### Book chapter

## On optimal solutions in a problem with two-dimensional bounded control

We consider an optimal control problem that is affine in two-dimensional control. The origin is a singular trajectory in this problem. We study the structure of optimal solutions in a neighborhood of the origin. We use the resolution of singularity via blow up and the invariant manifold theorems to find a family of optimal solutions.

### In book

The paper studies a problem of optimal insurer’s choice of a risk-sharing policy in a dynamic risk model, so-called Cramer-Lundberg process, over infinite time interval. Additional constraints are imposed on residual risks of insureds: on mean value or with probability one. An optimal control problem of minimizing a functional of the form of variation coefficient is solved. We show that: in the first case the optimum is achieved at stop loss insurance policies, in the second case the optimal insurance is a combination of stop loss and deductible policies. It is proved that the obtained results can be easily applied to problems with other optimization criteria: maximization of long-run utility and minimization of probability of a deviation from mean trajectory.

We formulate the optimal control problem for a class of nonlinear objects that can be represented as objects with linear structure and state-dependent coefficients. We assume that the system is subjected to uncontrollable bounded disturbances. The linear structure of the transformed nonlinear system and the quadratic quality functional let us, in the optimal control synthesis, to pass from Hamilton–Jacobi–Isaacs equations to a state-dependent Riccati Equation. Control of nonlinear uncertain object in the problem of moving along a given trajectory is considered using the theory of differential games. We also give an example that illustrates how theoretical results of this work can be used.

We consider a control problem for longitudinal vibrations of a nonhomogeneous bar with clamped ends. The vibrations of the bar are controlled by an external force which is distributed along the bar. For the minimization problem of mean square deviation of the bar we construct optimal solutions in the form of the Fourier series. To find Fourier coefficients we consider an optimal control problem in the space l^2. For the control problem in l^2 we show that in a certain neighborhood of the origin the structure of the optimal solutions is the following one: for the finite time the optimal nonsingular trajectory enters the singular surface with infinite numbers of control switchings, after that the optimal trajectory remains on the singular surface and attains the origin for the infinite 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.

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.

Proceedings include extended abstracts of reports presented at the III International Conference on Optimization Methods and Applications “Optimization and application” (OPTIMA-2012) held in Costa da Caparica, Portugal, September 23—30, 2012.