Equivalence of the probability measures generated by the solutions of nonlinear evolution differential equations in a Hilbert space,disturbed by Gaussian processes. Part I

Nonlinear evolution differential equations with unbounded linear operators of disturbance by Gaussian random processes are considered in an abstract Hilbert space. For the Cauchy problem for the differential equations, the sufficient existence and uniqueness conditions for their solutions are proved and the sufficient conditions for the equivalence of the probability measures generated by these solutions are derived. Moreover, the corresponding Radon–Nikodym densities are calculated explicitly in terms of the coefficients or characteristics of the considered differential equations.     

Optimal Control of the Dynamic Viscoelastic Stress-Strain State of a Composite Body

New optimal control problems for the dynamic viscoelastic stress-strain state of a composite body with quadratic cost functionals are considered. Existence theorems for unique optimal controls are proved for all the cases considered. __________ Translated from Kibernetika i Sistemnyi Analiz, No. 5, pp. 73–92, September–October 2005.     

Solving the Wiener-Hopf equation with a probabilistic kernel

Inhomogeneous Wiener-Hopf equations on a halfline with probabilistic kernels are considered. They are analyzed using the theory of isotonic and contraction operators. Necessary and sufficient conditions of existence and general sufficient conditions of uniqueness for the solutions of these equations are established. The conditions for and the rate of convergence of successive approximations of the solution are found. __________ Translated from Kibernetika i Sistemnyi Analiz, No. 2, pp. 39–47, March–April 2006.     

Optimal control of an elliptic system with a nonsymmetric state operator

New optimal control problems for elliptic systems with contact conditions and nonsymmetric state operators are considered. Existence theorems for optimal solutions are proved. High-accuracy schemes for approximating optimal controls are proposed for the case where the set of admissible controls coincides with a complete Hilbert space. __________ Translated from Kibernetika i Sistemnyi Analiz, No. 6, pp. 67–85, November–December 2005.     

Equivalence of the probability measures generated by the solutions of nonlinear evolution differential equations in a Hilbert space, disturbed by Gaussian processes. Part II

The paper continues the studies in the equivalence of the measures generated by the solutions of nonlinear evolution differential equations with unbounded linear operators perturbed by random Gaussian processes in a Hilbert space, in particular í. Two different nonlinear evolution differential equations perturbed by the same random Gaussian process in the right-hand side are considered in the space í. The sufficient existence and uniqueness conditions are established for the solutions of these equations, the equivalence of the measures generated by the solutions is proved, and explicit formulas for the Radon–Nikodym density of the respective measures calculated in terms of the coefficients of the considered equations are written.     

Optimal Strategies for Inventory Control Systems with a Convex Cost Function

Some controlled Markovian processes in discrete time in the context of optimization of inventory control systems are studied. Optimality of (s, S)-policies for the case of convex cost functions is proved using theorems on existence and uniqueness of a nonrandomized stationary optimal policy for Markovian processes with discrete time and a continuous control set for criteria characterizing mean costs per unit time and overestimated total costs and Bellman equations.     

An optimal control model for a system of degenerate parabolic integro-differential equations

An initial-boundary-value problem for a system of degenerate parabolic integro-differential equations is considered. The sufficient conditions for the existence and uniqueness of its generalized solution and for the existence of at least one optimal control for a given performance functional are obtained. A stable numerical solution to the initial-boundary-value problem is derived for a locally one-dimensional case and conditions are formulated for constructing a stable numerical algorithm of the optimal control problem on a class of piecewise-smooth control functions. __________ Translated from Kibernetika i Sistemnyi Analiz, No. 6, pp. 90–102, November–December 2007.     

Controllability and Optimization of Pseudohyperbolic Systems

The problems of singular optimization and controllability of systems described by the boundary-value problems for pseudohyperbolic equations are studied. The existence and uniqueness theorems for solutions to basic boundary-value problems for pseudohyperolic equations in the nonclassic Sobolev-like spaces under wide assumptions as to right-hand sides are proved. Based on these results, final impulse-point controllability of the systems and existence of optimal control are studied.     

Using categorical methods in computer science

Some categorical methods used in developing computer programs are considered. Relevant theorems are proved. Translated from Kibernetika i Sistemnyi Analiz, No. 4, pp. 3–11, July–August, 2000.     

Optimal control of the dynamic state of a thin compound plate

This paper addresses new optimal control problems for dynamic deflections of a thin compound plate with quadratic cost functionals. Theorems on the existence of optimal controls are proved for all the cases considered. __________ Translated from Kibernetika i Sistemnyi Analiz, No. 4, pp. 151–175, July–August 2006.     

Un metodo per la costruzione di smoothing splines naturali mono e bidimensionali

Sommario In questa nota vengono dapprima presentati i teoremi di esistenza ed unicità (in opportuni spazi) delle smoothing splines naturali mono e bidimensionali, soluzioni rispettivamente dei problemi (1,5)–(1,6) e (4,5)–(4,6). Viene quindi indicato un procedimento, basato sulla costruzione di basi locali di splines naturali, per la costruzione di tali soluzioni. Infine viene presentato un metodo generale per la costruzione di tali basi.

