A majorant criterion for the total preservation of global solvability of controlled functional operator equation

We prove the unique solvability of a nonlinear controlled functional operator equation in a Banach ideal space. We also establish sufficient conditions for the global solvability of all controls from a pointwise bounded set, provided that some majorant equation for the given family of these controls is globally solvable. We give examples of controlled boundary value problems reducible to the considered equation.     

A majorant-minorant criterion for the total preservation of global solvability of a functional operator equation

We study a nonlinear controlled functional operator equation in an ideal Banach space. We establish sufficient conditions for the global solvability for all controls from a given set, and obtain a pointwise estimate for solutions. Using upper and lower estimates of the functional component in the right-hand side of the initial equation (with a fixed operator component), we obtain majorant and minorant equations. We prove the stated theorem, assuming the monotonicity of the operator component in the right-hand side and the global solvability of both majorant andminorant equations. We give examples of the reduction of controlled initial boundary value problems to the equation under consideration.     

The Total Preservation of Unique Global Solvability of the First Kind Operator Equation With Additional Controlled Nonlinearity

For the Cauchy problem associated with an evolutionary operator equation of the first kind with an additional controlled term which nonlinearly depends on the phase variable, in a Banach space, we establish conditions for the total (on the set of admissible controls) preservation of unique global solvability under variation of the control parameter. We also establish the uniform bound for solutions. As examples, we consider initial-boundary value problems that are associated with a pseudoparabolic equation and a system of Oskolkov equations.     

Programmed iteration method and sets of positional absorption

A pursuit–evasion differential game is considered, and the programmed iteration method is used to construct a set of positional absorption corresponding to the Krasovskii–Subbotin alternative theorem. The case is considered where the set of positions determining the state constraints may not be closed (in the position space), but has closed sections corresponding to fixed times. Properties are established that are interpreted as the (one-sided) continuity of the positional absorption set from above, and the relation to the solution of the game in the class of set-valued quasi-strategies is shown.     

Weak Invariance of a Cylindrical Set with Smooth Boundary with Respect to a Control System

We consider the problem of constructing resolving sets for a differential game or an optimal control problem based on information on the dynamics of the system, control resources, and boundary conditions. The construction of largest possible sets with such properties (the maximal stable bridge in a differential game or the controllability set in a control problem) is a nontrivial problem due to their complicated geometry; in particular, the boundaries may be nonconvex and nonsmooth. In practical engineering tasks, which permit some tolerance and deviations, it is often admissible to construct a resolving set that is not maximal. The constructed set may possess certain characteristics that would make the formation of control actions easier. For example, the set may have convex sections or a smooth boundary. In this context, we study the property of stability (weak invariance) for one class of sets in the space of positions of a differential game. Using the notion of stability defect of a set introduced by V.N. Ushakov, we derive a criterion of weak invariance with respect to a conflict control dynamic system for cylindrical sets. In a particular case of a linear control system, we obtain easily verified sufficient conditions of weak invariance for cylindrical sets with ellipsoidal sections. The proof of the conditions is based on constructions and facts of subdifferential calculation. An illustrating example is given.     

Idealized program packages and problems of positional control with incomplete information

The paper is devoted to developing the method of program packages as a tool for investigating problems of positional control with incomplete information. The method is embedded in the field of guaranteed control theory and was stipulated by a number of constructions from this theory. Under the assumption that an a priori given set of initial positions of a controlled system is finite, it is established that the solvability of a guaranteed guidance problem in the class of program packages (or, which is the same, in the class of positional strategies) is equivalent to the solvability of this problem in the class of considerably simpler program operators, namely, in the class of idealized program packages.     

Double control problems of age-distributed population dynamics

This paper is concerned with some optimal control problems of an age-structured population model controlled by fecundity and culling rates. The first-order necessary optimality conditions of the following four problems are deduced by means of a Dubovitskii–Milyutin functional analysis approach: a problem with fixed horizon and terminal constraint set, a free horizon counterpart, a differential game and a mini–max problem with a non-differentiable objective functional.     

On a differential game in a nondamped distributed system

We study a differential game of approach in a system whose dynamics is described by a Sobolev‐type second‐order operator differential equation in Hilbert spaces. Solutions of the equation are represented by cosine and sine operator functions. To obtain solvability conditions of the game problem, we use support functionals of two sets, which defined by the behaviors of pursuer and evader. The results are applied to the investigation of conflicted‐controlled dynamics of bending waves in a rod.     

