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## Optimality of refraction strategies for a constrained dividend problem

We consider de Finetti’s problem for spectrally one-sided Lévy risk models with control strategies that are absolutely continuous with respect to the Lebesgue measure. Furthermore, we consider the version with a constraint on the time of ruin. To characterize the solution to the aforementioned models, we first solve the optimal dividend problem with a terminal value at ruin and show the optimality of threshold strategies. Next, we introduce the dual Lagrangian problem and show that the complementary slackness conditions are satisfied, characterizing the optimal Lagrange multiplier. Finally, we illustrate our findings with a series of numerical examples.

We consider the optimal bail-out dividend problem with fixed transaction cost for a Lévy risk model with a constraint on the expected present value of injected capital. To solve this problem, we first consider the optimal bail-out dividend problem with transaction cost and capital injection and show the optimality of reflected (c_1,c_2) -policies. We then find the optimal Lagrange multiplier, by showing that in the dual Lagrangian problem the complementary slackness conditions are met. Finally, we present some numerical examples to support our results.

17th IFAC Workshop on Control Applications of Optimization CAO 2018

Yekaterinburg, Russia, 15–19 October 2018

**The problem of optimal control is formulated for a class of nonlinear objects that can be represented as objects with a linear structure and parameters that depend on the state. The linear structure of the transformed nonlinear system and the quadratic functional of quality allow for the synthesis of optimal control, i.e. parameters of the regulator, move from the need to search for solutions of the Hamilton-Jacobi equation to an equation of the Riccati type with parameters that depend on the state.** **The main problem of implementing optimal control is related to the problem of finding a solution to such an equation at the pace of object functioning.** **The paper proposes an algorithmic method of parametric optimization of the regulator. This method is based on the use of the necessary conditions for the optimality of the control system under consideration. The constructed algorithms can be used both to optimize the non-stationary objects themselves, if the corresponding parameters are selected for this purpose, and to optimize the entire managed system by means of the corresponding parametric adjustment of the regulators. The example of drug treatment of patients with HIV is demonstrated the effectiveness of the developed algorithms.**

This paper is dedicated to optimal state-feedback control problem for discrete-time descriptor systems in presence of “colored” noise with known mean anisotropy level. Here “colored” noise stands for a stationary Gaussian sequence, generated by a linear shaping filter from the Gaussian white noise sequence. The control goal is to find a state feedback control law which makes the closed-loop system admissible and minimizes its a-anisotropic norm (mean anisotropy level a is known).

This collection of articles contain materials of the talks presented at the International Conference "Systems Analysis: Modeling and Control" in memory of Academician A.V. Kryazhimskiy, Moscow, May 31 - June 1, 2018

We consider a linear model of a rotating Timoshenko beam. We show that for some initial conditions, the solutions of the minimization problem for the deviation of the beam from the equilibrium state have Fuller singularities.

A model for organizing cargo transportation between two node stations connected by a railway line which contains a certain number of intermediate stations is considered. The movement of cargo is in one direction. Such a situation may occur, for example, if one of the node stations is located in a region which produce raw material for manufacturing industry located in another region, and there is another node station. The organization of freight traﬃc is performed by means of a number of technologies. These technologies determine the rules for taking on cargo at the initial node station, the rules of interaction between neighboring stations, as well as the rule of distribution of cargo to the ﬁnal node stations. The process of cargo transportation is followed by the set rule of control. For such a model, one must determine possible modes of cargo transportation and describe their properties. This model is described by a ﬁnite-dimensional system of diﬀerential equations with nonlocal linear restrictions. The class of the solution satisfying nonlocal linear restrictions is extremely narrow. It results in the need for the “correct” extension of solutions of a system of diﬀerential equations to a class of quasi-solutions having the distinctive feature of gaps in a countable number of points. It was possible numerically using the Runge–Kutta method of the fourth order to build these quasi-solutions and determine their rate of growth. Let us note that in the technical plan the main complexity consisted in obtaining quasi-solutions satisfying the nonlocal linear restrictions. Furthermore, we investigated the dependence of quasi-solutions and, in particular, sizes of gaps (jumps) of solutions on a number of parameters of the model characterizing a rule of control, technologies for transportation of cargo and intensity of giving of cargo on a node station.