to third-order equations of this type, we obtain an asymptotics of extremums, which is expressed through the asymptotics of extremums of a “quasiperiodic” solution. These results clarify the asymptotic formulas for the modules of extremums of solutions obtained by the author earlier.

For higher-order Emden–Fowler type equations with regular nonlinearity, the asymptotic behavior of their blow-up solutions is investigated. It is proved that for weakly superlinear differential equations, all such solutions have power-law asymptotic behavior.

For the equation $$\begin{aligned} y^{(n)}= p_0\,{\mid y \mid }^{k}\,\mathrm{sgn}\,y,\,\,\, n\ge 12,\,\,\, k>1,\,\,\, p_0>0, \end{aligned}$$ (1) the existence of positive solutions with nonpower-law asymptotic behavior is proved, namely $$\begin{aligned} y(x)=(x^*-x)^{-\frac{n}{k-1}}\ h(\log \,(x^*-x)), \ \ x\rightarrow x^*-0, \end{aligned}$$ (2) where h is a positive periodic non-constant function on \(\mathbb {R}\). To prove the existence, a useful modification of the Hopf bifurcation theorem is used.

While growing plants in industrial hothouses it needs to keep the temperature according to round-the-clock graph at the point of growth of plant located at the fixed height. Only small deviations are admitted. To obtain this it is possible to increase the temperature by heating the floor and to decrease the temperature by opening the ventilator windows at the ceiling. We propose and analyse the model based on the heat equation. Physical sense of this problem is that at one end of the infinitely thin rod of length l (the height of the hothouse) we keep during the time T the temperature ϕ(t) (control function), while at the other end we have the given heat flow (t). It requires to find the control function ϕ0(t) such that the temperature at the fixed point c be maximally closed to the given temperature z(t). For the estimation of the control quality we use a quadratic integral functional.

We consider the differential equation y (n) = p 0|y| k sgn y, where p 0 > 0 and 12 ≤ n ≤ 14, and prove that there exists k > 1 such that the equation has positive solutions with nonpower asymptotics y(x) = (x ∗ -x) -α h(ln (x ∗ -x)), x < x ∗ , where h is a nonconstant continuous positive periodic function. For n ≥ 2 we prove that such a solution exists, but with an oscillating periodic function h. Bibliography: 8 titles. Illustrations: 1 figure.

This paper is devoted to the problem of asymptotic equivalence of nth order differential equations with exponentially equivalent right-hand sides. With the help of this result asymptotic behaviour of solutions to nonhomogeneous differential equations is described

We study the problem of control over the temperature conditions in industrial hothouses. We consider a model based on the one-dimensional heat equation on a bounded interval with quadratic cost functional, examine the existence and uniqueness of a control function from a prescribed set, and study the structure of the set of accessible temperature functions.We study the problem of control over the temperature conditions in industrial hothouses. We consider a model based on the one-dimensional heat equation on a bounded interval with quadratic cost functional, examine the existence and uniqueness of a control function from a prescribed set, and study the structure of the set of accessible temperature functions.

We study the problem of control over the temperature conditions in industrial hothouses. We consider a model based on the one-dimensional parabolic equation on a bounded interval with quadratic cost functional, prove the existence and the uniqueness of a control function from a prescribed set, and study the structure of the set of accessible temperature functions.

The asymptotic behavior of all solutions to the fourth-order Emden– Fowler type differential equation with singular nonlinearity is investigated. The equation is transformed into a system on the three-dimensional sphere. By investigation of the asymptotic behavior of all possible trajectories of this system an asymptotic classification of all solutions to the equation is obtained.

For the equation $$\displaystyle{ y^{(n)} + p_{ 0}\,\vert y\vert ^{k}\ \mathrm{sign}\,y = 0, }$$ in the cases n = 3, 4, p0 > 0 or p0 1 and singular nonlinearity 0 < k < 1 asymptotic classification of all solutions are given.