The paper presents an extension of discrete $2\mathrm{D}$ systems of Roesser and Fornasini–Marchesini types for the continuous case. Formulas for solution representation of linear continuous $2\mathrm{D}$ systems are obtained, including the possibility of expanding the methods of operational calculus by constructing corresponding integral transformations with respect to the spatial variable. Some linear-quadratic optimization problems are considered and the corresponding optimality conditions based on the operator setting are stated. Also, some form of the optimal control representation is presented.

We study sensitivity and asymptotic behavior of solutions to parametric convex semi-infinite programming problems with compact polyhedral index sets. We analyze properties of optimal solutions, Lagrange multipliers, and active sets as the parameter approaches a nominal value. Under standard smoothness, convexity, and constraint qualification assumptions, we characterize the limits and first-order derivatives of the associated primal and dual mappings.

.E.Slutsky's theorem, which establishes a relationship between the comparative statics indicators of this theory. In production theory (of the firm), the main problems are similar to those of consumption theory. Therefore, it is of interest to obtain a similar result within the theory of the firm. Similar results can be found in the academic literature across various approaches to formulating the problem of the firm. In this paper, we obtain relationships between the comparative statics indicators of the theory of the firm, which are analogous to Slutsky's theorem in consumer theory as interpreted by J.Hicks.

Various type of stability for discrete nonstationary linear systems of Volterra equations are investigated, the peculiarity of which is the dependence of each subsequent state on the entire prehistory of the process. The solutions of the system of linear Volterra equations are represented through Cauchy matrices. Necessary and sufficient conditions for stability and asymptotic stability in the usual and interval statement are established. The presented conditions are expressed in the form of certain restrictions on the norms of Cauchy matrices and on the spectral radius of a special matrix. The stability problems of a perturbed system are considered and estimates are obtained based on the norms of perturbation matrices, which preserve asymptotic stability.

There are proved stability and asymptotic stability criteria for linear rough path differential systems controlled by rough signals with Holder exponent greater than $1/3$. We obtain sufficient conditions of stability in probability and asymptotic stability in probability of stochastic linear rough path differential equations controlled by rough signals with Holder exponent greater than $1/3$.

In this paper we consider a vector optimization problem with an uncertain preference relation. The uncertainty arises from the fact that only a family $\mathcal{P}$ of preference relations is known, one of which is actually realized, while the concrete realization is unknown. As an optimal solution of this problem, we propose to consider such admissible solutions that provide the objective mapping with the guaranteed (weak, strong) maximum relative to the family $\mathcal{P}$, i.e., those that provide the objective mapping with (weak, strong) maximum for any concrete realization of the preference relation from $\mathcal{P}$. Under some assumptions regarding the family $\mathcal{P}$, a theorem on the scalarization of this vector optimization problem by means of difference sublinear functions is proved. Then this theorem is used for deriving first- and second-order necessary and sufficient conditions for admissible solutions that provide the objective mapping with the guaranteed $\mathcal{P}$-maximum. The necessary conditions are established for solutions that provide the guaranteed weak $\mathcal{P}$-maximum, while the sufficient conditions are established for solutions that provide the guaranteed strong $\mathcal{P}$-maximum.

The object of the investigation is a two-dimensional autonomous differential system whose right-hand sides are homogeneous polynomials of the fourth degree. The subject of the study is local phase portraits of this type of systems. The Frommer method used in the study is briefly described as one of the effective tools for splitting multiple equilibrium points, and a brief review of the known results for nonlinear systems is provided. The goal of the study is to obtain not only the local phase portraits of the systems under investigation but also the coefficient conditions under which such phase portraits exist. The relevance of the work is justified by the incompleteness of obtaining this type of coefficient criteria due to the difficulties in solving an algebraic equation of the fifth degree, which arises when finding possible finite measures of curvature of the $O^+$-curves of the system under study. In the main part of the paper, local phase portraits of the system under study are obtained for some cases and coefficient criteria for their implementation are indicated. The results of the research are summarized in the conclusion. The obtained results can be used in further research of two-dimensional autonomous systems.

Two classes of optimal observation problems arising in the context of optimal robust control of linear systems under inaccurate and incomplete state measurements are investigated. Linear optimal observation problems are associated with the construction of optimal guaranteed open-loop controls and provide estimates of information sets that describe states consistent with measurements obtained. A priori optimal observation problems are related to the synthesis of multiple-closable control strategies and the determination of the worst-case future information sets. These problems are formulated as specific multi-level mathematical programming problems, for which a recurrent procedure for their reduction to large-scale linear programming problems is developed. The proposed algorithms are oriented towards real-time applications in optimal guaranteed control systems.

The problem of optimizing the transient process in a linear singularly perturbed system with two groups of fast variables is considered, which consists in finding multidimensional control with minimal intensity. It is shown that to construct an approximate (in the asymptotic sense) solution to this problem, it is sufficient to solve three unperturbed optimal control problems of a smaller dimension.

A criterion for the relative controllability of a linear time-invariant system with constant delay as well as its discrete-time analog is established by means of a limiting transition from the discrete-time system to the continuous-time one. The differences between the governing equations for the continuous-time and discrete-time systems are identified.

We consider an autonomous system on the plane involving a parameter $\lambda $, commonly referred to as a small parameter, and with a singular point (either a center or a focus) at the origin. It is assumed that the system is isochronous at this singular point. We also assume that its isochrones – defined as the pre-images of radial lines under the transformation from the system under consideration to a system with uniform isochronicity – are independent of the parameter $\lambda $. Thus, in this paper, the parameter $\lambda $ may take on any real value and, in that sense, is not actually small.

We construct systems with quadratic nonlinearities in the right-hand side that satisfy these requirements and determine their isochrones. Phase portraits of the resulting isochronous foci and their isochrones are also constructed.

We investigate the finite spectrum assignment problem for linear autonomous neutral-type systems using output-feedback controllers with lumped delays. We establish a connection between the solvability of this problem in the class of state feedback controllers and in the class of output feedback controllers. Finally, we consider an observer-based controller and validate the separation principle within the framework of the finite spectrum assignment problem.

The analytical properties of solutions of the generalized Rabinovich–Fabrikant system related to models of nonlinear physics and possessing certain qualitative properties at the values of its parameters are investigated.

Different notions of pointwise controllability are introduced, compared and analyzed for a linear time-invariant singularly perturbed system with state delay. These notions differ in the dependence of the terminal time and control on the small parameter, the boundedness of the control, and the accuracy of hitting prescribed points. A hierarchy of the introduced notions is established. The structure of the pointwise controllability operator is revealed, which justifies the relationship between robust and $\mu$-uniform controllability. Sufficient decomposition conditions for robust controllability, expressed in terms of the controllability matrices of the slow and fast subsystems, are proved. The theoretical results are illustrated by examples.