### 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 proceedings of the 8th International Conference ”Optimization and Applications” (OPTIMA-2017) that was held in Petrovac, Montenegro, during October 2-7, 2017. The Conference brought together researchers and practitioners working in the field of optimization theory, methods, software and related areas. Optimization is now a rapidly growing area in computer science. It is widely applied in various fields, e.g. data analysis and processing, image recognition, robotics, material sciences, design, logistics etc. Thus it was not surprising that OPTIMA-2017 attracted a significant number of participants. This year about 90 researchers participated in the conference.

In this paper we initiate the rigorous analysis of controlled Continuous Time Random Walks (CTRWs) and their scaling limits, which paves the way to the real application of the research on CTRWs, anomalous diffusion and related processes. For the first time convergence is proven for payoff functions for controlled scaled CTRWs and their position dependent extensions to the solution of a new pseudo-differential equation which may be called the fractional Hamilton-Jacobi-Bellman equation.

The functionals constructed on trajectories of the controlled semi- Markov process with a finite set of states are investigated. Theorems of functionals’ structure (dependences on the probability measures defining the Markov homogeneous randomized strategy of control) and of structure of probability measures on which the extremum of these functionals is reached, are formulated. Examples are given.

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.

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.

Optimal control problem for the system of partial differential equations of hyperbolic type is considered. By using the Fourier method this problem is reduced to the optimal control problem for the corresponding Fourier coefficients. For some special initial data we prove the existence of optimal solutions with a countable number of switchings on a finite time interval and optimal spiral-like solutions which attain the origin in a finite time making a countable number of rotations. The problem of controlling the vibrations of the Timoshenko beam is considered as an example of the optimal control problem for linear system of PDE.

This work is concerned with the optimal control problems governed by the 1D wave equation with variable coefficients and the control spaces $\mathcal M_T$ of either measure-valued functions $L^2(I,\mathcal M(\Omega))$ or vector measures $\mathcal M(\Omega,L^2(I))$. The cost functional involves the standard quadratic terms and the regularization term $\alpha\|u\|_{\mathcal M_T}$, $\alpha>0$. We construct and study three-level in time bilinear finite element discretizations for the problems. The main focus lies on the derivation of error estimates for the optimal state variable and the error measured in the cost functional. The analysis is mainly based on some previous results of the authors. The numerical results are included.

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.