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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

One considers the differential equation $$y{\prime\prime\prime}(x) + p\bigl(x, y(x), y{\prime}(x), y{\prime\prime}(x)\bigr) |y(x)|^{k-1} y(x) = 0,$$ where k > 1, the function p(x, y 0 , y 1 , y 2) is continuous and satisfies the inequalities $$ 0 < p_* \le p(x, y_0, y_1, y_2) \le p^* < \infty,$$ as well as the Lipschitz condition with respect to the last three arguments. Uniform estimates are obtained for the moduli of the solutions with a common domain.

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.

Using the factorization of a linear differential operator, we obtain uniform estimates of positive solutions with a common domain of definition, the oscillation criterion for all solutions of higher-order quasilinear differential equations, and estimates of modules of all solutions with a common domain of definition to higher-order quasilinear differential inequalities.

We obtain uniform estimates for positive solutions of higher-order quasi-linear differential equations that have a common domain and depend on estimates for the coefficients of the equation but not on the coefficients themselves.

For a higher order quasilinear differential equation, the existence of uniform estimates for positive solutions with common domain of definition is proved; these estimates depend on the estimates for the coefficients of the equation and do not depend on the coefficients themselves.

We obtain uniform estimates for positive solutions with the same domain to the equation $$y^{(n)} + \sum\limits_{i = 0}^{n - 1} {a_i (x)y^{(i)} } + p(x)\left| y \right|^{k - 1} y = 0$$ of even order n with k > 1 and continuous functions p(x) > 0 and a i (x). In the case where a 0(x) ≡ ⋯ ≡ a n−1(x) ≡ 0, the uniform estimates obtained depend on p = inf p(x) > 0 and are independent of the function p(x) itself.

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.

The following differential equation is considered in this paper: $$y^{(n)} + p(x)\left| y \right|^{k - 1} y = 0, k > 1, p(x) \geqslant p_* > 0$$ . For positive solutions of this equation defined on the closed interval [a, b], the author obtains an upper estimate using a power function of (x − a) with coefficient depending only on n, k, and p.

One considers the differential inequality $$y^n + \sum\limits_{j = 0}^{n - 1} {a_j (x)y^{(j)} \geqslant p * \left| y \right|^k } ,$$ , where a j (x) are continuous functions, p* > 0, n ≥ 1, k > 1, and its special case $$r_n (x)\frac{d}{{dx}}\left( {...\frac{d}{{dx}}\left( {r_1 (x)\frac{d}{{dx}}(r_0 (x)y)} \right)...} \right) \geqslant \left| y \right|^k ,$$ , where all r j (x) are sufficiently smooth positive functions. Uniform estimates are obtained for solutions defined in the same domain.

Sufficient conditions are established for the existence of nonoscillatory solutions to a quasilinear ordinary differential equation of higher order. For the equation with a positive potential, a criterion is established for the existence of nonoscillatory solutions with nonzero limit at infinity. In the case of even order, a criterion is obtained for all solutions at infinity to be oscillatory.

The existence of uniform estimates for positive solutions with the same domain to the even-order differential equation $$y^{(n)} + \sum\limits_{i = 0}^{n - 1} {a_i (x)y^{(i)} } + p(x)\left| y \right|^{k - 1} y = 0$$ with k > 1 is proved. The estimates for solutions depend on those for the continuous coefficients p(x) > 0 and ai(x), not on the coefficients themselves.

We study asymptotic properties of a higher-order, nonlinear, Emden-Fowler-type differential equation.We investigate asymptotics of all possible solutions of the equation in the cases of regular and singular nonlinearity for n=3 , 4.We use the method of change of variables,which allows one to reduce the initial equation of order n to a dynamical system on the (n-1)-dimensional compact sphere.

The paper is devoted to the existence of oscillatory and non-oscillatory quasi-periodic, in some sense, solutions to a higher-order Emden-Fowler type differential equation.

For differential equation of the third order with power nonlinearity, uniform estimates of solutions with the same domain are obtained. The existence of solutions with prescribed domain is proved.

We study the asymptotic behavior of solutions of an equation of a special form (with a root nonlinearity) completely for the third-order equation and partly for the higher-order equations.

For the equation the existence of positive solutions with non-power asymptotic behavior is proved, namely where is an arbitrary point, h is a positive periodic non-constant function on R. To prove this result, the Hopf bifurcation theorem is used.