### Book chapter

## Optimal Chattering Regimes in Nonhomogeneous Bar Model

We consider an optimal control problem of longitudinal vibrations of a nonhomogeneous bar with clamped ends. We assume that the density of external forces is a control function. Using the Fourier method we prove that the optimal control is the chattering control, i.e., it has an infinite number of switchings in a~finite time interval.

### In book

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.

As a standard, DEStech will submit the technical science conference proceedings to be indexed in EI (Citation and Compendex). DEStech will submit the technical science conference proceedings to EI, Clarivate Analytics (formerly Thomson Reuters Web of Science) and CNKI Scholar for worldwide online citation

We study an optimal control problem for a nonlinear spherical inverted pendulum on a movable base. As the cost functional, the mean-squared deviation of the pendulum from the upper equilibrium is considered, so optimal controls stabilize the pendulum at the unstable upper position. We show that the problem under consideration posses a singular point of the second order and there are spiral-similar solution which attains the singular point in finite time.

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.

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.

Book include abstracts of reports presented at the IX International Conference on Optimization Methods and Applications "Optimization and applications" (OPTIMA-2018) held in Petrovac, Montenegro, October 1 - October 5, 2018.