This is the second Special Issue on Lyapunov's Methods in Stability and Control of the journal Mathematical and Computer Modelling. It is devoted to further theoretical development and to applications that arise in optimal control, differential games, nonlinear mechanics, population dynamics, financial and economic processes, long term planning, and in mathematical modelling theory aimed at development and use of reliable models. The papers presented in this issue cover currently active areas in applied stability, optimal control, cooperative differential game theory with uncertainties, absence of arbitrage and the existence of equivalent martingale measures for complete markets, nonlinear hydrodynamics, artificial neural networks, stochastic and hereditary systems, multi-model adaptive systems for large scale projects, and multi-objective long term optimal planning. The results present new developments in the field and can be used in practical applications and in mathematical modelling of nonlinear processes. A new and promising approach is presented with the introduction of non-causal models for long term planning when economic agents' rational expectations about future policies affect current decisions.
The first paper is devoted to the Dirichlet problems for singularly perturbed Hamilton-Jacobi-Bellman equations (HJBEs). The research appeals to the theory of minimax solutions to the Dirichlet problem for HJBEs that are known to coincide with the value functions of time-optimal control problems for a system with fast and slow motions. Effective sufficient conditions based on the fact are suggested, and the key condition is the existence of a Lyapunov type function providing convergence for singularly perturbed characteristics of HJBEs to the origin. Moreover, convergence implies equivalence of the limit function and the value function of an unperturbed time-optimal control problem in the reduced subspace of slow variables.
In the second paper, dynamically stable cooperative solutions in randomly furcating stochastic differential games are considered. Analytically tractable payoff distribution procedures contingent upon specific random realizations of the state and payoff structure are derived. This new approach widens the application of cooperative differential game theory to problems where the evolution of the state and future environments are not known with certainty, of which important cases abound in regional economic cooperation, corporate joint ventures and environmental control.
In the third paper, the well known A. A. Lyapunov Theorem (1940) that the range of any n-dimensional vector measure is compact, and also convex if the measure is atomless, is used to revisit the case where the first fundamental theorem of asset pricing may fail when dealing with infinitely many trading dates. New results are obtained to retrieve the equivalence between the absence of arbitrage in financial markets and the existence of equivalent martingale measures for complete markets such that the Sharpe Ratio is adequately bounded.
The fourth paper studies a jump-diffusion type vorticity model, describing evolution of incompressible homogeneous viscous fluid in R 2 in terms of its rotation. The model arises from a particle systems perspective, adopted in the point vortex theory, and represents a measure-valued stochastic partial differential equation (SPDE) whose solution, under certain conditions, is an empirical process generated by a finite system of randomly moving vortices. The model provides natural stochastic approximation to the classical (deterministic) Navier-Stokes equation which loses its predictive power for large Reynold's numbers requiring extra "correction" terms which are terms that are naturally represented in the stochastic vorticity equation considered in the paper. The results provide the background for stochastic vorticity modelling. Generalizations are possible to compressible airflow with important applications to flight aerodynamics.