A. A. Davydov

Qualitative Theory of Control Systems

Implicit first-order differential equations Local controllability of a system Structural stability of control systems Attainability boundary of a multidimensional system References.

Smooth normal forms of folded elementary singular points

In this work the smooth and topological normal forms of the first-order implicit differential equations in the plane near its folded degenerate elementary singular point are found, and thereby, the smooth and topological classification of folded elementary singular points of these equations is completed. It is proved, for example, that the equation is equivalent near its folded singular point of saddlenode type to some equation (dy/dx+x2+Ax3)2=y, whereA is a real number, in some appropriate smooth coordinate system in the plane with the origin at this point. The numberA is the parameter of the normal form.

Simulating flows of incompressible and weakly compressible fluids on multicore hybrid computer systems

A logically simple algorithm based on explicit schemes for modeling flows of incompressible and weakly compressible fluids is considered. The hyperbolic variant of the quasi-gas dynamic system of equations is used as a mathematical model. An ingenious computer cluster based on NVIDIA GPUs is used for the computations.

Singularity theory approach to time averaged optimization

Using singularity theory tools we study the time averaged problem for control systems and profit densities on the circle. We show that the optimal strategy always can be selected within periodic cyclic motions and stationary strategies. When the problem depends additionally on a one dimensional parameter, we analyze generic transitions between these two types of strategies under change of the parameter and classify all generic singularities of the averaged profit as a function of the parameter.

Structural stability of simplest dynamical inequalities

The structural stability of families of orbits is proved for the simplest generic smooth dynamical inequality in the plane with bounded complement of the domain of complete controllability. Typical singularities of the boundaries of nonlocal transitivity zones for such inequalities are found. The stability of these singularities under small perturbations of the generic inequality is proved.

Existence and uniqueness of a stationary distribution of a biological community

For an integral equation describing stationary distributions of a biological community, we point out conditions on its parameters under which this equation has a unique solution that satisfies necessary requirements for such a distribution.

Local controllability bifurcations in families of bidynamical systems on the plane

We classify generic local controllability bifurcations in two-parameter families of bidynamical systems on the plane at points with nonzero velocity indicatrix.

Uniqueness of a Cycle with Discounting That Is Optimal with Respect to the Average Time Profit

For cyclic processes modeled by periodic motions of a continuous control system on a circle, we prove the uniqueness of a cycle maximizing the average one-period time profit in the case of discounting provided that the minimum and maximum velocities of the system coincide at some points only and the profit density is a differentiable function with a finite number of critical points. The uniqueness theorem is an analog of Arnold’s theorem on the uniqueness of such a cycle in the case when the profit gathered along the cycle is not discounted.

Normal Forms in One-Parametric Optimization

We deal with one-parametric families of optimization problems in finite dimension with a finite number of (in-)equality constraints. Its set of generalized critical points can be classified according to five types (cf. ). In this paper the corresponding normal forms of the problem near each of these five types of singular points are presented.

Learning Through Fictitious Play in a Game-Theoretic Model of Natural Resource Consumption

Understanding the emergence of sustainable behavior in dynamic models of resource consumption is essential for control of coupled human and natural systems. In this letter, we analyze a mathematical model of resource exploitation recently reported by the authors. The model incorporates the cognitive decision-making process of consumers and has previously been studied in a game-theoretic context as a static two-player game. In this letter, we extend the analysis by allowing the agents to adapt their psychological characteristics according to simple best-response learning dynamics. We show that, under the selected learning scheme, the Nash Equilibrium is reachable provided certain conditions on the psychological attributes of the consumers are fulfilled. Moreover, the equilibrium solution obtained is found to be sustainable in the sense that no players exhibit free-riding behavior, a phenomenon which occurs in the original open-loop system. In the process, via a Lyapunov-function based approach, we also provide a proof for the asymptotic global stability of the original system which was previously known to be only locally stable.