---

In this paper, we shall present the theorems of existence and uniqueness (in proper spaces) of the natural smoothing splines which are the solutions of the problems (1,5)–(1,6) for the one dimension case, and of the problems (4,5)–(4,6) for the two dimension case. We shall then suggest a method, based on construction of local bases of the natural splines, to determine these solutions. Finally, we shall indicate a general method to construct these bases.

---

---

Lavoro eseguito nell’ambito del gruppo di ricerca GNAFA del CNR e dei programmi di ricerca dell’Istituto per le Applicazioni del Calcolo del CNR in Roma     

Existence, uniqueness, and asymptotic behavior of solutions to a class of Zakai equations with unbounded coefficients

Conditions are given to guarantee the existence and uniqueness of solutions to the Zakai equation associated with the nonlinear filtering of diffusion processes. The conditions permit stronger than polynomial growth of the coefficients, and depend instead on the relative growth rates. The results are derived by adapting, through a sequence of exponential transformations, the classical existence and uniqueness theorems for parabolic PDE's due to Besala to the "robust" form of the Zakai equation. In this process we also obtain sharp estimates for the tail behavior of the conditional density. Examples, including observations through a polynomial sensor and estimation of the state of a "bilinear" system, are worked out in detail. Our results are compared to those of Fleming and Mitter, Pardoux, and Sussmann who, among others, have obtained existence and uniqueness theorems for a more limited class of problems by different methods.     

Robust dissipativity of discrete systems and its investigation with the help of a sequence of sets of Lyapunov functions

A generalized concept of robust dissipativity is introduced and theorems on the analysis of this property are formulated and proved with the help of the Lyapunov functions method. It is proposed to use a specially constructed sequence of sets of Lyapunov functions that can allow one to improve the initial estimate of the boundary set of a dissipative system up to the establishment of the asymptotic robust stability property in the limiting case. An example of investigating the dissipativity of a linear discrete system with unknown parameters under additive perturbation is given. __________ Translated from Kibernetika i Sistemnyi Analiz, No. 2, pp. 13–23, March–April 2008.     

Stability of solutions of dynamic systems with randomly structured aftereffect

The relation is established between the asymptotic stochastic stability of a linear functional differential equation and exponential stability of the trivial solution to this equation. The direct and inverse Lyapunov theorems on the stability of linear differential equations are proved. __________ Translated from Kibernetika i Sistemnyi Analiz, No. 2, pp. 31–38, March–April 2006.     

Quasi-copulas and copulas on a discrete scale

In the paper the structure of quasi-copulas and copulas on a finite discrete scale is studied. The possibility of construction of quasi-copulas (copulas) from given values at diagonal points is investigated. Moreover, the problem of the uniqueness of the existence of a quasi–copula (copula) with given diagonal section is solved. Several examples are included.     

The method of successive approximation applied to find the probability for an insurance company with random premiums

A risk process that describes the evolution of the capital of an insurance company is analyzed, random premiums and claims being available. Integral equations of nonbankruptcy probability as a function of the initial capital are derived. Necessary and sufficient conditions for the existence and uniqueness of solutions of these integral equations, and convergence conditions for the method of successive approximation for finding their solutions are established __________ Translated from Kibernetika i Sistemnyi Analiz, No. 1, pp. 112–127, January–February 2006.     

Theory of general linear automata

Linear and affine automata are considered in their general form. The concept of dimensions of a finite automaton is introduced and finite automata of maximal dimensions are shown to be possible. The state reachability problem in monomial form is proved to be undecidable for two-dimensional affine automata. An analogue of Moore's theorem and theorems on homogenous and diagnostic words are also proved. An application of linear automata to mathematical economics is considered. Translated from Kibernetika i Sistemnyi Analiz, No. 1, pp. 10–21, January–February 2009.     

Untersuchungen zur qualitativen Behandlung von linearen Differentialgleichungssystemen mit unstetiger rechter Seite mit Hilfe der Theorie der Distributionen

By means of the theory of distributions the possibilities to obtain qualitative statements concerning solutions of systems of linear differential equations with discontinuous right hand sides are investigated. A number of existence and uniqueness theorems is proved for a suitable defined initial value problem. Moreover the structure of solutions (continuity, periodicity etc.) is investigated more closely. It is shown that a number of theorems known from the classical theory of periodical solutions is applicable to differential equations in the space of distributions with only slight modifications.     

Necessary and sufficient conditions of existence and uniqueness of solutions to integral equations of actuarial mathematics

Necessary and sufficient conditions of existence and uniqueness of solutions of integral equations are established for ruin probability as a function of the initial capital of an insurance company. Several sufficient conditions and correctness conditions of the problem of finding ruin probability are also established. A general method of successive approximations for solution of this problem is substantiated. __________ Translated from Kibernetika i Sistemnyi Analiz, No. 5, pp. 157–164, September–October 2006.     

Stability of diffusion stochastic functional differential equations with Markov parameters

The paper justifies the second Lyapunov method for diffusion stochastic functional differential equations with Markov parameters, which generalize stochastic diffusion equations without aftereffect. Analogs of Lyapunov stability theorems, which generalize the results for systems with finite aftereffect, are proved. __________ Translated from Kibernetika i Sistemnyi Analiz, No. 1, pp. 74–88, January–February 2008.