### Book chapter

## Suboptimal Control of Nonlinear Object:Problem of Keeping Tabs on Reference Trajectory

An optimal control problem is formulated for a class of nonlinear systems which can be presented by system with linear structure and state-depended coefficients (SDC). The system being under the influence of uncontrollable disturbance is supposed. The linearity of the transformed system structure and the quadratic functional make it possible to pass over from the Hamilton–Jacoby–Bellman equation (HJB) to the state dependent Riccati equation (SDRE) upon the control synthesis. In thus paper the optimal control problem by nonlinear system in a task of Keeping Tabs on Reference Trajectory we decide in a key of differential game. The presented example illustrates the application of the proposed control method.

This volume contains refereed proceedings of the IX International Conference Optimization and Applications (OPTIMA 2018) held in Petrovac, Montenegro, October 1–5, 2018. The previous conferences during 2009–2017 years attracted a significant number of students, researchers, academics, and engineers working in the field of optimization theory, methods, software, and related areas. The Conference was organized by five institutions: • The Montenegrin Academy of Sciences and Arts (Montenegro); • Federal Research Center "Computer Science and Control" of Russian Academy of Science (Russia); • University of Montenegro (Montenegro); University of Evora (Portugal); • Institute of Information and Computational Technologies (Kazakhstan). The Conference covered many optimization related areas ranging from pure theoretic studies to software and applications. The broad scope of OPTIMA made it an excellent collaboration platform for researchers from various domains related to optimization. The conference allowed specialists from different fields to present their work and discuss both theoretical and practical aspects of their research. Another important aim of the conference was to stimulate scientists and people from industry to benefit from the knowledge exchange and identify possible grounds for fruitful collaboration.

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The chapter studies a dynamic risk model defined on infinite time interval, where both insurance and per-claim reinsurance policies are chosen by the insurer in order to minimize a functional of the form of variation coefficient under constraints imposed with probability one on insured's and reinsurer's risks. We show that the optimum is achieved at constant policies, the optimal reinsurance is a partial stop loss reinsurance and the optimal insurance is a combination of stop loss and deductible policies. The results are illustrated by a numerical example involving uniformly distributed claim sizes.

The collection represents proceedings of the XVIII international conference “Problems of Theoretical Cybernetics” (Penza, 19–23 June, 2017), that is sponsored by Russian Foundation for Basic Research (project N 17-01-20217-г). The conference subject area includes: control systems synthesis, complexity, reliability, and diagnostics; automata; computer languages and programming; graph theory; combinatorics; coding theory; theory of pattern recognition; mathematical programming and operations research, mathematical theory of intelligence systems; applied mathematical logic; functional systems theory; optimal control theory; applications of cybernetics in natural science and technology. For scientists and specialists in areas of mathematical cybernetics, discrete mathematics, computer science and their applications.

A new approach to the transformation of solutions of optimal control problems based on the special form of relaxation of complementary slackness conditions is presented. The proposed approach is tested on the Russian banking system model, which is derived as a solution of a linear nonautonomous optimization problem with mixed constraints. It is shown that the use of this method regularizes the model in a sense it becomes applicable for the forecasting of the main Russian banking indicators.

We propose a model of the Russian banking system. It is based on the problem of a macroeconomic agent ”bank” which is modelled according to the principles of aggregated description, optimality and perfect foresight. To derive the equations of the model, we use the original method of relaxation of complementary slackness conditions. The model successfully reproduces main indicators of the banking system,

We introduce a longevity feature to the classical optimal dividend problem by adding a constraint on the time of ruin of the firm. We extend the results in [HJ15], now in context of one-sided Lévy risk models. We consider de Finetti's problem in both scenarios with and without fix transaction costs, e.g. taxes. We also study the constrained analog to the so called Dual model. To characterize the solution to the aforementioned models we introduce the dual problem and show that the complementary slackness conditions are satisfied and therefore there is no duality gap. As a consequence the optimal value function can be obtained as the pointwise infimum of auxiliary value functions indexed by Lagrange multipliers. Finally, we illustrate our findings with a series of numerical examples.

This book constitutes the refereed proceedings of the 9th International Conference on Optimization and Applications, OPTIMA 2018, held in Petrovac, Montenegro, in October 2018.The 35 revised full papers and the one short paper presented were carefully reviewed and selected from 103 submissions. The papers are organized in topical sections on mathematical programming; combinatorial and discrete optimization; optimal control; optimization in economy, finance and social sciences; applications.

We introduce SANgo (Storage Area Network in the Go language)—a Go-based package for simulating the behavior of modern storage infrastructure. The software is based on the discrete-event modeling paradigm and captures the structure and dynamics of high-level storage system building blocks. The flexible structure of the package allows us to create a model of a real storage system with a configurable number of components. The granularity of the simulated system can be defined depending on the replicated patterns of actual system behavior. Accurate replication enables us to reach the primary goal of our simulator—to explore the stability boundaries of real storage systems. To meet this goal, SANgo offers a variety of interfaces for easy monitoring and tuning of the simulated model. These interfaces allow us to track the number of metrics of such components as storage controllers, network connections, and hard- drives. Other interfaces allow altering the parameter values of the simulated system effectively in real-time, thus providing the possibility for training a realistic digital twin using, for example, the reinforcement learning (RL) approach. One can train an RL model to reduce discrepancies between simulated and real SAN data. The external control algorithm can adjust the simulator parameters to make the difference as small as possible. SANgo supports the standard OpenAI gym interface; thus, the software can serve as a benchmark for comparison of different learning algorithms.