Best mean-quasiconformal mappings

We look for best mean-quasiconformal mappings as extremals of the functional equal to the integral of the square of the functional of the conformality distortion multiplied by a special weight. The mapping inverse to an extremal is an extremal of the same functional. We obtain necessary and sufficient conditions for the Petrovskii ellipticity of the system of Euler equations for an extremal. We prove the local unique solvability of boundary values problems for this system in the 2-dimensional case. In the general case we prove the unique solvability of boundary value problems for the system linearized at the identity mapping.     

The construction of stable bridges in differential games with phase constraints

A differential approach-and-evasion game in a finite time interval is considered [1]. It is assumed that the positions of the game are constricted by certain constraints which represent a closed set in the space of the positions. In the case of the first player, it is necessary to ensure that the phase point falls into the terminal set at a finite instant of time and, in the case of the second player, that this terminal set is evaded at this instant [1]. A method is proposed for the approximate construction of the positional absorption set, that is, the set of all positions belonging to a constraint from which the problem of approach facing the first player is solvable. Relations are written out which determine the system of sets which approximates the positional absorption set. The main result is a proof of the convergence of the approximate system of sets to the positional absorption set and the construction of a computational procedure for constructing the approximate system of sets.     

The stability defect of sets in game control problems

The stability property in a game problem of the approach of a conflict-controlled system to a goal set at a fixed terminal moment is investigated. The notion of a stability defect is introduced for sets in the space of game positions.     

On the convexity of global solvability sets for controlled initial-boundary value problems

We consider a controlled functional-operator equation that is a convenient form for describing of controlled initial-boundary value problems. For this equation, considered in a Banach ideal space, we define the set Ω of global solvability as the set of all admissible controls for which the equation has a global solution. We show that Ω is convex under the conditions imposed on the right-hand side of the equation. For each control in a given segment in Ω, we obtain a two-sided pointwise estimate for the corresponding solution under the abovementioned assumptions. We prove the theorem on the convex continuous dependence of the solution on the parameter specifying the displacement along the segment.     

Linear stabilization of the programmed motions of non-linear controlled dynamical systems under parametric perturbations

A non-linear controlled dynamical system that describes the dynamics of a broad class of non-linear mechanical and electromechanical systems (in particular, electromechanical robot manipulators) is considered. It is proposed that the real parameter vector of a non-linear controlled dynamical system belongs to an assigned (admissible) constrained closed set and is assumed to be unknown. The programmed motion of the non-linear controlled dynamical system and the programmed control that produces it are assigned (constructed) by using an estimate, that is, the nominal value of the parameter vector of the non-linear controlled dynamical system, which differs from its actual value. A procedure for synthesizing stabilizing control laws with linear feedback with respect to the state that ensure stabilization of the programmed motions of the non-linear controlled dynamical system under parametric perturbations is proposed. A non-singular linear transformation of the coordinates of the state space that transforms the original non-linear controlled dynamical system in deviations (from the programmed motion and programmed control) into a certain non-linear controlled dynamical system of special form, which is convenient for analysing and synthesizing laws for controlling the motion of the system, is constructed. A certain non-linear controlled dynamical system of canonical form is derived in the original non-linear controlled dynamical system in deviations. The transformation of the coordinates of the state space constructed and the Lyapunov function methodology are used to synthesize stabilizing control laws with linear feedback with respect to the state, which ensure asymptotic stability as a whole of the equilibrium position of the non-linear controlled dynamical system of canonical form and dissipativity “in the large” of the non-linear controlled dynamical system of special form and of the original non-linear controlled dynamical system in deviations. In the control laws synthesized, the formulae for the elements of their matrices of the feedback loop gains do not depend on the real parameter vector of the non-linear controlled dynamical system, and they depend solely on the constants from certain estimates that hold for all of its possible values from an assigned set. Estimates of the region of dissipativity “in the large” of the non-linear controlled dynamical system of special form and the original non-linear controlled dynamical system in deviations closed by the stabilizing control laws synthesized are given, and estimates for their limit sets and regions of attraction are presented.     

On the structure of a solution set of controlled initial-boundary value problems

For a controlled nonlinear functional-operator equation of the Hammerstein type describing a wide class of controlled initial-boundary value problems, we obtain simple sufficient conditions for the convexity, pointwise boundedness and precompactness of the set of solutions (the reachability tube) in the Lebesgue space. As for boundedness and precompactness, we mean certain conditions of the majorant but not Volterra type requirements which give also the total (with respect to the whole set of admissible controls) preservation of solvability of mentioned equation. As some examples of reduction of a controlled initial-boundary (boundary) value problem to the equation under investigation, and verification of the proposed hypotheses for this equation, we consider the first initial-boundary value problem associated with a semilinear parabolic equation of the second order in a rather general form, and also the Dirichlet problem associated with a semilinear elliptic equation of the second order.     

