The morphology of the floodplain–channel complex of the Moscow River and its development history before the beginning of intensive human intervention in its functioning are considered. The Moscow ...River valley reflects the lithological structure of the river basin in its various parts, as well as the relationship with the morphostructural plan of the upper and lower parts of the basin. According to the general morphology and floodplain–channel complex, the entire river valley can be divided into several morphologically homogeneous sections: Mozhaisk and Tuchkovo areas in the upper reaches, Zvenigorod–Moscow in the middle reaches, and an area with different widths of floodplains in the lower reaches. Each area is dominated by its own relief of floodplain–channel complexes: large macromeanders, the embedded nonfloodplain part, again macromeanders, and a set of different forms of floodplain relief in the alternating valley narrowing and widening in the lower area. The Late Glacial and Holocene history of the river was reconstructed based on the floodplain relief features in different parts of the valley and based on earlier studies. The radiocarbon dating of floodplain (oxbow) deposits recovered from different morphological sections of the floodplain–channel complexes of the river helped to reconstruct the natural development stages of the river valley during these periods: high water content and runoff coefficient in the Late Glacial, low water content in the early Holocene, floodplain rivers with many channels in the Late Holocene, and the current stage of active interaction of the natural and anthropogenic valley- and riverbed-forming processes. It is obvious that the first three stages ultimately served as a natural basis which over the past hundreds of years has been actively influenced by human activity. Meanwhile, traces of micromeanders found in the floodplain only in the lower reaches, marking a certain stage at the beginning of the sub-Atlantic period of the Holocene, have not yet been explained in the evolutionary series of the valley, floodplain, and riverbed of the Moscow River.
This paper continues the author’s research on the problem of preserving the solvability of controlled operator equations. As a preliminary result (which is of independent interest) for a general ...operator
acting on an arbitrary Banach space
, new theorems on the existence and uniqueness of a fixed point are obtained. Here the well-known concept of the cone norm is used:
, where
is, generally speaking, another Banach space semi-ordered by the cone
. These theorems are based on the assumption that the operator analog of the local Lipschitz condition with respect to the cone norm
is satisfied and generalize the result by A.V. Kalinin and S.F. Morozov (
,
). The role of an analog of the Lipschitz constant on a given bounded set
is played by a bounded linear operator
, depending on this set, with spectral radius
. In addition, M.A. Krasnosel’skii’s lemmas on the equivalent norm are used. Based on the statements obtained, we prove theorems on local and total (over the set of admissible controls) preservation of the solvability of the controlled operator equation
,
, where
is the control parameter from, generally speaking, an arbitrary set
. The abstract theory is illustrated by examples of a controlled nonlinear operator differential equation in a Banach space as well as a strongly nonlinear pseudoparabolic equation.
For the Cauchy problem associated with a controlled semilinear evolution equation with an unbounded maximal monotone operator in a Hilbert space, sufficient conditions are obtained for exact ...controllability to a given final state. Here a generalization of the Browder–Minty theorem and results on the total global solvability of this equation obtained by the author earlier are used. As an example, a semilinear wave equation is considered.
An optimal control problem is investigated for an abstract semilinear differential equation of the first order in time in a Hilbert space with an unbounded operator and control involved linearly in ...the right-hand side. The cost functional is assumed to be additively separated with respect to state and control, with a rather general dependence on the state. For this problem, the existence of an optimal control is proved and the properties of the set of optimal controls are established. The author’s previous results on the total preservation of unique global solvability (totally global solvability) and on solution estimation for such equations are developed in the context of the nonlinearity of the equation under study. The indicated estimate is found important for the present study. A nonlinear heat equation and a nonlinear wave equation are considered as examples.
The paper is a continuation of author’s research on the existence of an ε-equilibrium in the sense of piecewise program strategies in two-person zero-sum games associated with a nonlinear ...nonautonomous controlled differential equation in a Banach space and a general cost functional. As in the preceding paper on this subject, the main result consists in sufficient conditions for an ε-equilibrium. The difference is that this time we investigate a game without discriminating the players or fixing a Volterra chain. The results are illustrated as applied to a game associated with a nonlinear pseudoparabolic partial differential equation governing the evolution of an electric field in a semiconductor.
The paper studies the problem of optimizing the lowest coefficient understood as a function with values in a Banach space, which enters linearly into an abstract semilinear pseudoparabolic ...evolutionary differential equation in a Banach space. For this problem, an existence theorem for an optimal control is proved. Due to the nonlinearity of the equation under study, the author uses his previous results on the total preservation of the unique global solvability (on the totally global solvability) and on the estimation of solutions for similar equations. This estimate turns out to be significant in the course of the study. As an example, the Oskolkov’s hydrodynamic system of equations is considered.
The subject of the paper is finite-dimensional concave games, i.e., noncooperative
-person games with objective functionals concave in “their own” variables. For such games, we investigate the ...problem of designing iterative algorithms of searching for a Nash equilibrium with convergence guaranteed without requirements concerning objective functionals such as smoothness and convexity in “alien” variables or any other similar hypotheses (in the sense of weak convexity, quasiconvexity, and so on). In fact, we prove some assertion similar to the theorem on convergence of an
-Fejér iterative process for the case in which the operator preserves a finite-dimensional compact set and the closeness to the objective set is measured with the help of an arbitrary continuous function. Then, based on this assertion, we generalize and develop an approach to searching for a Nash equilibrium in concave games formerly suggested by the present author. The latter approach can be regarded as a cross between the relaxation algorithm and the Hooke–Jeeves method of configurations (but taking into account the specific nature of the discrepancy function to be minimized). In addition, we present and discuss results of numerical experiments.
We prove the existence of a Stackelberg equilibrium (in the style of M.S. Nikol’skii) for a nonlinear Volterra functional operator equation controlled by two players with the help of ...finite-dimensional program controls with integral objective functionals. On this way, we use our formerly proved results on the continuous dependence of the state and functionals on finite-dimensional controls and also the classical Weierstrass theorem. The property of being a singleton for the minimizer set of the first player is proved by M.S. Nikol’skii’s scheme applied earlier to a linear ordinary differential equation.
The paper deals with obtaining sufficient conditions for existence of an
-equilibrium in the sense of piecewise program strategies in zero-sum games associated with a time-varying nonlinear ...controlled differential equation in a Banach space and a cost functional of a sufficiently general form. The concept of piecewise program strategies in such a game is defined on the base of a concept of Volterra chain for the operator on the right-hand side in the corresponding integral equation controlled by the opponent players and according to a given partition of the time interval. As an example we consider the game associated with a nonlinear pseudoparabolic partial differential equation governing the evolution of electric field in a semiconductor.
We analyse the
p
- and
hp
-versions of the virtual element method (VEM) for the Stokes problem on polygonal domains. The key tool in the analysis is the existence of a bijection between Poisson-like ...and Stokes-like VE spaces for the velocities. This allows us to re-interpret the standard VEM for Stokes as a VEM, where the test and trial discrete velocities are sought in Poisson-like VE spaces. The upside of this fact is that we inherit from Beirão da Veiga et al. (Numer. Math. 138(3), 581–613,
2018
) an explicit analysis of best interpolation results in VE spaces, as well as stabilization estimates that are explicit in terms of the degree of accuracy
p
of the method. We prove exponential convergence of the
hp
-VEM for Stokes problems with regular right-hand sides. We corroborate the theoretical estimates with numerical tests for both the
p
- and
hp
-versions of the method.
PDF