The stable set of the social conflict game with commitments: existence,uniqueness, and efficiency

We investigate the stable sets of social conflict games by employing the framework of the (abstract) system by Greenberg (Theory of social situations: an alternative game theoretic approach. Cambridge University Press, Cambridge, 1990). The social conflict game is a class of strategic games that includes the prisoners’ dilemma and the chicken game. We first show that the stable set generally fails to exist in a system that is directly derived from the social conflict game. In this system, the stable set exists if and only if the strong equilibrium exists in the underlying game. If the stable set exists, it coincides with the set of the strong equilibria that is equivalent to the core for the system. Then, we turn to a modified system where the players are allowed to make commitments. In the system with commitments, the stable set always exists, and it consists of efficient outcomes with a certain property. We also discuss the relationship between the core and the stable set for the system with commitments.     

Programmed iteration method in packages of spaces

An abstract version of the programmed iteration method (used in the theory of differential games) is considered that is intended for the problem of observing phase constraints on a nonempty subset of the real line. It is assumed that trajectories and uncontrolled disturbances are realized in packages of mutually inconsistent spaces. This consistency is postulated to investigate solvability conditions for the problem in the class of quasi-strategies. As a result, the set of successful solvability is constructed and the structure of resolving strategies is determined.     

Continuously stable strategies,neighborhood superiority and two-player games with continuous strategy space

The continuously stable strategy (CSS) concept, originally developed as an intuitive static condition to predict the dynamic stability of a monomorphic population, is shown to be closely related to classical game-theoretic dominance criteria when applied to continuous strategy spaces. Specifically, for symmetric and non symmetric two-player games, a CSS in the interior of the continuous strategy space is equivalent to neighborhood half-superiority which, for a symmetric game, is connected to the half-dominance and/or risk dominance concepts. For non symmetric games where both players have a one-dimensional continuous strategy space, an interior CSS is shown to be given by a local version of dominance solvability (called neighborhood dominance solvable). Finally, the CSS and half-superiority concepts are applied to points in the bargaining set of Nash bargaining problems. R. Cressman thanks the referee for pointing out further connections between neighborhood superiority and other dominance concepts in the literature. This research was supported by a Natural Sciences and Engineering Research Council of Canada Individual Discovery Grant.     

Nonlinear set-valued boundary value problems with continuous feedback

We obtain sufficient conditions for the existence of at least one absolutely continuous solution of a nonlinear functional-differential inclusion in a finite-dimensional space with nonlinear set-valued functional boundary conditions. The set-valuedness of the dynamics may be due to the presence of a control. In addition, we consider the case in which the set-valuedness of the system dynamics and the boundary conditions is specified by inequalities. We assume the presence of a continuous feedback and impose the requirements of solvability of the open-loop problem. Statements of problems of this type arise in connection with the analysis of conflict-control systems.     

Stability estimates in identification problems for the convection-diffusion-reaction equation

Identification problems for the stationary convection-diffusion-reaction equation in a bounded domain with a Dirichlet condition imposed on the boundary of the domain are studied. By applying an optimization method, these problems are reduced to inverse extremum problems in which the variable diffusivity and the volume density of substance sources are used as control functions. Their solvability is proved for an arbitrary weakly lower semicontinuous cost functional and particular cost functionals. An analysis of the optimality system is used to establish sufficient conditions on the input data under which the solutions of particular extremum problems are unique and stable with respect to small perturbations in the cost functional and in one of the functions involved in the boundary value problem.     

Degenerate Linear Evolution Equations with the Riemann–Liouville Fractional Derivative

We study the unique solvability of the Cauchy and Schowalter–Sidorov type problems in a Banach space for an evolution equation with a degenerate operator at the fractional derivative under the assumption that the operator acting on the unknown function in the equation is p-bounded with respect to the operator at the fractional derivative. The conditions are found ensuring existence of a unique solution representable by means of the Mittag-Leffler type functions. Some abstract results are illustrated by an example of a finite-dimensional degenerate system of equations of a fractional order and employed in the study of unique solvability of an initial-boundary value problem for the linearized Scott-Blair system of dynamics of a medium.